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Algebra: letters, machines and patterns

Hi, I am Bea! Algebra is just maths with a mystery number in it. We use a letter or a shape to hold the place of the number we do not know yet. Let us crack the code together.

1. Cheat sheet: Algebra

Everything you need on one page. Press the button to print it on one sheet of A4.

Function machines

  • A machine does steps in order.
  • Opposites (inverses): + and −, × and ÷.
  • To go backwards, undo the LAST step first.

Think of a number

  • Write the steps in order.
  • Start from the answer and undo each step.
  • Check by going forwards again.

Expressions

  • A letter is a number we do not know yet.
  • 3n means 3 × n. n + 4 means 4 more than n.
  • c pence for one pen: 5 pens cost 5c pence (5 × c).

Formulae

  • Swap each letter for its number, then work it out.
  • P = 2l + 2w with l = 11, w = 6 gives P = 34.

Sequences

  • Term-to-term rule: what you do to get the next term.
  • 4, 9, 14, 19: add 5 each time.
  • nth term: 5n − 1 (stretch, beyond KS2).

Two unknowns

  • Sum S, difference D:
  • bigger = (S + D) ÷ 2
  • smaller = bigger − D
  • Symbol puzzles: take the same from both sides.

Combinations

  • 3 tops and 4 skirts: 3 × 4 = 12 outfits.
  • List in a grid or by an order, so you miss none.

Words to know

  • Term: one number in a sequence.
  • Expression: letters and numbers, no =.
  • Equation: has an = sign.
  • Formula: a rule using letters.
  • Inverse: the opposite operation.

2. Mini lessons

Function machines and working backwards

ALG-01ALG-02

A function machine follows a rule on every number that goes in. To find the number you started with, begin at the answer and undo each step, last step first, using the opposite operation.

Animation: do it forwards, then undo it backwards

ForwardBackward: undo in reverse order×3+451519−4÷319155

Start with 5. 5 goes in. 5 × 3 = 15. 15 + 4 = 19. Now work backwards from 19. 19 − 4 = 15 (undo the +4 first). 15 ÷ 3 = 5 (then undo the ×3). We are back at 5. Undoing the steps in reverse order brings you home.

Worked example: I think of a number, multiply it by 3, then add 4. I get 19. What was my number?

  1. The steps were: × 3, then + 4.
  2. Undo the last step first. Opposite of + 4 is − 4: 19 − 4 = 15.
  3. Undo the first step. Opposite of × 3 is ÷ 3: 15 ÷ 3 = 5.
  4. Check: 5 × 3 = 15, 15 + 4 = 19. Correct!

Answer: 5

Expressions and formulae

ALG-03ALG-04

A letter stands for a number. If one pencil costs c pence, then 3 pencils cost 3c pence. A formula is a rule written with letters. To use it, swap each letter for its number (this is called substituting).

l = 11 cmw = 6 cmP = 2l + 2w
A rectangle: P = 2l + 2w. The letters l and w stand for numbers.

Worked example: The perimeter of a rectangle is P = 2l + 2w. Find P when l = 11 cm and w = 6 cm.

  1. Swap in the numbers: P = 2 × 11 + 2 × 6.
  2. Work out each part: 2 × 11 = 22 and 2 × 6 = 12.
  3. Add: 22 + 12 = 34.

Answer: 34 cm

Linear sequences

ALG-05ALG-06

A linear sequence goes up or down by the same amount each time. That amount is the term-to-term rule. The nth term is a rule that jumps straight to any term. It is stretch work, beyond KS2, so it is fine to use the term-to-term rule if you are not sure yet.

Animation: a sequence is a set of equal jumps

4914192429+5+5+5+5+5

The first term is 4. Jump +5 to reach 9. Jump +5 to reach 14. Jump +5 to reach 19. Jump +5 to reach 24. Every jump is +5, so the term-to-term rule is "add 5": 4, 9, 14, 19, 24, 29.

Worked example: The sequence is 4, 9, 14, 19, ... What is the 10th term? (The nth term part is stretch, beyond KS2.)

  1. The jumps are + 5 each time, so the nth term starts with 5n.
  2. For n = 1, 5n gives 5 but the first term is 4. So subtract 1: the nth term is 5n − 1.
  3. For the 10th term: 5 × 10 − 1 = 49.
  4. Check by listing: 4, 9, 14, 19, 24, 29, 34, 39, 44, 49.

Answer: 49

Two unknowns and symbol puzzles

ALG-07ALG-08

When two numbers are mixed up in one puzzle, use what you know about their total and their difference. With shape puzzles, think of a balance: whatever you take off one side, take off the other.

Animation: a balance scale

442▲ + 3 = 11Take 3 off each side: 2▲ = 8Halve each side: ▲ = 4

Two triangles and 3 squares balance 11 squares. This is the puzzle: 2▲ + 3 = 11. Take 3 squares off BOTH sides. It still balances: 2▲ = 8. Split the 8 into two equal groups, one for each triangle. Each triangle is worth 4. A balance stays level if you do the same to both sides.

Worked example: Two numbers add up to 44. One is 6 more than the other. Find them.

  1. Add the difference to the total: 44 + 6 = 50.
  2. Halve it to get the bigger number: 50 ÷ 2 = 25.
  3. The smaller number is 25 − 6 = 19.
  4. Check: 25 + 19 = 44 and 25 − 19 = 6.

Answer: 25 and 19

Listing all the combinations

ALG-09

To count choices, pair each item from one group with each item from the other. Listing in a grid or in a fixed order makes sure you do not miss any or repeat any.

