Cheat sheet: Problem solving
Probability as a fraction
Chance = number of ways you want ÷ total number of equally likely ways.
3 red in 7 equal sections: 3/7.
The probability scale
- Impossible = 0
- Even chance = 1/2
- Certain = 1
- Never more than 1
Not always equal
Two dice totals are not equally likely: total 7 has 6 ways out of 36, total 2 has 1 way out of 36.
Systematic listing
- Fix one thing, change the next, then move on.
- Tree or table: 3 × 2 = 6 meals.
- Digits 3, 6, 9 used once each: 6 numbers.
Logic puzzles
- Draw a grid. Tick yes, cross no.
- Use the surest clue first.
- Fewest coins: biggest coin first, then check.
Working backwards
- Undo the last step first.
- Opposites: + and −, × and ÷.
- Always check by going forwards.
Age puzzles
- Everyone ages the same number of years.
- The gap between ages never changes.
- Use a bar model or a simple equation.
Words to know
outcome, likely, unlikely, certain, impossible, fair, combination, systematic, clue, inverse (opposite).
Mini lessons
Probability as a fraction
Skills: PROB-01. Stretch: usually taught in Year 7, but some 11+ tests include it.
Probability tells us how likely something is. Write it as a fraction: the number of ways you want, over the total number of equally likely ways. A chance of 0 is impossible and 1 is certain. Be careful: some outcomes are more likely than others, as the dice grid shows.
7 equal sections, 3 of them red. The chance of red is 3 out of 7, or 3/7.
A bag has 4 red, 3 blue and 3 green counters. What is the chance of picking a red counter?
- Count all the counters: 4 + 3 + 3 = 10.
- Red counters are what we want: 4.
- Chance = 4 out of 10 = 4/10.
- Simplify: 4/10 = 2/5.
Answer: 2/5
Systematic listing
Skills: PROB-02
Systematic means in an organised order, so you do not miss anything or list something twice. Keep one thing fixed while you change the next, then move on to the next fixed thing. A tree diagram or a table helps.
3 sandwiches × 2 drinks = 6 different meals.
A café has 3 sandwiches (cheese, ham, egg) and 2 drinks (juice, milk). A meal is one sandwich and one drink. How many different meals are there?
- Fix cheese: cheese + juice, cheese + milk (2 meals).
- Fix ham: ham + juice, ham + milk (2 meals).
- Fix egg: egg + juice, egg + milk (2 meals).
- Add them: 2 + 2 + 2 = 6, which is 3 × 2.
Answer: 6 meals
Logic and optimisation puzzles
Skills: PROB-03
In a logic puzzle, every clue rules something out. Draw a grid and cross off what cannot be true. When only one choice is left in a row or column, it must be right. Optimisation means finding the best, such as the fewest coins.
A tick means “yes”, a cross means “no”. Each row and column has exactly one tick.
Ali, Bo and Cy each have one pet: a cat, a dog or a fish. Bo’s pet lives in water. Ali does not have the dog. Who has which pet?
- Bo’s pet lives in water, so Bo has the fish. Cross out fish for Ali and Cy.
- Ali does not have the dog, and the fish is taken, so Ali has the cat.
- Only the dog is left, so Cy has the dog.
Answer: Ali has the cat, Bo has the fish, Cy has the dog
Age and working-backwards puzzles
Skills: PROB-04
When you know the end result, work backwards. Start from the answer and undo each step in reverse order, using the opposite operation. For age puzzles, remember that everybody gets older by the same amount.
In 5 years’ time Tom will be 12. How old was Tom 3 years ago?
- Tom is 12 in 5 years, so now he is 12 − 5 = 7.
- 3 years ago he was 7 − 3 = 4.
- Check: 4 + 3 = 7, and 7 + 5 = 12 ✓
Answer: 4 years old
Worked examples across the levels
Level 1 is the easiest. Level 5 is a real challenge. Try each one yourself before you read the steps.
Level 1 (Year 3): one step
A shop has 2 hats (a woolly hat and a sun hat) and 2 scarves (red and blue). You choose one hat and one scarf. List every pair. How many are there?
- Woolly hat with the red scarf, woolly hat with the blue scarf.
- Sun hat with the red scarf, sun hat with the blue scarf.
- That is 4 pairs.
Answer: 4 pairs
Level 2 (Year 4): one or two steps
A school offers 2 clubs on Monday (chess or art) and 3 clubs on Tuesday (choir, football or drama). A child picks one club each day. How many different pairs of clubs are possible?
