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15 = 7 + 8 and 10 = 1 + 2 + 3 + 4. Can you say which numbers can be expressed as the sum of two or more consecutive integers?

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Arrange the numbers 1 to 16 into a 4 by 4 array. Choose a number. Cross out the numbers on the same row and column. Repeat this process. Add up you four numbers. Why do they always add up to 34?


The well known Fibonacci sequence is 1 ,1, 2, 3, 5, 8, 13, 21.... How many Fibonacci sequences can you find containing the number 196 as one of the terms?

Quadratic Patterns

Age 11 to 14
Challenge Level

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 -

the two numbers multiplied together had a difference of 2,
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.