1. How many 1 x 1 squares are in each stage of this pattern?
2. What might stage 5 of this pattern look like? How many 1 x 1 squares would be in stage 5?
3. Write an expression that describes the number of 1 x 1 squares in stage n of the pattern. Justify
your answer geometrically by referring to the pattern.
4. How much does the number of squares change from stage 1 to stage 2 of the pattern?
5. How much does the number of squares change from stage 2 to stage 3 of the pattern?
6. How much does the number of squares change from stage 3 to stage 4 of the pattern?
7. What do your answers to 4-6 tell you about the rate of change of the number of squares with
respect to the stage number?
8. You have previously worked with linear and exponential functions. Can this pattern be
expressed as a linear or exponential function? Why or why not?
1. How many 1 x 1 squares are in each stage of this pattern?
Stage
Number of squares
1
1
2
3
3
7
4
13
2. What might stage 5 of this pattern look like? How many 1 x 1 squares would be in stage 5?
There will be 21 squares in stage 5
3. Write an expression that describes the number of 1 x 1 squares in stage n of the pattern. Justify
your answer geometrically by referring to the pattern.
See A, B, C, and D below.
Encourage students to write their expressions according to how they see the pattern, not to
write any particular form of the expression. If students have trouble writing an expression
symbolically, encourage them to verbally express how they found the number of squares in
each stage and how would expect to find the next number of squares.
4. How much does the number of squares change from stage 1 to stage 2 of the pattern? 2
5. How much does the number of squares change from stage 2 to stage 3 of the pattern? 4
6. How much does the number of squares change from stage 3 to stage 4 of the pattern? 6
7. What do your answers to 4-6 tell you about the rate of change of the number of squares with
respect to the stage number?
Because the amount of change in the number of squares does not change by the same
amount per each stage, the rate of change is not constant, but instead it is a variable rate of
change.
8. You have previously worked with linear and exponential functions. Can this pattern be
expressed as a linear or exponential function? Why or why not?
No. A linear function would have a constant rate of change, and an exponential function
would have a constant ratio between successive values. This has a changing rate of change
that is different from what we have seen before.
Note: This is the time to introduce the term quadratic and let students know that you will be
studying a new type of function, quadratic functions, in this unit. It is not necessary to go
into more depth at this point, as students will engage in more in-depth study in the next few
tasks.
As most groups finish, you should expect expressions of the following forms:
A. n(n – 1) +1
B. n2 – n +1
C. n2 – (n – 1)
D. n + (n – 1)(n – 1)
You should wrap up this task with a discussion in which different groups of students share
their solutions, particularly to question 3. Invite students to share why these expressions are
equivalent, and check to see if students know how to expand or simplify to demonstrate
algebraically that the expressions are equivalent. If students do not yet know how to multiply
binomials, this is a good opportunity to move to area models and discuss this skill. It is
essential that students be able to multiply binomials prior to engaging in the rest of the
activities of this unit. If students need additional instruction, a good resource is the use of
Algebra Tiles. Many sites are available online with additional resources for this.
Additionally, be sure to discuss question 8 and the differences between linear and exponential
functions, which they studied in Coordinate Algebra, and quadratic functions.
This task is adapted from one discussed on page 55 in the following book:
Boaler, J. & Humphreys, C. (2005). Connecting mathematical ideas: Middle school video
cases to support teaching and learning. Portsmouth, NH: Heinemann.