## Saturday, January 31, 2009

### 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.

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?

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.

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?

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.