Elaboration of Georgia Performance Standards: Mathematics

Double Number Line Diagrams

2012-01-30T18:54:00.000-08:00

6.RP.3. Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.

I just created a 9-minute video on the basic idea of double number line diagrams. Although double number line diagrams are specifically mentioned in a Grade 6 standard, if we want students to use them to solve ratio and rate problems in Grade 6, they really need to be familiar with the representations by then. That means they should be really introduced in elementary schools.

Anyway, I hope to create additional videos to elaborate how double number lines may be used to represent students' own reasoning, and eventually become their own thinking tools.

Measurement in K-2

2011-12-02T14:09:00.000-08:00

K.MD Describe and compare measurable attributes
1.MD Measure lengths indirectly and by iterating length units
2.MD Measure and estimate lengths in standard units

I wrote about teaching of measurement in primary grades almost 3 years ago (Dec. 2008), In the post, I stated that there are 3 related yet distinct goals while teaching measurement:
* understanding the attribute being measured
* process of measurement
* how to use measuring instruments.

Also, there is a general consensus that teaching of measurement should proceed along the following instructional sequence:
1. Direct comparison
2. Indirect comparison
3. Measuring with non-standard units
4. Measuring with standard units

From this perspective, I wrote that the GPS was unclear about steps 2 and 3. In contrast, the Common Core follows the suggested sequence explicitly, at least with the attribute of length. For other attributes like (liquid) volume, mass, area, angle, etc., the CCSS appears to jump right in with step 4, measuring with standard units. For example, 3.MD.2 states, "Measure and estimate liquid volumes and masses of objects using standard units of grams (g), kilograms (kg), and liters (l)." Although I certainly do not advocate going through the 4 steps of measurement instruction with every attribute, I am not sure if experiences with one attribute (length) is enough for children to generalize the process of measurement. In the typical Japanese curriculum, children study length and capacity (liquid volume) in Grades 1 and 2 (the first 2 years of elementary schools in Japan), and they go through these 4 steps with each attribute. They will also include some direct comparison activities with comparison of areas before they study how to calculate area. As we move ahead with the implementation of the CCSS, we may want to include some comparison activities as well as measuring with non-standard units for some of the attributes. Furthermore, explicit discussions on the process of measurement (selecting a unit, using a unit to iterate/cover the object, count the number of units) so that with the later attributes, we can start with the question of what we should use as a (standard) unit.

Number and Operations in Base Ten (1.NBT.4~6)

2011-10-09T07:10:00.000-07:00

In the Grade 1 Number and Operations in Base Ten domain, you will see the following cluster.

Use place value understanding and properties of operations to add and subtract.

4. Add within 100, including adding a two-digit number and a one-digit number, and adding a two-digit number and a multiple of 10, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used. Understand that in adding two-digit numbers, one adds tens and tens, ones and ones; and sometimes it is necessary to compose a ten.
5. Given a two-digit number, mentally find 10 more or 10 less than the number, without having to count; explain the reasoning used.
6. Subtract multiples of 10 in the range 10-90 from multiples of 10 in the range 10-90 (positive or zero differences), using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.

As I discussed briefly in the previous post, these standards are based on students' understanding of ten as a unit, which are the major focus of 1.NBT.2. While in Kindergarten, students considered 10 as a "benchmark" or a "mile marker" to understand numbers 11 through 19, in Grade 1, students learn to count ten, that is, ten as a unit. Thus, extending the idea of composing and decomposing number further, instead of thinking 64 as just 60+4, we want students to think of 60 as 10+10+10+10+10+10, or six 10's. Being able to coordinate two units, i.e., ones and tens, is not a trivial task for young children. Many children at this age can recite the sequence of decade number words correctly, "ten, twenty, thirty, forty, ..." However, some children will have to count by ones to answer, "what is 10 more than 36?" For many young children, the decade number word sequence is just a memorized set of words with no numerical significance. They do not yet understand when they go from "thirty" to "forty," the number increased by 10. Thus, these standards, although they are based on 1.NBT.2, are not necessarily something that comes after students master 1.NBT.2. As students think about addition of 2 multiples of 10 or subtract a multiple of 10 from another multiple of 10, we want to encourage students to think in terms of 10's. So, questions like, "how many 10's are in __?" and "how many 10's are we adding (or taking away)?" must be an important part of teachers' questioning repertoire.

These standards also illustrate an important pedagogical idea that seems to come up several times in the CCSS, that is, as students encounter new forms of calculations, they first take advantage of the structures of numbers and model numbers using concrete materials or visual representations before they formalize them into written procedures. Thus, while "adding a two-digit number and a one-digit number, and adding a two-digit number and a multiple of 10" students should think about the base-10 structure of the numbers. For example, if they are adding 38 + 7, they might think, "well, 38 is 3 tens and 8. But 2 more will make another ten, so that will be 4 tens and 5, so 45 is the answer." Or, if they are adding 38 + 40, they might think "we are adding 4 more tens to 3 that we already have. So there will be 7 tens and 8, so 78." They might use base-10 blocks to think along. However, the important reason for using base-10 blocks is not so that students can find the answer using the blocks but so that students can think in terms of unit of ten (i.e., long's). So, asking students to think about how they might model the addition with base-10 blocks but without actually using them might be a useful activity.

Finally, the last part of 1.NBT.4, "Understand that in adding two-digit numbers, one adds tens and tens, ones and ones; and sometimes it is necessary to compose a ten." raises a question about the type of addition (and perhaps subtraction) that should be the focus in Grade 1. Specific types of addition mentioned by 1.NBT.4 are "adding a two-digit number and a one-digit number, and adding a two-digit number and a multiple of 10." Thus, addition problems like 38 + 7 or 38 + 40. However, there is this little word, "including," in this standard. So, does that mean addition of two general 2-digit numbers should be taught in Grade 1? Since addition problems like 38 + 7 or 38 + 40 do not require students to do both addition of ones and addition of tens, it might be a bit difficult for children to develop this understanding. It might be technically true that when children add 38 and 40, they are doing 3+4 and 8+0, I doubt many children will see it that way. Perhaps the statement, "adding a two-digit number and a one-digit number, and adding a two-digit number and a multiple of 10," was included to suggest that these types of addition problems should be discussed before addition of two general 2-digit numbers is addressed. 1.NBT.6 seems to be much clearer in determining the type of subtraction problems to be discussed in Grade 1. A similar, more specific indication on this matter will be helpful for teachers and curriculum developers.

Number and Operations in Base Ten (K.NBT.1 & 1.NBT.2)

2011-09-05T05:38:00.000-07:00

Number and Operations in Base Ten (K.NBT.1 & 1.NBT.2)
At the end of the last post, I briefly touched upon the idea of composing and decomposing numbers 11 through 19. This idea is discussed in both Kindergarten and Grade 1.

K.NBT.1 Compose and decompose numbers from 11 to 19 into ten ones and some further ones, e.g., by using objects or drawings, and record each composition or decomposition by a drawing or equation (e.g., 18 = 10 + 8); understand that these numbers are composed of ten ones and one, two, three, four, five, six, seven, eight, or nine ones.

1.NBT.2 Understand that the two digits of a two-digit number represent amounts of tens and ones. Understand the following as special cases:
a. 10 can be thought of as a bundle of ten ones—called a “ten.”
b. The numbers from 11 to 19 are composed of a ten and one, two, three, four, five, six, seven, eight, or nine ones.
c. The numbers 10, 20, 30, 40, 50, 60, 70, 80, 90 refer to one, two, three, four, five, six, seven, eight, or nine tens (and 0 ones).
What is important to note here is the slight difference in these two standards. In Kindergarten, students are thinking of numbers 11 through 19 as "ten ones and some further ones" while in Grade 1, students need to develop an understanding of 10 as "a 'ten.'" In other words, in Grade 1, students need to develop 10 as a unit - at the same time it is a collection of ten ones. Research has shown that this understanding is a major shift, and some might argue that this expectation is not developmentally appropriate for most first graders. Children can easily learn to recite the number word sequence, "ten, twenty, thirty, ... ninety," but just as simply reciting "one, two, three, four, ..." does not necessarily indicate an understanding of numbers, the ability to recite the decade number words in order does not indicate the understanding of ten as a unit (1.NBT.2.c).

In historical numeration systems, the idea of grouping by 10's, 100's, etc. appears fairly early. In those systems, 20, 30, 40, ... were recorded with multiple symbols of 10's instead of saying how many 10's. Even in the systems that utilized place values like the Babylonian System, 20, 30 , 40 were recorded with multiple symbols of 10's just as simpler additive systems did. Thus, even in those systems, 30, for example, meant 10+10+10, not three 10's (or 3x10). This shift, although it might look rather simple for those of us who already understand the base-10 numeration system, is not that obvious for children. For them, "10" doesn't naturally mean 1 tens and 0 ones. Rather it is just like a word "cat" spelled with multiple letters. "10" is just "ten" spelled with 2 numerals "1" and "0." Thus, it is not logical that twenty should be spelled as "20" - even if they understand twenty is made up of 2 tens. After all, there is no logical connection (in how they are written) going from 1 to 2 ones. Although this standard puts this understanding of ten as a unit in focus, we should keep in mind that students will not develop this understanding in one single lesson. In fact, this understanding will probably take months to develop - perhaps stretching into Grades 2 and 3. We should keep this in mind as we look at other NBT standards in Grade 1 - they are, in part, serving to achieve this standard even though their focus may be elsewhere.

Kindergarten: Operations and Algebraic Thinking (2)

2011-07-25T17:28:00.000-07:00

Kindergarten: Operations and Algebraic Thinking (2)
In the previous post, I discussed the difference in the meaning of subtraction between the CCSS and the current GPS and its potential implications. In this post, I would like to begin the discussion of the five specific standards in the cluster. Those standards are as follows:

1. Represent addition and subtraction with objects, fingers, mental images, drawings, sounds (e.g., claps), acting out situations, verbal explanations, expressions, or equations.
2. Solve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects or drawings to represent the problem.
3. Decompose numbers less than or equal to 10 into pairs in more than one way, e.g., by using objects or drawings, and record each decomposition by a drawing or equation (e.g., 5 = 2 + 3 and 5 = 4 + 1).
4. For any number from 1 to 9, find the number that makes 10 when added to the given number, e.g., by using objects or drawings, and record the answer with a drawing or equation.
5. Fluently add and subtract within 5.

There is a footnote for the term "drawings" in Standards 1. The footnote states, "Drawings need not show details, but should show the mathematics in the problem." It is easy to read this statement and think that we are making things simpler for children since we are asking them to do less (not showing details). However, for those of us who work with primary students know that this is not quite as simple as it may sound. In fact, many of us have experiences of watching children draw very detailed drawings when they are asked to "draw pictures" to help them solve word problems. In order for children to draw pictures that "show the mathematics in the problem," children must understand first what features of problems are and are not relevant to the mathematics in the problem. If children are drawing pictures to help them solve word problems, they may not understand what the mathematics in the problem is. If so, how can they know what features are or are not relevant to the mathematics? Thus, helping children become able to draw pictures that "show the mathematics in the problem" is itself a major teaching goal in Kindergarten. At the same time, we also want to help students develop an understanding/disposition that drawings are useful thinking tools. So, how might we achieve this goal? One potentially useful strategy used in many Japanese elementary school mathematics textbooks is to use problem contexts in which objects in the problems are fairly simple objects. Thus, when children draw their pictures, drawings will not be overly complicated. Moreover, it will be useful for children to share their drawings. By examining and reflecting on different drawings their friends made, children can begin to think what features of their drawings are essential for doing mathematics.

Another major change from the GPS to the CCSS is the idea of representing with equations. The GPS does not emphasize the formal representations with numerals and mathematical symbols in Kindergarten. However, the CCSS begins the use of the formal/symbolic representations in Kindergarten. Some people may disagree that such an expectation is developmentally appropriate. However, the expectation is there, and we must teach Kindergarteners about the formal representations. As we do, I hope we will emphasize both representing and interpreting. Thus, not simply asking children to represent addition or subtraction situations using equations, we should ask them to come up with different situations for a given equation. Moreover, even from the beginning, we should remember that the "=" sign indicate that the two quantities on both sides are equal, not "calculate." Thus, from time to time, we should write "5 = 3 + 2," not just "3 + 2 = 5."

2011-02-23T06:05:00.000-08:00

Kindergarten: Operations and Algebraic Thinking (1)

In the domain of Operations and Algebraic Thinking in Kindergarten, there is only one cluster - "Understand addition as putting together and adding to, and understand subtraction as taking apart and taking from." This cluster statement makes it quite clear what meaning Kindergarteners are to give to the arithmetic operations of addition and subtraction. The current GPS (MKN2a) states, "Use counting strategies to find out how many items are in two sets when they are combined, separated, or compared." Table 1 of the CCSS explain what is meant by "putting together," "adding to," "taking apart," and "taking from." In my previous post on MKN2a (link), I discussed how the GPS's classification was based on the framework developed by the Cognitively Guided Instruction (CGI). The categories used by the CCSS are comparable to the CGI categories as well, but labeled differently. Thus, "adding to" is equivalent to "combine," "taking from" is equivalent to "separate." "Putting together" and "taken apart" are the "part-part-whole" category of the CGI - with the "putting together," the whole is unknown while in "taken apart," a part is unknown.

At this point, one major difference between the CCSS and the current GPS should be obvious. In the CCSS there is no comparison meaning of subtraction is addressed in Kindergarten. Instead, the CCSS includes the part-unknown case of the part-part-whole structure for subtraction. How significant is this difference? This might turn out to be a pretty significant difference. One of the findings from the CGI research is that children approach these word problems using different strategies - usually counting and/or direct modeling of the problem situations as the first step. Gradually, children will move toward the strategies that involve more advanced counting or the use of previously learned facts.

An example of "taken apart" problem included in the Appendix is this:
Five apples are on the table. Three are red and the rest are green. How many apples are green?From an adult's perspective, we might think it is simple to use counting or direct modeling for this problem. You just need to count on from 3 till you reach 5, or start counting from 5 down to 3. However, I discussed in the previous post that counting-on requires a major cognitive development. Moreover, the CGI research seems to show that "counting down to" is a more advanced counting strategy (than simply counting back 3 times). In order to model the situation, children must be able to anticipate the result - you can't start with 5 because it is made up of the known quantity and the unknown quantity. Thus, that is a more advanced thinking as well.

Comparison problems, on the other hand, are easier to model. Children can model both quantities, and they can make one-to-one correspondence between the two groups. The ones without matches are the difference. So, from a developmental perspective, comparison situations seem to be more "primitive" type. When mathematics educators discuss subtraction, we often talk about three different ways we can think of subtraction: subtraction as a take away, subtraction as comparison, and subtraction as missing addend. The "taken apart" (or part-part-whole with part unknown) seems to relate more to the last type, and we can see, from other standards, the CCSS emphasizes that way of thinking subtraction. Perhaps a careful and thoughtful teaching with that focus might help students make the necessary cognitive advances. But, I think it is critical teachers are aware of the non-trivial challenges students are expected to overcome.

Kindergarten: Counting and Cardinality (K.CC)

2011-02-04T18:43:00.000-08:00

Kindergarten: Counting and Cardinality (K.CC)

In the Counting and Cardinality domain in Kindergarten, there are 7 standards in 3 clusters (Know number names and count sequence; Count to tell the number of objects; and Compare numbers). Those standards are as follows:1. Count to 100 by ones and by tens.
2. Count forward beginning from a given number within the known sequence (instead of having to begin at 1).
3. Write numbers from 0 to 20. Represent a number of objects with a written numeral 0-20 (with 0 representing a count of no objects).
4. Understand the relationship between numbers and quantities; connect counting to cardinality.a. When counting objects, say the number names in the standard order, pairing each object with one and only one number name and each number name with one and only one object.
b. Understand that the last number name said tells the number of objects counted. The number of objects is the same regardless of their arrangement or the order in which they were counted.
c. Understand that each successive number name refers to a quantity that is one larger.5. Count to answer “how many?” questions about as many as 20 things arranged in a line, a rectangular array, or a circle, or as many as 10 things in a scattered configuration; given a number from 1–20, count out that many objects.
6. Identify whether the number of objects in one group is greater than, less than, or equal to the number of objects in another group, e.g., by using matching and counting strategies.
7. Compare two numbers between 1 and 10 presented as written numerals.
Compared to the current GPS, there are some similarities, but there are some differences, too. Like the current GPS, the CCSS expects students to write numbers up to 20 and be able to compare two (or more) sets - the CCSS has an additional expectation that students be able to compare two written numbers (between 1 and 10) without actual objects. One difference that might stand out is that the CCSS expects students to be able to count up to 100 by ones and tens while the current GPS expects students to be able to count up to 30 objects in Kindergarten. In the current GPS, the range of numbers is expanded to 100 in Grade 1, as well as counting by ones and tens. In contrast, in the CCSS the range of numbers are expanded to 120 in Grade 1. On the surface, this difference (up to 30 or up to 100) appears rather significant. On the other hand, there is an obvious number word patterns in counting from 20 through 99. So, from a language perspective, this difference might not be too significant - other than learning additional number words for 40 through 90 and 100.

Perhaps a bigger question is what is meant by the phrase, "by ones and tens." The CCSS does not provide any elaboration, but if this is limited to simply knowing the decade number words (ten, twenty, thirty, ... ninety) in sequence, it is probably not a major concern. However, the CCSS expects students to be able to count beginning with numbers other than 1. If this expectation also applies to counting "by tens," then that may not be developmentally appropriate. This idea (start counting from number other than one, or counting on) involves a major cognitive development. For many young children, numbers exist only as a result of counting. Thus, numbers do not exist without counting from 1. In order to start counting from numbers other than 1 meaningfully, or to count on from a given number, require a different way of understanding of numbers. Moreover, research seems to be clear that understanding of ten as an iterable unit is a major step that even some 2nd graders are not ready to make. I hope that there will be further elaboration and articulation of what these standards are expecting in terms of children's understanding of ten.

The CCSS seems to articulate various aspects of counting much more explicitly and in details (Standard 3). These ideas are implicit in the GPS as I discussed this matter previously (here). However, the CCSS does not appear to place much emphasis on counting (other than expanding the range of numbers to 120) in Grade 1. However, I believe counting is not something children just "master" in one grade level. Rather, it should be an important activity in primary grades for children to build number concepts. Although we do not want children to become dependent on counting to complete simple arithmetic, counting is nevertheless an important foundational activity for children to construct their number concepts. So, I hope primary grade teachers will continue to engage their students in appropriate counting activities.

Mathematical Practice

2011-01-08T10:28:00.000-08:00

Mathematical Practice

As I mentioned previously, the State of Georgia has adopted the mathematics standards developed by the Common Core State Standards Initiative. These Standards will become the new state standards starting in the school year 2012-13. So, in this blog, I will try to discuss the specific CCSS standards and compare/contrast with the current GPS.

In the GPS, there are two sets of standards: content standards and process standards. The process standards are the five standards that are discussed at the end of each grade and relate directly to the process standards discussed in the NCTM Standards - Problem Solving, Reasoning, Connection, Communication, and Representation. The CCSS mathematics standards, in contrast, include a set of standards on mathematical practice. According to the CCSS, mathematical practice is a variety of "expertise that mathematics educators at all levels should seek to develop in their students," and the eight expertise are:
1. make sense of problems and persevere in solving them
2. reason abstractly and quantitatively
3. construct viable arguments and critique reasoning of others
4. model with mathematics
5. use appropriate tools strategically
6. attend to precision
7. look for and make use of structures
8. look for and express regularity in repeated reasoning
Some of the items in this list sound very similar to the current GPS process standards while others appear to be new and different. For example, the idea of persevering to solve problems is not explicitly stated in the current GPS, but if students were to learn from problem solving, it is essential that students persevere. On the other hands, some of the current GPS process standards are much more obviously related to the eight expertise while others may appear to be forgotten. However, a more detailed look at the mathematical practice does suggest that even those standards are still important. For example, the connection standards seem to be absent from the list of mathematical practice. However, the description of "modeling with mathematics" include the following:
Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose.
These descriptions of "mathematically proficient" students clearly suggest students must be able to connect their understanding of mathematics to things both within and outside of mathematics, and both within and outside of classrooms.
One of the main concern as we move forward with the CCSS is that these standards on mathematical proficiency will receive less attention just as the process standards of the current GPS do. In some ways, it is understandable as it is rather difficult to imagine these mathematical practice standards in action. Moreover, it is not quite clear how these standards will be assessed. Thus, it is natural for some teachers to focus on things that will be assessed. The authors of the CCSS, however, offers a suggestion that can guide us as we grapple with the content standards:
Expectations that begin with the word “understand” are often especially good opportunities to connect the practices to the content. Students who lack understanding of a topic may rely on procedures too heavily. ... In short, a lack of understanding effectively prevents a student from engaging in the mathematical practices.
In this respect, those content standards which set an expectation of understanding are potential “points of intersection” between the Standards for Mathematical Content and the Standards for Mathematical Practice. These points of intersection are intended to be weighted toward central and generative concepts in the school mathematics curriculum that most merit the time, resources, innovative energies, and focus necessary to qualitatively improve the curriculum, instruction, assessment, professional development, and student achievement in mathematics.
As I continue discussing the specific standards, I will try to keep this suggestion in mind. I would also like to encourage you to keep thinking about the mathematical practice standards as we go through this time of transition.

M2N3a - Writing multiplication equations

2010-11-02T04:09:00.000-07:00

M2N3. Students will understand multiplication, multiply numbers, and verify results.
a. Understand multiplication as repeated addition.Now that Georgia has adopted the mathematics standards developed by the Common Core State Standards Initiative, I will incorporate the CCSS in my discussion of Georgia standards. I have previously discussed M2N3a in a June 2007 post. In that post, I raised an issue of treating multiplication as repeated addition.

In the CCSS, multiplication is introduced in Grade 3 in the domain of "Operations and Algebraic Thinking." The first standard in the cluster related to multiplication, the CCSS states the following:
1. Interpret products of whole numbers, e.g., interpret 5 × 7 as the total number of objects in 5 groups of 7 objects each. For example, describe a context in which a total number of objects can be expressed as 5 × 7.
In the public draft released in spring, 2010, there was a statement about multiplication as repeated addition, just like M2N3a. However, that statement has been removed. Instead, the CCSS focuses on the meaning of multiplication as an operation to find the total amount when objects are arranged in equal groups. Then, in Grade 5, the CCSS states that the meaning of multiplication to be expanded and consider multiplication as scaling (or re-sizing). I think the approach the CCSS has taken is much more appropriate than what the current GPS states. For those of you interested in the discussion of whether or not multiplication is repeated addition, I encourage you to read a series of columns written by a Stanford University mathematician, Keith Devlin (June 2008, July-August 2008, and September 2008).

Today, however, I would like to focus on the implicit idea in the CCSS - and the current GPS does not even touch upon this idea. Toward the end of my previous blog, I discussed the order in which you write multiplication sentences. In it, I made it clear that my preference is to write the multiplicand, i.e., the number of objects in a group, first then the multiplier - I might argue that is THE correct way mathematically. However, the CCSS actually suggests we write multiplication sentences in the opposite order. Thus, 5x7 is interpreted as "the total number of objects in 5 groups of 7 objects each." Although I have stated in the past that what is important is we have an agreement on the order, I have run into several situations recently that revealed writing the multiplicand first is the way to go.

But, let's first start with how we state/write/read multiplication sentence. A common way teachers and students read multiplication sentence is "5 times 7 is 35." However, if the sentence is representing the situation with 5 groups of 7 in each, a mathematical way of reading the sentence is "7 multiplied by 5 is 35." When we use the phrasing, "N is multiplied by M," it is clear that M is the number of groups - that is, N is taken M times. Thus, one surface level issue is that the order in which we read multiplication sentences and how they are written may not align. Some might argue that this is a non-issue. After all, the same thing happens with division, too. We say "35 divided by 7," but we also say, "how many times does 7 go into 35?" When we write division problem on paper, the divisor may follow the division symbol or it may be outside of the long division symbol (thus to the left of the dividend).

To me, however, the issue is fundamental, and writing the multiplier first creates some difficulties in mathematical discourses. Let me share some examples. In the 4th grade CCSS standard, student are expected to understand multiplication of fractions by whole numbers. The CCSS document is very careful to remain consistent with the order, so in the examples they include always have the multiplier in front, such as 3 x (2/5). Once we agree that we write the multiplier first, problems such as (2/5) x 3 are treated in Grade 5. [There is actually a similar distinction with respect to multiplication and division of decimal numbers in the current GPS. Students learn about multiplying and dividing decimal numbers by whole numbers in Grade 4, and multiplication and division by decimal numbers are discussed in Grade 5.]

Then, in Grade 5, the CCSS treats multiplication by fractions: "Interpret the product (a/b) × q as a parts of a partition of q into b equal parts; equivalently, as the result of a sequence of operations a × q ÷ b." So, (2/3) x (4/5) is interpreted as taking 2/3 of 4/5. Thus, we first divided 4/5 by 3 (thus students must have learned how to divide a fraction by a whole number before this topic), then we take 2 groups of it. Thus (2/3) x (4/5) = 2 x (4/5 ÷ 3). Thus, the multiplier, 2/3, gets split around the multiplicand, 4/5. If you write the multiplicand first, taking 2/3 of 4/5 will be written as (4/5) x (2/3) = (4/5) ÷ 3 x 2 = (4x2)/(5x3). This seems to be much easier to connect to the formula (a/b)(c/d) = ac/bd. [Another issue here is why there is no parentheses around q ÷ b in the CCSS. It seems like you must first find the "partition of q into b equal parts," but the order of operations says we go from left to right. Without parentheses, the statement "a x q ÷ b" means multiply q by a, then partition the result into b equal parts.]

Another example is when discussing how to create equivalent fractions. Again, the CCSS is very careful about the order in which multiplication is written. Thus, they say, "Explain why a fraction a/b is equivalent to a fraction (n × a)/(n × b) by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size." Typically, this idea is discussed as "if you multiply both the numerator and the denominator of a fraction by the same number, the value of fraction remains the same." More often than not, this relationship is written in textbooks as a/b=an/bn - the multiplier written after the multiplicand. But, this notation would not match our agreement on how to write multiplication.

