Work backwards strategy
Start from the final result and reverse the steps.
Quick answer
Work backwards means starting with the answer and reversing the steps to find the beginning of a problem. It's like going from the end of a story back to the start!
Learning outcome
By the end of this page, you will be able to solve problems by starting with the final result and working backwards to find the beginning.
- You can identify when a problem can be solved by working backwards
- You can reverse the steps of a problem to find missing information
- You can check your answer by working forwards from your solution
- You can explain your thinking when solving backwards problems
Why this matters
Working backwards is a super helpful way to solve tricky problems! Sometimes it's easier to start from the end and go back than to start from the beginning and go forward. This strategy helps you solve puzzles, word problems, and even games.
Think about it like solving a mystery: if you know the final clue, you can work backwards to find out what happened earlier. This skill will help you in maths tests, in board games, and even when you're trying to figure out how to share your snacks fairly!
How it works
Here's how to work backwards:
- Start with what you know - Look at the final answer or result
- Think about what happened - What steps must have led to this result?
- Reverse the operations - If you added, subtract; if you multiplied, divide
- Check your answer - Work forwards to make sure it makes sense
It's like being a detective! You look at the clues (the final result) and figure out what happened before.
Examples you can see
Let's try a simple example together!
Example 1: The Birthday Cake
Mum baked a cake. She cut it into 8 equal pieces. After the party, there were 3 pieces left. How many pieces were eaten?
Working backwards:
- We know: 3 pieces left
- We know: cake was cut into 8 pieces
- To find: how many pieces were eaten
- Reverse thinking: 8 - 3 = 5 pieces eaten
Check: 5 pieces eaten + 3 pieces left = 8 pieces total
Example 2: The Number Machine
A number machine does this: it multiplies by 3, then adds 5. If the final result is 20, what was the starting number?
Working backwards:
- Final result: 20
- Machine did: multiply by 3, then add 5
- To find starting number: reverse the operations
- Reverse: 20 - 5 = 15, then 15 ÷ 3 = 5
Check: 5 × 3 = 15, then 15 + 5 = 20 ✓
Try it as a mini adventure
The Treasure Hunt Challenge
You are going on a treasure hunt in your garden! Here's your clue:
- Start at the big oak tree
- Walk 10 steps forward
- Turn right and walk 5 steps
- Find a red rock
- Walk back 5 steps
- Turn left and walk 10 steps
- Find the treasure!
But wait! You're not the one who made the clue. You're the one who has to figure out where the treasure is. You only know the final result: "The treasure is at the big oak tree."
Work backwards:
- Final result: treasure at oak tree
- Step 6: Walk 10 steps left (reverse of right)
- Step 5: Walk 5 steps forward (reverse of back)
- Step 4: Red rock is where you were
- Step 3: Turn left (reverse of right)
- Step 2: Walk 5 steps back (reverse of forward)
- Step 1: Walk 10 steps back (reverse of forward)
So you started at the treasure location and worked backwards!
Common mistakes
- Mistake: Forgetting to reverse operations correctly
Why it happens: Kids might add when they should subtract, or multiply when they should divide.
Fix: Always think "What operation did the machine do?" and then do the opposite. If it added 5, subtract 5 to reverse it.
- Mistake: Not checking the answer by working forwards
Why it happens: It's easy to get excited and forget to double-check.
Fix: After solving backwards, work forwards to see if you get the same answer.
- Mistake: Thinking all problems can be solved backwards
Why it happens: Some problems need you to work forwards, not backwards.
Fix: Look at the problem carefully. If you know the end result and need to find the beginning, work backwards. If you know the beginning and need to find the end, work forwards.
Parent and carer guide
Helping your child with working backwards is easy and fun!
What to say:
- "Let's look at what happened at the end first"
- "What would have to happen before this?"
- "Can you check your answer by working forwards?"
- "What's the opposite of adding?"
How to help:
- Start with simple problems that involve 1 or 2 steps
- Use everyday examples like cooking, sharing snacks, or games
- Ask your child to explain their thinking
- Don't worry if they make mistakes - that's how they learn!
- Practice with short, fun problems
What not to worry about:
- They might get confused at first - that's normal
- They might forget to check their answers
- They might try to work backwards on problems that don't need it
Remember, practice makes progress!
Practice questions
A teacher had some pencils. She gave 7 away and now has 9 left. How many pencils did she start with?
Answer: 16 pencils
A number machine multiplies by 4 and subtracts 3. The final result is 25. What was the starting number?
Answer: 7
Tom had some stickers. He gave 4 to his friend and then bought 6 more. He now has 12 stickers. How many did he start with?
Answer: 2 stickers
A farmer had some apples. He sold 5 and then picked 8 more. He now has 15 apples. How many did he start with?
Answer: 12 apples
Anna had some money. She spent £3 and then saved £4. She now has £9. How much did she start with?
Answer: £8
FAQ
What does "work backwards" mean?
It means starting with the final answer and reversing the steps to find what came before.
When should I use this strategy?
Use it when you know the end result and need to find the starting point. It's great for puzzles, word problems, and games.
How can I check if my answer is right?
Work forwards from your answer to see if you get the same result as the problem gave you.
Can I use this strategy for all problems?
No, not all problems work backwards. Sometimes you have to work forwards from the start to the end.
Why is this strategy useful?
It helps solve tricky problems that might be hard to start from the beginning. It's like being a detective!