This document provides ideas and resources for teaching mathematics concepts by thinking backwards. It suggests having students solve problems by working backwards from the answer to the question. Some specific examples mentioned are: solving percentage increase and decrease problems by working backwards from the new price to the original; factorizing algebraic expressions by multiplying out factors; and applying laws of logarithms and calculus concepts from right to left as well as left to right. The document contains hyperlinks to further resources on topics like number operations, factoring, percentages, and puzzles that can be approached by thinking backwards.

1. Good Mathematicians Can
Go Backwards!
Colleen Young

2. Going Backwards
There is so much opportunity for thinking
backwards when we teach – a great
learning opportunity and also a problem
solving strategy.

3. Going Backwards
The following slides include ideas and
resources for thinking backwards!
Many of the slides have hyperlinks for
further information and resources.

4. Tables
Learn them
backwards too.
Try Transum’s Fast
Factors

5. Number Operations
Manipulate numerical expressions

6. Percentages
The price of a dress is increased by 10%,
the original price is £40, what is the new
price?
Solve, then pose the same question
backwards:
After an increase of 10% a dress costs
£44, what was the original price?

7. Factorising
If you can multiply out brackets, you can
factorise
4(x+y) = 4x+4y
6a−6b = 6(a-b)
(x+4)(x+2) = x2+6x+8
x2-5x+6 = (x-2)(x-3)

8. Standards Unit – Build an Equation
pdf file

9. From Nrich
Working backwards
at KS2; the ideas
here could also be
used at KS3.
See this Nrich article
from Liz Woodham
on Developing
Problem-solving skills
which includes the
link above

10. Arithmagons
Perfect for any topic
for thinking
backwards.
See the ideas and
resources here –
everything from
simple arithmetic to
Calculus!
Arithmagons from Flash Maths

11. Problems
See for example the
AQA problems
included here.
This type of
backwards problem
really helps students
think deeply.

12. Here’s the diagram..
For a variation on
here’s the answer,
what’s the question?
Try, here’s the
diagram what’s the
question?
Or try Algebra
Snippets

13. Puzzles
• Sudoku, see these Nrich resources
• Ken Ken
• Find the Factors

14. Laws of logs
Learn everything from right to
left as well as left to right!

15. Learn it both ways!
Learn everything from right to left as well
as left to right!
If you know your laws of indices you should
recognise that x10 = x1 × x9
For every derivative you know, you also
know an integral!
.......and so on...and on....and on!

16. Mathematics for Students Mathematics, Learning and Technology