One final example is not explicitly mentioned in the CCSS nor the GPS. However, once we learn division by fractions, we often makes the statement, "division is the same as multiplication by the reciprocal of the divisor." When we use mathematical notations, we will typically write (a/b) ÷ (c/d) = (a/b) x (d/c). However, if division is the same as multiplication "by the reciprocal," that is, the reciprocal is the multiplier, it should really be written as (a/b) ÷ (c/d) = (d/c) x (a/b).

In general, many familiar ways we discuss/write multiplication assumes that the multiplier comes after the multiplicand. Thus, "5x7" should not be "5 times" of 7, but rather 5 "7 times." Perhaps because I am not a native English speaker, I also get a different sense when I hear "5 times 7" in one breath and "5" (pause) "times 7." The former gives me the sense of 5 groups of 7 but the latter makes me think of 7 sets of 5. Anyway, I wish we can eventually agree to write the multiplicand first - perhaps in the next revision of the CCSS.

By the way, some people might wonder about how this idea works in algebra. After all, when we think of simplifying "5x + 3x," it is much easier to think of this as having 5 x's and 3 x's altogether, thus 8 x's. On the other hand, we want students to understand the slope of a linear function, like "3" in y=3x+2, as the "rate of change." A rate, however, is typically amount per unit. Thus, "3x" in this context suggests we have x units of 3. In algebra (and other higher level mathematics), I believe there are actually two competing conventions. One is the order in which we write multiplication and the other is the convention of writing numbers (and constants) before variables in a term. In algebra, I believe, the latter convention wins perhaps because it makes manipulation of algebraic expressions simpler. But, I think it is still very important that students pay close attention to how we write multiplication when they first learn it in elementary grades.

M6M4 ab: Surface area formulae

2010-10-21T03:31:00.000-07:00

M6M4. Students will determine the surface area of solid figures (right rectangular prisms and cylinders).
a. Find the surface area of right rectangular prisms and cylinders using manipulatives and constructing nets.
b. Compute the surface area of right rectangular prisms and cylinders using formulae.Surface area is simply the sum of the area of all the faces of a solid. Thus, as long as we can calculate the area of each face, there is nothing really new involved in this standard. The only tricky part here is to figure out the shape of the lateral face of a cylinder. But, students learn about the nets of various solids including cylinders in Grade 4. Thus, the surface area of a cylinder is the sum of the area of the two bases (circles) of the cylinder, and the area of the lateral face which is the rectangle with the dimensions equal to the height of the cylinder and the circumference of the base. We can summarize it in a formula like this:

Surface Area = 2 x (Area of base) + (Circumference of the base) x Height

Although I don't think it is that critical that students know this formula or the formula for prisms, it may be useful to have students explore the surface area of (rectangular) prisms not just as the sum of the areas of the faces. In fact, the first indicator discusses the use of nets in determining the surface area. If the surface area is simply the sum of the areas of the faces, there is really no need to use a net. So, what might be the reason for using nets to calculate the surface area of (rectangular) prisms?

We know that there are many different nets for a prism. However, a common net of a prism has all the lateral faces forming a "train" of rectangles and the two bases on the opposite sides of this "train" like this one.

Instead of calculating the area of each of the faces, you can consider the "train" of the lateral faces as one big rectangle, like this:

The length (vertical side in the drawing above) is equal to the height of the prism. The width (horizontal side) is actually the perimeter of the base. Thus, we can calculate the sum of the areas of the lateral faces as (Perimeter of the base) x height, too. But, then, the calculation of the surface area of a prism can be summarized in this formula:

Surface Area = 2 x (Area of base) + (Perimeter of base) x Height.

Perhaps investigating the surface area of prisms from this perspective allows us to use the same formula for all prisms and cylinders. However, I still don't think it is that important for students to know the formula...

2010-09-21T18:34:00.000-07:00

M6M3. Students will determine the volume of fundamental solid figures (right rectangular prisms, cylinders, pyramids and cones).
a. Determine the formula for finding the volume of fundamental solid figures.
b. Compute the volumes of fundamental solid figures, using appropriate units of measure.
There is actually a standard in Grade 5 that discusses the volume:
M5M4. Students will understand and compute the volume of a simple geometric solid.
c. Derive the formula for finding the volume of a cube and a rectangular prism using manipulatives.
d. Compute the volume of a cube and a rectangular prism using formulae.So, what are the difference between these two standards? There are two obvious differences in these two standards. First, the Grade 5 standard involves the volume of "simple geometric solids," while the Grade 6 standard deals with "fundamental solid figures." Specifically, in Grade 6, students are expected to determine the volume of cylinders, pyramids, and cones in addition to cubes and rectangular prisms learned in Grade 5. So, the Grade 6 standard deals with a wider range of solids than the Grade 5 standard does.

Another difference is that, in Grade 5, students are to derive the formula using manipulatives while the Grade 6 standard does not mention the use of manipulatives. So, how do we expect Grade 6 students to derive the formula?

In Grade 5, students may determine the volume of cubes and rectangular prisms by filling them with unit cubes. Those experiences parallel what students might have done as they determine the area of squares and rectangles using unit squares. From these experiences, students learn that the dimensions of cubes and rectangular prisms can tell us the number of unit cubes that fit in each dimension. Thus, they can conclude that the volume of a rectangular prism can be calculated by multiplying its length, width and height.

The solids students explore in Grade 6 cannot be filled with unit cubes because of their shapes. So, how can students determine the formula for those solids? One important step is to re-visit the formula for the volume of cubes and rectangular prisms. When we determine the number of unit cubes inside a rectangular prism, we typically figure out the number of unit cubes in one layer, then multiply the result with the height, which signifies the number of layers. However, the first product, the number of unit squares in a single layer is equal to the area of the rectangular base. Thus, we can express the formula for calculating the volume of a rectangular prism as (Area of Base) x height, instead of length x width x height.

When we consider the volume formula for a rectangular prism as (Area of Base) x height, a natural question is whether or not this formula can be applied to prisms whose bases are something other than rectangles. Students can explore this question with triangular prisms and other prisms. Through such an exploration, they will find that the formula applies to any prism - and cylinders.

The volume formula for pyramids (and cone) is slightly different. It may be difficult to derive the volume formula for pyramids/cones directly. In fact, what we need to do is to relate the volume of a pyramid/cone to the related prism/cylinder, which has the congruent base and the same height as the pyramid/cone. A common way to establish this relationship is to have students actually fill up both a pyramid and the related prism (there are commercially made sets available for this purpose) with water or rice grains. Through such experimentations, students can establish the relationship that the volume of a pyramid/cone is a third of the volume of the related prism/cylinder. Thus, the volume formula for a pyramid is simply (Area of the base) x height ÷ 3 - if students have already learned multiplication of fractions before this unit, the formula can be written as (1/3) x (Area of the base) x height.

It may be useful to have students actually cut out (or the teacher demonstrate cutting) a cube into 3 congruent square pyramids like this - I apologize the poor quality of my 3-D drawing, and I hope you get the idea from this picture.

Note that these pyramids are different from most pyramids students seen in K-8 curriculum. Pyramids students study typically has the vertex that is not on the base to be directly above the center of the base. These pyramids, in contrast, has the vertex directly above one of the vertices of the base.

Clearly, such a demonstration does not establish the 1:3 relationship of the volume of any pyramid to the volume of the related prism. However, it may still be a worthwhile experience for students to have. There is, I believe, a commercially made puzzle that asks you to make a cube out of 3 congruent pyramids.

M7G3 - Proportional Relationships (7)

2010-09-04T06:05:00.000-07:00

M7G3. Students will use the properties of similarity and apply these concepts to geometric figures.
b. Understand the relationships among scale factors, length ratios, and area ratios between similar figures. Use scale factors, length ratios, and area ratios to determine side lengths and areas of similar geometric figures.
This standard is another example of how proportional relationships play an important role in the middle school mathematics curriculum. We say two figures are similar if one can be made to overlap the other exactly through a combination of translation (slide), rotation (turn), reflection (flip), and dilation (magnification). The parts of two similar figures that match up are called corresponding angles, sides, etc. In a pair of similar figures, we know that corresponding angles are congruent and the ratios of corresponding segments are constant - and the value of this ratio is the scale factor. For example, the two quadrilaterals shown below are similar.

Therefore, angles A and E, B and F, C and G and D and H are congruent, respectively. Moreover, the ratios of the lengths of sides, AB: EF, BC:FG, CD:GH, and DA:HE, are constant, and in this case the ratio is 1:2. The scale factor depends on which of the two figure we consider as the base of the comparison. So, if we consider quadrilateral ABCD as the base, the scale factor, in this case, is 2. On the other hand, if we consider quadrilateral EFGH as the base, the scale factor is 1/2.

Suppose AB = 4 cm, BC = 2 cm, CD = 5 cm, and DA = 6 cm. Then, EF = 8 cm, FG = 4cm, GH = 10 cm, and HE = 12 cm. Let's organize these lengths in a table.

Now, you see that as the length in ABCD doubles and triples (from 2 cm to 4 cm or 6 cm), the length of EFGH also doubles and triples. Even when the length becomes 2.5 times as long, from 2 cm to 5 cm, in ABCD, the corresponding length also becomes 2.5 times as long, 4 cm to 10 cm. Thus, the lengths in these two figures are in a proportional relationship. In general, if two figures, X and Y, are similar, the lengths in these two figures are in a proportional relationship. Thus, we can apply all the tools we discussed previously in representing this relationship. So, if we use a double number line, the relationship can be represented something like this:

Thus, if we know a side in Figure X is 15 cm and the corresponding side in Figure Y is 6 cm, we can use that relationship to determine the length of any side can be determined if we know the length of the corresponding sides. Suppose, we know another side in Figure X is 20 cm, the relationship can be represented in a double number line like this:

On the other hand, if you know the length of a side in Figure Y is 4 cm, the relationship will be represented like this:

Another feature of a proportional relationship is that the quotients of corresponding quantities are constant. So, if we divide the lengths in Figure X by the corresponding lengths in Figure Y, the quotients are constant. We can also use this relationship to represent the two situations above like this:

These tables basically show the four values (including the missing value represented by a ?) from the double number line representations above. Note that in the second table, the columns are in the reverse order. A number line has a particular direction, i.e., as you move to the right, the numbers become larger, However, a table does not have such an inherent directionality. So, for students, it might be more natural if we place the relationship as they are presented.

In any event, since 15 x 0.4 = 6, ? = 20 x 0.4 -- 0.4 is the scale factor if we consider Figure X as the base. For the second problem, we can say that since 6 x 2.5 = 15, ? = 4 x 2.5 -- 2.5 is the scale factor if we consider Figure Y as the base.

As is the case with the conversion of measurements from one unit to another, what is important is to help students develop an understanding that mathematics is a web of relationships. The focus of this standard is not just for students to find the missing lengths in similar figures. We also want them to understand that what they have learned previously, namely proportional relationships, can be used to represent, interpret, and investigate new situations.

M6M1 - Proportional Relationships (6)

2010-08-14T05:13:00.001-07:00

M6M1. Students will convert from one unit to another within one system of measurement (customary or metric) by using proportional relationships.

Some people consider the idea of proportional relationship as the culmination of the elementary school mathematics and the cornerstone of the middle school mathematics. This standards is one example of how proportional relationships play a role in different parts of the middle school mathematics.

Let's think about a situation of converting linear measurements between inches and feet.

You can easily see that as the numbers for inches become 2, 3, 4,... times as much, the numbers for inches also become 2, 3, 4, ... times as much. Therefore, these numbers are in a proportional relationship. Thus, we can use all the tools we have discussed previously to convert from one unit to another.

Suppose you want to find out how many inches 34 feet may be, you can set up the double number line representations in this way.

This representation shows that we know the per-one (or per-unit) quantity and you want to know the number corresponding to 34 units. So, you can use multiplication to find the missing number: ? = 12 x 34.

Going the other direction, for example, converting 104 inches to feet, can be represented in the same way.

Again, we know the per-unit quantity, and you want to know how many units would correspond to 104. Thus, this is a quotitive (measurement) division situation. So, you can find the missing number by division: ? = 104 ÷ 12.

To solve all these unit conversion problems, students do need to know (or be able to look up) one relationship between the two units - and it doesn't have to be 1 to something else. If you know that 2 feet = 24 inches, that's good enough to set up a double number line representation. You can solve it like you do with other proportion problems.

In principle, the situation remains the same whether you are converting within or across different measurement systems. If you know that 1 inch is approximately 2.5 cm, that is enough information for students to convert between inches and centimeters - approximately, but all measurements are approximation, anyway. Although students in earlier grades should be able to convert measurements from one unit to another in some simple cases, once students learn about proportional relationships, they no longer have to think of it in isolation. The idea of proportional relationships, thus, unifies many of the ideas students have learned previously. And, helping students to revisit some of those ideas and look at them from a new perspective is something we need to emphasize, not just the procedure of solving proportional problems.

Proportional Relationships (5)

2010-07-28T05:16:00.000-07:00

M6A2. Students will consider relationships between varying quantities.
c. Use proportions (a/b=c/d) to describe relationships and solve problems, including percent problems.
While discussing "segment/tape diagrams," I discussed how those diagrams can be used to solve problems involving percents. In the last post, while discussing models for proportional problems, I discussed how double number line may be used to represent problems involving proportional relationships.

Percents describes the relative size of quantities compared to the base quantity. It turns out that "percents" and the actual quantities are in a proportional relationship. For example, suppose the base quantity is 80. The table below summarizes the relationship between the size of quantities being compared and corresponding percentages.

You can see that they are in a proportional relationship because as the quantity becomes 2, 3, 4, ... times as much, the percentages also become 2, 3, 4, ... times as much.

So, if quantities and percentages are in a proportional relationship, then we can also use double number lines to represent problems involving percents, too. So, here are 3 examples.

The first double number line representation may be for a problem like the following:
At Jackson Elementary School, there were 80 fifth grade students this year. Next year, they anticipate that the fifth grade class to be 115% of this year's fifth grade class. How many fifth graders will there be?
The second represents a problem like this:
At Jackson Elementary School, there were 80 fifth grade students this year. Next year, they are expecting 115 fifth grade students. What percents of this year's fifth grade class will the next year's class be?
Finally, the third one represents a problem like this one:
At Jackson Elementary School, they are expecting 115 fifth graders next school year. This is 115% of this year's fifth grade class. How many fifth graders are there this year?
While discussing Process Standards 5, I shared how a segment/tape diagram to represent and solve problems involving percents. The double number line is a different representation. Double number lines representing multiplication or division problems always included a "1" on one of the number lines. In these situations, there is no "1," but by placing a "1" on either number line, a solution approach that combines division and multiplication - the approach discussed in the previous entry - may become apparent.

M6A2b - Proportional Relationships (4)

2010-06-12T06:56:00.001-07:00

M6A2. Students will consider relationships between varying quantities.
b. Use manipulatives or draw pictures to solve problems involving proportional relationships.
In the last three posts, we considered two proportional situations. They are,
c) the distance traveled and the time of travel (at a constant speed), and
f) the amount of meat and the price of meatThe tables below show the values of these quantities:

"Problems involving proportional relationships" from these contexts might be something like the following:
Jim can walk 9 miles in 3 hours. If he maintained the same speed, how far can he walk in 6 hours?

4 pounds of meat cost $18. How much will 10 pounds of the same meat cost?So, what kinds of pictures might we draw to solve these problems? Actually, you may find it rather difficult to draw pictures for these problems. We can draw pictures that might represent the contexts of the problems, but those pictures may not be too helpful in actually solving the problems. What about manipulatives? What manipulatives would you use to solve these problems? I am not sure what I would use.

If it is difficult to use a picture or manipulative to solve these problems, what is this standard talking about? Perhaps "pictures" here are really referring to diagrams. One particular form of diagrams is double number lines. I used double number lines extensively to talk about multiplication and division of decimal numbers (November 2008). But they can be useful to represent problems involving proportional relationships. Here are the double number line representations of the two problems above.

When students are familiar with double number lines with multiplication and division, they will notice the difference between these double number line representations and those of typical multiplication and division problems. Here are examples of multiplication and division double number line representations:

Do you notice the difference? In the two representations of the problems involving proportional relationship, there is no "1" in the representation. If we put a "1" in the representation, then we can see a solution path. For example, let's use the second problem. If we put a "1" on the top number line, it will look like this:

Now, the left side part of the representation,

is really a partitive division situation. Thus, by dividing 18 by 4, we can find that # = 4.5. Now, double number line representation looks like this:

Now, if we can ignore the middle part of this representation, it will look like this:

and this is a multiplication situation, isn't it. So, multiplying 4.5 by 10, we can obtain the missing quantity.

So, double number line representations can not only represent problems involving proportional relationships, they can also suggest ways to solve the problems, too. Of course, if we want students to be able to use double number lines as their thinking tool at this stage, they do need to be familiar with double number lines. Thus, it is important for teachers of different grade levels to discuss what representations they want to emphasize. It is important for students to be able to use multiple representations. But, if there is any representation, like double number line, that may be used across grades, then that representation should be consistently introduced/developed/used across grades.

M6A2ae: Proportional Relationships (3)

2010-05-23T03:34:00.000-07:00

M6A2. Students will consider relationships between varying quantities.
a. Analyze and describe patterns arising from mathematical rules, tables, and graphs.
e. Graph proportional relationships in the form y = kx and describe characteristics of the graphs.
In the previous post, we analyzed the ways two quantities that are in a proportional relationship using a table. Today, we want to look at the idea of analyzing proportional relationships - and identifying features that are unique to proportional relationships - using graphs. So, let's consider one of the relationship we talked about last time:

When you graph this set of data, it will look like this:

Since these quantities are both continuous quantities, we can actually connect the data points using a line (actually a ray):

Let's compare this graph to graphs of three other situations. The first one is the siblings' ages.

The second situation is the candle situation: the length of candles burned and the length of the remaining candle.

The last case is the length and the width of rectangles with a fixed area measurement.

Since the first situation involves the discrete quantities - and since I don't know how to graph the curve for the last one, I am just going to plot these data points.

When you compare these graphs to the graphs of the proportional relationship from earlier, you immediately notice that the graph of the inverse proportional situation isn't a straight line. However, the other three situations seem to result in straight lines. Although it is not really appropriate to use a line to represent the siblings' ages data with a straight line, I'm going to do so to illustrate the similarities and differences - and I'm showing all three lines on the same coordinates.

From these graphs, we noticed that one difference seems to be that the graph of the proportional situation goes through the origin, but not the other two. As it turns out this is indeed unique to proportional situations. The other two cases, constant sum and constant difference situations, result in a straight line. One commonality among the three situations is that the rate of change is constant. Thus, the characteristic of the data sets that are represented in straight lines. I think this might be an idea that is worth discussing explicitly in Grades 7 and 8 when linear equations/functions are studied more formally.

By the way, the fact that the graphs of proportional relationships go through the origin relates to the fact that double number lines we use to represent multiplication and division situations are "hinged" at 0 - in other words, both quantities will go to 0 at the same time. In fact, proportional relationships are assumed in all multiplication and division situations. In middle grades, that fact should become explicit instead of being an implicit assumption.

6A2a: Proportional Relationships (2)

2010-05-08T14:47:00.000-07:00

M6A2. Students will consider relationships between varying quantities.
a. Analyze and describe patterns arising from mathematical rules, tables, and graphs.
In the previous post, I discussed how we can analyze situations where two quantities are changing simultaneously. From that analysis, we defined what a proportional relationship was - two quantities are in a proportional relationship if the quantities change in such a way that their quotient stays constant. This relationship may be represented as y÷x = k, or y = kx, where k is the constant.

Let's think about how else this relationship may be seen by looking at a couple of specific instances. The two proportional situations we discussed last time were:
c) the distance traveled and the time of travel (at a constant speed), and
f) the amount of meat and the price of meat
The tables below show the values of these quantities:

So, what commonalities do you notice about the way quantities are changing in these tables? One thing students might see quickly is that, in both situations, the quantity are changing by the same amount. In this case, both time and amount are increasing by 1 unit as you go from left to right. The distance is increasing by 3 miles while the price is increasing by $4.50. Of course, this observation is really the function of the way we listed these quantities. We could have skipped some instances like this:

Or, we could have listed these pairs unordered:

So, one thing student can learn, more generally about collecting and displaying data, is that when we organize them systematically, we might be able to observe patterns more easily. But, is there any relationship we can observe in these tables even if the data are not organized as neatly?

Let's look at the way the distance changes as the time goes from 4 hrs to 8 hours, 5 hours to 10 hours, 15 hours and 30 hours. In other words, what happens to the distance as the time doubles? What about the price as the amount of meat doubles? What if we the time changed from 10 hours to 30 hours, or 30 hours to 90 hours - i.e., if the time becomes 3 times as long? What if the amount of meat changes from 1 pound to 4 pounds, 2 pounds to 8 pounds, 3 pounds to 12 pounds - i.e., if the amount of meat becomes 4 times as much?

In these proportional situations, when one quantity becomes 2, 3, 4,... times as much, the other quantity is also becoming 2, 3, 4,... times as much. Let's see if that is also the case in other situations. Since the constant quotient relationships is an increase-increase situation, we really don't have to consider an increase-decrease situation. So, the only other increase-increase situation was the constant difference situation. So, let's look at the ages of two siblings shown in the table below:

So, when Ariel becomes twice as old, will Desmond also becomes twice as old? For example, if Ariel's age goes from 10 years old to 20 years old, what happens to Desmond's age. When Ariel is 10 years old, Desmond is 13 years old. That tells us that Desmond is 3 years older than Ariel. So, when Ariel is 20 years old, Desmond will be 23 years old. Clearly 23 is not the double of 13. So, what we noticed about the proportional relationships above is indeed unique. In fact, in most, if not all, Japanese textbooks, proportional relationship is defined using this characteristic: Two quantities are in a proportional relationship if as one quantity becomes 2, 3, 4, ... times as much, the other quantity also becomes 2, 3, 4, ... times as much.

In the same way, Japanese textbooks define inverse proportional relationships this way: Two quantities are in an inversely proportional relationship if as one quantity becomes 2, 3, 4, ... times as much, the other quantity becomes 1/2, 1/3, 1/4, ... times as much. As I stated last time, it is important that students compare and contrast these various situations from the same angle so that they can identify what characteristics are unique to proportional relationships. So, I think it would be useful for students to analyze a variety of situations from this particular perspective, i.e., when one quantity becomes 2, 3, 4, ... times as much, what happens to the other quantity.

M6A2 Proportional Relationship (1)

2010-04-25T05:51:00.000-07:00

M6A2. Students will consider relationships between varying quantities.
a. Analyze and describe patterns arising from mathematical rules, tables, and graphs.
d. Describe proportional relationships mathematically using y = kx, where k is the constant of proportionality.
There are many quantities around us that vary in relationship to each other. For example, here are some examples of pairs of quantities that vary simultaneously:
a) ages of two siblings on January 1 each year
b) the number of pages of a book that have been read and the number of pages to be read
c) the distance traveled and the time of travel (at a constant speed)
d) the speed and the time it takes to travel a fixed distance
e) the length of a candle that has been burned and the remaining length
f) the amount of meat and the price of meat
g) the length and the width of a rectangle with a fixed area
h) time of the day in Atlanta and Los Angels
Let's look at these situations a little more carefully. How are the ways the quantities change similar or different? One thing you notice is that in situations a, c, f, and h, as one quantity increases the other also increases. We can cal these increase-increase situations. In contrast, in situations b, d, e, and g, as one quantity increases, the other decreases. So at one level, we can sort these situations into increase-increase and increase-decrease situations.

But, let's dig a little deeper. Let's look at each group more carefully. How are the ways quantities changing different from each other? Let's look at the increase-decrease situations (b, d, e, and g) first. As the two quantities in each of these situations change, is there anything that is not changing - mathematically, the idea of "invariance" is a very important one. You notice that in situations b and e, the sum of the two quantities remain the same. For example, the total number of pages in a book is the sum of the number of pages already read and the number of pages to be read. In contrast, in situations d and g, what stays constant is the product of the two quantities, the distance traveled in d and the area in g.

Now, let's look at a, c, f, and h. As the two quantities in each situation change, is there anything that is staying the same. In situations a and h, what stays the same is the difference between the two quantities. For example, the difference between the ages of two siblings on January 1 will always be the same no matter how old they become. In contrast, in situations c and f, what stays the same is the quotient of the quantities.

So, these situations can be sorted into four categories based on what stays constant in each situation: constant sum, constant difference, constant product, and constant quotient. Based on this way of sorting, we can also express the relationship between the two quantities using mathematical equations in the following ways (k is a constant):
constant sum: x + y = k
constant difference: x - y = k
constant product: x*y = k
constant quotient: y÷x=k
Of these four ways two quantities change simultaneously, we call the last situation, i.e., constant quotient, a proportional relationship. This relationship can be written in mathematical equation as y÷x = k, or y = kx (M6A2 d & e). Moreover, the constant product relationship, xy = k, or y = k÷x, is called an inverse proportional relationship.

When we want students to understand a new concept, it is very important and useful if we provide situations to compare and contrast several cases - examples and non-examples. Clearly there are many other quantities that change in relationship to each other that do not necessarily fit into these four categories - for example, the amount of time you study for a test and your score on a test. Thus, restricting the situations to examine to these four types may be a bit arbitrary. However, sometimes we may want to investigate only those situations that will allow us to analyze them in a particular way. It does not mean that we should investigate other, more messy situations. However, we may not need non-examples that are too complicated.

M7N1c - Integers

2010-04-13T03:01:00.000-07:00

M7N1. Students will understand the meaning of positive and negative rational numbers and use them in computation.
c. Add, subtract, multiply, and divide positive and negative rational numbers.
I usually don't venture into the 6-8 standards. But, since we discussed the compensation strategies recently, I thought I would discuss how two of those strategies can be used to derive the methods of calculations with integers.

Recall that the equal addition principle of subtraction states that if we add (or subtract) the same number to both the minuend and the subtrahend, the difference stays the same. Thus, 93 - 18 = (93 + 2) - (18 + 2) = 95 - 20. Another property of subtraction students encounter early on is that subtracting 0 will not change the number, that is A - 0 = A. By combining these two properties of subtraction, we can think about a problem like 8 - (-3) this way:
"We know subtracting 0 does not change the number. So, what can I do to change the subtrahend (-3) to 0? Add 3. But, the equal addition principle of subtraction says I have to add the same number to the minuend to keep the difference the same. So,
8 - (-3) = (8 + 3) - (-3 + 3) = (8 + 3) - 0 = 8 + 3."
Thus, you can see that subtracting a negative number is the same as adding the opposite.

