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.