Animation: list every meal, row by row

JuiceMilkCheeseCheese + juiceCheese + milkHamHam + juiceHam + milkEggEgg + juiceEgg + milk3 sandwiches × 2 drinks = 6 meals

Cheese, ham or egg sandwich. Juice or milk. Take each sandwich in turn and pair it with each drink. Six different meals. A neat grid means you cannot miss one or count one twice.

Worked example: Ayesha has 3 hats and 4 scarves. How many hat-and-scarf pairs can she make? Then she adds a choice of 2 pairs of gloves. How many outfits now?

  1. Hats and scarves: 3 × 4 = 12 pairs.
  2. Each of the 12 pairs can have either pair of gloves: 12 × 2 = 24.

Answer: 12 pairs, then 24 outfits

3. Worked examples from easy to hard

Level 1 is Year 3, one step. Level 5 is a challenge with 3 to 5 steps. Try each one yourself before you read the steps.

Level 1One step

A machine multiplies by 5. What comes out when 9 goes in?

  1. Do the rule once: 9 × 5 = 45.

Answer: 45

Level 2One or two steps

A machine doubles a number, then adds 5. What comes out when 12 goes in?

  1. Double 12: 12 × 2 = 24.
  2. Add 5: 24 + 5 = 29.

Answer: 29

Level 3Two steps, working backwards

I think of a number, multiply it by 6, then subtract 8. I get 40. What was my number?

  1. Undo the − 8 with + 8: 40 + 8 = 48.
  2. Undo the × 6 with ÷ 6: 48 ÷ 6 = 8.
  3. Check: 8 × 6 = 48 and 48 − 8 = 40.

Answer: 8

Level 4Exam standard

Tom and Asha have 46 stickers altogether. Tom has 8 more than Asha. How many stickers does Tom have?

  1. Add the difference to the total: 46 + 8 = 54.
  2. Halve it to get Tom's stickers: 54 ÷ 2 = 27.
  3. Check: Asha has 27 − 8 = 19 and 27 + 19 = 46.
  4. The question asks about Tom, so the answer is 27, not 19.

Answer: 27

Level 5Challenge

A square (■) and a circle (●) stand for numbers. ■ + ● = 15 and ■ + ■ + ● = 22. What is ● + ● + ■?

  1. Compare the two lines. The second has one more ■, and the total is 22 − 15 = 7 more. So ■ = 7.
  2. Use the first line: ● = 15 − 7 = 8.
  3. Now work out ● + ● + ■ = 8 + 8 + 7 = 23.
  4. Check line 2: 7 + 7 + 8 = 22.

Answer: 23

4. Common mistakes

Lots of children make these slips. Spot them early and you will win marks.

Treating a letter as a thing

ALG_LETTER_AS_OBJECT

The mistake: Reading a letter as an object (b for bananas), or thinking an answer like 8 + g is not finished.

Why it is wrong: A letter stands for a NUMBER we do not know yet, not a thing. 8 + g is a perfectly good answer: it is 8 and some more.

How to avoid it: Say the letter as "the number of". Use a function machine: put in g, add 8, out comes g + 8.

✗ Wrong: Ria has 8 more bananas than Tom, who has b. She has "8b".
✓ Right: Ria has b + 8. If Tom has 5, she has 5 + 8 = 13.

Stopping after one undo

TECH_STOPPED_AT_STEP_1

The mistake: Undoing only the first step when working backwards.

Why it is wrong: The question needs ALL the steps undone. The number after one undo is still a middle number.

How to avoid it: Count the steps in the story, then count your undo steps. They must match.

✗ Wrong: "×3, then +4 gives 19": 19 − 4 = 15, answer 15.
✓ Right: Undo again: 15 ÷ 3 = 5, answer 5.

Answering the other number

TECH_MISREAD_QUESTION

The mistake: Finding both numbers but writing down the wrong one.

Why it is wrong: Sum and difference problems have two answers. The question says which one it wants.

How to avoid it: Underline the name or word in the question, then circle that answer.

✗ Wrong: "How many stickers does Tom have?" Writes 19 (Asha's).
✓ Right: Tom has the bigger share: 27.

5. Quick check

Five questions, easy to hard. Have a go on paper, then open the answer.

  1. A machine subtracts 9. What comes out when 35 goes in?

    Show answer

    26

    35 − 9 = 26.

  2. I think of a number, triple it and add 4. I get 55. What was my number?

    Show answer

    17

    Undo + 4: 55 − 4 = 51. Undo × 3: 51 ÷ 3 = 17. Check: 17 × 3 + 4 = 55.

  3. The sequence is 3, 10, 17, 24, ... What is the 8th term?

    Show answer

    52

    It goes up by 7 each time. From the 4th term, 24, there are 4 more jumps: 24 + 4 × 7 = 52. (Rule: 7n − 4, and 7 × 8 − 4 = 52.)

  4. C = 5n + 3. Find C when n = 7.

    Show answer

    38

    5 × 7 = 35, then 35 + 3 = 38.

  5. Two numbers add up to 74. They differ by 8. What are they?

    Show answer

    41 and 33

    (74 + 8) ÷ 2 = 41. Then 41 − 8 = 33. Check: 41 + 33 = 74.

6. For grown-ups

  • Ask "What would you do to undo that?" before you give any hint. Children who say the inverse out loud get working backwards quickly.
  • Make letters concrete: "n is the number of sweets in the bag. If I add 5 sweets, how many now?" Answer n + 5.
  • Play "think of a number" in the car. Take turns to be the thinker and the solver, and always ask for a check by going forwards.