Area model as a reasoning tool (2)

In the last entry, which was posted more than 3 months ago, I discussed how area models might be used to make sense of and solve some challenging mathematics problems. I ended the entry with 2 problems for you to solve using area models.

The first problem was this one:

Suppose there is a simple coin-toss game. On each toss, you receive 7 points if you get a Heads and 3 points if you get a Tails. After you played the game 20 times (that is, 20 coin tosses), your score was 104 points. How many times did you get Heads? How many times did you get Tails?

In this problem the points you earn is the sum of the points you earn by getting Heads and the points you earn by getting Tails. The point you earn by getting Heads is the product of 7 and the number of Heads, and the point you earn by getting Tails is the product of 3 and the number of Tails. Since a product can be represented by an area model, you can draw the following area model to represent this situation.

Since the total number of tosses is 20, (Number of Heads) + (Number of Tails) = 20. Moreover, the total points earned was 104 points, so the total area must be 104. So, this is just like the chicken and pig problem. So, by using the same reasoning we discussed last time, you can use one of the following approaches.

From the picture on the left we can see that (Number of Tails) × 4 = 36. So, the number of tails must have been 9, thus the number of heads was 11. Or, from the picture on the right, we can see that (Number of Heads) × 4 = 44, and we can get the number of heads to be 11.

The second problem was a bit different.

There are some candies and children. If we give each child 4 candies there will be 18 candies left. However, in order to give each child 6 candies, we will need 12 additional candies. How many candies and how many children are there?

The number of candies must be somewhere between (Number of children) × 4 and (Number of children) × 6. If we represent the number of children as the horizontal side of the area model, then the two rectangles that represent these two products will have the same "width" but the different height.

So, we can represent the total number of candies by a L-shape like this:

Moreover, since we know that there will be 18 candies left when we give each child 4 candies, we can show that fact like this.

Finally, since we know that we will be short 12 candies if we try to give each child 6 candies, we can represent that fact like this.

Since the vertical dimension of the two colored rectangles is 2, that is, the difference between 6 candies and 4 candies, we can easily tell that there were 15 children. The total number of candies can be calculated either 4 × 15 + 18 = 78 or 6 × 15 − 12 = 78.

I hope these two entries piqued your interest in area model as a reasoning tool. I encourage you to try to find other problems that can be represented and solved arithmetically using this tool.

Area model as a reasoning tool

4.NBT.5
Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.

Area models play an important role in the CCSS-M, particularly in relationship to understanding multiplication. For example, 4.NBT.5 (above) suggests that area models may be useful to explain the process of multiplication calculation. I have previously discussed how area models may play a role in developing the whole number multiplication algorithm (Sept. – Oct. 2009) as well as how area models may be used to model division of fractions.

Although the CCSS-M does not necessarily discuss area models as a problem solving tool, I want to illustrate how area models may be used to make sense of and solve problems in certain situations.

Let’s consider the following problem (probably very familiar to most readers except the specific numbers used in the problem may be different).

A farmer has both pigs and chickens on his farm. There are 78 feet and 27 heads. How many pigs and how many chickens are there?

One way to solve this problem is to set up an equation, by letting the number of chickens as x. Then, we have the following equation: 4(27 – x) + 2x = 78. Alternately, we can use an area model to represent this problem situation. The number of feet of the pigs will be 4 × (number of pigs) and the number of feet of the chickens is 2 × (number of chicken). Since area models can be used to represent a product, we can draw two rectangles for the number of feet of the pigs and the number of feet for chickens as shown below.



In these rectangles, the vertical dimension represents the number of feet of an animal while the horizontal dimension represents the number of an animal. The area of each rectangle represents the number of feet.

Since the total number of feet is given as 78, if we draw these rectangles side by side to form an L-shape (see below), we know that the area of this L-shape represents the total number of feet, while the total of the horizontal dimension must be 27.



This diagram suggests a couple of possible solution approaches. One possibility is to note that the “area” of the top left corner of the L-shape (shaded in the figure below) must be 24 (= 78 – 54). Since the vertical dimension of the shaded region is 2 (feet), we can easily tell that the number of pigs must be 12. Then, since the total number of animals is 27, we can find that the number of chickens is 15 (= 27 – 12).



