Quadratic Patterns
Surprising numerical patterns can be explained using algebra and diagrams...
Problem
Quadratic Patterns printable worksheet
Charlie has been playing with calculations again...
$2 \times 4 + 1 = 9$
$4 \times 6 + 1 = 25$
$5 \times 7 + 1 = 36$
$9 \times 11 + 1 = 100$
What do you notice?
Click below to see what Charlie said:
"If you multiply two numbers that differ by 2, and then add one, the answer is always the square of the number between them!"
Can you explain what's happening?
Click below to see Charlie's explanation:
$ n(n+2) +1 = n^2 +2n+1 = (n+1)^2$
or alternatively,
$(n-1)(n+1)+1 = n^2-n+n-1+1=n^2$
Alison drew a diagram to explain the results. Click below to see:
Can you make sense of Charlie's method and Alison's diagrams?
Here are some more number patterns to explore. Some have been expressed numerically, some in words, and some algebraically.
Can you represent each pattern in all four ways,
in words (so you can describe the pattern),
algebraically (so you can prove the pattern continues),
and using a diagram (to explain the pattern)?
- $2 \times 3 + 3 = \, ?$
$5 \times 6 + 6 = \, ?$
$4 \times 5 + 5 = \, ?$
$9 \times 10 + 10 = \, ?$
What do you notice?
- Choose three consecutive numbers, square the middle one, and subtract the product of the other two.
Repeat with some other sets of numbers.
What do you notice?
- $3 \times 3 - 1 \times 1 = \, ?$
$8 \times 8 - 6 \times 6 = \, ?$
$7 \times 7 - 5 \times 5 = \, ?$
$10 \times 10 - 8 \times 8 = \, ?$
What do you notice?
- $n(n+1) - (n-1)(n+2) = \, ?$
$(n+1)(n+2) - n(n+3) = \, ?$
$(n-3)(n-2) - (n-4)(n-1) = \, ?$
What do you notice?
- $3\times 5 + 1= \, ?$
$5\times 7 + 1= \, ?$
$7\times 9 + 1= \, ?$
$9\times 11 + 1= \, ?$
What do you notice?
- Choose three consecutive numbers and add the product of the smallest two to the product of the greatest two.
Repeat with some other sets of numbers.
What do you notice?
With thanks to Don Steward, whose ideas formed the basis of this problem.
You may be interested in the other problems in our Factorise This! Feature.
Getting Started
You could start by taking a look at Perimeter Expressions and Seven Squares, which offer a good introduction to describing patterns verbally, numerically and algebraically.
Teachers' Resources
This problem is available as a printable worksheet: Quadratic Patterns
Why do this problem?
This problem offers students the opportunity to engage with quadratic number patterns using multiple representations - in words, numerically, algebraically, and with diagrams.
By moving between different representations for different purposes, students gain a deeper understanding of the underlying structure of quadratic sequences.
Possible approach
Write up Charlie's four calculations on the board:
$2 \times 4 + 1 = 9$
$4 \times 6 + 1 = 25$
$5 \times 7 + 1 = 36$
$9 \times 11 + 1 = 100$
"Here are four calculations. Take a moment to look at them and see what you notice."
"Now share your ideas with your partner."
Invite a few pairs to describe what they noticed -
Charlie always added 1,
the answer was always a square number...
"It looks as though if you multiply two numbers that differ by 2, and then add one, the answer is always the square of the number between them!
I wonder if we can prove that it will always work?"
Give students some time in pairs or small groups to explore and think of ways of convincing themselves that the pattern continues. While they are working, circulate and listen to the ideas they come up with.
In particular, look out for students explaining the pattern algebraically or with a diagram, like Charlie and Alison did in the problem.
After a while, bring the class together and invite any students with an algebraic or diagrammatic way of explaining the pattern to come out to the board and share their method, or if no-one used Charlie's or Alison's method, show it to them.
Once students are comfortable with the different representations, they could work through the problems on this worksheet, proving their findings with algebra and a diagram each time.
Key questions
How can we represent the patterns algebraically?
How can we interpret the product of two numbers geometrically?
Possible support
Perimeter Expressions and Seven Squares offer students a good introduction to describing generic patterns verbally, numerically and algebraically.
Possible extension
Pair Products and Hollow Squares offer more opportunities for observing, conjecturing and thinking about proof in the context of quadratic relationships.