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

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)

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."

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)

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

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.

3.OA.1 and M2N3a - Writing multiplication equations

3.OA.1 Interpret products of whole numbers, e.g., interpret 5 × 7 as the total number of objects in 5 groups of 7 objects each.

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

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...

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)

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)

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)

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 92 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 92 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)

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)

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)

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)

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

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

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

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.