M6A2 Proportional Relationship (1)

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

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

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

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

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

M7N1c - Integers

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

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

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

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

M*P5: Tape diagrams

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

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

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

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

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

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

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

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

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

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