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
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 |
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
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
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’