Algebra
A number hides behind a box or a letter, like □ + 3 = 10 or 5n + 5 = 20. Here’s how to find it every time, work out sums in the right order, and spot the traps in the wrong answers.
Chapter 1
What are algebra questions?
A number is hiding. A box or a letter stands in its place, like □ + 3 = 10 or 5n + 5 = 20, and you find it. Some questions have no letter at all, just a sum like 28 − 4 × 3 that you must work out in the right order.
The letter. It stands for one number. Here n is the number you’re finding.
The equation. 5n + 5 = 20 says both sides are equal: work out the left and you get 20.
5n + 5 = 20
What is n?
The options
- Option 1: 15
- Option 2: 3
- Option 3: 4
- Option 4: 5
A Maths question from Rush, level 3.
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A number next to a letter means times
5n is five lots of n: 5 × n. If n is 3, then 5n is 15. Trap: reading it as 53, or as 5 + 3 = 8.
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+undoes−
×undoes÷
Opposites undo each other
Take away undoes add, and divide undoes times. For □ × 7 = 28, undo × 7 with ÷ 7: 28 ÷ 7 = 4. Trap: 21, from taking the 7 away.
Chapter 2
Solve it in 3 steps.
Here’s the question, one step at a time.
The question
5n + 5 = 20
What is n?
The options
- Option 1: 15
- Option 2: 3
- Option 3: 4
- Option 4: 5
A Maths question from Rush, level 3.
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Step 1
List what was done to n.
An equation is a machine in a line. 5n means n was times 5. Then 5 was added, and out came 20.
Two steps: × 5, then + 5.
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Step 2
Undo them, last step first.
Start at 20 and work back. The last step was + 5, so take 5 off: 20 − 5 = 15. That’s 5n. Then undo the × 5: 15 ÷ 5 = 3.
The undo row runs backwards.
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Step 3
Put your answer back in.
Run 3 forwards through the machine: 3 × 5 = 15, then 15 + 5 = 20. It makes 20, so it’s right.
3 makes 20, so it fits.
Three mistakes to spot
| The mistake | Do this instead |
|---|---|
| Stopping halfway. 15 is 5n, the number after the first undo. | One more undo: 15 ÷ 5 = 3. |
| Skipping the + 5. 4 is 20 ÷ 5. It undoes the × 5 but never takes the 5 off. Check: 5 × 4 + 5 = 25. | Undo every step, starting with the last. |
| Adding when you should take away. 5 is (20 + 5) ÷ 5: the + 5 done again. Check: 5 × 5 + 5 = 30. | Undo + 5 with its opposite, − 5. |
So n is 3. (20 − 5) ÷ 5 = 3, and 5 × 3 + 5 makes 20 again.
Remember: Last step done, first undone.
Chapter 3
Everything else Rush asks.
Each card is one kind of question in Rush’s Order & Algebra game: the rule, one worked example, and the trap.
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- 28 − 4 × 3
- = 28 − 12
- = 16
× and ÷ before + and −
Brackets first, then powers, then × and ÷, then + and −. Its name is BIDMAS.
28 − 4 × 3: do 4 × 3 = 12 first, then 28 − 12 = 16. Trap: 72, from doing the take-away before the times (24 × 3). A power like 3² means 3 × 3.
× and ÷ are partners, and so are + and −. When partners meet, work from the left: 12 − 4 + 2 = 10, and 8 ÷ 2 × 4 = 16.
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- 45 ÷ (3 + 2)
- = 45 ÷ 5
- = 9
Brackets first
Work out the brackets before anything else, even × and ÷.
45 ÷ (3 + 2): 3 + 2 = 5, then 45 ÷ 5 = 9. Trap: 17, from 45 ÷ 3 + 2, which ignores the brackets. And 5 is only the bracket.
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A sum on both sides
Work out the side with no box, then undo.
□ × 9 = 117 − 45. The right is 72, so □ × 9 = 72 and □ = 8. Traps: 72, the other side, not the box, and 13, from 117 ÷ 9, which forgets the − 45.
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Function machines
Do the steps in the order shown. Going backwards, undo them, last step first.
Put 7 in: + 6, then × 3. 7 + 6 = 13, and 13 × 3 = 39. Traps: 13, which stops halfway, and 27, from doing the × 3 first.
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Find one n first
Divide to find one n, then multiply.
3n = 33. What is 6n? One n is 33 ÷ 3 = 11, so 6n = 6 × 11 = 66. Or see that 6n is double 3n. Traps: 11 (just n), 33 (3n again) and 18 (3 × 6).
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- 3n − 9 = 15
- 3n = 24
- n = 8
Two letters? Put in what you know
Swap in the numbers you’re given, then undo as usual.
m = 3n − p. If m = 15 and p = 9, what is n? Put them in: 3n − 9 = 15. Undo the − 9: 3n = 24. Undo the × 3: n = 8. Traps: 24, which is 3n, and 2, from taking the 9 away instead of adding it.
