Pair Products
**This problem was adapted from http://nrich.maths.org**
Objectives:
 Students
 Students
 Students
 Students
will
will
will
will
be able to identify and extend patterns through critical thinking.
be able to effectively apply problem solving steps and strategies.
use algebraic reasoning to justify their answer.
explore an application of the distributive property.
Standards Addressed: (8th grade New Colorado State Standards)
#2: Patterns, Functions, & Algebraic Structures
 Use the distributive, associative, and commutative properties to simplify algebraic expressions.
 Represent the distributive property in a variety of ways including numerically, geometrically, and
algebraically.
The Pair Products Problem:
Choose 4 consecutive whole numbers. Find the product of the first and last numbers and then find the
product of the second and third (2nd to last) numbers. What is the connection? Does this work for any four
consecutive whole numbers? How do you know?
Choose 5 consecutive whole numbers. Find the product of the first and last numbers and then find the
product of the second and fourth (2nd to last) numbers. What is the connection? Does this work for any five
consecutive whole numbers? How do you know?
What happens when you take n consecutive whole numbers and compare the product of the first and
last numbers with the product of the second and second to last numbers? Justify your findings.
Implementation:
**Students should have some previous experience with the problem solving steps as well as with using their
problem solving bookmark during the problem solving process.**
The teacher(s) will…
…provide each student with a problem solving bookmark and a copy of the “pair products” problem.
…ask guiding questions (see below) as students problem solve individually and in small groups.
…facilitate a large group discussion during which students share their own solutions to the problem.
The students will…
…problem solve individually for 10 minutes by completing the first two – three problem solving steps
as well as answering one question from their bookmark for each step.
…share their ideas in small groups and begin completing the justifying step.
…participate in a large group discussion during which students share different solutions to the problem.
…complete all five steps of the problem solving process.
Possible Guiding Questions:
 What vocabulary words are important for this problem? Do you need to find the meaning of any of
those words?
 Have you thought about drawing a picture?
 How can you organize your thinking more effectively?
 Have we learned anything recently that might help you solve this problem?






Have you considered representing the problem or solution using symbols?
Why do you think this works?
Have you compared the answers you found for 4 consecutive numbers and 5 consecutive numbers?
Have you investigated the answer for 6 or 7 consecutive numbers? How can this help you solve n
consecutive numbers?
How can you explore whether or not your idea works in all situations?
Would anyone be able to understand your written explanation without talking to you?
Possible Solutions to the Problem:
4 consecutive numbers – the product of the first and last number is always 2 less than the product of the
second and third numbers.
Algebraic Proof – if you have 4 consecutive numbers: x, x+1, x+2, and x+3, compare the product
x(x+3) with the product (x+1)(x+2).
x(x+3) =
(x+1)(x+2) =
which is 2 more than the equation for x(x+3)
Geometric Reasoning –
Consider 4, 5, 6, & 7.
4 by 7 can be represented by
and 5 by 6 can be represented by
…so the rectangles match with 2 extra squares
For any x, x+1, x+2 and x+3
x by x+3 can be represented by
𝑥
x
x
x
and x+1 by x+2 can be represented by
𝑥
x
x
x
5 consecutive numbers – the product of the first and last number is always 3 less than the product of the
second and fourth numbers.
Algebraic Proof – if you have 5 consecutive numbers: x, x+1, x+2, x+ 3 and x+4, compare the
product x(x+4) with the product (x+1)(x+3).
x(x+4) =
(x+1)(x+3) =
which is 3 more than the equation for x(x+4)
Geometric Reasoning –
For any x, x+1, x+2, x+3 and x+4
x by x+4 can be represented by
and x+1 by x+3 can be represented by
𝑥
x
x
x
x
𝑥
x
x
x
x
n consecutive numbers – the product of the first and last number is always n – 2 less than the product of
the second and second to last numbers.
Algebraic Proof – if you have n consecutive numbers: x, x+1, x+2, … and x+(n – 1), compare the
product x(x + n – 1) with the product (x+1)(x + n – 2).
(
)
x(x + n – 1) =
(
)
(x+1)(x + n – 2) =
which is n – 2 more than the equation for x(x + n – 1)
Geometric Reasoning –
For any x, x+1, x+2,… and x + n – 1
x by x + n - 1 can be represented by
and x+1 by x + n – 2 can be represented by
𝑥
x
Scaffolding Ideas:
x…x
𝑥
x
(n – 1)
x
x…x
X
(n – 2)
Extension Ideas:
 The “n consecutive numbers” proof is an extension for 8th graders. They should be able to recognize
that the solution is a difference of n – 2 but proving this is true will be challenging.

You could explore this problem using fractional numbers such as
.
Scaffolding Ideas:
 Graph paper might be a useful tool to offer students.
 For students who are having trouble recognizing a pattern you could provide a chart, like the one
below, to help them organize their thinking:
4 Consecutive #s


Product of
First & Last
Product of
2nd & 3rd
Comments / Observations
After using this chart, you could ask students to create their own chart for 5 consecutive numbers, etc.
It might be helpful to give students a list of guiding questions like, “what do you think the pattern will
be for 6 consecutive numbers? …for 7?” This may help them make the jump to n consecutive numbers.