We noted that there is a parallel between the compensation strategies for subtraction and division. We can actually use the equal multiplication principle of division to think about division of fraction problems, by combining it with another parallel property, dividing by 1 does not change the number. So, if you are given 3/5 ÷ 2/3, we can think like this:
"We know dividing by 1 does not change the number. So, how can we change (2/3), the divisor, into 1? Multiply by its reciprocal, of course. But the equal multiplication principle of division says I will have to multiply the dividend by the same number, too. So,3/5 ÷ 2/3 = (3/5 x 3/2) ÷ (2/3 x 3/2) = (3/5 x 3/2) ÷ 1 = 3/5 x 3/2."
Thus, we see that the division of fractions is the same as the multiplication by the reciprocal of the divisor.

Of course, strictly speaking, there is a minor glitch in both of these arguments. We established the four compensation strategies with whole numbers. But, we don't know if they still hold if we expand the range of numbers to integers/rational numbers. So, there is a circularity in these arguments. So, I'm not advocating these strategies to establish the computation algorithms, specially since there are other ways where students can meaningfully develop algorithms. However, I think these mathematical relationships are still interesting.

M*P5: Tape diagrams

2010-04-01T17:47:00.000-07:00

M*P5. Students will represent mathematics in multiple ways.

In the recently released draft of the Common Core Standards, there is a noticeable emphasis on linear models such as number lines. I have discussed how many Japanese textbooks use double number lines to discuss multiplication and division of fractions, as well as some proportional problems. However, the Common Core Standards also include a model that is called "tape diagram." In their glossary, "tape diagrams" is defined as follows:
Drawings that look like a segment of tape, used to illustrate number relationships. Also known as strip diagrams, bar models or graphs, fraction strips, or length models."
In an earlier post, I discussed how a tape diagram may help children represent addition and subtraction situations. The primary purpose of such diagrams is to help students decide the appropriate operation, that is addition or subtraction. However, Japanese textbooks also use tape diagrams, or segment diagrams, to deal with problems in upper grades, too.

Consider a problem like this one:
A fifth grade class counted the number of cars that went by the front entrance of the school between 9 o'clock and 10 o'clock. The total number of cars counted were 156. There were 3 times as many passenger cars as trucks. How many passenger cars and how many trucks were counted?
For this problem, you can use a diagram like the following:

From this diagram, we can see that the total number of cars are made up of 4 equal segments, one of which is equal to the number of trucks and the other three are equal to the number of cars. Since the four segments are equal, we can divide 156 by 4 to find out how many cars each segment represent.

Here is another problem:
There are 3.5 times as many fifth graders at School A as School B. There are 115 more students at School A than at School B. How many students are there at School A and at School B?
This problem can be represented as follows:

From this diagram, students can determine that 115 is made up of 5 equal segments since the last short segment is a half of the other segments, each of which is equal to the number of students at School B. So, 115÷5=23 represents a half of School B. Thus, the number of students at School B is 46 students. The number of students at School A is 23x7=161.

You might notice that these problems can be easily solved if we use algebra, but having diagrams such as tape/segment diagrams, students can develop the foundation for solving these problems algebraically.

There are other types of problems for which tape/segment diagram may be useful. Consider this problem:
At Jimmy's school, there were 475 students last year. This year, there are 24% fewer students. How many students are at Jimmy's school this year?
You can represent this problem using a tape/segment diagram like this:

From this diagram, we can tell that the number of students this year should be 100-24=76% of last year's student population, 475. Thus, we can find the answer by multiplying 475 by 0.76. Alternately, we can subtract 475x0.24 from 475, too. [Click here for a discussion on how double number lines may be used with problems involving percents.]

What we need to keep in mind about these representations is that they are supposed to be students' thinking tools, not just teachers' explanation tools. In order to help make these representations as their own thinking tools, these representations have to be carefully taught. In the Japanese textbooks, they start building these linear models starting in Grade 2 and help students experience increasingly more complicated representations gradually and systematically. I believe the emphasis on linear models in the Common Core Standards is important, but just showing these models to students will not automatically produce positive results.

M2N2e: ways to compensate

2010-03-23T16:44:00.000-07:00

M2N2. Students will build fluency with multi-digit addition and subtraction.
e) Use basic properties of addition (commutative, associative, and identity) to simplify problems (e.g. 98 + 17 by taking two from 17 and adding it to the 98 to make 100 and replacing the original problem by the sum 100 + 15).
Students, and adults, often use different mental computation strategies. The one that is discussed in this standard is often explained by using the associative property of addition: 98 + 17 = 98 + (2 + 15) = (98 + 2) + 15

However, we can also explain it slightly differently. "98 + 17" means we are putting together 98 and 17. If we pretended 98 were 100, that means we actually have 2 more than we are supposed to. So, if we don't want to change the final answer, we have to make 17 smaller by 2. In other words, 98 + 7 = (98 + 2) + (17 - 2). In general, if we added a number to one of the addends, we have to subtract the same number from the other addend to compensate.

What about subtraction? Let's think about 83 - 18. Subtracting 20 mentally is much easier. But if we subtract 20 instead of 18, we will be taking away 2 more than we are supposed to. So, to compensate for that, we must make the starting number bigger by 2, too. That is, 83 - 18 = (83 + 2) - (18 + 2). Alternately, you might think if we make 83 into 89, then there will be no re-grouping needed. But, in that case, you are starting with 6 more. So, if we want to keep the answer the same, we must take away 6 more than 18 as well. Thus, 83 - 18 = (83 + 6) - (18 + 6). As it turns out, for subtraction, if we add (or subtract) the same number to both the minuend and the subtrahend, the difference stays the same. This idea is sometimes called the equal addition principle of subtraction.

What about multiplication? How do we compensate? Let's think about 35 x 16. If we had 70, it might be easier to multiply mentally. But if we realize that 35 x 16 means 16 groups of 35 [I'm using the Japanese convention of writing the number in a group first]. So, if we make 35 into 70, you are actually putting 2 of those 35's together, and there will be only 8 groups. Or 70 x 8. Thus, we see that 35 x 16 = (35 x 2) x (16 ÷ 2). In general, if we multiply a factor by a number, then we must divide the other factor by the same number to keep the product the same.

For division, let's think about 112 ÷ 14. One way to interpret 112 ÷ 14 is to figure out how many in each group if we split 112 into 14 equal groups. The answer should be the same if we only consider 7 groups with a half as many total. So, 112 ÷ 14 = 56 ÷ 7. In general, if we multiply (or divide) both the dividend and the divisor by the same number, the quotient does not change. In the GPS, this particular idea is actually explicitly mentioned in M4N3(d). I sometime call this relationship the equal multiplication principle of division. Probably the most common place where we see the use of this principle is with problems like 2400 ÷ 400.

When you look at these four ways of making compensations, you notice that there are parallels between addition/multiplication and subtraction/division. With addition and multiplication, we do "opposite" to the two numbers to keep the result the same. However, with subtraction and division, we do the same to both numbers. Although only the division situation is mentioned explicitly in the GPS, looking at these compensation strategies may be useful in helping students develop a deeper understanding of the four arithmetic operations and how they may relate to each other.

M1N3 f - Mastering the basic addition and subtraction

2010-03-14T19:41:00.000-07:00

M1N3. Students will add and subtract numbers less than 100, as well as understand and use the inverse relationship between addition and subtraction.
f. Know the single-digit addition facts to 18 and corresponding subtraction facts with understanding and fluency. (Use strategies such as relating to facts already known, applying the commutative property, and grouping facts into families.)
Many of today's elementary school mathematics textbooks discuss a variety of thinking strategies children can use to figure out the basic addition facts. Some textbooks even organize their addition units according to those strategies: add 1/2, doubles, doubles plus/minus 1, make 10, etc.. Young children often "invent" these strategies. In fact, these strategies are the results of children's developing number sense. Enriching children's number sense, for example, composing and decomposing numbers (MKN2b, M1N3c), is a major emphasis in primary grades. Thus, this particular standards has two purposes: helping children master basic facts and helping them further their number sense.

Clearly, we want children to be able to recall the basic addition facts quickly, and some people may wonder why we need to bother with these different strategies. There are many reasons to include students' invented strategies in primary grades mathematics instruction, but to me the following three are the major reasons. First, these strategies are natural for children. If we take the idea of "starting with where children are," then we should think about how to take advantage of children's natural thinking processes. Another reason is that these invented strategies are the results of and promote further development of children's number and operation senses. I believe that the ability to see numbers and calculations flexibly is a powerful mathematical tool. If that is the case, it seems to make little sense to squash children's natural ability to think and force them to memorize the basic facts first then try to teach these flexible ways of thinking later. Such an approach seems to be rather inefficient. Finally, I believe that a major reason we teach mathematics in elementary schools is to help students become better thinkers. Thus, we should be always encouraging students to think. Quick recall is a goal, but if we want students to continue developing their thinking ability, we must dedicate some time in mathematics classrooms that focuses on children's thinking.

Anyway, although these strategies should be discussed as children naturally "invent" them, there is one particular strategy that should be treated intentionally. That strategy is the make-10 strategy. For example, 9 + 6 can be thought of as (9 + 1) + 5 = 10 + 5 = 15. For subtraction, like 13 - 8, children can think 13 - 8 = (10 - 8) + 3 = 2 + 3 = 5, or 13 - 8 = (13 - 3) - 5 = 10 - 5 = 5. 10 is such an important number in our numeration system. Thus, developing the ability to think with 10 systematically must be a major goal of mathematics teaching. For some of the invented strategies, I don't think it is necessary for all children to be able to use them. However, the make-10 strategy is mathematically so significant that all children should understand and be able to use it effectively. This way of thinking also helps students to go beyond the counting-by-one approach. If we consider older students counting on their fingers a problem, we have to offer them an alternative that can be just as effective and perhaps more efficient. The make-10 strategy is one such strategy.

M3N5 - What are decimal fractions?

2010-03-08T18:05:00.000-08:00

M3N5. Students will understand the meaning of decimal fractions and common fractions in simple cases and apply them in problem-solving situations.

When the GPS was first released, some people wondered what the phrase found in this standard, "decimal fractions," meant. If you research the Internet, you will find that "decimal fractions" are fractions with powers of 10 as denominators. This interpretation was emphasized in the 2008 revision of the GPS. Thus, M3N5(b) states, "Understand that a decimal fraction (i.e. 3/10) can be written as a decimal (i.e. 0.3)." The corresponding standards in the original GPS, M3N5(c), stated, Understand a one place decimal fraction represents tenths, i.e., 0.3 = 3/10."

However, I believe this was an unnecessary change which actually made the revised GPS a bit incoherent. It seems clear that the phrase, "decimal fractions," in the original GPS was used to mean "decimal numbers." Although the phrase "decimal fractions" isn't commonly used in the existing literature, when it is used, it typically means decimal numbers - or fractional quantities expressed in decimal format. Clearly, it is important for students to understand the equivalence of 3/10 and 0.3, but separating out fractions with powers of 10 as denominator seems to make a little sense mathematically. Furthermore, there are other statements in the GPS where this interpretation of "decimal fractions" creates some problems.

For example, the first sentence describing Grade 3 Number and Operations states, "Students will use decimal fractions and common fractions to represent parts of a whole." By examining the actual standards, we notice that students are also introduced to decimal numbers in Grade 3, but if we interpret "decimal fractions" as fractions with powers of 10 as denominators, then there is no reference to decimal numbers in the description of the standard. Similarly, the description of Grade 4 Number and Operations states, " Students will further develop their understanding of addition and subtraction of decimal fractions and common fractions with like denominators." However, students are to learn addition and subtraction of decimal numbers, M4N5.

In fact, everywhere except in M3N5, the GPS makes much better sense if we interpret "decimal fractions" to mean "decimal numbers." This is a great example how a simple phrase plays an important role in interpreting the standards. I really wish the state DOE will actually publish a document that will further elaborate what they meant.

M2N2a Addition/Subtraction Algorithms

2010-02-25T17:57:00.000-08:00

M2N2. Students will build fluency with multi-digit addition and subtraction.
a) Correctly add and subtract two whole numbers up to three digits each with regrouping.

Understanding the addition and subtraction algorithms is an important goal for Grade 2 GPS. However, what is the point in teaching algorithms? For that matter, what does it mean to teach algorithms? Some believe that we are to teach children the algorithms that are currently conventional in the United States. Others believe that teaching algorithms mean to help students make their own methods into a written procedures. This perspective doesn't mean teach children different algorithms. Many of today's textbooks include "alternative" algorithms, and some teachers will teach their students each one of them, thinking children can pick what is the easiest to them. However, the source of the algorithms is still outside of the children. Thus, although they are given some choices, algorithms are still imposed on them. Helping children make their own methods into a written procedure means the source of the algorithm is within children.

Take, for example, subtraction. Ask children to make 85 with base-10 blocks. They have no problem representing this number with 8 longs and 5 units. If you then ask them to give you 32 from what they have, they will give you 3 longs and 2 units, in that order. Ask them to make 82. Then, ask them to give you 46, they often will give you 4 longs and then pause. They see that they don't have enough units to give you. At that point some will ask if they can trade one of the remaining longs with 10 units. Once they trade the long for 10 units, they can give you 6 units, leaving them 3 longs and 6 units.

This process can be made into a written procedure like this:

This "algorithm" will work with any numbers, however long they are. If the purpose of teaching algorithms is for students to have a reliable and efficient computational method, there is nothing wrong with this algorithm. So, is that the only reason we teach algorithms?

Actually, I think there is another very important point we should help students understand when we teach addition and subtraction algorithms. That idea starts when we study simple addition and subtraction problems like 40 + 30 or 80 - 20 in Grade 1 as I alluded in the last entry. Addition problems like 40 + 30, 400 + 300, 4000 + 3000 are all related to 3 + 4 because in each case we are putting together 3 of something with 4 of the same thing. On the other hand, 300 + 40 does not relate to 3 + 4 because even though we may have 3 of 100 and 4 of 10, 3 and 4 are referring to different units. One of the important ideas of addition and subtraction is that you can only add or subtract numbers if they are referring to the same unit - we cannot add apples and oranges. To make the paper and pencil addition/subtraction easier, we arrange the numbers vertically, one on top of the other. When we do this, we also line up the place values of each number. By doing so, we know we can add or subtract the numerals in each column because they are referring to the same unit.

Although many textbooks will include the vertical notation even while studying the basic addition and subtraction facts, the importance of the notation is about this idea of lining up the place values so that we can add or subtract like numerals. With the basic facts, such an idea is rather implicit.

If we can help students make this understanding explicit, they can use it when they study addition and subtraction of decimal numbers and fractions. When we "line up the decimal points," what we are really doing is to line up the place values. Thus, we are simply following the same idea. When we add or subtract fractions, we need a common denominator, because the numerators will tell us how many while the denominators tell us what we are counting. Thus, in order to add two fractions, we have to have the two numerators referring to the same unit, or the same denominator. And, when we add, what is being counted do not change, thus the denominator stays the same. Thus, teaching of addition/subtraction algorithms is when we lay this important foundation - not just teaching them an efficient and reliable computational strategies.

M1N2 - Understanding "place values"

2010-02-14T18:03:00.000-08:00

M1N2. Students will understand place value notation for the numbers 1 to 99. (Discussions may allude to 3-digit numbers to assist in understanding place value.)

I have written about this standard previously. In that entry, I discussed different rules of our numeration system. In this post, I want to discuss a bit about what it means to understand "place value."

When you ask young children problems like 24 + 32 before they learn addition of two-digit numbers formally, they would often say something like this: "I know 20 and 30 is 50 and 4 and 2 is 6. So, the answer is 56." So, does this child understand "place value"? It is difficult to say. English number words beyond 20 has a very distinct and easily recognizable pattern. 21 is read "twenty one," 22 "twenty two," etc.. Young children easily notice that "twenty" and "one." Thus, they can easily "decompose" the number words into "twenty" and "two," but that is not enough to say they understand our number system. Understanding of our number system requires not only recognizing 21 is made up of 20 and 1, but also 21 is made up of "2 tens and 1." Because children are often familiar with the decade number words, "ten, twenty, thirty, forty, fifty, sixty, ..." they can determine that "twenty and thirty is fifty." Children who understand "place value" can say that 20+30 is the same things as 2 tens plus 3 tens, thus 2+3=5 tens.

Clearly understanding of "place value" is important for children's understanding of computational algorithms starting in Grade 2. However, this understanding is one of the important goals when we have children think about how to solve problems like 20+30 in Grade 1 (M1N3g). The focus of M1N3g is not to develop computational strategies but really to deepen their understanding of our number system.

M5M1g - Developing Area Formulas (9)

2010-02-06T14:22:00.000-08:00

M5M1. Students will extend their understanding of area of geometric plane figures.
g. Derive the formula for the area of a circle.
In the last post, we established the relationship between the diameter and the circumference of a circle, i.e. the circumference of a circle is always π times as long as the diameter of the circle. Unlike the area formula for polygons we have looked at previously, to establish the formula for the area of circles, we need to know something about the circumference as it will become clear a little later.

But, before we get to the formula, let's investigate the area of a circle a bit more intuitively. The picture below shows a circle with an inscribed square (black) and a circumscribed square (red):

If you compare the area of these two squares to the square that has the radius of the circle as a side (shaded), you see that the inscribed circle has the area twice of the shaded square, and the area of the circumscribed square is 4 times of the shaded square.

It should be clear that the area of the circle is greater than the area of the inscribed square but less than the area of the circumscribed square. Since the area of the shaded square is (radius)^2 (I can't figure out how to make "2" into a superscript...), we can say the following about the area of circle:

[Note: the exponentiation notation isn't discussed in elementary schools, so it is probably better to write it (radius x radius).] If you draw a circle on a grid paper (let's say with the radius of 10 cm), you can refine this approximation even further. The area of circle turns out to be a little more than 3 times of (radius)^2.

In everyday situation, knowing that the area of a circle is a little more than 3 times of (radius)^2 may be good enough. However, sometimes you may want to know more exact value. So, how can we derive the formula for the area of circles? How can we change a circle into a familiar shape? There are a few different possible routes, but I will discuss the most common (I think) approach.

Suppose you have some pizzas left over. Your refrigerator is too full to put the whole box in. What would you do? One thing you might do is to re-arrange the slices like the picture below shows:

When we rearrange, we know that we still have just as much pizza as we did before. But the resulting shape looks more like familiar shapes we have seen before, in this case, a trapezoid. So, we try to use this idea as we derive the formula for the area of a circle.

Suppose we subdivide a circle into 6 equal sectors, then re-arrange those sectors. It will look like this:

If we cut the same circle into 8 sectors then re-arrange the sectors, it will look like this:

If we keep increasing the number of sectors, each sector will get thinner and thinner. Here are the pictures showing what happens when the same circle is cut into 12 sectors and then 24 sectors:

As the number of sectors increases, the shape we get after we re-arrange them will get closer and closer to a rectangle:

Because the re-arranging of sectors do not change the area, the area of this rectangle is equal to the area of the original circle. So, we just need to calculate the area of this rectangle. The area of rectangle can be calculated by multiplying its length and width. By paying attention to where these parts came from the original circle, we see that the length (vertical dimension in this picture) is equal to the radius of the circle, while the width is a half of the circumference of the original circle. Therefore,

But, we already know that,

By substituting the second one in our formula for the area of a circle (because we only want a half of circumference), we end up with the formula for the area of a circle,

The process of changing the given shape into a familiar one used here is a bit different from what we did with polygons. In fact, it will probably be very difficult for children to actually cut circles and re-arrange the sectors - it can be done by providing circles with the sectors already drawn. However, even when we provide pre-drawn sectors, realistically, we can cut the circle into 12 or so sectors at most. At that point, the result may not look like a rectangle - it may look much more like a parallelogram. Since we already know the formula for the area of a parallelogram, we can also use that idea, too.

Deriving the area formula for circles is definitely more complicated than deriving other formulas. Unlike other formulas, there may have to be more "demonstration" than actual hands-on activities. Some may question whether or not it is an appropriate learning goal for Grade 5. However, it is a Grade 5 standard in the GPS, and we need to think about how we can make the formula more meaningful to students. What we discussed here is just one approach to deriving the formula. I encourage readers to investigate other ways of approaching this topic.

M5M1 - Developing Area Formula (8)

2010-01-23T10:35:00.001-08:00

M5M1. Students will extend their understanding of area of geometric plane figures.
g. Derive the formula for the area of a circle.

Deriving the area formula for circles is a bit more complex than deriving the formulas for polygons. The fundamental idea, "When you are given an unfamiliar shape, you may be able to calculate its area if you can make a familiar shape (or a collection of familiar shapes)," is still applicable. However, we can't really make a polygons out of a circle. So, the process typically requires an additional step.

However, before we get to the derivation of the area formula, let's think about another idea first. Here is a little puzzle for you. I'm sure many of you have seen racquetball balls sold in a canister like this one:

So, which do you think is loner, the height of the canister or the distance around the canister, or are they about the same? I encourage you to actually find a canister and test it yourself.

Sometimes, they come in a package of 3 balls in a canister like this one:

Again, which is longer, the height or the distance around it, or are they about the same?

Tennis balls usually sold in canisters, sometimes with 3 balls and others may have 4.

Again, which do you think is longer, the height or the distance around it, or are they about the same?

I have once seen a set of 3 softballs being sold in a canister like these, too. Again, which do you think is longer, the height or the distance around it, or are they about the same?

It turns out that whenever there are 3 balls in a canister, the height and the distance around the canister are about the same, no matter what kind of balls they are - the distance around is actually a bit longer but they are pretty close. So, what is happening? A simplified picture of the question will be like this:

In other words, the question was asking you to compare the length around a circle (we call it circumference) and the sum of three diameters. What we observed in the canisters of balls is that the circumference of a circle is about three times of its diameter no matter how big a circle is. In other words, the ratio of the circumference to the diameter is constant, and the value of this ratio is a little more than 3. This ratio is called the ratio of circumference, and we often use the Greek letter, π (pi). This number is an example of irrational number, that is, a number that cannot be expressed exactly as a fraction - you may have seen people use 22/7 as an approximation, but it is not exactly equal to pi. Another, and perhaps more common, approximation of the ratio of circumference is 3.14, and this is mentioned in M5M1h.

What we just observed can be summarized in the following equations:

Because the diameter is twice the radius, we can also express the relationship between the circumference and the radius this way:

It turns out the circumference plays an important role when we are trying to calculate the area of a circle. I will discuss the area of the circle in the next post.

M5M1 d - Developing Area Formulas (7)

2010-01-08T17:32:00.000-08:00

M5M1. Students will extend their understanding of area of geometric plane figures.
d. Find the areas of triangles and parallelograms using formulae.

The most common difficulty I observe when children (or adults) are asked to find the area of triangles or parallelograms isn't that they can't remember the formulas, but rather they don't seem to have a clear understanding of what "b" (base) and "h" (height) are supposed to be. This is particularly telling when students are asked to find the area of parallelogram like the one shown below:

Although there are sputtering of errors such as adding all measurements, the most common error is multiplying the lengths of two adjacent sides, which is an overgeneralization of the area formula for rectangles. Some students may even write, "a = bh," on the paper, yet still multiply two adjacent sides. Even if we help students derive the formulas, it will be a good idea to provide students to deepen their understanding of the formulas. When you give problems, you may want to include extra information like the one above. Another potentially useful activity is to have students find all triangles with the area of 2 square units on a regular 5 by 5 geoboard.

Many students will find the following triangles:

Note that the one on the bottom right is really the same as the one in the top middle, just in different orientation.

If you ask them what strategy they used, many will say that the product of the base and the height must be 4. So, the possible combination is either 2 and 2 or 1 and 4 (or 4 and 1). If students are having difficult time going beyond those found above, you may want to focus on the two triangles on the left and ask if there is any other triangle you can make with the base of 2 units and the height of 2 units. You may even want to draw both triangles on the same base:

Eventually, someone will notice that you can "slant" the triangle a bit further:

It is important to discuss here if the height is still 2 units for this triangle. Once they agree that the height of this new triangle is 2 units, others may find another one.

If you put all of these triangles (and a mirror image of one of them) on the same base, you can see a picture like this:

Students may notice that the "top" vertex is moving along a line, and if the geoboard was larger, we could slant the triangle even further by moving the vertex further and further to the right along the same line. If you ask them to describe the relationship between the line and the base of the triangle, students will notice that they are parallel. From here, we can generalize that the area of triangle does not change if we move a vertex along the line that is parallel to the opposite side. We can then re-emphasize that this is indeed the case because all of these triangles have the same height, and the height is really the distance between the base and the parallel line containing the opposite (from the base) vertex. The distance between the two parallel lines is measure by the length of perpendicular segments from one to the other, so the height of a triangle can be measured anywhere although some are more commonly used in textbook drawings than others:

You can do a similar activity with parallelograms, as well. Through these questions, you can deepen students understanding of the formulas - in particular the understanding of the height in the triangle (and parallelogram) formula.

M5M1 & M5A1 - Developing Area Formulas (6)

2009-12-19T07:31:00.000-08:00

M5M1. Students will extend their understanding of area of geometric plane figures.
M5A1. Students will represent and interpret the relationships between quantities algebraically.

The topic of deriving the area formulas lie in the intersection of the four of the five content strands in the GPS: Number & Operations, Geometry, Measurement, and Algebra. Obviously, since we are determining the area, we are measuring the size of various shapes. Although we can determine the area by covering the given shape with unit squares or drawing it on a grid paper, we are going beyond measurement based on counting. Rather, we are calculating the area. Therefore, students will have to utilize their understanding of number and operations - and students can practice their skills as they study this topic as well. As we try to calculate the area of new shapes, students will have to somehow make familiar shapes for which they know how to calculate the area. In that process, they will have to utilize their knowledge and skills of geometry. Finally, we are not just interested in calculating the area. We want to generalize the process and derive formulas that can be applied to other shapes of the same type. Thus, students are utilizing and developing their understanding and skills of algebra as they generalize and express the relationship among length measurements of different parts of the given shape and its area.

