Using a context of chairs around square tables, students will be exposed to three different linear patterns in this lesson. The patterns vary slightly from situation to situation, and the third situation allows students to determine a solution in multiple ways, in the end leading to an intuitive understanding of perimeter.
Present the following situation to students:
At Pal-a-Table, a new restaurant in town, there are 24 square tables. One chair is placed on each side of a table. How many customers can be seated at this restaurant?
Show an arrangement of one table with four chairs. If your room contains large square tables at which students work in groups, use them as a demonstration. If not, you can draw a picture on the chalkboard, or you can use pattern blocks or other transparent manipulatives on the overhead projector.
When all students understand how chairs are placed, ask, "If there were 24 tables in a room, how many chairs would be needed?" Depending on students’ understanding of multiplication, they may immediately realize that the answer is 24 × 4 = 96. If not, work with the class to complete a table, as follows:
| Tables | Chairs |
| 1 | 4 |
| 2 | 8 |
| 3 | 12 |
| 4 | 16 |
| 5 | 20 |
From this table, students should realize that the number of chairs is equal to four times the number of tables. Alternatively, they might recognize that each time a table is added, four chairs are added. If there are some students who use each approach, this is a good opportunity to reinforce the connection between multiplication and repeated addition. That is,
| 2 × 4 | = | 4 + 4 |
| 3 × 4 | = | 4 + 4 + 4 |
| 4 × 4 | = | 4 + 4 + 4 + 4 |
| 5 × 4 | = | 4 + 4 + 4 + 4 + 4 |
and so on.
Ask students to explain their observations. "What is the pattern? How can you find the number of chairs for any number of tables?" [Multiply the number of tables by 4. If there are 24 tables, for instance, the number of chairs is 96. If there are n tables, the number of chairs is 4n.]
After the original problem has been solved ("How many people can be seated at this restaurant?"), explain to students that the restaurant needs some additional help. Pose the following problem:
Pal-a-Table has a problem. For large groups, they must push some of the tables together to make a longer table. As before, they place one chair on each side of the table. How many tables would be needed for a group of 18 people?"
Again, show students an example of the situation. Explain that for just one square table, again four chairs are needed, but when two tables are pushed together, six chairs are needed.
Ask, "How many chairs would be needed if three square tables were pushed together? What about four tables? …five? How would you determine the number of chairs needed for any number of tables?"
At this point, divide students into groups. Allow them to explore various arrangements of tables and chairs using pattern blocks or the Chairs Around a Table applet. This online activity allows students to explore the various situations described in this lesson.
As students work, they should keep an organized record of their data. Before they begin to explore, you may wish to discuss how a table or chart could be used to keep their data organized. Allow students to investigate the relationship between tables and chairs, and circulate as they explore. Make sure that every group is working toward finding a general relationship between the number of chairs and the number of tables.
After the exploration, conduct a class discussion to reveal the relationship between chairs and tables. Ask, "If you know the number of tables, how can you determine the number of chairs?" Allow students to suggest relationships, and keep a record of their suggestions on the overhead projector. Then, allow others to agree or disagree with the suggestions. Continue until the class reaches a consensus on which relationships are correct. (If students believe that a suggested relationship is correct, they should be able to show that it is true for the data they collected or provide an explanation of how they know it is true.)
As students suggest answers using words, convert their answers to algebraic expressions using variables. For instance, if a student says, "You find the number of chairs by multiplying the number of tables by 2, and then adding 2," then you might write either of the following on the overhead projector:
chairs = 2 × tables + 2
or
c = (2 × t) + 2
Although students in grades 3‑5 are not expected to understand symbolic manipulation, these examples will lay the conceptual foundation for understanding the use of variables.
Students may suggest several correct rules for determining the number of chairs. One student may realize that there are two chairs on the sides of each table, as well as an additional chair on each end, leading to the rule c = (2 × t) + 2. This is shown in the first diagram below. Another student may realize that there are three chairs around each end table, and there are two chairs on the sides of each middle table, leading to the rule c = 3 × 2 + 2 × (t - 2). This is shown in the second diagram below. Although students may not be able to verify that these relationships are algebraically equivalent, they should realize that both yield the same solutions.
Once students have discovered acceptable relationships, they should use them to determine the number of chairs when the number of tables is known, and vice versa. To ensure student understanding, ask the following two questions:
- How many chairs would be needed for 24 tables?
[Two chairs are needed on the sides of each table, so that is 2 × 24 = 48 chairs. An additional two chairs are needed, one on each end, for a total of 48 + 2 = 50 chairs.]
- How many tables would Pal-a-Table have to push together for a group of 18 people?
[When tables are pushed together, one person can sit on each end. That leaves 18 ‑ 2 = 16 people to be seated on the sides of the tables. Two chairs can be placed on either side of each table, and 16 ÷ 2 = 8. Therefore, 8 tables are needed for a group of 18 people.]
Finally, allow students to consider arrangements in which square tables are connected to form the outside border of a rectangle. Pose the following problem to students:
Customers at Pal-a-Table like that tables can be combined for larger groups, but they don’t like that tables are only arranged end-to-end to form a long chain. One patron suggests that tables should instead be arranged in a rectangular pattern with chairs placed around the outside.
Demonstrate some examples on the overhead projector or chalkboard. An arrangement showing 14 tables and 18 chairs is shown below.
Ask, "How many chairs would you need when tables are arranged in rectangular patterns like this?" Again, allow students time to explore various arrangements in their groups using pattern blocks or the Patch Tool.
As with the previous problem, students may discover various relationships that allow the number of chairs to be determined. If students are familiar with perimeter, then they may realize that the number of chairs is dependent on two variables, namely length and width. The arrangement above can be thought of as a rectangle with length 5 and width 4. The length contributes five chairs on two sides, and the width contributes four chairs on two sides, so the total number of chairs is 2(5) + 2(4) = 18. (Coincidentally, the perimeter of a 5 × 4 rectangle is 18 units.) In general, students should realize that the number of chairs for such an arrangement is equal to twice the sum of the length and width.
Alternatively, students may approach this situation like the previous one and realize that each of the four corner tables has two chairs next to it, but all other tables have only one chair. That gives a total of 4(2) + 10(1) = 18 chairs.
Similarly, students could realize that all tables have one chair next to them, but the four corner tables have an "extra" chair, giving a total of 14(1) + 4(1) = 18 chairs.
If the number of chairs in the length and width of the arrangement are represented by m and n, these three approaches lead to the following symbolic rules, respectively:
c = 2m + 2n
c = 8 + 2(m - 2) + 2(n - 2)
c = t + 4
Note that the last of these symbolic rules indicates that the number of tables (t) is four less than the number of chairs. But the number of tables is given by t = 2m + 2n ‑ 4, so the last expression could be represented by:
c = (2m + 2n ‑ 4) + 4
A whole-class discussion or individual group presentations are important after this exploration. Because there are various ways that students could represent the relationship between chairs and tables, it is important that all students are exposed to other students’ strategies.