Another approach is to note that the “area” of the top right corner (shaded in the figure below) must be 30 (= 4 × 27 – 78). Once again, since the vertical dimension of this (shaded) rectangle is 2, we can easily find that the number of chickens must be 15. Then, by simple subtraction, we can find the number of pigs.



These solution approaches correspond to common solution strategies for this type of problems. The first approach is parallel to the idea of “let’s pretend all the animals were chicken.” Then, we should have only 54 feet, so, the extra 24 must be from the 2 extra feet each pig will bring to the total. The second approach, on the other hand will pretend if all the animals were pigs. Thus, 30 feet that are short must be due to the presence of chickens each of which contribute 2 fewer feet to the total. Perhaps the area models more clearly illustrate what 24 and 30 represent in the problem context.

Let’s see if you can use the area model to solve a similar problem by making use of area models.

Suppose there is a simple coin-toss game. On each toss, you receive 7 points if you get a Heads and 3 points if you get a Tails. After you played the game 20 times (that is, 20 coin tosses), your score was 104 points. How many times did you get Heads? How many times did you get Tails?

Here is another problem, but this one may require you to think a little more carefully about the situation.

There are some candies and children. If we give each child 4 candies there will be 18 candies left. However, in order to give each child 6 candies, we will need 12 additional candies. How many candies and how many children are there?

The answers to these two questions will be in the next post.

How should we read 5 × 7?

3.OA.1 makes it clear that 5 × 7 should be interpreted as 5 groups of 7 objects in each group. In other words, 5 is the number of groups and 7 is the group size. But, how should we read this multiplication expression? Interestingly, the CCSS never explains how it should be read. I suspect most people will read “5 × 7” as “five times seven.” However, “times” is not a mathematical term, and another, and perhaps more formal, way of reading a multiplication expression is “__ (is) multiplied by __.” So, is 5 × 7 “five (is) multiplied by seven” or “seven (is) multiplied by five”?

Some people might wonder why we need to worry about this question. In a way, it is a trivial issue. On the other hand, there is at least one instance in the CCSS where this issue is critical. In Grade 4, students are expected to study “multiplying a fraction by a whole number” and in Grade 5, they learn “multiplying a fraction or whole number by a fraction.” So, for example, is 5 × ¾ a Grade 4 topic or Grade 5? How about ¾ × 5? Unless we have an agreement on how to read a multiplication express like “5 × 7,” we can’t answer this question.

Although the CCSS does not discuss explicitly how to read multiplication expressions, there are places in the CCSS and accompanying Progressions documents that suggest what the authors were thinking. For example, 4.NF.1 states: “Explain why a fraction a/b is equivalent to a fraction (n × a)/(n × b) by using visual fraction models, …” Progressions for the Common Core State Standards in Mathematics: Grades 3-5 Numbers and Operations – Fractions provides this explanation:

“Grade 4 students learn a fundamental property of equivalent fractions: multiplying the numerator and denominator of a fraction by the same non-zero whole number results in a fraction that represents the same number as the original fraction” (p. 6, emphasis added).

Since n in the expression, (n × a)/(n × b), is “the same non-zero whole number,” the numerator, for example, should be read as “a multiplied by n.” Since in the expression, n × a, n is the number of groups and a is the group size, the number following “by” should be the number of groups. In other words, “(group size) multiplied by (number of groups)” is the way to read a multiplication expression. Thus, 5 × 7 should be read as “7 multiplied by 5.”

This interpretation is consistent with the explanation Progressions document provide about Grade 4 “multiplying a fraction by a whole number.” According to their explanation, 5 × ¾ is a Grade 4 topic while ¾ × 5 is a Grade 5 topic. You will also see that the writers of the CCSS tried to pay close attention to this interpretation as you read 5.NF.4.a: “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.” In other words, ¾ × 5 means “3 parts of a partition of 5 into 4 equal parts,” or, written in equation, 3 × 5 ÷ 4. In this expression 3 must be written in front of the multiplication symbol because it is the number of groups (or units).