Although the GPS does not expect students to derive area formulas for other polygons, deriving formulas for other polygons may be helpful for students to deepen not only their understanding of area formulas but also advancing their understanding of algebraic thinking and representations. Two possible polygons to consider are trapezoids and kites. The formula for calculating the area of trapezoids may be familiar to many people, so I will focus on the formula for the area of kites today.

Students can derive the formulas for these shapes using what they already know. The general steps are:
(1) Find the area of various trapezoids (or kites) by making a familiar shape (or a collection of familiar shapes) using the strategies discussed before.
(2) Determine what measurements (or the given trapezoid/kite) will be needed to calculate the area.
(3) Determine which strategy of calculating area might be useful for generalization.
(4) Derive the formula - i.e., summarize the process of calculation in an equation using variables (words or letters).

Note that a kite is a quadrilateral with two distinct pairs of equal adjacent sides. Squares and rhombuses are special types of kites because they have four equal sides. One of the properties of rhombuses is that their diagonals are perpendicular to each other and they intersect at their mid-points. The diagonals of kites are also perpendicular, but they intersect at the mid-point of one of the diagonals but not necessarily the other. In the figure below, the diagonals are all perpendicular. However, in (a), they are intersecting at the mid-point of both diagonals while in (b) ~ (e), they are intersecting at a mid-point of one of the diagonals but not the other. However, since they all have two distinct pairs of equal adjacent sides, they are all kites. Since all four sides are equal, (a) is also a rhombus.

So, how can children find the area of a kite? One of the big idea from earlier is that, when we are given an unfamiliar shape, we may be able to find the area by making a familiar shape (or a collection of familiar shapes). We have also seen that there are three general strategies that can be used to make a familiar shape: (1) sub-dividing, (2) making-it-bigger, and (3) cutting-and-moving. For example, here are three different ways the area of a kite can be calculated:

In (1), we are using the sub-dividing method. Since we already know how to calculate the area of triangles, we can divide the kite into two triangles. The area of the kite is the sum of the area of the two triangles. In (2), we surround the kite by a rectangle. We can see that the length and the width of this rectangle are the length of the two diagonals of the kite. In (3), we cut and moved the parts of the kite to form a rectangle. One of the dimensions of the rectangle is the same as the length of one of the diagonals while the other dimension is a half of the other diagonal.

With each of these methods, we need to know the lengths of the diagonals. Thus, in order to calculate the area of kites, the length of diagonals are needed. However, in method (1), if we use the diagonal whose mid-point is the point of intersection of diagonals (as this example above shows), we will need to know exactly how the other diagonal is split to calculate the area of the two triangles.

For method (3), we can make a new rectangle differently, yet still with the dimension of one diagonal of the kite by a half of the other diagonal of kite as shown below.

However, it may be difficult for some children to understand what the vertical dimension (in this example) should be a half of the diagonal.

Therefore, although each of these methods may be generalized to derive a formula, method (2) may be simpler one to derive a formula. With method (2), you simply enclose the kite with a large rectangle, and the new rectangle you create has the area that is twice the area of the given kite:

Therefore, the area of the kite can be calculated by dividing the area of the rectangle. Thus, the formula for the area of kites might look like this:
Area of Kites = Diagonal 1 x Diagonal 2 ÷ 2.

What's important is not the actual formula but understanding the general process of generalizing and deriving the formula. This can be said of the formulas for triangles and parallelograms, too. We should emphasize the reasoning involved in the process of deriving the formulas. If students understand the reasoning, they will be able to re-derive the necessary formula even if they can't simply recall it.

M5M1 c - Developing Area Formulas (5)

2009-12-06T06:46:00.000-08:00

M5M1. Students will extend their understanding of area of geometric plane figures.
c. Derive the formula for the area of a triangle.
In some textbooks, the area of triangles is studied before the area of parallelograms. In those cases, students typically examine how to calculate the area of right triangles. They notice that the area of a right triangle is a half of the rectangle you can make by using two copies of the right triangle.

They will then investigate how the area of more general triangles may be calculated, for example a triangle like this one:

Typically, students will split the triangle into two right triangles and apply the same method as before:

From here, those textbooks often generalize the way to calculate the area of triangles as:

However, sometimes students develop a misconception that the base must be the side of triangle that can be split to form 2 right triangles. In particular, they may not be able to calculate the area of the following triangle:

They want to split the triangle into two right triangles as shown, but they cannot determine the length of "base" and "height" in this case:

What they could do is to use the make-it-bigger approach and form a larger right triangle by adding on a smaller right triangle as shown below:

You can not only calculate the area of the given triangle by using this method, this method can be generalized to support the derivation of the formula. However, the challenge for students to derive (or verify the previously developed) formula is that they have to apply the distributive property as shown below:

Students may be used to applying the distributive property over addition, but they may not have had much experiences with the distributive property of multiplication over subtraction. Moreover, they have to treat the expression 4 ÷ 2 as a quantity.

Although it is possible to discuss the area of triangles before the area of parallelograms, my preference is to discuss parallelograms first. If parallelograms are "familiar" shapes, students can use a variety of methods to find the area of triangles. Here are just a couple students have come up on their own:

From the method on the left, we can see that the area of the triangle is the half of the area of the parallelogram, and the area of the parallelogram may be calculated by multiplying the "base," which is one side of the triangle, and the "height," which is the distance between the base and the parallel line containing the third vertex. From the method on the right, we can see that the area of the triangle is equal to the area of the new parallelogram. The area of the new parallelogram may be calculated by multiplying the "base," which is a side of the given triangle, and a half of the "height" of the triangle.

Either way, we can generate the formula, Area = Base x Height ÷ 2. Just as was the case with parallelograms, it is important that students understand that any of the three sides of the triangle may serve as the base and for each base, there is a corresponding height. The height is the distance between the base and the third vertex, which is the same thing as the length of perpendicular segment from the third vertex to the base.

By the way, it is probably important to write the formula with "÷ 2" since students only learn to model fraction multiplication in Grade 5 according to the GPS.

M5M1 b - Developing Area Formulas (4)

2009-11-14T13:49:00.000-08:00

M5M1. Students will extend their understanding of area of geometric plane figures.
b. Derive the formula for the area of a parallelogram.
As discussed in the previous post, through activities like finding the area of L-shaped region, students can develop the understanding that "when we are given an unfamiliar shape, we may still be able to calculate its area by somehow making a familiar shape (or a collection of familiar shapes)." Moreover, students can develop the following strategies to make a familiar shapes:
* divide the given shape up into several familiar shapes
* cut and re-arrange to make a familiar shape
* make-it-bigger
Now they are ready to tackle this standard.

In many textbooks, students are asked to find the area of parallelograms like the one shown below using what they already know:

Some students will count the number of unit squares, making appropriate adjustments when only a part of a unit square is inside the parallelogram. Other students will try to change the parallelogram to a rectangle, a familiar shape they already know how to calculate the area of. The typical way that this is accomplished by cutting a triangular segment from one end of the parallelogram and moving it to the other side, as shown below:

Since this rectangle is 6 cm wide and 4 cm long, area can be calculated by 6 x 4, or 24 cm2.

In most textbooks, this method is then generalized to derive the formula for calculating the area of parallelograms: Area of Parallelograms = Base x Height. So, is this the end? Have we successfully addressed this particular standard? I argue that the formula at this stage is an overgeneralization. Students at this point may have difficulty calculating the area of parallelograms like the following:

Some students will try to create a new rectangle like before and notice that "the height (in red) stops here!"

Others might try to turn the figure and make a rectangle like this:

Unfortunately, they can't determine the length and the width of this new rectangle other than actually measuring them, which isn't possible if the figure isn't drawn to scale. Even if the figure is drawn to scale, actually measuring the length and the width will introduce measurement errors. So, what can students do? Actually, there are a lot of things they can do using the understanding they developed through the L-shape lesson. Here are some possibilities:

Note that (a), (c) and (d) use the "cut and re-arrange" strategy, (b) uses the "divide up" strategy, and (e) uses the "make-it-bigger" strategy. In (b), (c) and (d), the "familiar" shape students created are parallelograms that can be changed to rectangles by cutting and re-arranging right triangles.

Some of you may be wondering about (e) since students have not learned how to calculate the area of triangles. In this case, instead of calculating the area of each triangle, this student actually pushed together the two triangles that were used to make a bigger rectangle. The two triangles will make a rectangle whose dimensions are 5 cm by 6 cm.

Actually, some students may use this make-it-bigger strategy with the first parallelograms. If they did, then, this "slanted" parallelograms do not pose any challenge to them since they can use exactly the same strategy to this one as well. This strategy could have been used to derive the formula for calculating the area of parallelograms, too. Look at the figure below:

The area of the original parallelogram (un-shaded part in the figure on the left) can be calculated by subtracting the area of shaded rectangle (in the middle figure) from the large rectangle. However, this difference is really the area of the yellow rectangle in the figure on the right. That means that the area of the parallelogram is the same as the area of rectangle you can build on the base whose length is the distance between the base and its opposite side, or more accurately, the distance between the parallel lines containing the base and its opposite side. If we consider this distance between the base and its opposite side as height, we still have the same formula, Area of Parallelogram = Base x Height.

The important idea here, though, is what constitute as the height. The height of a parallelogram is the distance between the base and its opposite side, and the distance between two parallel lines is the length of a perpendicular segment connecting them. It is not the length of the adjacent side to the base. In case of a rectangle, which is a special type of parallelograms, the adjacent side may be used as the height because it is perpendicular to the base. However, that is not generally the case in parallelograms. Thus, understanding what the height of a parallelogram is may be the most important aspect of deriving the formula. Unfortunately, students don't understand this idea because they aren't asked to grapple with parallelograms like the second one we saw above, or derive the formula through the make-it-bigger strategy. I hope you will seriously consider giving your students this challenge as they try to derive the formula for calculating the area of parallelograms.

M5M1 - Developing Area Formulas (3)

2009-11-07T07:29:00.000-08:00

M5M1. Students will extend their understanding of area of geometric plane figures.

As we discussed in the previous post, the GPS expects students to determine the area of rectangles and squares by counting or calculation. Then, in Grade 5, students are expected to derive and use formulas to determine the area of parallelograms, triangles, and circles. Interestingly, there is nothing about area mentioned in Grade 4. It is listed as one of the "Concepts/Skills to Maintain," but there is no specific standard about the area measurement in Grade 4. Many people might wonder about the feasibility of fifth graders actually deriving the area formulas of parallelograms and triangles on their own. Do they have enough background knowledge? What background knowledge do they need to increase the likelihood of their deriving those formulas?

In a previous post on the idea of teaching through problem solving (April, 2009), how children can learn through problem solving new mathematical ideas. Those mathematical ideas are the ones that will serve as the bridge between M3M4 (area of rectangles and squares) and M5M1 (area of parallelograms, triangles, and circles). As we will see shortly, those specific understandings will be used over and over to derive the formulas. So, in Grade 3, finding the area of L-shapes may be simply a complex application of what they learned, but, in Grade 5, the focus should be on ways of thinking involved in calculating the area. If those understandings are made explicit, students are much more likely to be successful in deriving the area formulas. So, I encourage you to read that post again (or for the first time, if you have not read it before).

By the way, element (f) of this standard says, "Find the area of a polygon (regular and irregular) by dividing it into squares, rectangles, and/or triangles and find the sum of the areas of those shapes." Actually, this element is simply one of the strategies developed in the L-shape lesson, that is, sub-dividing the given unfamiliar shape into a collection of familiar shapes. The only difference is what shapes are available to students as familiar shapes. When students work on the L-shape problem, they only knew how to calculate the area of rectangles and squares. However, after students have learned the formulas for the area of parallelograms and triangles, students can also use those figures. So, in case you are wondering if you can afford to spend an extra time to discuss something that is not explicitly mentioned in the GPS, the L-shape lesson does address the GPS directly, too.

M3M4 - Developing Area Formulas (2)

2009-11-03T15:02:00.000-08:00

M3M4. Students will understand and measure the area of simple geometric figures (squares and rectangles).
a. Understand the meaning of the square unit and measurement in area.
b. Model (by tiling) the area of a simple geometric figure using square units (square inch, square foot, etc.).
c. Determine the area of squares and rectangles by counting, addition, and multiplication with models.

Once students understand that area is the amount of space inside any geometric figures, we are ready to start thinking about ways to measure the area of various shapes. The next step is to pick a unit and actually "cover" shapes to see how many units will be needed. So, what should we use as a unit? Although we will eventually use squares as units, we may want to think about using anything that can cover the plane without a hole or an overlap. Also, using a familiar objects might be helpful to focus students' attention on the process of area measurement. One such familiar object might be index cards. Students can measure the area of the surface of desks or any other large rectangular regions.

If students have many index cards available to them, they will cover the rectangular region in many different ways. Here are three possibilities.

In this particular example, no matter how you cover the rectangle, it takes 24 small rectangles. So, we can say that the area of the rectangle is 24 units.

After measuring the area by actually covering rectangles with units, many students will realize that some ways of covering the given shape is easier to count than others. For example, the arrangements like the one on the left requires us to actually count all of the units to determine how many units were used. On the other hand, since the other two arrangements will result in equal groups (either rows or columns), we can use multiplication to find the area (either 4x6 or 8x3).

At this point, you might want to give students only 3 or 4 unit pieces to see if they can think about ways of calculating the area. A common error at this stage is to do the following:
and .

So, the area is 4x3=12 units. It is important for students to understand here why they cannot rotate the unit as they measure how many units will fit in each dimension of the rectangle. What we are trying to do when we measure the second dimension is how many rows (in this example) of 4 units there are. If we turn the unit as shown on the right, we are no longer counting the number of rows of 4 units.

You may want to ask students what we can do to avoid this type of confusion. Some students will realize that if we use a square as a unit, then it doesn't matter whether we rotate it since squares have 4 equal sides. You can then introduce that the standard units of area measurement are squares with unit length on each side, e.g., 1 cm, 1 in, 1 ft, etc.. Each unit square is said to have the area of 1 cm2, 1 in2, 1 ft2, etc., respectively. Actually, I am not sure exactly how the GPS wants these standards units of area to be handled. Unlike the units for volume, these area units are not mentioned in the GPS. However, it seems strange not to talk about the units when we are talking about the area of rectangles.

By using unit squares, we can also make it easier to determine the number of units that fit along each dimension of a rectangle by simply measuring their lengths. So, if a rectangle is 5 inches wide and 8 inches long, that means we can fit 5 1-inch squares along one row and there will be 8 rows. Therefore, we can multiply 5 and 8 to get 40 cm2. It is important that students understand that when 2 lengths are multiplied together, the product mysteriously becomes the area measurement. The two lengths we are measuring are simply telling us how many unit squares will fit along each side of the given rectangle.

Also note that students are not introduced to letters as variables until Grade 5, the formula should be written (if it is to be written at all) as, Area of Rectangle = Length x Width. Again, it is important to emphasize that this formula is to calculate the area of rectangles. Some students (and adults, unfortunately) will say that area is "length x width," but it is only a formula for a specific shape. Area is the amount of space inside a shape, no matter what the shape is.

M3M4 & M5M1 - Developing Area Formulas (1)

2009-10-24T09:58:00.000-07:00

M3M4. Students will understand and measure the area of simple geometric figures (squares and rectangles).
M5M1. Students will extend their understanding of area of geometric plane figures.

In the past several posts, I discussed important ideas involved in helping students develop multiplication and division algorithms. With today's post, I want to start a new series on how to help students develop various area formulas (M3M1 & M5M1). In today's post, however, I want to focus my attention on the basic ideas about teaching and learning of area measurement.

I have discussed previously (December, 2008) some basic ideas of teaching measurement. As we teach measuring of any attribute (e.g., length, weight, area, angle, etc.), we must first help students understand the particular attribute we are trying to measure. Without understanding the attribute, measuring it will not make any sense. For that purpose, comparison activities are very useful. Typically, we start with direct comparisons, then move on to indirect comparisons. After students understand the concept, we can start thinking about quantifying the amount of the attribute, i.e., measuring the object. Many people suggest that we start with non-standards units first. One reason for this suggestion is that if we start with standard units, students will have to learn the idea of using units to quantify the attribute and the idea of standard units. Moreover, when we try to measure with standard units, we typically use measurement instruments, such as rulers and protractors. So, students will also have to learn how to use measurement instruments. If we start with non-standard units, students can focus on the notion of quantifying, or measuring, first. Once students understand how a particular attribute can be measured using a non-standard unit, they can use that understanding to both measuring with standard units and learning how to use measurement instruments. After all, the notion of "standard" units probably does not make sense unless you have some experiences with "non-standard" units.

So, what does this all mean when we are teaching the area measurement? Obviously, the first focus should be on helping students understand what area is about. To do so, it seems like we should have students engage in some comparison activities. So, for example, let's ask students to compare the following two rectangles (one is actually a square, but we know that squares are rectangles, don't we?).

Note that I am just showing you the drawing of rectangles, but you should give children cut out pieces to compare. Moreover, I included grid lines to show the dimensions of the rectangles, but you may not want to do so when you give these shapes to students.

Anyway, when students are given these two shapes and asked "Which is bigger?" you see generally two different ways students will compare these shapes. Some children will compare the shapes by overlapping them (on the left below). Others will put the shapes next to each other (on the right).

We can ask students if these two ways of comparing are actually comparing the same attribute (we may not want to use this particular word with 3rd graders). Students may not know, but they can understand that if the conclusions we get from these two ways are different, then, they couldn't be comparing the same attribute. So, have them try comparing these two shapes using both ways. For example, if we put the shapes next to each other and rotate one of them around the other shapes, you see that the two shapes are the same size:

When you overlap shapes, we see that the square is actually "bigger" than the rectangle:

Since the results are different, these two ways of comparison are indeed comparing two different attributes. We know that the first comparison was comparing the length around these shapes, or perimeter, while the second comparison is about area, i.e., the amount of space inside the shapes.

I am not saying that children will understand the difference between the perimeter and the area by doing this one activity. However, it is important for children to have a number of comparison activities to compare the length around and comparing the amount of space inside. When two same objects give different results like the one above several times, students might develop a better sense of the difference between these two attributes.

Once students understand what the area as an attribute is about, we can now move into the discussion of measuring it. So, in the next post, we will discuss M3M1, in which students think about the area of rectangles and squares.

M4N3 - Developing multiplication algorithms (7)

2009-10-17T06:34:00.000-07:00

M4N3. Students will solve problems involving multiplication of 2-3 digit numbers by 1 or 2 digit numbers.

So far, we have discussed the following:
(1) extending the multiplication table to 10x10
(2) multiplying multiples of 10 and 100 by 1-digit numbers
(3) multiplying 2- and 3-digit number by 1-digit numbers
(4) multiplying by multiples of 10.

Now, we are ready to tackle multiplication of 2- and 3-digit numbers by 2-digit numbers. Before we get started, I wanted to say that, to me, teaching of an algorithm means helping students make their own strategies into written procedures instead of imposing a specific algorithm upon students. Of course, that doesn't mean "anything goes." Rather, teachers must think carefully about how to influence students' thinking naturally. Moreover, it may be possible for teachers to sequence students' experiences in such a way that the algorithm students develop "naturally" is something very similar to, or exactly the same as, the conventional algorithm. For that purpose, the area model of multiplication can play a very important role. Therefore, the use of the model along with base-10 blocks before reaching this point is an integral part of the process. So, how do we help students expand their written methods into multiplication of 2- and 3-digit numbers by 2-digit numbers?

Let's think about 12x23 first. How can students use what they have learned so far to think about ways to calculate this problem? There are at least three possible ways. At the most abstract level, students might be able to think of 12x23 as 12x20+12x3 - i.e., 23 groups of 12 can be split into 20 groups of 12 and 3 groups of 12. Then, each of 12x20 and 12x3 are already discussed. If students can think about this way, they can record the process using the vertical notation,
or .

The notation on the right is basically the standard algorithm for multiplication.

Another possibility is for students to go back to the area representation of multiplication. 12x23 means that we are making a rectangle with the dimension of 12 units by 23 units. The product is represented by the area of this rectangle. So, if you construct this rectangle using base-10 blocks, and using the fewest number of blocks (i.e., use large blocks whenever possible), you can make a rectangle like this one:

By examining the arrangement, we see that there are 1 by 2 rectangles made of flats (200), 2 by 2 rectangles of longs (40), 1 by 3 rectangles made of longs (30) and 2 by 3 rectangles of units (6). So, the product is 200+40+30+6=276. After students have become comfortable with the area model representation with base-10 blocks, you may want to encourage students to move toward drawing instead of using actual base-10 blocks. Sometimes you can make this transition simply by giving students multiplication problems with larger factors. Students will realize that actually making rectangles using base-10 blocks is too tedious.

When students become comfortable with drawing rectangles, they might realize that it is still rather tedious. This is when you may be able to suggest if they could use an adaptation of a notation that we used when we were multiplying 2- or 3-digit number by 1-digit number. Some students may be able to start at this point, without going all the way back to using base-10 blocks. That judgment must be made by teachers, using their knowledge of students. Anyway, the notation might look like this for 12x23:

Again, after students have become fluent with this notation, you might want to bring their attention to the four products (in the example here, 200, 40, 30, and 6). Noticing that these are the products of the two tens digits, the tens digit and the ones digit (in both direction) and the two ones digits. So, you can introduce a new notation that records the same information as this diagram does:

You can then negotiate with your students a consistent order in which you calculate these four products (typically called "partial products) so that we can make sure that we have accounted for all of them. If you really want students to understand the conventional multiplication algorithm, you will start with the ones digit of the multiplier (the bottom number) and multiply the ones and then the tens digits of the multiplicand (the top number). You will then multiply the tens digit of the multiplier with the ones and then the tens digits of the multiplicand. So, this problem would look like this:

If you combine the first two partial products and the last two partial products, you will have:

Note that the example we used, 12x23, did not involve any re-grouping. In a way, this is the most "basic" situation. As students move from one notation to another, you may want to consider moving back to a basic situation. Once students become comfortable with the notation (area model, symbolic notation, or whatever), then you want to look at other situations such as those involving re-grouping and a 0 in the factor/product.

When extending the multiplicand to 3-digit numbers, for example, 587x34, you may want to go back to the diagram notation - it will be rather difficult to actually model these multiplication with base-10 blocks. From the diagram, you can move to the notation that will explicitly record all partial products, then eventually to the conventional algorithm.

As usual, you do want to pay close attention to the numbers (factors) you use. Some students have difficulty with 0's - either in the factors or in the product/partial products, so you want to pay particular attention to those situations.

M3N3d - Developing multiplication algorithms (6)

2009-10-09T02:39:00.001-07:00

M3N3. Students will further develop their understanding of multiplication of whole numbers and develop the ability to apply it in problem solving.
d.Understand the effect on the product when multiplying by multiples of 10.

This standard talks about multiplying by multiples of 10, for example 37x30. This situation is different from multiplying multiples of 10, 100, etc. (which we have discussed in a previous post) because we now have 30 groups of 37. Now, if we study this idea after students have already developed a paper-and-pencil algorithm, these problems can be considered as a special case where there will be a 0 in the product. So, procedurally, there are different ways to deal with these problems. Some will carry out the calculation exactly in the same manner as they do with other multipliers:

After students get used to this calculation, they might try to combine the steps to make it more efficient:

From this perspective, this multiplication isn't much different from something like 35x18. The important idea is that we have to write a 0 in the ones place as a place holder.

However, M3N3d states that students must understand "the effect on the product when multiplying by multiples of 10." Moreover, according to the GPS, students do not study how to multiply by 2-digit number until Grade 4 (next post). So, it seems rather odd to talk about multiplying by multiples of 10, which are 2-digit number, at this point. If students' don't know how to multiply by 2-digit number, then we can't focus on the procedural aspect discussed above. Rather, we want students to understand what is going on when we multiply by multiples of 10. Although we cannot use the idea of 10 as a unit in the same way as we did when we were multiplying multiples of 10, we can still use the idea of 10 as a unit when the multipliers are multiples of 10. For example, you can think of 37x30 as 37x3x10. Alternately, you can think of 37x30 as 37x10x3. Either way, multiplying a 2- or 3-digit number by 3 is something students have already learned. What students may not have studied is multiplying 2- (or 3-) digit number by 10. So, that seems to be the primary focus of this standard.

As we explore multiplying 2- and 3-digit numbers by 10, we may again want to go back to the area model of multiplication. For example, if students are to model 17x10 using base-10 blocks, they might at first construct something like this by simply extending what they have done previously:

At this point, some might notice that we can actually use a flat on the left side since there are 10 longs. Moreover, on the right side, since there are 10 rows of units, we can replace each column by a long, resulting in an arrangement like this:

Students can also record the process more abstractly like this, too:

They can also consider cases like 40x10 by extending their thinking of 40 as four 10's. If you have 10 groups of four 10's, you can think of that as 4 groups of ten 10's as well, or 100x4.

From these exploration, students may notice that when you multiply 2- and 3-digit numbers by 10, the product will contain the same set of numerals in the same order but every numeral is moved one place to the left - and there is a 0 in the ones place as a place holder.

Although we may be able to consider multiplying by multiples of 10 as a special case of multiplying by 2-digit numbers, students still need to learn the effect of multiplying by 10 before they can explore multiplying by 2-digit numbers. Moreover, once you study the effect of multiplying by 10, extending it to multiplication by multiples of 10 may be useful to help students deepen their understanding of multiplication operation. Although the formal study of properties of multiplication is done in Grade 4, Grade 3 students can use the associative property to reason about the effect of multiplying by multiples of 10. I believe that's the point behind this standard, not just the procedural fluency.

M3N3c - Developing multiplication algorithms (5)

2009-09-30T08:06:00.000-07:00

M3N3. Students will further develop their understanding of multiplication of whole numbers and develop the ability to apply it in problem solving. c. Use arrays and area models to develop understanding of the distributive property and to determine partial products for multiplication of 2- or 3-digit numbers by a 1-digit number.
In the previous post, I discussed how students can develop a paper-and-pencil algorithm for multiplying 2-digit numbers by 1-digit numbers. Let's consider how we can help students extend the procedure to multiplication of 3-digit numbers by 1-digit number.

