M4A1 - To use tables or not?

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

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

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

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

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

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

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

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

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

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

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

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

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

Method 1

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

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

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

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

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

M4A1bc & M5A1a - symbols such as □ and Δ

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

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

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

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

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

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

M4A1 - Mathematical expressions (2)

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

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

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

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

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