Some of you may be a bit disturbed by the fact that 5 × 7 can be read as “5 times 7” or “7 is multiplied by 5,” reversing the order the factors appear in the expression. Perhaps in the revision of the CCSS, they might choose to use the convention that the first factor is the group size. If 5 in 5 × 7 is the group size, then we can read it as “5 times 7” or “5 multiplied by 7.” The formula for creating equivalent fractions would look like, a/b = (a × n)/(b × n), which might be more familiar. However, we do need to keep in mind that there are multiple ways we describe arithmetic calculations. For example, we can say “7 take away 4” or “7 minus 4,” but we also say “subtract 4 from 7,” again reversing the order of numbers. So, a part of teaching in elementary grades must be to help students become familiar with different ways to describe the same calculation in words. What we do need to avoid is to use different wording without the assumption that it should be obvious to students.

1.OA.6 Add and subtract within 20, demonstrating fluency for addition and subtraction within 10.

1.OA.6 Add and subtract within 20, demonstrating fluency for addition and subtraction within 10. Use strategies such as counting on; making ten (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14); decomposing a number leading to a ten (e.g., 13 – 4 = 13 – 3 – 1 = 10 – 1 = 9); using the relationship between addition and subtraction (e.g., knowing that 8 + 4 = 12, one knows 12 – 8 = 4); and creating equivalent but easier or known sums (e.g., adding 6 + 7 by creating the known equivalent 6 + 6 + 1 = 12 + 1 = 13).

I teach mathematics content courses for prospective elementary school teachers. Some of my students are parents of elementary school children. Sometimes I hear them complain about the "Common Core math." One complaint I often hear is that their children are made to practice different strategies - strategies like those discussed in 1.OA.6. However, I believe such teaching practice misses the point of 1.OA.6. The standard expects students to add and subtract within 20 (fluency within 10) using strategies. The standard does not say that each students needs to master or be fluent with each strategy.

Let's think about 8 + 9. There are a variety of strategies students can use. Here are some that I can think of:

* count all
* count on (from 8)
* count on (from 9)
* "I knew 8 + 8 is 16, so 1 more is 17."
* "I knew 9 + 9 is 18, so 1 less is 17."
* "I knew 10 + 10 is 20. Since 10 is 2 more than 8 and 1 more than 9, I took away 3 from 20, and the answer is 17."
* "I took 2 from 9 and 8 + 2 = 10. Then added 7 more to get 17."
* "I took 1 from 8 and 9 + 1 = 10. Then added 7 more to get 17."
* "I knew 5 + 5 = 10, and 3 + 4 = 7. 10 + 7 = 17."

I'm sure there are others. When students figure out 8 + 9 using their own reasoning, what is the point we want to emphasize? Are these strategies equally good? If not, how do we decide which strategy is "better than" others? Better in what sense?

If the focus is getting the correct answer to 8 + 9, perhaps all of these strategies are equally good. However, if our goal is more about helping students to think about ways to calculate, then perhaps these strategies have "good" in different ways. For example, those who recognize counting on from the larger added has begun to focus on the more efficient way to find the sum. Those who use doubles, either 8 + 8 or 9 + 9, realize that the known sums can be used to figure out an unknown sum. They have also figured out that if an added increases (or decreases) by 1, the sum also increases (or decreases) by 1. Figuring out such a relationship is an important mathematical practice - perhaps noticing and making use of mathematical structure. Using 10 + 10 also requires the realization that known facts may be useful to figure out unknown sums. In addition, these students are beginning to focus on 10 as a useful benchmark in our number system. Perhaps we can say the similar thing with those who use 5 + 5. Finally, the other two strategies, 8 + 9 = 8 + 2 + 7 or 8 + 9 = 7 + 1 + 9, are not only focusing on 10 as an important benchmark, they are also taking advantage of the way numbers between 10 and 20 can be thought of as "10 and some more."

So, these strategies have different strengths, and it makes no sense to force all students to be fluent with all of these strategies. Moreover, if the strength of a double strategy is the fact that students are beginning to develop the disposition to seek what they already know to figure out something they don't know yet, giving students several problems to solve using this particular strategy seems to be totally inappropriate.