How can we multiply 312 x 3? How can students use what they have learned so far to calculate this? One possibility is to think of 312 as 300+12. Then, we can multiply 300x3 and 12x3. Both of these are already learned ideas. If students have already understood how to multiply a 2-digit number by a 1-digit number using a paper-and-pencil method, they can then combine their learning and record this multiplication something like this:

When extending the multiplicand from 2-digit to 3-digit, therefore, there isn't really any new concept involved. Even the idea of looking at 312x3 as 300x3+12x3 is really the same idea as looking at 12x3 as 10x3+2x3, i.e., the distributive property of multiplication, which will be formally studied in Grade 4.

One important thing to think about when we study multiplying 3-digit numbers by 1-digit numbers is different situations where re-grouping must take place, or when there is a 0 (or more) in either the multiplicand or the product. The example we just saw, 312x3, does not involve re-grouping and there is no 0 in the multiplicand nor the product. So, in a way, it is a "general" case of multiplying 3-digit numbers by 1-digit numbers. But, here are some of other cases:
Re-grouping is involved
• 227x3
• 227x5
• 162x3
• etc.

0 is involved
• 406x7
• 365x4
• 527x4
• etc.
I encourage you to think about other cases. As teachers, we must also think about how we want to deal with them. We can carefully sequence those cases and have students think about how they can adapt the written procedure they developed those situations. As you do, it will be helpful if you explicitly ask students what is different about each case compared to the most general one that we start with.

As we look at those special cases, it is important that students understand what is actually happening when we are multiplying 3-digit numbers by 1-digit numbers. For that, it might be useful to go back to the notation system that we used when we developed when we were multiplying 2-digit numbers by 1-digit numbers. For example, let's think about 427x4. Since we can think of 427x4 as 400x4+27x4, and we can use a pictorial notation like this:

Or, we can use more symbolic notation like this (with the previous agreement that we start recording with the partial product of the ones digits first):

We can combine some of the steps involved in this notation and develop a notation like this:

No matter how you approach this topic, what we cannot do is to start with the standard algorithm, which is the most sophisticated way of recording the processes. Help students extend what they have previously learned, which may be the standard algorithm for multiplying 2-digit numbers by 1-digit number by thinking about the structure of numbers and the meaning of operations. If necessary, go back to the intermediate notations that were used while developing the algorithm for multiplying 2-digit number by 1-digit numbers. By experiencing this extension, students can then think about how they can extend the algorithm for multiplying 3-digit numbers by 1-digit numbers to multiplying 4-digit (or even longer) numbers by 1-digit numbers. They have not only the experiences of multiplying two numbers but also the experience of "extending" their procedure from one case to another. So they can ask themselves not just "How did I multiply 2- or 3-digit numbers by 1-digit numbers?" but also "How did I extend the algorithm for multiplying 2-digit numbers to 3-digit numbers?" Therefore, when teaching multiplication of 3-digit numbers by 1-digit numbers, what is important is not the procedure but the idea of how to extend the previously learned procedure (2-digit multiplicands) to a new situation (3-digit multiplicands).

M3N3c - Developing multiplication algorithms (4)

2009-09-25T02:36:00.000-07:00

M3N3. Students will further develop their understanding of multiplication of whole numbers and develop the ability to apply it in problem solving. c. Use arrays and area models to develop understanding of the distributive property and to determine partial products for multiplication of 2- or 3-digit numbers by a 1-digit number.
This is the fourth in a series of posts in which I am discussing the development of multiplication algorithms. Up to this point, students were calculating mentally. The focus has been more on consolidating students' understanding of our number system and the meaning of multiplication by using those understanding to figure out multiplication beyond the basic facts. Today's standard is the first step toward developing paper-and-pencil algorithms. As I begin my post, let me emphasize that teaching of an algorithm for any operation should focus on helping students develop the algorithm on their own. In other words, we need to move away from the show-and-tell approach where teachers show students how to multiply using the multiplication algorithm and then have them practice over and over. Practice is important, but students should first develop the algorithm themselves. Of course, that does NOT mean that we just leave students on their own. Rather, teachers must plan carefully to guide students' thinking.

One useful idea in developing a multiplication algorithm is the area model of multiplication. In Grade 3, students learn about area of rectangles and squares. When students cover a rectangle with unit squares, they notice that they are arranged rows and columns of equal sizes. Because all rows (or columns) are equal, we can use multiplication to efficiently determine the area. This idea can be used to model multiplication where the two factors are represented by the two dimensions of a rectangle and the product is represented by the area. So, for example, 4x6=24 can be modeled as shown below.

Notice that since you can turn the rectangles around without changing the area, this is also a useful model to show why the commutative property of multiplication is true. It is also useful to model the distributive property of multiplication.

When you model multiplication problems like 14x7 using base-10 blocks, you can certainly try to make 7 groups of 14 (1 long and 4 units). However, we want to encourage students to organize the model more systematically using the area model. The area model representation will make it much easier to determine the product by observation if the same type of blocks are grouped together.

Eventually, we want to help students move beyond modeling with actual base-10 blocks. One useful approach to do so is to have students draw what they would have done with base-10 blocks. Thus, drawing the picture like the one above. Grid papers can be very helpful in that process. However, as they become comfortable with drawing pictures, they realize that drawing can be rather tedious. Given our goal is to determine the product, what we want to know is how many longs and how many units we have. Thus, we can model the multiplication explicitly showing only the information we need. Here is an example for 14x7.

Once students become comfortable with modeling multiplying 2-digit number by 1-digit number this way, we can ask if they can think of a way to represent this model using a vertical notation like we did with addition and subtraction. Here are two possibilities:

Students can see that 70+28 and 28+70 are the same. Thus, we can write it either way. At this point, it is ok to suggest that we agree to write the product of the ones digit first. Again, after students practice this notation, they might notice that the ones digit for the partial product of the tens digit on the multiplicand and the multiplier is always 0. Therefore, the ones' digit of the product is always the ones digit of the partial product of the ones digit of the multiplicand and the multiplier, 8 in the example above. Then, we have to add the tens digits of the partial products to find the product. This process can be combined if you use a notation like this:

This may be a slightly different notation than some of us are used to, where the tens digit of the partial product above the tens digit of the multiplicand. That notation sometimes causes students to add the re-grouped digit and the tens digit of the multiplicand before multiplying by the multiplier - that is students end up doing (2+1)x7 instead of 1x7+2. Writing the re-grouped digit below the horizontal bar (the equal sign) might minimize that error.

In the next post, I will discuss how this procedure may be extended to multiplying 3-digit numbers

M3N3 Developing multiplication algorithms (3)

2009-09-19T08:50:00.000-07:00

M3N3. Students will further develop their understanding of multiplication of whole numbers and develop the ability to apply it in problem solving.

This is the third in a series of posts in which I discuss the development of multiplication algorithms in Grades 3 and 4. If you haven't read the first two, I encourage you to do so, either before or after reading this post.

One idea that is important as children continue to develop multiplication algorithms yet not explicitly mentioned in the GPS is the idea of multiplying multiples of 10 and 100, such as 40x7 and 600x3. Note, multiplying by multiples of 10 and 100 is a Grade 4 standard.

So, how can students make sense of multiplying multiples of 10 and 100? So, let's think about 40x7. Again, just a reminder that I am following the Japanese convention of writing the multiplicand (number in a group) first. Therefore, this problem is asking us to find the total amount when there are 40 groups in each and there are 4 groups.

An important idea here is the understanding of 10 and 100 (and 1000) as unit. When students first learned simple addition and subtraction of 2-digit numbers such as 30+40 and 70-20 in Grade 1, they used the idea of 10 as a unit. Since 30 and 40 are made up of 3 and 4 tens, putting those two numbers together meant there are seven 10's, or 70. In a similar way, we can think of 40x7 using 10 as a unit. Since there are 7 groups of 40, or 7 groups of four 10's, we see that there are 7x4=28 tens altogether. Therefore, the product is 280. Similarly, you can think of 600x3 as 3 groups of six 100's, or 3x6=18 hundreds, i.e., 1800.

The idea of 10 and 100 (and 1000) as units was the focus of M2N1(b):
Understand the relative magnitudes of numbers using 10 as a unit, 100 as a unit, or 1000 as a unit. Represent 2-digit numbers with drawings of tens and ones and 3-digit numbers with drawings of hundreds, tens, and ones.
Although the second part of the statement emphasizes looking at 3-digit numbers as composed of hundreds, tens, and ones, it is also important for children to understand numbers like 280 as twenty-eight 10's. This notion of relative magnitude ("relative size" in M3N1(b)) plays an important role in students' mathematics learning in the future. Therefore, it is important to help them deepen and consolidate their understanding as we teach multiplication of multiples of tens and hundreds.

M3N3(b) - Developing multiplication algorithms (2)

2009-09-13T05:40:00.000-07:00

M3N3. Students will further develop their understanding of multiplication of whole numbers and develop the ability to apply it in problem solving. b. Know the multiplication facts with understanding and fluency to 10 x 10.
The new idea here is multiplication with 10 as a factor, either the multiplicand or the multiplier. Let's first look at the cases where 10 is used as the multiplicand, i.e., 10x1, 10x2, 10x3, .... These can be interpreted as one 10, two 10's, three 10's, ... respectively. In Grade 1, when students studied the numbers up to 100, this is something they should have encountered. Thus, they can use that particular prior knowledge to figure out what these facts will be, i.e., 10x1=10, 10x2=20, 10x3=30, ....

What about 10 as the multiplier, i.e., 1x10, 2x10, 3x10, .... These means, respectively, ten 1's, ten 2's, ten 3's, .... For adults, these are obvious and we might think they should be obvious to children, too. However, although young students can answer the problem on the left very quickly, many of the same students have much more difficult time with the problem on the right.

So, how can students think about problems like 3x10? Hopefully, when they were constructing their multiplication table, they have used the idea that when the multiplier increases by 1, the product increases by the multiplicand. For example, the answer for 3x6 should be 3 (the multiplicand) more than 3x5. This idea, then, can be used to think about 3x10. The answer to 3x10 should be 3 more than 3x9, which is a part of the basic fact they have learned in Grade 2. This idea is really a particular case of the distributive property, which students will formally study in Grade 4. However, the distributive property plays an important role as students think about how to multiply by larger numbers. Therefore, it may be useful if this idea is discussed explicitly in classrooms.

Some of us grew up memorizing the multiplication table up to 12x12. Even today, some teachers/schools/districts still make their students consider the multiplication table up to 12x12. Although it may have some usefulness in everyday situations to know the multiplication facts of 11's and 12's, there is really no particular mathematical reason for expanding the multiplication table to 12x12. Once students develop an algorithm for multiplying by 2-digit numbers, they can calculate anything beyond 10x10 using the algorithm. On the other hand, students can also use the property of multiplication and think of 11's and 12's as simply 10's and 1's or 10's and 2's. Thus, 7x12 can be thought of as 7x10+7x2 and 12x8 can be thought of as 10x8+2x8. Perhaps it is much more important for students to develop that form of flexible thinking than simply memorizing those facts.

M3N3/M4N3 -- developing multiplication algorithms

2009-09-06T11:03:00.000-07:00

M3N3. Students will further develop their understanding of multiplication of whole numbers and develop the ability to apply it in problem solving.
M4N3. Students will solve problems involving multiplication of 2-3 digit numbers by 1 or 2 digit numbers.

In the previous 2 posts, I discussed division algorithms. So, in the next few posts, I would like to discuss multiplication algorithms. Today, as the first entry on multiplication algorithm, I want to discuss an overview of teaching and learning multiplication algorithms.

Students are introduced to multiplication in Grade 2. The GPS (M2N3) states that students should construct the multiplication table and correctly multiply 1-digit numbers. What is not quite clear is where multiplication involving 0 as a factor (either the multiplicand or the multiplier) should be discussed. Many US textbooks introduce multiplication with 0 and 1 as factors fairly early on in their discussion of multiplication. In contrast, in the Japanese textbooks, multiplication with 1 as the multiplicand, i.e., 1x1, 1x2, 1x3,..., are discussed AFTER students study the 9's facts. [Note: in the Japanese notation, the first number is the multiplicand, i.e., the number in a group.] They do not discuss 0 as a factor until the 3rd grade. They do this because the emphasis in Grade 2 is developing the meaning of multiplication first. For children, considering 1, or even 0, item as a "group" may be strange. From the equal group perspective of multiplication, therefore, 0 and 1 as the multiplicand are special cases. Therefore, they start with more general cases first (2's through 9's), then discuss the special cases (1's and 0's). Textbooks often treat 0's and 1's early because getting the answers is easy. However, if our focus is on the meaning of multiplication, that may not be a wise choice.

Anyway, after students study 1x1 through 9x9 in Grade 2 (and possibly 0's), students are expected to learn to multiply larger numbers in Grades 3 and 4 (M3N3 and M4N3). So, by the end of the 4th grade, we want students to be able to calculate problems like 512 x 43. Using the conventional algorithm, we can calculate this problem as shown below:

With this algorithm, we can calculate this problem by performing 6 basic multiplication and 5 basic addition. In fact, with our base-10 numeration system, once we learn the basic addition and multiplication facts, we can perform the basic 4 operations with any size numbers. Although this is not explicitly spelled out in the GPS, we would like students to understand this merit of our number system as a result of learning the computational algorithms.

Of course, this is by the end of Grade 4, and we have to think about how to help students go from knowing only the 1-digit multiplication facts to that point. So, how should we organize our instruction? What are some important mile markers in this endeavor?

Here are some important understandings students need.
* We can think of 512x43 as 512x40+512x3.
* 512x3 can be thought of as 500x3+10x3+2x3.
* 512x40 can be thought of as 512x10x4.

The first idea involves the use of the distributive property. Although the formal study of the properties of operations is in Grade 4, students use the distributive property as they construct the multiplication table. For example, they might have thought of 7x6 as7x5+7. Or, they thought of 8x7 as 8x5+8x2. So, this is not a completely new idea. However, multiplying 512x3 certainly is. So, they need to learn how to multiply 2- and 3-digit numbers by 1-digit number. The third idea uses the associative property of multiplication. Students my have used it to find something like 7x4 as 7x2x2. So, the use of property itself may not be new, but 512x10 certainly is. Students must learn how to multiply numbers by 10 before they think about multiplying a number by multiples of 10.

So, from this example, we can see 5 important mile markers of multiplication instruction in Grades 3 and 4.
a. Expand the basic multiplication up to 10x10. [M3N3b]
b. Understand how to multiply multiples of 10 and 100 by 1-digit number (e.g., 30x8, 400x6, etc.).
c. Understand how to multiply 2- and 3-digit numbers (but not multiples of 10 and 100) by 1-digit numbers. [M3N3c]
d. Understand how to multiply by multiples of 10. [M3N3d]
e. Understand how to multiply be general 2-digit numbers [M4N3]

As you can see, all but one mile marker is explicitly noted by the GPS. Starting next entry, I will discuss these 5 mile markers in more details.

M4N4b - Developing division algorithms (2)

2009-08-29T06:58:00.000-07:00

M4N4. Students will further develop their understanding of division of whole numbers and divide in problem solving situations without calculators.
b. Solve problems involving division by 1 or 2-digit numbers (including those that generate a remainder).In the previous post about the long division algorithm, I mentioned that the partitive (fair sharing) division may be more useful to develop that algorithm. What students might do to model a partitive problem with concrete materials like base-10 blocks will match up very nicely with the paper-and-pencil algorithm you are trying to help students develop. What will happen if we have a quotitive (measurement, or repeated subtraction) division problem? Let's look at Problem 2 from the last post:
Problem 2: There are 72 sheets of construction paper. If you make bundles of 4 sheets, how many bundles can you make?To solve this problem using concrete materials, students will make groups of 4. Often times, adults (or students) will describe the first step of the long division by saying, "how many times can 4 go into 7?" However, since we are dealing with 72, "7" is actually 7 rods. To match up the algorithm, what we are asking ourselves is, "how many groups of 4 rods can we make with 7 rods?" The fact that you can make 1 group of 4 rods, actually suggests that we can make 10 groups of 4 units. So, if you already know the long division algorithm, you can make the process match the algorithm. However, for children who are learning the algorithm for the first time, that task isn't as straightforward as it will be with the partitive division.

However, this alternative way of looking at division may be useful when you are actually dividing by large numbers - like when you have to divide by a 2-digit number in Grade 4. Suppose you have the following problem:
Problem 3: There are 1950 sheets of construction paper. If you make bundles of 38 sheets, how many bundles can you make?So, you will ask, "how many times can 38 go into 195?" To estimate this partial quotient, you may round up 38 and think about, "how many times can 40 go into 195?" 40x5 is 200, and that's too big. So you estimate the tens digit of the quotient is 4. You multiply 38x4 and subtract it from 195 and get the difference of 43! So, the tens digit of the quotient must be 5, not 4. So, you have to re-calculate.

Instead of doing this, you can think about the problem differently. The question is to determine how many groups of 38 you can make with 1950. If you have a reasonable number sense, you can see that the double of 38 will be 76. So, you can easily make 20 groups, or use 760 sheets. 1950 - 760 = 1190, so you can make another set of 20 groups. 1190 - 760 = 430, so we can make 10 more groups. 430 - 380 =50, and that's one more group. 50 - 38 = 12, so we can't make any more group. Therefore, we made 20 + 20 + 10 + 1 = 51 groups, with 12 sheets left over.

This process can be made into a written process like this:

This algorithm is sometimes called the Scaffold algorithm. Others may call it a "forgiving method," as it doesn't require the best estimate of the partial quotient. It is useful in some situations, like when the divisor becomes large. Should all students know this algorithm? I am not so sure. One of the important ideas of teaching students computational algorithms is that students understand that with our numeration system, we can look carry out calculation by focusing on one place value at a time. This algorithm treats the numbers (divisors) as a whole.

On the other hand, it does have some usefulness as we can see. For me, teaching of an algorithm means helping children make their own procedures (with concrete materials or thinking strategies) into a written procedure. So, if children aren't thinking this way, then imposing a method doesn't seem to be too productive. Of course, by asking students to think about quotitive (measurement) division problems, you can increase the likelihood of students thinking this way, too.

If we are to teach this algorithm, I think it is important for students to realize when this method might be more useful than the long division algorithm. We want students to make intelligent decisions about how to calculate - which algorithm to use, whether or not an estimation is good enough, etc. So, if this algorithm is included in your curriculum, I encourage you to help your students understand its merits so that they can use different methods flexibly.

M3N4e - Developing division algorithms (1)

2009-08-16T19:22:00.000-07:00

M3N4. Students will understand the meaning of division and develop the ability to apply it in problem solving.
e. Divide a 2 and 3-digit number by a 1-digit divisor.A lot of people seem to have a very negative feeling toward "long division," the division algorithm which is commonly used in the US today. Some people even call for eliminating long division since we can use calculators. Clearly, the writers of the GPS did realize the foolishness of such a recommendation. Thus, in M4N4, they specifically state that students can divide without calculators. So, why is "long division" so disliked by many?

One of the reasons is probably the multiple steps involved in the procedure. There are many different mnemonics that is supposedly help children remember the sequence of those steps. How else is the long division algorithm different from other algorithms? One difference is that the long division is the only common algorithm that goes left to right. With addition, subtraction and multiplication, we are taught to start with the ones place - of course, it is perfectly possible to go left to right, but that's a different story. Constance Kamii and other researchers have pointed out that children, when they are asked to think about numbers, naturally start with the largest places. So, many first graders, when asked to find the sum of 23 and 31, they would think like, "20 and 30 make 50, and 3 and 1 make 4, so the answer is 54." Many adults, when they are estimating the sum or difference of 3- or 4-digit numbers mentally, they find it much easier to go from left to right. For example, with 584+279, you would think, "500 and 200 is 700, 80 and 70 is 150, so 850 altogether, 4 and 9 is 13, so the answer is 863." So, in a way, we can argue that the long division algorithm is the only common algorithm that aligns with our natural way of thinking.

So, how can we help our students more naturally develop the long division algorithm? One of the keys is how we organize our instruction. First, let's think about the meaning of division. We know that students are introduced to two types of division situations (M3N4b): partitive (fair sharing) and quotitive (repeated subtraction). Which should we use when teaching the long division algorithm? Some might say it would not make any difference, but I argue that it is much easier to work with partitive situations if you want students to develop the long division algorithm. Thus, we should start with Problem 1, instead of Problem 2:
Problem 1: There are 72 sheets of construction paper. If you share them among 4 students, how many sheets will each student receive?

Problem 2: There are 72 sheets of construction paper. If you make bundles of 4 sheets, how many bundles can you make?
Second, let's think about what learning tools students should use. For teaching and learning of the long division algorithm, I think base-10 blocks are very useful. Of course, if base-10 blocks are to serve as students' thinking tools, they have to be comfortable with them before they start using to work on problems like Problem 1. If students are familiar with base-10 blocks, how might they solve this problem? It is not unreasonable to think that they will first make 72 by using 7 rods and 2 unit cubes. They will then give one rod to each of the 4 groups. At that point, they will trade in the remaining 3 rods to get 30 unit cubes, and they now have 32 unit cubes altogether to share among 4 groups. They will then distribute 8 unit cubes to each group, with no remainder. Thus, the quotient is 18.

Once students gotten used to solving division problems with base-10 blocks, it's time to help them move beyond the blocks. You can have them draw what they would have done with the blocks, instead of actually using blocks. So, with Problem 1, students will draw 7 rods and 2 units, and 4 circles for the groups.

You can cross out 4 rods and give 1 rod to each group.

Then, you have to cross out the remaining 3 rods and draw 30 units.

Now, you can give 8 units to each.

After while, students will feel this is too much drawing, and that's when you can suggest a couple of things. First, you can suggest that you really don't need all four groups since the final answer is how much is in each group. The second suggestion is to write numerals instead of pictures of blocks using a place value mat. So, for Problem 1, you would write something like this:

Now, when you give 1 rod to each group, you used 4 rods, so you have to take away 4 of the 7.

Now you have to exchange those 3 remaining rods with 30 units, but since you already had 2 units, you now have 32 units.

After giving each group 8 units, you used 32 units and 32-32=0.

You see how similar these notations are to the actual long division algorithm. Once students get used to using this notation, you can probably show the long division algorithm and ask students if they can explain what is happening at each step.

Finally, you want to consider what kinds of numbers you use. When thinking about a 2-digit number divided by a 1-digit number, you want to think about each of the numerals in the 2-digit number in relationship to the divisor. Do you want that to be greater than, equal to, or less than the divisor? If it is less than, that means the tens place in the quotient will be empty - which may be a bit too much for the opening problem. If it is equal, that means there is no left over after all rods are shared. Basically, you can divide each numeral by the divisor. Again, that is a special case, and you may wonder about whether or not starting with a special case is good Something like 72 and 4 as in Problem 1 where the tens digit is greater than the divisor may be a good starting point.

After students develop the division algorithm, that's when we might want to think about those special cases. In addition to the case when the leading digit is equal to the divisor, we must think about those situations when there is an empty place in the quotient - the case when the leading digit is less than the divisor is one such case, i.e., 0 in the tens place (or the leading place). Other cases are when 0 is in the ones place and 0 is in the middle, when dividing 3- or longer digit numbers divided by 1-digit numbers.

With these ideas in mind, the long division algorithm can be learned more naturally. As stated earlier, the long division algorithm may be more "natural" of the four algorithms. However, that doesn't mean that students will automatically develop the algorithm. It takes careful planning by teachers.

M3N4d - Meaning of a remainder in division

2009-08-09T07:04:00.000-07:00

M3N4. Students will understand the meaning of division and develop the ability to apply it in problem solving.
d. Explain the meaning of a remainder in division in different circumstances.
In an earlier post (August, 2007), I discussed the relationship between the two division situations, fair sharing and measurement (discussed in element b) and multiplication situations. When students are first introduced to division, they must understand that the division is an operation needed when you are making equal groups with the given amount. We can use the division to find the number of groups given the number in each group (measurement division), or we can use the division to find the number in each group given the number of groups (fair sharing). We want students to develop a unified understanding of these two situations as "division."

In the introductory stage of division instruction, students focus on those division problems that are the inverse of the basic multiplication facts. Thus, they develop the strategy to find the quotient of division by looking for the related multiplication facts. For example, to solve 48 ÷ 6, students think about 6 and what multiplied together will equal 48. Since 6x8=48 (or 8x6=48, depending on the type of division), we can say the answer is 8. Thus, if we are giving each person 6 candies, 48 candies can be shared by 8 people.

What if we had 50 candies? There is no multiplication fact with 6 as a factor that will give the product of 50. Most children can solve this problem if they are allowed to use concrete materials to manipulate. As division with remainders is introduced, it is important that students initially use concrete materials to model the problem situation. They should compare and contrast the situation with earlier division situations (without remainder) to realize that these situations are also creating equal groups, thus it is appropriate to represent it with division sentences. Furthermore, implicit in division problem is to maximize the quotient (either the number of people sharing or the number of items for each person) - that is, the point isn't just to make equal groups but to use up as many of the given amount. The "remainder" is the amount left over when the maximum amount of the given amount is used up.

When students simply rely on computation (multiplication) to find the remainder, sometimes you see mistakes like 50÷6=9 remainder 4. This occurs because when you look for the quotient by checking the 6's multiplication facts in order, you recognize the quotient only after the product exceeds the total. Thus, some children mistakenly think that the quotient is 9.

If students do not understand the meaning of the remainder clearly, the opposite can also happen. Some students may say that 50÷6=7 remainder 8. Those students do not understand that we have to use up as many of the given amount. Since this is not a computation error (in the sense of getting an incorrect product or incorrect difference), the answer checking algorithm (Dividend = Divisor x Quotient + Remainder - Gr. 4 GPS) will not detect it. To avoid this error, students must clearly understand the meaning of remainders. Furthermore, since we are using up as many of the given amount, the remainder cannot be greater than the divisor - in either type of division situation. If the left over amount is greater than the divisor, that means we can give (at least) one more person his/her share (measurement division) or (at least) one more item to each person (fair sharing). But, this relationship, Remainder is less than Divisor, is the result of the often implicit requirement of division that we must use up as many of the given amount as possible.