- Each Monday club can go with 3 Tuesday clubs.
- 2 × 3 = 6.
Answer: 6 pairs
Level 3 (Year 5): two steps
How many different 3-digit numbers can you make using the digits 3, 6 and 9 once each? How many of them are more than 600?
- List in order: 369, 396, 639, 693, 936, 963. That is 6 numbers.
- Numbers over 600 start with 6 or 9: 639, 693, 936, 963.
- That is 4.
Answer: 6 numbers, and 4 of them are more than 600
Level 4 (exam standard): several steps or working backwards
I think of a number, multiply it by 3, subtract 4, then halve it. The answer is 13. What was my number?
- Work backwards. The last step was ÷2, so undo it: 13 × 2 = 26.
- Undo the −4: 26 + 4 = 30.
- Undo the ×3: 30 ÷ 3 = 10.
- Check: 10 × 3 = 30, 30 − 4 = 26, 26 ÷ 2 = 13 ✓
Answer: 10
Level 5 (challenge): three to five steps
Nina is twice as old as Omar. Four years ago she was three times as old as he was. How old are they now?
- Let Omar be a years old now. Nina is 2a.
- Four years ago: Omar was a − 4 and Nina was 2a − 4.
- Nina was 3 times as old: 2a − 4 = 3 × (a − 4) = 3a − 12.
- So 12 − 4 = 3a − 2a, which gives a = 8.
- Omar is 8 and Nina is 16. Check four years ago: 4 and 12, and 12 = 3 × 4 ✓
Answer: Omar is 8 and Nina is 16
Common mistakes
Thinking every outcome is equally likely
Mistake code: PROB_EQUIPROBABILITY
The mistake: Saying that with two dice a total of 2 is as likely as a total of 7 because “they are both just one total”.
Why it is wrong: A total of 2 can only happen one way (1 and 1). A total of 7 can happen 6 ways (1+6, 2+5, 3+4, 4+3, 5+2, 6+1).
How to avoid it: Write out all the possible results in a grid first. Count the ways, then compare.
Stopping before the last step
Mistake code: TECH_STOPPED_AT_STEP_1
The mistake: In the red counter question, working out the total of 10 counters and giving 10 as the answer.
Why it is wrong: The total is only the first step. The question asks for a chance, so you must go on and write the fraction.
How to avoid it: Write “total = …” first, then finish with the fraction and simplify.
Missing or repeating items in a list
The mistake: Writing outfits or numbers in a random order and counting 5 when there are 6.
Why it is wrong: Without an order it is easy to forget one or write one twice.
How to avoid it: Be systematic. Fix the first choice, list every second choice, then change the first choice.
Using the same operation when working backwards
The mistake: A machine does +5 and then ×2, and the answer is 30. Working back by doing 30 + 5, or by undoing the +5 first.
Why it is wrong: To undo a step you need its opposite, and you must undo the last step first.
How to avoid it: Write the steps in a row, then go from right to left using opposites.
Quick check
Five questions, easy to hard. Use paper and have a go before you open the answer.
A fair spinner has 5 equal sections numbered 1 to 5. What is the chance of landing on 4?
Show answer
1/5
One section is a 4, and there are 5 equal sections.
A bag has 1 red, 2 yellow and 7 green counters. What is the chance of picking a counter that is not green?
Show answer
3/10
Total = 1 + 2 + 7 = 10. Not green = 1 + 2 = 3. So the chance is 3/10.
A pizza comes in 2 sizes with a choice of 4 toppings. Each pizza has one size and one topping. How many different pizzas are there?
Show answer
8
2 sizes × 4 toppings = 8. You can list them to check.
A number machine multiplies by 3 and then subtracts 7. The answer is 29. What was the starting number?
Show answer
12
Backwards: 29 + 7 = 36, then 36 ÷ 3 = 12. Check: 12 × 3 = 36, 36 − 7 = 29.
You have 1p, 2p, 5p, 10p, 20p and 50p coins. What is the fewest coins you need to make 63p?
Show answer
4 coins
Biggest first: 50p + 10p + 2p + 1p = 63p. That is 4 coins, and no way uses fewer.
For grown-ups
- Play games with dice, cards or coloured counters, and ask “what are the chances?” before each turn. Then try it out many times and see what happens.
- For puzzles, ask “what do we know for sure?” and “could we draw it?” before any calculating. A drawing or a list is a perfectly good method.
- Praise careful checking. Ask your child to put their answer back into the puzzle to see if it works.