Of these strategies, only the make-10 strategies are the strategies we want to make sure that all students understand and be able to use. So, perhaps assigning some homework problems where students practice these strategies are appropriate. However, we should remember that whether or not students themselves will realize the usefulness of these strategies really depend on how easily they can compose/decompose numbers to 10 and their understanding that numbers between 10 and 20 can be thought of as 10 and some more. If they don't have that prerequisite understanding, they may not see the point of these strategies. On the other hand, if they have that prerequisite understanding, they may not need much "practice" to master these strategies. It is perfectly appropriate to assign some practice problems, but we should be careful about what we want students to practice.

3.NF.3.d ... Recognize that comparisons are valid only when the two fractions refer to the same whole...

3.NF.3 Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size.

d) Compare two fractions with the same numerator or the same denominator by reasoning about their size. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model.

In the previous post, I discussed a common misconception that might arise from comparing fractions with the same numerator. In this post, I want to discuss a different part of this standard: "Recognize that comparisons are valid only when the two fractions refer to the same whole." I know the authors were thinking about situations like the following:
In this situation, 1/3 looks greater than 1/2. Thus, in order to compare these fractions, we must have the same whole. But, when we compare the two shaded parts above, are we comparing fractions, or are we comparing fractional pieces? Are fractions and fractional pieces the same? I tend to think not. Even in the picture above, when we are comparing "fractions" we are comparing the act of partitioning a whole into 2 or 3 equal parts and taking one of them. We are comparing the actions, not the results of the actions. The question becomes, how can we compare two actions and say one is "greater" than the other. In a way, we cannot compare "sizes" of actions. However, one way we can compare may be to look at the effects when the actions are applied to the same whole. Another way is to compare equivalent actions. So, taking 1 of 2 equal parts is the same as taking 3 of 6 equal parts while taking 1 of 3 equal parts is the same as taking 2 of 6 equal parts. Then, we can compare the number of parts we take.

Fundamentally, I think we should not be using drawings like the one above when we are comparing fractions. We compare fractions because they are numbers, and fractions as numbers always refer to the whole of 1. If fractions are numbers, then the whole is always the same. Therefore, if we are comparing fractions as numbers, perhaps the model we should be using is a number line.

3.NF.3.d

3.NF.3.d Compare two fractions with the same numerator or the same denominator by reasoning about their size. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model.

One of common misconceptions students (and adults) have about fractions is, "the closer the numerator is to the denominator, the larger the fraction." [Note, we are restricting our discussion to proper fractions.] They will say something like, 7/9 is greater than 2/5 because 7/9 is only 2 from the whole but 2/5 is 3 from the whole. What they do not realize is that they are confusing "closer to the denominator" with the idea, "closer to the whole." How close the numerator is to the denominator is indicated by subtraction, denominator - numerator. However, this difference simply indicates the number of unit fractions that are missing from the whole. How much is missing is determined not only by the number of unit fractions missing but also the size of those unit fractions. So, even though you may be missing 10 unit fractions from the whole, if each unit fraction is small, the total amount missing may be much less than 2 of larger unit fractions. So, 3/5, which is missing 2 1/5-units is missing more from the whole than 91/101, which are missing 10 1/101-units because 1/101-unit is much smaller than 1/5-units.

The 3.NF.3 standard states, "Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size." The intent of the standard, I believe, is for students to understand the idea that a (non-unit) fraction is composed of unit fractions. For example, 3/5 is made of 3 1/5-units. Thus, if you have two fractions with the same numerator, then those fractions are made of the same number of unit fractions. The sizes of those fractions depend on the sizes of the unit fractions. Since a unit fraction with a smaller denominator is greater than a unit fraction with a larger denominator, the fraction with the smaller denominator is made of the same number of larger units. Therefore, if the two fractions have the same numerators, the one with the smaller fraction is greater.