By the way, in many Japanese elementary mathematics textbooks, the long division notation is introduced when students are learning division with remainders - after students understand the meaning of division, remainder, and the relationship between the remainder and the divisor. This notation is introduced to ease the mental demand involved in division with remainders. For example, in the case of 50÷6, children must identify the first multiplication facts with 6 as a factor that exceeds 50, 9, then subtract 1 from it to make 8 as the quotient, find the product of 6x8, then subtract the product, 48, from 50 to find the remainder. The long division notation can provide a way for children to record the intermediate steps and procedures more explicitly. However, it is important to remember that we are not really teaching children the long division algorithm here. The notation is simply introduced as a way to deal with simple division with remainders (that is, those division that requires the application of the multiplication facts only once). I will discuss the development of the long division algorithm, element e, in a separate post.

M1N2 - Understanding place value notation

2009-08-02T07:00:00.000-07:00

M1N2. Understand place value notation for the numbers between 1 and 100. (Discussions may allude to 3-digit numbers to assist in understanding place value.)

Most, if not all, elementary school teachers know that understanding of place value is the key to success in elementary school mathematics. But what does it mean to understand place value? How did we get stuck with such a complex notation system? What benefits are there that this particular system offers that other system didn't?

Probably the simplest number notation system is the tally system. It may have started with people carrying around some pebbles (or acorns or whatever), but eventually became a written system. If you see "|||||||||||||" you can actually "see" the number. Unfortunately, when the number gets large, it becomes difficult to distinguish numbers. Soon, people started to come up with simplified system, such as noting 5 as ||||. A further extension is a system like the Egyptian system where they used a new notation for 10, 100, etc. With those notations, it became easier to distinguish large numbers, but numbers themselves are no longer "visible."

One of the shortcomings of a system like the Egyptian system is that you need, in theory, infinite number of symbols in order to express very large numbers. Other people, like the Babylonians and the Mayans, came up with a system in which where a numeral is written also contributed in the way the number is written. The picture below shows how the Mayan system worked:

Each group of symbols actually represents a number between 0 and 19 inclusively. However, where it is written makes difference - so, "17" in the second from the bottom is in the 20's place, so it actually represents 340. Therefore, this system is like ours in that they used place values.

So, why didn't this system survive? It used only 3 symbols: 1 (dot), 5 (horizontal bar), and 0 (sea shell). Our system requires 10 numerals (0 through 9). One of the reason why this system did not survive is probably because of this economy of the symbols. When you have 5 in one place and another 5 in the adjacent place, it is difficult to distinguish that from 10 in one place.

So, it appears that our history can be characterized as the search for a balance between simplicity and complexity. Our current numeration system, typically called the Hindu-Arabic system, eliminated the confusion of the Mayan (and the Babylonian) system by using more symbols but a smaller exchange rate for adjacent places. So, here are the major "rules" of our number system:
1. Where a numeral is written matters, i.e., "1" in 31 and 15 represents different numbers.
2. Any pair of adjacent places have 10-to-1 relationship, i.e., you need 10 of the smaller place value to exchange with 1 of the larger value - therefore, the place values are all powers of 10.
3. The total value of a number is determined by multiplying the numeral by the place value and finding their sum.
4. There must be one and only one numeral in each place.
5. Because of (4), we must use "0" as a place holder - except for the leading 0's (and trailing 0's in decimal numbers).
(4) and (5) are often learned in the process of learning how to record addition/subtraction using written algorithms. Most teachers have seen children who tried to write "12" in the ones place when they add 35 + 17. Up till that point, (4) is not made explicit so children do what is most natural thing to do. In fact, (4) is the reason why we have to worry about re-grouping. And now we seemed to have introduced another complexity to our numeration system.

So, what are the merits of our number system? Probably the biggest merit of our number system is that calculation is simple. Wait a minute! we just said because of a rule of our system, we had to worry about re-grouping, a difficult idea for many children.

Yes, it is true that re-grouping is difficult, but with our system, if we know the basic addition and multiplication facts, we can do any calculation. Just think about this for a minute. If you know 100 addition facts (0+0 through 9+9) and 100 multiplication facts (0x0 through 9x9), you can do ANY calculation, no matter how large or how small numbers are. Just imagine how you would calculate 34x72 using the tally system, the Mayan system, the Roman numerals, etc.. So, one important, and often implicit, goal of teaching children computational algorithms is to help students understand this merit of our number system.

M5N3c - Multiplying and divising by numbers less than one

2009-07-26T10:32:00.000-07:00

M5N3. Students will further develop their understanding of the meaning of multiplication and division with decimals and use them. c. Multiply and divide with decimal fractions including decimal fractions less than one and greater than one.
Consider a problem like the following:
A ribbon costs $ 1.80 for one meter. If you want to buy 0.8 meter of the ribbon, how much will it cost?
When children are asked what operation they would use to solve this problem, many will pick division. Some might think those children simply do not understand multiplication and division. However, that is not the case. Those children pick division as the operation because they know that "multiplication make bigger and division makes smaller." Research has shown that many children, and adults, hold this misconception.

Actually, calling it a "misconception" may be inappropriate. Rather, it is an overgeneralization children make based on their experiences. While students are working only with whole numbers, the only exception to this generalization is multiplication by 0 and 1. However, once they go beyond the "basic facts" stage of multiplication learning, practically all experiences involve multiplying by a number greater than one. The same can be said of division. The only time division does not result in a number less than the dividend is when it is divided by one. However, once again, practically all children's experiences before this point are division by a number greater than one.

Once the range of numbers is expanded to include decimal numbers and fractions, however, there are many cases where we do multiply or divide by numbers less than one. Therefore, it is an important goal of mathematics teaching that our students overcome this overgeneralization. A potentially powerful tool for this purpose is double number lines. If we represent the ribbon problem, it will look like this:

From this diagram, you can easily see that if the multiplier (represented on the bottom number line) is less than 1, the product (? mark) will be on the left of the multiplicand (1.80, i.e., amount corresponding to 1). On the other hand, if the multiplier is greater than 1, the product will be to the right of the multiplicand. Therefore, we can generalize:
If multiplier is greater than 1, multiplicand < product.
If multiplier is less than 1, product < multiplicand.
Similarly, you can use double number lines to contrast the situations when the divisors are less than 1 and those cases where the divisors are greater than 1.

However, the most difficult part for students (thus for teachers) is to help them understand that these situations are indeed situations where multiplication is the appropriate operation. For some students, double number line may not be sufficient. Another possible tool is to write mathematical expressions using words to describe the relationship among the quantities involved. In the ribbon problem, there are three quantities: cost of 1 meter of ribbon, total length of ribbon, and the price. The relationship among these three quantities can be expressed as
Price = [Cost of 1 meter] x [Total Length].Thus, for this problem, ? = 1.80 x 0.8.

An implicit, yet very important, goal of teaching multiplication and division of fractions and decimal numbers is to expand students' understanding of these operations. In early elementary grades, these operations are considered in equal group situations. Thus, when the multiplier or the divisor (in the case of fair-sharing division, the quotient in the case of measurement division) becomes something other than whole numbers, students have difficulty interpreting what it means. Through teaching of multiplication and division of decimal numbers and fractions, we want students to develop more proportional understanding of these operations. For example, A x B = C, should be interpreted as "A is to 1, C is to B," or "C is B times as much as A." Although this is not an explicitly stated goal in the GPS, it is something all teachers must keep in mind.

MKN1 h, i, j - Coins

2009-07-18T09:50:00.000-07:00

MKN1. Students will connect numerals to the quantities they represent.
h. Identify coins by name and value (penny, nickel, dime, and quarter).

i. Count out pennies to buy items that together cost less than 30 cents.

j. Make fair trades using combinations involving pennies and nickels and pennies and dimes.

Quite frankly, I really don't understand why money and clock reading are in the mathematics curriculum. Those ideas should be learned in everyday contexts in which they mean something. The names and the values of each coin aren't mathematical concepts. However, whether or not I like these topics in the mathematics curriculum really matters much as teachers are expected to teach them. I heard many teachers say that money is a difficult idea for children. There may be a number of reasons for children to have difficulty with money. For one thing, being able to exchange a merchandise with a piece of metal or paper cannot be really "natural" to children. Another major source of difficulty for young children is the notion of exchanging coins. It's not quite logical why 5 shiny pennies can be exchanged with one, slightly larger coin, for example. Neither of these is mathematical ideas, and I'm not really sure how you can help young children with these ideas.

However, I do want to say something about exchanging coins. A part of the difficulty for young children is, I believe, because they have yet to develop a very sophisticated understanding of numbers. Kindergarten teachers are familiar with an exchange like the following:
Teacher: How many blue counters do you see?

Child: (counting to herself) One, two, three, four. Four !

Teacher: How many red counters do you see?

Child: (counting to herself) One, two, three. Three!

Teacher: So, how many counters are there altogether?

Child: One, two, three,...

Teacher: (interrupting). Wait a second. How many blue counters?

Child: One, two, three, four. Four.

Teacher: How many red counters?

Child: One, two, three. Three.

Teacher: So, how many counters altogether?

Child: One, two, three, four, five, six, seven. Seven!
Many adults are simply puzzled why a child will have to count all the counters when they know that there are four blue counters and three more red ones. However, for many young children, numbers exist only after they count - "four" cannot exist by itself. Furthermore, for many of them, four simply means four ones. Children must develop an understanding that four can be considered as an entity, or a unit, in itself before they can count on. In the previous post, I discussed the idea of five and ten as a benchmark. Children cannot think of five as a benchmark unless they can think of five as five ones and, simultaneously, one five. For adults, this is so obvious, and it is difficult to even fathom anyone (including children) not understanding it. However, research clearly shows that children don't automatically understand this idea.

So, if a child is still "counting all" stage, it is probably not reasonable to expect him/her to be able to make an exchange of five pennies with one nickel with understanding. One way to help children overcome difficulty with money is to help them develop good number sense - the ability to see numbers flexibly. Without number sense, money is much more difficulty to make sense.

MKN1f - Five and ten as benchmarks

2009-07-10T05:24:00.000-07:00

MKN1. Students will connect numerals to the quantities they represent.
f. Estimate quantities using five and ten as a benchmark. (e.g. 9 is one five and four more. It is closer to 10, which can be represented as one ten or two fives, than it is to five.)

Although this indicator includes the words "estimate," what it is talking about isn't really estimation in the sense of "about how big" a number is. Rather, it is more about looking at a number from different perspectives. Thus, 8 isn't just eight ones, but rather, it is three more than 5 and two less than 10 as well. From that perspective, this standard relates very closely to MKN2 b, "Build number combinations up to 10 (e.g., 4 and 1, 2 and 3, 3 and 2, 4 and 1 for five) and for doubles to 10 (3 and 3 for six)." Using five and ten as a benchmark is in a way a special case of this indicator. Furthermore, being able to look at numbers from multiple perspectives is something that is continuously emphasized in the elementary GPS. According to Elementary School Teaching Guide for the Japanese Course of Study, the ability to see a number as a sum, a difference, a product, or a quotient of other numbers is an important foundation for algebraic thinking.

One common tool that is often used to help students develop this idea of five and ten as a benchmark is a ten frame:

It is just a table with 2 rows of 5 cells. Different numbers can be represented by filling up these cells with counters. However, when you represent numbers 6, 7, and 8, it is important that a row is filled up completely so that those numbers are represented as 5 and some more,

not like

The latter representation is useful if we want children to develop the understanding that 8 can be represented as 4 and 4 (MKN2b). Ten frames are very versatile tools, but that means we, as teachers, must be very intentional about how we use them to help students develop a specific understanding.

M4N2 a - Rounding

2009-06-19T04:16:00.000-07:00

M4N2. Students will understand and apply the concept of rounding numbers.

a. Round numbers to the nearest ten, hundred, or thousand.

Rounding is a specific technique to approximate numbers. Some teachers in primary grades actually teach their students rounding when they want students to "estimate." However, "estimation" and "rounding" aren't the same idea. In fact, as an approximation technique, it is probably better to teach rounding when students are working with larger numbers.

As you know, rounding a number to the nearest designated place means to look at the numeral to the right of the place to which we are rounding. If the numeral is 4 or less, we will round down (i.e., simply change all places to the right of the designated place 0's) and if it is 5 or above, we round up (i.e., increase the numeral in the designated place by 1 and change all numerals to the right 0's). So, when 45,542 is rounded to the nearest thousands place, it will be 46,000, and when it is rounded to the nearest hundreds, it will be 45,500.

One question students sometime ask is why we round up with a "5" even though 5 is right in the middle (of 0, 1, ..., 9). Some teachers will simply say it's just a rule. But, is it?

Let's consider 45,542. If we want to round this number to the nearest thousands place, we are really asking is it closer to 45,000 or 46,000. According to the procedure, we will be checking the numeral in the hundreds place. So, what numbers between 45,000 and 46,000 have a 5 in the hundreds place? Well, 45,500 is definitely one. But there are a lot more: 45,501, 45,502, 45,503, ... 45, 598, 45,599. Altogether there are actually 100 numbers in this range with a 5 in the hundreds place? So, which of these numbers are closer to 45,000? 46,000? Right in the middle? Well, it's obvious that all but one of these numbers are actually closer to 46,000, and the one exception is right in the middle. If that's the case, in general, does it make sense to round a number with a 5 in the hundreds place up or down?

The problem with "5 is right in the middle" comes up only when you are rounding to the nearest tens (and only if we are looking at whole numbers). Since approximate numbers are used when we have very large numbers of very small numbers, perhaps trying to teach rounding, a specific approximation procedure, with such small numbers may not make any sense.

M7n1 a - Meaning of 0

2009-06-09T17:47:00.000-07:00

M7N1. Students will understand the meaning of positive and negative rational numbers and use them in computation.
a. Find the absolute value of a number and understand it as the distance from zero on a number line.

I usually don't get many comments on my blog entry (and I would be happy to hear from more of you), but on May 30, PJGould said that he had come across a child who started his counting with zero. Of course, he noted, that made his counting always off by one. After all, zero is not a counting (natural) number. But what does zero mean?

In elementary (K-5) curriculum, there are 3 meanings of zero - perhaps it is more accurate to say 3 ways zero is used. First, zero indicates the cardinality of an empty set - that is, zero means 'nothing.' This is probably the most commonly used meaning of zero in elementary school. Another place zero is used is as a place holder in a written numbers, such as 3042. Of course, this is a slight extension of the first meaning in that there is no unit of one-hundred in this written number. So, it is still pretty close to the first meaning.

The third usage of zero in elementary schools is the starting point of a number line. In some textbooks, a number line actually starts with zero as shown below.

In other textbooks, the tick mark for zero is not at the end of a number line, implying that there may be something to the left of zero as well.

As students study positive and negative number, one of the important understanding students have to make is the meaning of zero as a referent point, or the origin, on the number line. As long as students are stuck with the idea that zero means 'nothing,' some will have difficulty making sense of numbers that is 'less than nothing.' Rather, students must look at zero as a referent point on a number line, and those number to the right of zero are positive and those on the left are negative. The distance from zero, whether on the right or left is the absolute value of the number.

For those of us who already understand positive and negative numbers, this way of looking at zero is not a major issue. However, we should be aware that this meaning of zero isn't something students are familiar with. In most elementary curriculum, very little explicit discussion takes place about the role of zero on a number line. Thus, when we introduce positive and negative numbers, we do have to keep this shift in understanding of zero in our mind.

M3N5 a - Modeling decimal numbers

2009-06-02T02:07:00.000-07:00

M3N5. Students will understand the meaning of decimal fractions and common fractions in simple cases and apply them in problem-solving situations.
a. Understand a decimal fraction (i.e., 0.1) and a common fraction (i.e., 1/10 represent parts of a whole.

According to the GPS, decimal numbers are introduced in Grade 3. In Grade 3, though, students only consider decimal numbers only in the first decimal place (or the tenths place, if students have already learned fraction terminology). Decimal numbers to the 2nd decimal place and beyond are studied in Grade 4 and above.

I have heard many teachers who say they use money as the model for decimal numbers. However, if we are limiting decimal numbers to be discussed in Grade 3 to the first decimal place, we cannot use money as an appropriate model as money amounts are shown to the second decimal place. Thus, money as a model for decimal numbers is appropriate only starting in Grade 4, and only with decimal numbers with 2 decimal places. Some people argue, and I agree, that money is not a good model for decimal numbers.

So, why isn't money a good model for decimal numbers? First, as we saw above, it is a very limited utility as a model - only in Grade 4 (and above) and only when we are dealing with decimal numbers with 2 decimal places. Although it is true that a 0 may be annexed to a decimal number with only 1 decimal place, e.g., 0.4 = 0.40, looking at all decimal number as 2-digit decimal numbers may not be the most helpful habit to develop.

Perhaps more serious problem with money as a model is that students (and adults) don't really have to think about money amounts as decimal numbers. Rather, they are really combinations of two monetary units, dollars and cents. By using two different units, we can simply work with two whole numbers. For example, we don't consider $2.35 as two and 45 hundredths dollars. Rather, it is TWO dollars and THIRTY-FIVE cents. If we get additional $3.18, we simply add TWO and THREE dollars and THIRTY-FIVE and EIGHTEEN cents. Therefore, we are not really considering those numbers as decimal numbers - they only use notations similar to decimal numbers. Mathematically speaking, there isn't really that much difference between monetary amounts and durations expressed in hours and minutes. If you spend 2 hours and 35 minutes watching TV and 3 hours and 18 minutes playing computer games, then you wasted 2+3=5 hours and 35+18=53 minutes!

So, if money isn't a good model, what other models can we use? Base-10 blocks are always an option - we just have to designate something other than unit cubes as "1." We can also use paper strips, too, just as you might use them to model fractions. No matter what model you decide to use, an important idea we want children to develop is the unitary perspective of decimal numbers. For example, 0.4 is made up of 4 0.1-units. This is very similar to the unitary perspective of fractions I discussed in the last post. This way of looking at decimal numbers will allow students to bridge decimal numbers to whole numbers. So, it is very important for us to think about models to use, but we should also keep in mind the goal understanding we want our students to develop.

M3N5 - Simple cases of fraction addition/subtraction

2009-05-21T03:44:00.000-07:00

M3N5. Students will understand the meaning of decimal fractions and common fractions in simple cases and apply them in problem-solving situations.e. Understand the concept of addition and subtraction of decimal fractions and common fractions with like denominators.
Addition and subtraction of fractions are discussed in three different grades (M3N5e; M4N6b; M5N4g). Both this current standard and M4N6b involve fractions with like denominators. For M4N6b, there is a note stating that denominators should not exceed 12. So, what is the difference between M3N5e and M4N6b? If one of the reasons for developing the GPS was to minimize repetitions, why is this topic repeated in Grade 4?

One of the differences is that in Grade 3, the sum or the minuend must be less than or equal to 1 as students will not be studying improper fractions and mixed numbers until Grade 4. Thus, 2/5 + 1/5 is appropriate in Grade 3 but not 4/5 + 2/5. However, the most important reason for discussing simple addition and subtraction in Grade 3 is to help students understand fractions as numbers, just like whole numbers.

Fractions are often introduced as parts of a whole. Although this way of looking at fractions is relatively easy for students to grasp, research also shows that this is a very limiting view of fractions. In other words, if students can consider fractions only as parts of a whole, they will have difficulty dealing with fraction arithmetic. Part of a whole is a relationship, and we cannot perform arithmetic operations on relationships. We can only add, subtract, multiply, and divide numbers. Thus, students must understand fractions as numbers in order to make sense of fraction arithmetic. So, how do we help students to see fractions as numbers? Well, one way is to help students experience situations where fractions are added or subtracted. From those experiences, students can realize that fractions are numbers because they can be added or subtracted. It sounds like a circular argument, and it probably is. However, I would like to think this relationship more of reflexive, i.e., neither one is a prerequisite for the other, and an understanding of one can actually promote and deepen the understanding of the other.

What is important, though, is that experiences students will encounter are something that they can determine as addition/subtraction situations. For example, we can ask students what is the total length of a tape if a 2/5-meter segment and 1/5-meter segment are put together end to end. They can see that this situation is an addition situation - you would use addition if the lengths of the segments were 2 meters and 1 meter, respectively.

Another key idea is the unitary view of fractions. In other words, students should understand 2/5-meters as made up of 2 1/5-meter segments. Then, 2/5 + 1/5 is really 2 1/5-units and 1 1/5-unit put together, or 2+1 1/5-units. By recognizing that fractions may be added or subtracted, and having a way to reason through to find the answers, students can develop the understanding of fractions as numbers. With this knowledge as the starting point, students in Grade 4 can explore fraction addition and subtraction more formally.

M2N1 - Number lines

2009-05-06T06:16:00.000-07:00

M2N1. Students will use multiple representation of numbers to connect symbols to quantities.

When I visit primary grade classrooms, I often see a large number line posted above the whiteboard in the front of the room. Sometimes I also see number lines taped on students' desks. A variety of experts, including the National Math Panel, state that number lines are powerful representation tools and mathematics instruction should develop students' proficiency with number lines. Singapore elementary mathematics textbooks are famous, in part, because of their use of "tape diagrams" to help students deal with complicated mathematical problems. As I have discussed previously, double number lines can be powerful thinking tools to support students' comprehension of multiplication and division of rational numbers. So, on the surface, the display of number lines in the primary grades (K-2) seems to be a sound teaching practice. But, is it?

When you examine Japanese elementary mathematics textbooks, the formal term, "number line" does not appear until Grade 3. However, that does not mean number lines are not used in Grades 1 and 2 (there is no Kindergarten in Japanese elementary schools). As usual, Japanese textbooks carefully and gradually develop number line representations. Thus, students' first encounter with something like number line is simply placing number cards 1 through 10 in order going from left to right. They will be asked to fill in the missing number in a sequence like 3 - 4 - [ ], or 7 - [ ] - 9. A little later on, once the range of numbers has been extended up to 20, there is a question which asks how far a space alien character hopped along a number line, starting at 0. Missing number problems may also involve number cards sequenced in backward (from large to small). When students are studying numbers up to 100, students are asked to locate given numbers on a number line, and similar questions are asked in Grade 2 when the range of numbers is extended to 1000.

What is conspicuously absent in the Japanese primary mathematics textbooks is the use of number lines to deal with addition and subtraction. Rather, number lines are used to represent visually relative sizes of numbers. I recently heard that some people distinguish number paths and number lines. Number paths, as I understand it, simply string together numbers, 1, 2, 3, ... On a number path, numbers are represented more by their positions (or orders) whereas on a number line, a number is represented by the distance of the tick mark from the origin, i.e., 0. So, the way the Japanese textbooks introduce and use number lines are much more along the line of number paths.

So, why don't Japanese textbooks use number lines to represent addition and subtraction, as is often done in some US textbooks? There are at least a couple of reasons. The idea that a number is represented by the distance of the tick mark from the origin is a difficult one for students in primary grades. This is difficult, in part, because those students are still learning about measuring length. So, they really don't have the prerequisite knowledge to interpret number lines in that manner. What they tend to do is to simply count the tick marks. However, when students count, they start with "1," and this is another reason number lines are complicated for young children. I have yet to meet a child who started his/her counting by saying, "zero... one, two, three, ..." For many young children the role of 0 (the origin) on a number line is mysterious. So, when they have to use number line to solve 5+3, they will start with the tick mark labeled "5," some will point to "5" and say, "one." Most, if not all, teachers of primary grades have seen young children line up their rulers starting at "1." It's the same problem.

Some people suggest that number lines are inappropriate for primary students, and we should not use number lines. However, I do think it is important that number lines are introduced in primary grades. However, we should be careful about how we use them. We can use them to think about relative sizes of numbers. However, it is probably a good idea to wait to use number lines as a tool for arithmetic. We can use something like tape diagram for that purpose. But, developing the idea that numbers can be represented on number lines is an idea that should start in primary grades, and we should guide students to understand how numbers are represented (as distance from the origin) on number lines, perhaps connecting to the study of linear measurement.

P1a - Teaching THROUGH problem solving

2009-04-17T21:07:00.000-07:00

P1. Students will solve problems (using appropriate technology). a. Build new mathematical knowledge through problem solving.

I'm going to write about the same process standard as the last entry; however, this time, I want to focus on the actual indicator, "build new mathematical knowledge through problem solving."

Teaching through problem solving has been a major emphasis in mathematics education over the last (at least) 2 decades - the emphasis on problem solving was there in the 1980 NCTM document. So, it is not necessarily a new idea, but it's not quite clear what this might actually look like in a real classroom. Some people have discussed the three related ideas:
* teaching for problem solving
* teaching about problem solving
* teaching through problem solving

Teaching for problem solving is exemplified by the common textbook organization where students are taught various rules and formulas in a unit, and at the end of the unit is the lesson(s) titled "applications." Students are taught necessary tools, so to speak, and they are given numerous problems for which those tools may be useful.

Teaching about problem solving typically means teaching various problem solving strategies such as guess and check, draw a diagram, look for a simpler problem, make a table, etc. Some textbooks will include a mini-unit on these strategies throughout their textbook, and students are asked to solve problems using the specified strategy.

However, neither approach really produces new mathematical knowledge by solving problems. Teaching through problem solving means students will solve a problem, using only what they have previously learned. Then, by examining their solution strategies, they will generate a new idea/rule/formula. Let's take a look at an example.

In the GPS, students are expected to learn how to determine the area of rectangles and squares by multiplying their dimensions in Grade 3 (M3M4c). Then in Grade 5, students are expected to derive the formulas for calculating the area of parallelograms and triangles (M5M1 b & c). Somewhere in between, students are often asked to find the area of L-shape like the one shown below.

If you ask students to find the area of this shape in many different ways, they may come up with solutions like the ones shown below.

All of these methods will determine the area of the L-shape. However, if you are teaching through problem solving, your real lesson starts once these solution strategies are shared because the goal of the lesson is NOT to determine the area of the L-shape. Rather, you may ask students, "What is common about ALL of these strategies?" One conclusion students may reach is that all of the strategies are somehow using rectangles and squares, shapes for which they already know how to calculate the area. Thus, by discussing that question, students may reach a new understanding that "when we are given an unfamiliar shape, we may still be able to calculate its area by somehow making a familiar shape (or a collection of familiar shapes)."

