** Surprising numerical patterns can be explained using algebra and diagrams…**

Charlie has been playing with calculations again…

2×4+1=9

4×6+1=25

5×7+1=36

9×11+1=100

**What do you notice?**

Click below to see what Charlie said:

**Can you explain what’s happening?**

Click below to see Charlie’s explanation:

or alternatively,

(n−1)(n+1)+1=n²−n+n−1+1=n²

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,

**numerically**(so you can spot the pattern),

**in words**(so you can describe the pattern),

**algebraically**(so you can prove the pattern continues),

and

**using a diagram**(to explain the pattern)?

- 2×3+3=?

5×6+6=?

4×5+5=?

9×10+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? - 3×3−1×1=?

8×8−6×6=?

7×7−5×5=?

10×10−8×8=?

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? - 3×5+1=?

5×7+1=?

7×9+1=?

9×11+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.*

Age 11 to 14