However, I wonder if comparison of fractions with the equal numerator also promotes some to develop the misconception I discussed above. For example, if you have 3/4 and 3/8, 3/4 is greater because the unit is greater. However, it is also the case that the numerator is closer to the denominator in 3/4 than in 3/8. I have no empirical evidence that this does happen, but it may be something 3rd grade teachers should keep in mind of this potential pitfall. The challenge is how can we help 3rd graders understand that the difference between the numerator and the denominator is not a reliable indicator for the size difference. There are some pairs of fractions where the one with the smaller difference is actually less than the one with the greater difference, for example, 1/2 vs. 4/6, 2/3 vs. 6/8, etc.. However, would that be enough? Are most 3rd graders developmentally ready to make sense of how deductive reasoning works. Do they understand that just because the less difference in 1/2 and 4/6 does not mean 1/2 is greater than 4/6, "the less difference" cannot be used as the rationale for concluding one fraction is greater than another? As 3rd grade teachers try to address this standard, it will be useful if they can make careful observations of their students and share their observations.

4.NF.1 Creating equivalent fractions

4.NF.1 in the Common Core says, "Explain why a fraction a/b is equivalent to a fraction (n x a)/(n x 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. Use this principle to recognize and generate equivalent fractions." When I first saw the expression (n x a)/(n x b), I thought that was odd. Usually I see math textbooks write this relationship as a/b = (a x n)/(b x n). Then I realized that the authors of the Common Core were trying to be consistent with the way they write multiplication expressions. "n x a" means "a is multiplied by n." Since we describe the process of creating equivalent fractions as "multiplying both the numerator and the denominator by the same non-zero number," it does make sense to write the expression as (n x a)/(n x b).

However, as I was thinking about how we might explain the process of equivalent fractions, I realized something else. One way, we can explain the process goes something like this - using 2/3 as an example. 2/3 is made up of 2 1/3-units, which is one of 3 equal parts of 1. So, if you use a diagram and a number line, it looks something like these:



Now, if we partition (split) each 1/3-unit into 4 equal pieces, we will have partitioned 1 into 12 equal parts, or 1/12-units. Pictorially, it will look like these:



Now, each 1/3-unit is made up of 4 1/12-units. So, 2 1/3-units are made up of 2 sets of 4 1/12-units. So, the number of 1/12-units in 2/3 is 2 x 4, and the number of 1/12-units in the whole is 3 x 4. So, 2/3 = (2 x 4)/(3 x 4), which is consistent with the conventional notation, a/b = an/bn.

I've been thinking about how we can use (n x a)/(n x b), but I haven't been successful, yet. I wonder if this is another instance where our language suggests the order of multiplication expression should be (group size) x (# of groups).

Recognize area (3.MD.7.d), angle (4.MD.7), and volume (5.MD.5.c) as additive

According to the CCSS, students are expected to recognize that area, angle, and volume are additive. But what does it mean for these attributes to be additive? If a measurable attribute is additive, that means that the measurement of the whole is the sum of the measurements of the (non-overlapping) parts. Thus, the area of an L-shape can be calculated by subdividing the shape into two rectangles or making a large rectangle then subtracting the area of the small rectangle that was added from the area of the large rectangle.

Actually, most measurable attributes studied in elementary schools are additive. In fact, the measurement process - select a unit, 'cover' the object with the unit without a hole or an overlap, and count the number of the unit - only works when an attribute is additive. Thus, other measurement attributes discussed in the K-5 standards - length, capacity (liquid volume), and elapsed time - are also additive. However, attribute such as speed and density studied in upper grades are not additive and their measurements are actually ratios of two other measurement. Therefore, one of the first ideas students need to understand as they study attributes that are ratios of two other measurements is focusing on one attribute is not sufficient.

As elementary school students learn different attributes are additive, we want them to understand about the measurement process explicitly. Whether that happens with area in Grade 3 or later will be a curricular decision.

It is a little curious that the CCSS does not explicitly state that students understand length as additive. However, when 2.MD.4 says, "Measure to determine how much longer one object is than another, expressing the length difference in terms of a standard length unit," implicit in this standard is that we can perform subtraction of two length measurements. Thus, students are indeed dealing with the fact that length is additive.