Your lesson may not stop there. You may want to ask students to sort these strategies - "which strategies are alike?" Often times, students will come up with the following three categories:
* divide the given shape up into several familiar shapes
* cut and re-arrange to make a familiar shape
* make-it-bigger
Thus, students can learn some specific strategies for creating familiar shapes by critically analyzing these strategies.

So, what can we say about teaching through problem solving? One important idea is that the discussion after various solution approaches are shared is the meat of the lesson. That means we must make sure that we leave sufficient amount of time for such discussion. Too often, we see lessons where very little time is left after the last solution is shared. Sometimes this happens because teachers lost track of time as they circulate around the classroom. Other times teachers feel that students need more time to solve the problem. However, I think it is very important for us to remember that the goal is not the answer to the problem. Rather, even if students have not completed their solution, perhaps their incomplete answer may still be sufficient for conducting productive discussion.

Teaching through problem solving is extremely challenging. It requires teachers to have deep understanding of mathematics they are teaching. It also requires teachers to understand their students' mathematical knowledge so that they can anticipate various solution strategies might come up. Furthermore, teachers must have a plan on how to orchestrate the discussion once strategies are shared. Few teachers, if any, can naturally do this; however, it is something teachers can learn, too. Japanese teachers continuously sharpen their craft of mathematics teaching through a process called lesson study. You can learn more about lesson study and also watch some interesting lessons by clicking here.

P1 - Technology and manipulatives

2009-04-06T17:53:00.000-07:00

P1. Students will solve problems (using appropriate technology).

This process standard includes the parenthetical statement, "using appropriate technology." Although the use of computers and other technologies are generally accepted by teachers and general public, the use of hand-held calculators, particularly in elementary classrooms, continue to be controversial. On the other hand, most people seem to endorse the use of concrete materials (manipulatives) in elementary school classrooms. So, what's the difference between technology (not just calculators) and manipulatives? Although some people may say both technology and manipulatives are both simply learning tools, I believe there is a fundamental difference in their nature.

Let's consider how long division algorithm may be taught using a very commonly found manipulatives, base-10 blocks. A typical instructional sequence will start with problems where students are asked to solve sharing problems using base-10 blocks. As students continue to solve these problems using base-10 blocks, teachers may encourage students to start drawing the picture of blocks and modify the picture as blocks are manipulated. Eventually, teachers will ask students to simply draw pictures of what they would do with base-10 blocks without actually working with the blocks. As students continue solving problems by drawing pictures, teachers will encourage students to use numerals to record the process - instead of drawing 4 flats (hundreds), students can simply write "4" under the heading of "flats (or hundreds)." Eventually students can organize the record using the familiar long-division notation. [See, for example, two activities Sharing Base-10 Blocks and Doing and Recording on my university web page: http://science.kennesaw.edu/~twatanab/.]

Now, here is a calculator game that may help students to develop mathematical thinking. It is called "NIM with Calculator." It is a 2-player game. Clear the calculator so that "0" is shown in the display. Players take turn adding 1, 2, or 3. The winner is the player who gets the sum of 21 after his/her turn. It is a very simple game and children do not have problem remembering the rules. When students become comfortable with the game, you may want to ask if there is a "winning strategy" for either player - the player who goes first or the player who goes second. It turns out there is a winning strategy for the player who goes first - that is, if you know the strategy, you can be 100 % sure that you will win if you go first. I encourage you to figure out the strategy.

Once you figure out the strategy, an interesting extension question is how you may be able to figure out the winning strategy if you change the goal number - for example, you can make the player who gets the sum of 24 to be the winner. You can then determine the relationship between the goal number and the winning strategy (in particular, the first number you must enter).

All of these activities - base-10 block division activities and NIM with Calculator - may be appropriate in elementary classrooms at the appropriate time. However, the roles these tools (base-10 blocks and calculators) play are very different in nature. With base-10 blocks, teachers ultimate goal is to help their students go beyond base-10 blocks. That's the reason teachers will start asking children to draw base-10 blocks or imagine what they might do with base-10 blocks. It is possible with base-10 blocks, and other manipulatives, for children to imagine what they would do and what the results of their actions might look like. Thus, they can examine the effects of their actions without having to use manipulatives. Of course, it is essential that children have opportunities to physically manipulate the blocks BEFORE they start imagining what they might do or what the results of their actions might look like.

On the other hand, calculators and other technological tools are often suited for such an instructional step. Children might be able to imagine which calculator keys to push, but it isn't always possible for students to imagine what the results of their actions may look like - in other words, they don't always know what they will see after they hit the "=" key. Similar point can be made with graphing calculators, dynamic geometry software, or productivity software like spreadsheet.

Teachers should be aware of this difference in the nature of these tools. Manipulatives are useful tools for students to make sense of different processes so that they don't have to use manipulatives to figure out the results. On the other hand, technological tools are to be used to help students think about mathematical relationships that might exist among different numbers and shapes in the problem context. We are not interested in "weaning" students from technology - we want our students to become better at using technology. Judicious use of technology requires us to pay attention to this difference.

M6G1 a - Symmetry

2009-03-26T14:31:00.000-07:00

M6G1. Students will further develop their understanding of plane figures.
a. Determine and use lines of symmetry.

In the last entry, I mentioned that the topic of odd/even numbers is one of the topics some teachers are surprised to see discussed so much later than they used to. Another topic that some teachers have expressed their surprise because of the lateness of the treatment is the idea of symmetry. Many teachers of primary grades have children explore (reflective) symmetry through paper folding. They will have students make symmetrical shape by cutting a folded papers, or have them fold symmetric figures so that the two sides will coincide.

Clearly, young children can explore, and enjoy exploring, symmetries through such activities. However, as valuable as such informal experiences may be, they are still "informal" explorations. It is important for children to consider and understand symmetry as a mathematical idea, too. Such study of symmetry is the focus of this particular standard.

According to this standards, students are supposed to "further develop their understanding of plane figures" by studying symmetry. Thus, the purpose of studying symmetry isn't just about learning symmetry. Rather, using symmetry as a new perspective to review those shapes that have been previously studied. So, for example, what kinds of triangles are symmetric? From this perspective, isosceles triangles and equilateral triangles are in one group, symmetric triangles.

Students can also explore which types quadrilaterals have reflective (line) symmetry. Parallelograms, with the exception of those which are also rectangles, do not possess reflective symmetry. Many children (and adults) think that the line that are parallel and in between a pair of sides will serve as the line of symmetry. When they actually fold a parallelogram, they are surprised that the two sides do not match up. That experience, in turn, can help students understand that the line of reflection must be the perpendicular bisector of the segments connecting corresponding points. [The notion of "corresponding points" follows from their study of congruent figures in Grade 5 -- because the two sides of a symmetric figures are congruent, there are corresponding points. As a result, the formal study of symmetry must follow the study of congruence.] With that understanding, children can now determine the line of symmetry (M6G1a) without having to actually fold the paper, or simply eye balling it. This knowledge will also allow them to complete a figure when one side of the figure and the line of symmetry are given.

Many Japanese mathematics teachers consider the study of symmetry in Grade 6 as the culminating point of the study of geometry in elementary schools (in Japan, elementary schools cover grades 1 through 6). Children not only learn about symmetry, but they also learn to use symmetry as a perspective to re-analyze shapes they have learned. Most Grade 6 classrooms in Georgia are in middle schools. So, perhaps we can position the study of symmetry as an entry point into a more formal study of geometry in secondary schools.

M5N1 a - Even and odd numbers

2009-03-10T03:45:00.000-07:00

M5N1. Students will further develop their understanding of whole numbers.
a. Classify the set of counting numbers into subsets with distinguishing characteristics (odd/even, prime/composite).
When teachers examine the GPS, there are (at least) a couple of topics they are surprised to see so much later in elementary schools than they are used to. The topic of odd/even numbers is one of those topics. Teachers are surprised that a topic that they used to discuss in the second grade (or even in the first grade) is now delayed until Grade 5. Some may be tempted to include this topic in an earlier grade. So, why does the GPS wait to discuss this topic until Grade 5? Although I cannot speak for the committee who developed the GPS, I can share with you the Japanese perspective.

There is no question that we can teach second grade children to distinguish odd/even numbers. We can connect to skip counting or simply tell students that all numbers in the sequence, "2, 4, 6, 8, 10, ..." are called even numbers and the rest are odd numbers. For larger numbers, they can simply use the rule, "if a number ends with a 0, 2, 4, 6, or 8, it is an even number." However, the point is NOT identifying even/odd numbers. Let's look at the GPS statement:
Classify the set of counting numbers into subsets with distinguishing characteristics (odd/even, prime/composite).
It is important to note that what we want students to understand is that "counting numbers" (whole numbers?) can be classified into different subsets by paying attention to various distinguishing characteristics. So, what is the distinguishing characteristics for even/odd numbers? It is the divisibility by 2. Even numbers are those numbers that are divisible by 2, while odd numbers are those that cannot be divided (with a whole number quotient) by 2. So, the emphasis is not about identifying even/odd numbers, but understanding ways to sort whole numbers. Even/odd numbers are just an example of one such classification schemes.

To help students focus more on ways to classify whole numbers, teachers may want to engage students with a task that require students to sort whole numbers in a similar way - by focusing on the remainder when divided by a number. You can watch a videotaped lesson from Japan, in which the teacher posed an interesting question involving the idea of classifying numbers by looking at the remainders when they are divided by 4.
http://tiny.cc/rkB1X

M4N7 - Order of operations and writing math sentences

2009-02-23T19:30:00.000-08:00

M4N7. Students will explain and use properties of the four arithmetic operations to solve and check problems.
b. Compute using the order of operations, including parentheses.

As I look back on my own school experiences in Japan, I really don't remember explicitly learning about the order of operations. As I examine the current Japanese elementary school mathematics textbooks, there is really no unit titled "Order of Operations." Of course, that does not mean that Japanese children do not learn the order of operations. What is important to notice is that how they study order of operations is more from the perspective of writing mathematical expressions.

For example, in Grade 4, students are given the following problem:
Makoto had a 1000-yen bill. He bought a 460-yen notebook and a 140-yen pair of scissors. How much change did he get back.However, the focus here is not on solving this problem as the solution is actually given on the textbook page. In fact, there are two possible solutions given. Here is Naoko's method:
1000 - 140 = 860
860 - 460 = 400Makoto, on the other hand, solved the problem this way:
140 + 460 = 600
1000 - 600 = 400The task given to students is to think about how these sentences may be combined into one math sentence. They are also given a math sentence with words:
[Money Paid] - [Total Price] = ChangeFrom here, students are expected to understand that Makoto's 140 + 160 is one quantity, namely [Total Price] in the math sentence with words. Therefore, they learn that the two sentences may be combined into: 1000 - (140 + 460). Since (140 + 460) represents one quantity, it has to be calculated first.

A little later in the unit, students are given the following problem:
Let's make one math sentence for each of the following problems, then find the answers.
a) You have a 100-yen coin. You buy 3 sheets of paper, and each sheet costs 25 yen. How much change will you get?
b) You buy a 500-yen pencil case and a half dozen pencils. A dozen of pencils cost 480 yen. How much will you pay?The textbook simply tells students that, in math sentences, multiplication and division (i.e., products and quotients) can be considered as one quantity and no parentheses is needed. Thus (a) can be represented by the math sentence, 100 - 25 x 3, while (b) can be represented as 500 + 480 ÷ 2.
Finally, after a few practice problems, the textbook summarizes the rules about the order of operations:

Generally goes from left to right.
If there are any parentheses, we calculate inside the parentheses first.
Multiplication and division are performed before addition and subtraction.

Clearly, knowing the order of operations is important. However, what I observe in these pages from the Japanese textbooks is that they place just as much, if not more, emphasis on writing and interpreting mathematical symbols/expressions/equations. Mathematical symbols/expressions/equations are the language of mathematics, and it is important for students to be able to communicate their ideas using the language of mathematics effectively. Furthermore, it is important for children to understand that we can create compound "sentences," too. Mathematical sentences allow us to concisely represent quantitative relationships and our own thinking processes. As we teach the order of operations, perhaps we should also keep in mind that it isn't just about calculation students are learning. We should consider including questions such as writing and interpreting compound sentences as well.

M4A1 - To use tables or not?

2009-01-31T05:56:00.000-08:00

M4A1. Students will represent and interpret mathematical relationships in quantitative expressions. a. Understand and apply patterns and rules to describe relationships and solve problems.

One of the important aspect of the study of algebra at the elementary school level is the idea of patterns. As a result, many curricula include problems like the following.

A square table can seat 1 person on each side. For an outside picnic, we are going to make a long "train" of tables like the picture below.

a) How many seats will there be if we make a train of 8 tables?
b) How many seats will there be if we make a train of 24 tables?
c) How can we easily calculate the number of seats if you know the number of tables?

Often times, children are encouraged to make a table and find out how many seats there will be for 1, 2, 3, ... tables. Then, they are encouraged to find patterns so that they can answer questions (b) and (c). Most children will have no problem coming up with the table like the following. [I'm having a little problem formatting this page -- please scroll down further.]

Tables	Number of Seats
1	4
2	6
3	8
4	10
5	12
6	14
7	16
8	18

Children can easily notice that every time we add a table, we increase the number of seat by 2. So, some will try to extend this pattern to find the number of seats when there are 24 tables. Others may notice that 24 is 3 times as many as 8, so they think that the number of seats must also be 3 times as many as 18, the number of seats with 8 tables. Of course, the answers will be different, and there could be a very productive discussion about what is going on here. Such a discussion may be particularly important in Grade 6 when students are studying proportional relationships. It is only in proportional situations where the latter reasoning process will work. In fact, in the Japanese curriculum, a proportional relationship is defined as the following. When one quantity becomes 2, 3, 4, ... times as much, the other quantity also becomes 2, 3, 4, ... times as much. Then, we say those two quantities are in proportion. So, this table problem is an example of relationship something other than proportional.
In order to answer (c), some may suggest students modify the way the number of seats are expressed slightly.

Tables	Number of Seats	details	summarized
1	4	4	4+2x0
2	6	4+2	4+2x1
3	8	4+2+2	4+2x2
4	10	4+2+2+2	4+2x3
5	12	4+2+2+2+2	4+2x4
6	14	4+2+2+2+2+2	4+2x5
7	16	4+2+2+2+2+2+2	4+2x6
8	18	4+2+2+2+2+2+2+2	4+2x7

From this table, we can see that: Seats = 4 + 2 x (Tables - 1). Other children may notice that the number of seats is always 2 more than the double of the number of tables. Therefore, they will come up with Seats = 2 x Tables + 2. However, many of these students will not be able to explain why we multiply by 2 (in both cases) or add 2 (in the second case). Some may say that +2 in the second case signifies the fact that the number of seats increases by 2 every time we add a table. But, is it?

Let's think about how children might approach this problem if we changed the way we pose this problem slightly.

A square table can seat 1 person on each side. For an outside picnic, we are going to make a long "train" of tables.
a) Think about different ways you can count (or calculate) the total number of seats when we make a train of 8 tables as shown below.

b) How many seats will there be if we make a train of 24 tables?
c) How can we easily calculate the number of seats if you know the number of tables?

Clearly, some will count seats going around this train one by one. However, there are many other possibilities. Here are four ways children might determine the number of seats.

Method 1

There are 2 seats on each table, top and bottom in the picture above (in alternating colors), and 2 more on the ends of the train. Therefore, with 8 tables, 2x8+2=18, or 18 seats.
In general, Seats = 2 x Tables + 2
Method 2

There are 3 seats on each of the tables on the end, and all other tables have 2 seats. So, 6+2x(8-2)=18, or 18 seats.
In general, Seats = 6 + 2 x (Tables -2).
Method 3

Each table can seat 4 people by itself. However, every time we put a table together with another, we loose 2 seats for each "joint." With 8 tables, there are 7 joints. So, 4x8-2x7=18, or 18 seats.
In general, Seats = 4 x Tables - 2 x (Tables - 1).
Method 4

There are 4 seats on the first table. Every time we add a table, one seat at the joint is shifted to the end of the train, but that seat really came from the first table. Then, each new table will add 2 additional seats, top and bottom in the picture above. So, 4+2x7=18, or 18 seats.
In general, Seats = 4 + 2 x (Tables - 1).
Although the final generalizations may be all different, it is not unreasonable to expect these students to be able to understand what each number and operation means in other students' equations.
From Method 1, we can tell that "+2" in Seats = 2 x Tables + 2 really comes from those 2 seats on the ends of the train. In other words, "2" in "+2" is actually the constant in this situation, not the 2 seats that will be added every time a table is added to the train. The 2 additional seats are actually represented by the coefficient of Tables. From an algebraic perspective, this makes sense because 2 new seats for each new table is actually the rate of change, or the slope of the line. Thus, it should be the coefficient of the independent variable, in this case Tables.

So, what was different about these two situations? Obviously the way the problems were posed was different, but how did that difference influence the outcomes? In the original problem, students created the table and try to find number patterns in the table. However, in the second situation, students were asked to focus on the way they came up with the number of seats. The students in the first situations might have counted the number of seats in many different ways. However, I suspect most children will simply count the number of seats around the train. Furthermore, once they notice "+2" relationship in the number of seats, some may even skip the counting step and simply fill in the table. On the other hand, in the second situation, students' focus was on their actions. What they were doing was to mathematically express, or represent, their actions. Particularly in elementary schools, mathematical expressions should represent the way quantities relate to each other, and that relationship often becomes explicit in students' actions. Thus, we need to encourage them to reflect on their actions. This is not to say looking for patterns in a table is unimportant. We want students to develop their number sense and mathematical reasoning with numbers abstractly as well. Such an ability may be particularly critical in science. However, in the fourth grade when students are first learning about expressing quantitative relationships using mathematical notations, perhaps we may want to emphasize students' reflection on their own actions so that they can understand the meaning behind each part of the mathematical expressions.

M4A1bc & M5A1a - symbols such as □ and Δ

2009-01-17T08:43:00.000-08:00

M4A1. Students will represent and interpret mathematical relationships in quantitative expressions.
b. Represent unknowns using symbols, such as □ and Δ.
c. Write and evaluate mathematical expressions using symbols and different
values.
M5A1. Students will represent and interpret the relationships between quantities algebraically.
a. Use variables, such as n or x, for unknown quantities in algebraic
expressions.

In the last entry, I discussed mathematical expressions as the language of mathematics and how important it is for students to learn to write and read mathematical expressions, starting when students start studying addition formally in Grade 1. I also discussed it may be possible for students to represent situations involving missing addend situations using mathematical expressions using a box, like 5+[ ]=8 (or 5+?=8, 5+__=8, etc.). In grades 1-3, those symbols are used as place holders for particular values. However, in Grade 4, students begin the next phase of using symbols to represent numbers and quantities, that is, the concept of variables.

As we consider teaching of this complex idea (variables), it may be worth noting a progression across grades:
Grades 1 - 3: writing math sentences with numbers, and occasionally with place holders
Grade 4: writing math sentences with symbols like □ and Δ.
Grade 5: writing math sentences using letters as symbolsSome people may wonder what's the point of using symbols like □ and Δ in Grade 4. Why not just use letters since that's what is typically done in higher math? Although there are probably many reasons for using symbols like □ and Δ, one possible reason is the principle I have observed in many Japanese curriculum materials: do not introduce a new representation and a new concept simultaneously. Although the idea of using letters to stand for numbers may be straightforward to those of us who already learned the concept, I'm also sure that you have heard people say how they were confused by the idea of using letters in math sentences. This suggests that use of letters to represent numbers and quantities isn't that simple. So, it may not be a good idea to introduce both letters as representations and the concept of variables at the same time. However, since we do need symbols to talk about variables, the natural choice seems to be to use something familiar, symbols like □ and Δ.

Now, there are a couple of implications from the previous paragraph. First, it is important that symbols like □ and Δ are familiar to the 4th grade students - that means they should be introduced to the use of those symbols in math sentences before Grade 4. The other implication is that the primary focus on Grade 4 is, then, on developing the concept of variables, not necessarily about using the symbols like □ and Δ. In fact, to develop the concept of variables, in some cases, you may not want to use symbols like □ and Δ. Instead, you may want to write mathematical expressions using words. For example, in the third grade, children learn about calculating the area of rectangles and squares. The GPS (M3M4) isn't quite clear whether or not the formulas should be developed in Grade 3. However, it may not be a bad idea to develop the formulas in the context of studying the concept of variables. In Grade 3, students learned that the area of rectangles and squares can be calculated by multiplying the lengths and the widths. Thus, we can express the relationship with a mathematical expression using words like this: Area = Length x Width. [Moreover, it is important to note that we cannot write the formula as A = lw yet since the use of letters is a fifth grade standards!] You can probably think of many other situations that will be appropriate for Grade 4 students, for example, Change = Amount Paid - Price, Number of Children = Boys + Girls, etc.. In fact, as students explore different patterns and rules to describe relationships (M4A1a), they can use mathematical expressions with appropriate words to represent the patterns and rules.

When students are comfortable with mathematical expressions with words, you may want to suggest using symbols like □ and Δ in some cases. In fact, having some experiences with mathematical expressions with words, may help students' transition to the use of letters as variables in Grade 5. In those situations, instead of using letters like x, y, a, b, etc., you may want to start with the initial of the words used in the expressions (thus A = lw).

Evaluating mathematical expressions (perhaps derived by students) by substituting different values (M4A1c) is also an important activity to help students understand the concept of variables. Again, it is important that we keep in mind that the main focus here is the concept of variables. By substituting different values, students are learning that the variables (words, symbols, or letters) stand for quantities that can vary.

M4A1 - Mathematical expressions (2)

2009-01-03T07:10:00.000-08:00

M4A1. Students will represent and interpret mathematical relationships in quantitative expressions.
b. Represent unknowns using symbols, such as □ and Δ.
c. Write and evaluate mathematical expressions using symbols and different values.
M5A1. Students will represent and interpret the relationships between quantities algebraically.
a. Use variables, such as n or x, for unknown quantities in algebraic expressions.
In the last entry, I discussed mathematical expressions as the language of mathematics and how important it is for students to learn to write and read mathematical expressions, starting when students start studying addition formally in Grade 1. I also discussed it may be possible for students to represent situations involving missing addend situations using mathematical expressions using a box, like 5+[ ]=8 (or 5+?=8, 5+__=8, etc.). In grades 1-3, those symbols are used as place holders for particular values. However, in Grade 4, students begin the next phase of using symbols to represent numbers and quantities, that is, the concept of variables.

As we consider teaching of this complex idea (variables), it may be worth noting a progression across grades:
Grades 1 - 3: writing math sentences with numbers, and occasionally with place holders
Grade 4: writing math sentences with symbols like □ and Δ.
Grade 5: writing math sentences using letters as symbols
Some people may wonder what's the point of using symbols like □ and Δ in Grade 4. Why not just use letters since that's what is typically done in higher math? Although there are probably many reasons for using symbols like □ and Δ, one possible reason is the principle I have observed in many Japanese curriculum materials: do not introduce a new representation and a new concept simultaneously. Although the idea of using letters to stand for numbers may be straightforward to those of us who already learned the concept, I'm also sure that you have heard people say how they were confused by the idea of using letters in math sentences. This suggests that use of letters to represent numbers and quantities isn't that simple. So, it may not be a good idea to introduce both letters as representations and the concept of variables at the same time. However, since we do need symbols to talk about variables, the natural choice seems to be to use something familiar, symbols like □ and Δ.

Now, there are a couple of implications from the previous paragraph. First, it is important that symbols like □ and Δ are familiar to the 4th grade students - that means they should be introduced to the use of those symbols in math sentences before Grade 4. The other implication is that the primary focus on Grade 4 is, then, on developing the concept of variables, not necessarily about using the symbols like □ and Δ. In fact, to develop the concept of variables, in some cases, you may not want to use symbols like □ and Δ. Instead, you may want to write mathematical expressions using words. For example, in the third grade, children learn about calculating the area of rectangles and squares. The GPS (M3M4) isn't quite clear whether or not the formulas should be developed in Grade 3. However, it may not be a bad idea to develop the formulas in the context of studying the concept of variables. In Grade 3, students learned that the area of rectangles and squares can be calculated by multiplying the lengths and the widths. Thus, we can express the relationship with a mathematical expression using words like this: Area = Length x Width. [Moreover, it is important to note that we cannot write the formula as A = lw yet since the use of letters is a fifth grade standards!] You can probably think of many other situations that will be appropriate for Grade 4 students, for example, Change = Amount Paid - Price, Number of Children = Boys + Girls, etc.. In fact, as students explore different patterns and rules to describe relationships (M4A1a), they can use mathematical expressions with appropriate words to represent the patterns and rules.

When students are comfortable with mathematical expressions with words, you may want to suggest using symbols like □ and Δ in some cases. In fact, having some experiences with mathematical expressions with words, may help students' transition to the use of letters as variables in Grade 5. In those situations, instead of using letters like x, y, a, b, etc., you may want to start with the initial of the words used in the expressions (thus A = lw).

Evaluating mathematical expressions (perhaps derived by students) by substituting different values (M4A1c) is also an important activity to help students understand the concept of variables. Again, it is important that we keep in mind that the main focus here is the concept of variables. By substituting different values, students are learning that the variables (words, symbols, or letters) stand for quantities that can vary.

M3A1 - Mathematical expressions (1)

2008-12-23T07:31:00.001-08:00

M3A1. Students will use mathematical expressions to represent relationships between quantities and interpret given expressions.

One of the things that some people are surprised (or even get upset) about is the fact that algebra is included as a content strand for elementary school students (grades 3-5). Unfortunately, there are some even well-educated people who mistakenly think that this means we will be teaching algebra as they experienced in high schools in elementary schools. Clearly, that's absurd. What is being expected, however, is that children begin developing some foundational ideas about algebra and algebraic reasoning. Of course, that raises the question, "What is algebra in elementary school?"

Even though the GPS is heavily influenced by the 1989 Japanese Course of Study, interestingly enough, there is no "algebra" strand in the Japanese standards. Instead, they have a strand titled, Quantitative Relations, in which student learn much of what we would typically include in Algebra and also Statistics (Data Analysis). In the elaboration document the Ministry of Education publishes, they state that two important "themes" in Quantitative Relations are learning about mathematical expressions and studying functional relationships. The current standard (M3A1) is clearly about mathematical expressions. In fact, this standard really needs to be considered as soon as we start teaching addition operation in Grade 1. Students should learn that 5+3=8 is a representation of a situation like where Johnny had 5 apples and his mom gave him 3 more to make the total number of apples to be 8. Mathematical expressions aren't about computation problems to be completed. They represent situations/physical phenomena/one's thinking, concisely and precisely.