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A rule from pairs
Find a rule for the first pair, then test it on the second.
(3 → 6) (12 → 15) (25 → ?). × 2 fits 3 → 6, but 12 × 2 is 24, not 15. + 3 fits both, so 25 + 3 = 28. Trap: 50, from × 2.
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A rule in words
A fixed charge is paid once. The rest is paid for each one.
Bike hire is £3, plus £5 for each hour. 5 hours cost 3 + 5 × 5 = £28. Traps: £40, which pays the £3 every hour, and £25, which leaves it off.
Chapter 4: In the exam
Write the algebra yourself.
The exam also asks you to turn words into algebra, put numbers into a formula, or solve a longer puzzle. Rush doesn’t ask these forms, but the same ideas do the work.
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Step 1
Put it all in one unit.
The pens are in pence, so change the £1 into pence too: £1 is 100p.
£1=100p
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Step 2
Build it one bit at a time.
4 pens at p pence each cost 4 × p, written 4p. Her change is what she paid, take away the cost: 100 − 4p.
Four pens, and the change is the rest.
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Step 3
Test it with a made-up number.
Say a pen costs 10p. 4 pens cost 40p, so the change is 60p. And 100 − 4 × 10 = 60. It works.
- p = 10
- 100 − 4 × 10
- = 60p
Three mistakes to spot
| The mistake | Do this instead |
|---|---|
| Taking away the wrong way round. 4p − 100 takes the £1 from the cost. With 10p pens, that’s 40 − 100: less than nothing. | Change is what you paid, take away the cost. The bigger amount comes first. |
| Mixing pounds and pence. 1 − 4p takes pence away from pounds. | Change everything to one unit before you start. |
| Stopping at the cost. 4p is what the pens cost, not her change. | Read the question again: it asks for the change. |
So it’s 100 − 4p. Test it: pens at 10p leave 60p change, and 100 − 4 × 10 = 60.
Remember: One unit. Build it. Test it.
Other ways the exam asks it
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- 2a + 1
- = 2 × 3 + 1
- = 7
Put the numbers in
Swap each letter for its number, then use BIDMAS.
If a = 3, what is 2a + 1? 2a is 2 × 3 = 6, so it’s 7. Trap: 24, from reading 2a as 23. And a² means a × a: 9, not 6.
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- A2
- B3
- C4
- D6
- E12
Letters for numbers
Work it out, then give the answer as a letter.
With this key, what is E − C × A? Times first: 4 × 2 = 8, and 12 − 8 = 4. 4 is C, so the answer is C. Trap: picking 4 when the answers are letters.
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- 42 ÷ 3 = 14
- 14 − 6 = 8
- 8 ÷ 2 = 4
Think of a number
Undo the steps from the end, as in chapter 2.
I think of a number, double it, add 6, then multiply by 3. I get 42. What was my number? Undo from the end: ÷ 3, then − 6, then ÷ 2. It was 4. Trap: 5, from undoing the steps in the order they were said.
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A letter on both sides
Do the same to both sides, so they stay equal.
3x + 4 = x + 12. Take x off both sides: 2x + 4 = 12. So 2x = 8, and x = 4. Check: 3 × 4 + 4 = 16, and 4 + 12 = 16. Trap: 8, which is 2x.
Chapter 5
Try it yourself.
Six real questions from Rush, easy to hard, 60 seconds on the clock.
Use the steps
- List what was done to n.
- Undo them, last step first.
- Put your answer back in.
Remember: Last step done, first undone.
For parents.
More on algebra, and what parents ask us.
More on algebra
Longer paper versions
Papers also set longer kinds. Some ask for an expression from words: "5 less than x" is x − 5, not 5 − x. Some put several values into a longer formula. Some give two numbers' total and difference: a total of 12 and a difference of 4 means 8 and 4. And some set related numbers three in a row, like (3 [12] 4), where the middle number comes from the outer two. Rush practises the order of operations, missing numbers, short formulas with one letter to find, rules given in words, one- and two-step equations and function machines. The longer forms are best practised on paper.
Questions parents ask
What algebra is in the 11+?
Missing numbers, function machines, short equations like 3n + 6 = 42, formulas with letters, and the order of operations. Algebra is a Year 6 topic in the national curriculum, so some of it may come up before school has taught it. Each step is undone with its opposite, so a child who knows the times tables and is sure of adding and taking away already has most of what they need.
What does 3n mean?
It means 3 × n. A number written next to a letter means multiply, so if n = 4, 3n is 12, not 34. In the same way, n/2 means n ÷ 2 and n² means n × n.
What is a function machine?
A number goes in, the machine does one or more steps in order, and a number comes out. To find the output, do the steps forwards. To find the input, start from the output and undo the steps backwards, last one first.
Is BIDMAS the same as BODMAS?
Yes. The I stands for indices and the O for orders, and both mean powers such as 3². Either word gives the same order of working.
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