Speaking of additivity of length, the word "perimeter" does not appear in the CCSS until Grade 3 when a cluster, Geometric Measurement, is introduced. In Grade 3, students are expected to "recognize perimeter as an attribute of plane figures and distinguish between linear and area measures." I often hear US teachers lamenting how students confuse area and perimeter. I think one reason for this confusion is because area and perimeter are often introduced simultaneously. However, the idea of perimeter should be discussed as soon as standard units for length are introduced in Grade 2. Just as students can determine how much longer one object is than another (2.MD.4), they can also find the total length by putting those objects end-to-end. Then, as a special application of this idea, students can think about the total length of sides around different geometric shapes they have seen. Perhaps the term "perimeter" can also be introduced. Then, in Grade 3, when students compare the "spaciousness" of two flat regions - for example, comparing two picnic blankets - we should help them realize explicitly that the perimeter is not an appropriate way to compare spaciousness. We can then introduce "area" as an attribute that measures how much space inside a flat figure. Having this experience at the time area concept is introduced will, I believe, reduce the amount of confusion students have about area and perimeter. Of course, if the focus of area instruction is just on how to calculate the area, then students' understanding of these ideas will continue to be limited.

2.MD.9

2.MD.9
Generate measurement data by measuring lengths of several objects to the nearest whole unit, or by making repeated measurements of the same object. Show the measurements by making a line plot, where the horizontal scale is marked off in whole-number units.

Recently, I was working on a project in which the above standard (2.MD.9) was one of the foci. Although I had a couple of different questions about this standard, I want to focus on the last part of the standard.

As I grew up in Japan and going through the Japanese elementary and secondary schools, I never came across a line plot as a way to represent data. Granted that the Japanese school mathematics curriculum back then placed a very limited emphasis on data analysis. However, I did learn bar graphs, broken line graphs, histograms, circle graphs, etc. which are pretty standard statistical graphs. In the CCSS are mentioned several times in Grades 3 through 5, and I wondered why. So, I read the progression document on K-5 Measurement & Data. In the document, the authors discuss that there are two paths in the K-5 MD (data analysis portion) standards. One path deals with categorical data and the other with measurement data. The authors state that in the categorical data path, they focus on bar graphs to represent and analyze such data. On the other hand, measurement data are represented on a line plot.

Although I understand the importance of students understanding the nature of data, I found their focus on bar graphs for categorical data and line plot for measurement data to be rather strange. Students should definitely understand that the nature of data influences what analysis is possible. For example, with categorical data, it is not possible to calculate the mean or the median. However, do we always represent measurement data on a line plot? Are there any cases where we might use a bar graph to represent measurement data?

Suppose a second grade class collected the data on the height of the students in the classroom (in nearest cm). In that situation, doesn't it make sense to represent the data (heights) using a bar graph - or anything other than a line plot? In fact, the mean of the data set is obtained when we even out the height of those bars (although second graders won't be calculating the mean). So, saying that a line plot is the primary representation of measurement data seems to be rather strange.

It seems like the distinction being discussed is more about whether or not we are looking at the actual measurements or the frequencies (counts) of measurement data. A line plot is used to represent the frequencies of measurement data - how many students are 134cm tall, how many are 135cm tall, etc.. Eventually, students may create intervals and represent the frequency distribution as a histogram. Or, if we collect data to see how (air) temperature changes during the school day. It seems perfectly reasonable to represent the data using a broken line graph. In a way, the distinction may be whether the measurement data are represented on the horizontal axis or the vertical axis. I am not sure if the distinction is categorical or measurement data. When I asked a statistician, she told me that when we create intervals to make a histogram, we are "categorizing" the data. So, in a certain sense, an interval is like a category, like "strawberry" as the favorite ice cream flavor.

Anyway, I do think it is important that students understand the distinction between categorical data and measurement data. I also think it is important for students to be mindful about whether or not we are interested in the actual values of measurement data or frequencies. However, when we represent frequencies of measurement data, there may be some things we shouldn't do. For example, with categorical data (like favorite ice cream flavors), it is perfectly appropriate to order the data from most frequent to the least frequent ones as we draw a bar graph. However, with measurement data on a horizontal axis (a line plot), such a manipulation is not appropriate.

Double Number Line Diagrams

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.

UPDATE (see below)

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.

UPDATE: February 4, 2012
I just uploaded another video on double number line. I apologize for some background noises. Also, at one point I said something like "to go from 0.8 to 0.1..." when I really should have said "to go from 0.8 to 1..." The app I'm using does not allow me to edit the video, and I didn't want to re-do the whole video. So, please excuse my errors.

Measurement in K-2

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)

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)

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.