Because they are representations of situations etc., it is also perfectly possible to write something like 8=3+5 to represent decomposition of 8 into 3 and 5, for example. Teachers should include this type of expressions from early on to help students understand that "=" means the two quantities on both sides are equal in magnitude. It does not mean "do something" to get the answer to be written on the right side. By understanding mathematical expressions as representations of situations etc., students can think about writing missing number situations using some place holders like a little box, for example 3+[ ]=8.

When you consider mathematical expressions as representations, and also tools for communications, there are some implications. One such implication is how you write multiplication expressions - I wrote about this in June, 2007 (M2N3a). Another implication is the last part of this standard, "interpret given expressions." If mathematical expressions are the language of mathematics, as I believe they are, then we have to not only worry about "writing" but also "reading." Ability to read/interpret given expressions must become an explicit focus of mathematics instruction, starting in Grade 1. Possible instructional activities may include having students tell stories (or write word problems, when students are old enough) that will match the given expressions and interpreting other students' thinking processes when they present their solutions using mathematical expressions.

Moreover, just as we sometimes "read in between the lines," mathematical expressions can be interpreted in different ways. For example, if we are given 5+3=8, we can simply interpret this statement to mean, "If you add 3 to 5, you get 8." However, we can interpret this statement even further. For example, 5 must be 3 less than 8 since you need to add 3 to 5 to get 8. This means that the difference between 8 and 5 is 3, or 8-5=3. Now, the original addition sentence can also be interpreted as "if you add 5 to 3 you get 8," or 3+5=8. Then, using the similar argument, we can also say that 3 is 5 less than 8, or the difference between 8 and 3 is 5, i.e., 8-3=5. In many US textbooks, students learn about "fact families." I have never heard of such a phrase while growing up in Japan. Instead of simply memorizing how numbers can be shifted around and the operation signs manipulated, it would be much better if our students can "read" math sentences like "5+3=8" and interpret all the relationships that are expressed by so-called fact families, wouldn't it?

MKM1, M1M1, & M2M1: Teaching Measurement in Primary Grades

2008-12-18T08:57:00.000-08:00

MKM1. Students will group objects according to common properties such as longer/shorter, more/less, taller/shorter, and heavier/lighter.
M1M1. Students will compare and/or order the length, weight, or capacity of two or more objects by using direct comparison or a nonstandard unit.
M2M1. Students will know the standard units of inch, foot, yard, and metric units of centimeter and meter and measure length to the nearest inch or centimeter.

When discussing teaching and learning of measurement, we need to keep in mind there are three different (yet clearly related) aspects that students must learn. They are,

understanding the attribute being measured
process of measurement
how to use measuring instruments.

The three standards above reflects the four-stage general sequence of teaching measurement:

Direct comparison
Indirect comparison
Measuring with non-standard units
Measuring with standard units

MKM1 is clearly in the first stage, and M2M1 is in the fourth and final stage. M1M1 explicitly indicate stages 1 and 3, but there is no mention of stage 2. Considering the general sequence of instruction, what should be happening is more in stages 2 and 3.

So, why is it important to follow these four stages as we begin our instruction on measurement? The major focus of the first two stages is to help students understand attributes that are being measured. After all, before we can measure anything, we really need to understand what it is that we want to measure. Thus, before we can measure length, we need to understand what length is. By putting two objects next to each other (direct comparison), students can determine which is longer/shorter. Through such experiences, students gain the understanding that length is about the amount of space between the two ends of an object. [Although we may use different words, "height" is not really an attribute. It is really length in the vertical orientation.] Of course, through direct comparison activities, students are gaining some fundamental understanding about how to measure an object as well. For example, when comparing the lengths of two objects, it is important that one end of the objects must be lined up. You cannot say the segment on top in the figure below is longer just because it "sticks out" farther to the right.

Students will also learn that the "amount of space" we are interested in is along a straight pat. Thus, we cannot simply compare the positions of the end points as shown in the figure below.

It should be obvious that these understanding play an important role in the process of measurement later on.

Unfortunately, not every two objects may be directly compared. In those situations, it is sometimes useful to use a third object that can be compared directly to each of the two objects that are being compared. Thus, if a door way is wider than your arm span but a second door way is narrower than your arm span, then you know that the first door way is wider than the second one. Indirect comparison provides more flexibility as you compare two objects. It also provides opportunities for children to experience an important mathematical property of relationships called transitivity, that is, if a > b and b > c, then a > c. Of course, the formal study of such property will not take place until much later. Perhaps more important reason for indirect comparison is that it sets the stage for the most fundamental idea about the measurement process - the use of a unit. When using a third object, it may not be in between the two objects - for example, a wooden stick may be much shorter than two door ways. In those cases, however, it may be possible to determine that one door way is taller than the three (of the same) wood sticks put end to end while the other one is shorter than three wood sticks. Now, we can say that the first door way is taller than the second one.

You can easily see that such experiences become the foundation for the idea of expressing an attribute in terms of the number of a third object, unit, necessary to "cover" it. When we move into this stage, we are now indeed "measuring" in the sense that we are assigning a number to an object in terms of how much of the attribute it has. There are many merits for expressing the amount of an attribute using numbers. Clearly, it simplifies the process of comparison as we no longer need to find different object to use as the reference. Comparison of multiple objects can be easily done by simply comparing numbers. Moreover, once we assign numbers, we can answer not only "which is longer?" but also "by how much?" In general, once we express the amount of attributes with numbers, arithmetic operations may be used to answer some questions. Although the GPS does not explicitly state those merits, I hope teachers help students experience and understand those merits.

Some people may argue that, once we get to this stage, we should just use standard units. This argument perhaps makes sense later in the elementary grades after students have learned about measuring three or four different attributes. However, at the primary grade level, it is also important to keep in mind that students are still learning about the process of measurement - pick a unit, then determine how many of the unit is necessary to equal the object you are measuring. For us, this is so obvious, but not so with children. Introducing standard units at this stage will require children to deal with two new ideas simultaneously - new units and new process. There are also other considerations. First, some units may be too small or too large so that the size of the resulting numbers may not be appropriate for children at this particular time. By using non-standard units, teachers can control the range of numbers students might obtain. Also, it is important to note that measuring with standard unit typically means measuring with various instruments. For example, if you are measuring with inches, you are most likely to be measuring with a ruler. However, learning to use a ruler is also a challenging task - this might be a third new idea students have to deal with if we are to introduce standard unit at this stage.

Although it may sound a bit paradoxical, the use of non-standard units is a useful experience for children to understand the need for having standard units. For example, if two students measure the width of the same door way using their pencils, they may get different results. They will soon realize that they cannot compare numbers unless their units are the same. This is when we can introduce standard units such as inches, feet, centimeters and meters.

Finally, learning how to measure with common instruments such as rulers is not as simple as adults might think. For that purpose, it may be useful if children had some experiences using their own measurement tools. For example, during the third stage (measuring with non-standard unit), students can tape together index cards to form their own measuring "tape." Initially, students may actually count the number of index cards, but eventually they may realize simply labeling the cards 1, 2, 3,... will make it simpler. Such experiences will allow them to understand that what we are counting on a measurement tape is the number of spaces between the tick marks, and the numerical label at a given tick mark indicates the total number of units up to that mark. Furthermore, as we learned in the first stage, the end (actually the starting point) of the measuring tape must be lined up with an end of the object, not the tick mark labeled "1." A variety of home-made measuring instruments can be made to measure length, capacity/volume, weight, and even angles. Making and measuring with home-made instrument may be a very fruitful experiences as students learn to measure with standard units.

Finally, it should be noted that weight is not formally studied until Grade 4. Thus, children's experiences in Grades K and 1 should be viewed within the context of teaching children more about the existence of different attributes. Weight is a difficult concept for children because we cannot "see" it - that is, some objects that look big may be light while others that look small may be quite heavy. Thus, direct and indirect comparison activities may be what we should focus on in Grades K and 1 with respect to weight.

M5N4(d) - Modeling Multiplication & Division of Fractions

2008-11-30T10:33:00.000-08:00

M5N4. Students will continue to develop their understanding of the meaning of common fractions and compute with them.
d. Model the multiplication and division of common fractions.

In the last three posts, I discussed multiplication and division of decimal numbers that do not depend on the knowledge of multiplication and division of fractions. That was necessary because in the GPS decimal multiplication and division are discussed prior to fraction multiplication and division. In this post, I would like to discuss multiplication and division of fractions. I have previously discussed this topic (November, 2007). In the post, I briefly discussed how the area model may be used to represent multiplication of fractions, as well as the double number line representation that can be used for both multiplication and division. So, in today's post, I want to focus on how to model division of fractions.

As students are introduced to division operation in Grade 3, they are expected to understand that "division m many equal parts of a given size or amount may be taken away from the who as in repeated subtraction, and the second is determining the size of the parts when the whole is separated into a given number of equal parts as in a sharing model" (M3N4b). We discussed how these interpretations must be extended as the number of "groups" become decimal numbers - whether as the divisor in a fair sharing problem or as the quotient in a measurement division problem.

The situation is basically the same with fraction multiplication and division. If the divisor is a whole number, we can use the fair sharing interpretation. When the divisor becomes a fraction, we must use either the extended meaning of fair sharing, that is, an operation that determines the per-one quantity, or the measurement interpretation. The measurement interpretation is much more easily modeled using manipulatives such as pattern blocks. We can model 3/4 divided by 1/6 this way: First, let's represent a whole using two hexagon pieces together. Then, 3 trapezoids will represent 3/4 and a blue rhombus will represent 1/6. The division question is asking how many blue pieces will fit in the 3 red trapezoids together. You can easily show that 4 blue pieces will fit completely inside the 3 red trapezoids. The remaining section is 1/2 of a blue rhombus. Therefore, the quotient is 4 1/2.

How else can we model division of fractions? In particular, how can we model fraction division that may also reflect the inverse relationship between multiplication and division? In my previous post on this standard, I mentioned how the area model of multiplication may be used to represent multiplication of fractions. In this model, the two dimensions of a rectangle represent the two factors and the product is represented by the area of the rectangle (in relationship to the unit rectangle). Thus, the figure below represents 1/3 x 2/3:

So, is there a way to represent division using the area model? For example, how can we model 3/4÷2/5? [I encourage you to think about how you may be able to represent this division using pattern blocks. You may find it a bit cumbersome.]

Since division is the inverse operation of multiplication, 3/4 must be the area of the rectangle, and the divisior, 3/4, is one of the two dimensions. Thus, we are trying to determine the other dimension of the rectangle so that the area will be 3/4. So, how can we model this? I'm sure that there are different ways, but here is one possibility.

Let's start by first representing 3/4:

Of course, this fraction has the dimension of 1 unit (vertically) by 3/4 units (horizontally). What we want is a rectangle that has the same area as the yellow rectangle but has 2/5 as one of the dimensions. If we say that the vertical dimension is 2/5 units, then we are looking for the horizontal dimension, as shown in the figure below:

So, how can we find the horizontal dimension? First, let's first draw in the segments showing the fifths in the yellow rectangle.

Now, we can see that another set of 2/5 by 3/4 rectangle (shown in green below) can be shifted to fit inside the rectangle whose vertical dimension is 2/5.

Now, two of the remaining 3 small rectangles (shown in blue) can be shifted.

Finally, the remaining small rectangle has to be split into two equal parts (shown in red).

So, how long is the horizontal dimension, which will be the quotient? Each of the small rectangle has the horizontal dimension of 1/4 unit. Clearly, we have 7 1/4-units. Finally, the red segment is a half of the small rectangle, or a half of 1/4. Thus, we have 7/4 and 1/8, or 15/8 altogether.

The figure below shows 1 2/3 ÷ 3/4.

Now, in this situation, the 2 small rectangles (in blue) had to be split into 3 equal parts, so the horizontal segment of the blue segment of the quotient is 2/3 of 1/3-unit, or 2/9. Thus, the quotient is 2 2/9.

Now, in this model, each small rectangle you obtain has the horizontal dimension which is the unit fraction with the denominator for the dividend (4 in the first example and 3 in the second). The total number of the small rectangles in the dividend is the product of the numerator of the dividend and the denominator of the divisor. The number of horizontal column of the unit fraction can be calculated by dividing the total number of the small rectangles by the numerator of the divisor. Thus, the quotient can be expressed as:

In other words, to divide a fraction by another fraction, you simply multiply the dividend by the reciprocal of the divisor. Of course, this generalization may be straightforward for us, but it is extremely important that we analyze what mathematical ideas are involved in making that generalization. Then, we can decide whether or not this generalization is accessible to our students.

In any event, it does raise some questions about why the GPS asks students to model division (and multiplication) of fractions in Grade 5 without specifying the development of the algorithm in the same grade level. As I stated earlier, I believe the appropriate interpretation of the current GPS is that the algorithms are to be developed (and mastered) in Grade 6. However, it seems rather strange to separate modeling from the algorithm development, which is the generalization based on the models.

Finally, I would like to emphasize that the area models are useful when we know the operation involved. The area model cannot help students determine which operation to use. For that purpose, models like double number line are much more suited.

M5N3 Multiplication & Division of Decimal Numbers (3)

2008-11-28T07:05:00.000-08:00

M5N3. Students will further develop their understanding of the meaning of multiplication and division with decimal fractions and use them.

OK, this is the third (and hopefully the last) in the series of posts discussing multiplication and division of decimal numbers. In the last two posts, we discussed multiplying and dividing decimal numbers by whole numbers and multiplying by decimal numbers. We are developing these ideas using only our understanding of whole number multiplication and a powerful idea about our numeration system, relative size of numbers. What is left for us now is dividing by decimal numbers. Let's go back to our problem:

Problem 4
A wire that is 2.4 meters long weighs 3.6 grams. How much will the same wire weigh if it is 1 meter long?

This problem requires us to divide 3.6 by 2.4. We already looked at dividing decimal numbers by whole numbers, but we have yet to consider division by decimal numbers. In some curricula, fraction arithmetic is discussed first, so we can change this division to division of fractions. However, that line of reasoning is not available if we follow the GPS. So, what can students do?

Whenever students encounter a new problem, we would like them to ask, "What do I know that I can use?" or "How is this problem similar to what I have studied previously?" Such a habit is an example of what the authors of Adding It Up (National Research Council, 200?) call productive disposition. Again, a diagram might help us think about this problem.

One possibility is to think about 2.4 as 24 0.1's as we did before. But, what do we get if we divide 3.6 by 24? Let's see what the diagram will show us:

We can tell from this diagram that the result of dividing 3.6 by 24 is the weight of a 0.1-meter wire. So, how can we find the weight of a 1-meter wire if we know that a 0.1-meter wire weighs 0.15 grams? Since 1 meter is 10 times as long as 0.1 meter, the weight should also be 10 times as much. So, to find the weight of a 1-meter wire, we just need to multiply the weight of a 0.1-meter wire by 10. So, a 1-meter wire will weigh 1.5 grams.

With Problem 3, we also had another approach that considered 10 times of the multiplier. What would a parallel reasoning in Problem 4 be like? If we make the divisor (2.4) into a whole number, what does it mean? That means we are looking at a 24-meter wire, instead of a 2.4-meter wire. Again, it's 10 times as long, therefore, it should weigh 10 times as much, i.e., 36 grams. But if we know that a 24-meter wire weighs 36 grams, we can find the weight of a 1-meter wire by simply dividing 36 by 24. We don't have to do anything with the result since we haven't changed the weight of 1-meter of wire when we considered the weight of the 24-meter wire. A diagram might show this approach clearly:

This second approach may be more useful to generalize a paper-and-pencil algorithm. Basically what we did was to multiply the divisor by a power of 10 to make it into a whole number. Then, the dividend must be multiplied by the same power of 10 - since the length of the wire is now that many times as long, it should weigh also that many times as much. Then, we can simply divide the new weight by the new length, we can find the weight for 1 meter. Therefore,

Another way of describing this process is to move the decimal point of the divisor (the number outside of the long division symbol) as many places as necessary to the right to make it into a whole number. Then, move the decimal point of the dividend the same number of places to the right as well - annexing 0's if necessary. Then, we can perform long division as we have done previously - either a whole number divided by a whole number or a decimal number divided by a whole number. Again, this is the familiar algorithm, isn't it?

As we saw in the three recent posts, the familiar multiplication and division algorithms can be meaningfully derived using only our knowledge of whole numbers and the idea of relative size of numbers. In the Japanese standards, they discuss decimal multiplication and division first because the algorithms are essentially the same as those of whole number multiplication and division. Thus, when students study multiplication and division, they can focus more on extending the meaning of multiplication and division. Then, when students study multiplication and division of fractions, they do not have to worry about dealing with the new meaning of operations AND the new algorithms. It is not clear if the GPS writers had the same intent, but I hope you see how students can develop multiplication and division algorithm for decimal numbers without knowing multiplication and division of fractions.

M5N3 Multiplication & Division of Decimal Numbers (2)

2008-11-09T16:57:00.000-08:00

M5N3. Students will further develop their understanding of the meaning of
multiplication and division with decimal fractions and use them.

In the last post, I discussed how the idea of relative size can be used to think about multiplying and dividing decimal numbers by whole numbers - M4N5(d). In this post, I want to continue to the next step, multiplying and dividing by decimal numbers. As I discussed in October, 2007, when the multiplier and the divisor is something other than a whole number, we must extend the meaning of division from an equal-group perspective to a more proportional one. Let's look at the two problems I left as "homework" last time.

Problem 3
One meter of wire weighs 2.4 grams. How much will 1.8 meters of the same wire weigh?

Problem 4
A wire that is 2.4 meters long weighs 3.6 grams. How much will the same wire weigh if it is 1 meter long?

Clearly, in Problem 3, we must multiply 2.4 by 1.8, while in Problem 4, we must divide 3.6 by 2.4. Since these situations involve a decimal multiplier and a decimal divisor, we can no longer use the equal group interpretation of multiplication and division - what does 1.5 or 2.4 groups mean? Rather, we must look at these situations more proportionally. In Problem 3, we are asking, if 2.4 is to 1, how much is to 1.8, and in Problem 4, if 3.6 is to 2.4, what is to 1? Alternately, if you use multiple comparison idea, Problem 3 asks how much is 1.5 times as much as 2.4, while Problem 4 asks 3.6 is 2.4 times as much as what?

Let's now think about how students can solve these problems using only what they have learned so far, which does not include how to multiply or divide by decimal numbers.

Problem 3
One meter of wire weighs 2.4 grams. How much will 1.8 meters of the same wire weigh?

One possible idea that students might use is to consider the multiplier, 1.8, in terms of the decimal unit using the idea of relative size. That is, 1.8 means there are 18 pieces of 0.1's. But what does that mean? A diagram might be helpful. Using a double number line (November, 2007), we can represent the problem like this:

When we say 1.8 is made up of 18 pieces of 0.1's, the diagram may look like this:

In other words, 1.8 meters can be thought of as a collection of 18 0.1 meter pieces. But, how does that help us find the missing number. We are not multiplying 2.4 by 18 - we don't have 18 groups of 2.4. What do we have 18 groups of on the top number line?

From this diagram, we can tell that what we have 18 of on the top number line is actually the weight of 0.1 meter wire. In other words, if we know how much a 0.1-meter wire weighs, then, we can find the answer. But, it's easy to see that the weight of a 0.1-meter wire can be determined by simply dividing 2.4 by 10, which is what students learned in Grade 4. Once we determine the weight of a 0.1-meter wire, i.e., 0.24 grams, then, we can multiply that by 18, which is also a Grade 4 idea. 0.24 x 18 = 4.32, so the weight of a 1.8-meter wire is 4.32 grams.

Here is another idea that students might come up with. Although we are looking for the weight of a 1.8-meter wire, let's first think about the weight of 18-meter wire, which is easy enough - simply multiply 2.4 by 18, a Grade 4 idea. However, since a 18-meter wire is 10 times as long a 1.8-meter wire, it should also weigh 10 times as much, too. So, in order to determine the weight of a 1.8-meter wire, we can simply divide that by 10 to find its weight. Since we already know how to divide decimal numbers by whole numbers, this last step should not be a problem. This line of reasoning may be represented on a number line like this:

Different students will feel more comfortable with different approaches. However, this second approach may be more useful to generalize into a written computation algorithm. In general, what we do in the first step is to make the multiplier into a whole number by multiplying it by an appropriate power of 10. Now, if the multiplicand is a decimal number, we end up multiplying it by a power of 10 to make it into a whole number as well (that's another way of thinking about the use of relative size). Now that we have two whole numbers, we can multiply them easily. However, this product is too big, and it must be divided by those powers of 10. For example,

Since multiplying by 10 means that the decimal point will move to the right one place while dividing by 10 means moving the decimal point to the left one place, we can describe what happened above this way: when we think of 3.7x4.26 as 37x426, we moved the decimal point 3 places to the right altogether, therefore, we have to move the decimal point to the left 3 places in the product of 37x246 to get the product for 3.7x4.26. And, this is (to us) the familiar multiplication algorithm for decimal numbers, isn't it?

Well, this has gotten a bit too long - of course, with actual 5th graders, you may need several lessons to get this much discussion done. Anyway, I think I must postpone the discussion of dividing by decimal number until next time. However, if you can think about how we solved Problem 3, you may find that Problem 4 can be solved in similar ways.

Revisiting M4N5(d) & M5N3 - Multiplication and Division of Decimal Numbers (1)

2008-11-03T16:27:00.000-08:00

M4N5. Students will further develop their understanding of the meaning of decimal fractions and use them in computations.

d. Model multiplication and division of decimal fractions by whole numbers.

M5N3. Students will further develop their understanding of the meaning of multiplication and division with decimal fractions and use them.

I have discussed these standards previously (October, 2007). However, I had an interesting conversation with a colleague from another university in Georgia recently. She also teaches mathematics courses for prospective elementary school teachers at her school. We were discussing how I incorporate Japanese curriculum materials to discuss decimal multiplication and division. She then shared with me that, when she teaches this topic, she relates it to fraction multiplication and division. Her approach is perfectly valid and viable; however, in the GPS, decimal multiplication and division are discussed before fraction multiplication and division. So, I feel it is important that our future elementary school teachers experience how decimal multiplication and division may be developed WITHOUT the knowledge of fraction multiplication and division. So, I want to discuss how multiplication and division of decimal numbers can be approached using the knowledge of whole numbers only.

Let's start with multiplying and dividing decimal numbers by whole numbers.

Problem 1

One meter of wire weighs 5.7 grams. How much will 3 meters of the same wire weigh?
In the last post, I discussed the idea of relative size. If you use that idea, 5.7 grams can be considered as a collection of 57 0.1 g pieces. Thus, in 3 meters, we have 3 sets of 57 0.1 g's. That means we can use the calculation 57x3 to find out the total number of 0.1 g's in 3 meters. If there are 171 pieces of 0.1 g's, again using the idea of relative size, we know that is the same as 17.1 grams.

In general, when you are multiplying a decimal number by a whole number, you can just consider the given decimal number in terms of its smallest decimal place value as the unit. For example, 0.37 is 37 0.01's, 0.824 is 824 0.001's, etc.. You will then have a whole number as the multiplicand (in terms of a decimal unit). Since the multiplier is a whole number, we can use multiplication to find the total number of pieces of the decimal unit. We can then convert the final result into a decimal number by using the idea of relative size. Using the familiar paper-and-pencil algorithm, it basically means that the decimal point for the multiplicand (the number on top) and the product are in the same place.

Now, let's look at division of decimal numbers by whole numbers.

Problem 2

A wire that is 3 meters long weighs 5.7 grams. How much will the same wire weigh if it is 1 meter long?
Again, using the idea of relative size, we can think of 5.7 grams as a collection of 57 pieces of 0.1 gram. Since 3 meters of this wire include these 57 pieces, and you want to know how much 1 meter will weigh, you simply need to divide 57 by 3, which is just a whole number division problem. The quotient, 19, tells us the number of 0.1 gram pieces in each group. Again, using the idea of relative size, we can conclude that 1 meter of this wire will weigh 1.9 grams.

In general, just as in the case of multiplication of decimal numbers by whole numbers, when you are dividing a decimal number by a whole number, you can consider the decimal dividend in terms of its smallest decimal place value as the unit. Then, we can simply use whole number division to find out how many pieces of the decimal unit will be in each group. The final quotient can be found by using the idea of relative size. Using the familiar long division notation, this suggests that the decimal point for the quotient and the dividend will be in the same place, i.e., the place values for the dividend and the quotient should line up, just as they did with whole numbers.

One point of complication we need to pay attention is what if the dividend (whole numbers pieces of a decimal unit) is not evenly divisible by the divisor. So, for example, what can we do if the weight of the 3-meter wire was 5.8 grams. When we divide 58 by 3, we have the remainder of 1. But, this is also the number of 0.1 gram piece. So, we can say that the answer to the division of 5.8 by 3 is 1.9 with the remainder of 0.1. In other words, the remainder is the number of the decimal unit. Thus, in the long division notation, the decimal point of the dividend and the decimal point for the remainder must also line up.

Of course, another option is to divide on - the remainder of 1 can be thought of as a collection of 10 0.1's (actually, the remainder is 0.1, so we are really talking about 10 0.01's), then we can keep dividing. Sometimes, this will result in a terminating decimal, while in other cases, you will have a repeating decimal. With this understanding of dividing on, when students learn about the quotient meaning of fractions they can then understand that every fraction can be re-written as a decimal number by simply dividing its numerator by the denominator.

OK, this post is already rather long. So, I will have to wait till the next time to discuss multiplication and division by decimal numbers. In the meantime, I encourage you to think about how students can reason about to solve the following problems, which involve multiplication and division by decimal numbers. Keep in mind that they have not learned how to multiply or divide by decimal numbers or fractions.

Problem 3

One meter of wire weighs 2.4 grams. How much will 1.8 meters of the same wire weigh?
Problem 4

A wire that is 2.4 meters long weighs 3.6 grams. How much will the same wire weigh if it is 1 meter long?