Hi, I am Pip! 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.
Pip’s tip: Undo the last step first when you work backwards. Check by putting your answer back in.
A machine multiplies its input by 3, then adds 4. What is the output for input 7?
7 × 3 = 21; 21 + 4 = 25. The picture shows the same steps.
Answer: 25.
Find n if n + 9 = 24.
15. 24 − 9 = 15; check 15 + 9 = 24.
Find 5a when a = 6.
30. 5a means 5 × 6 = 30.
Find the next term: 6, 11, 16, 21, …
26. Add 5 each time; 21 + 5 = 26.
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
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?
Answer: 5
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).
Worked example: The perimeter of a rectangle is P = 2l + 2w. Find P when l = 11 cm and w = 6 cm.
Answer: 34 cm
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
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.)
Answer: 49
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
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.
Answer: 25 and 19
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
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?
Answer: 12 pairs, then 24 outfits
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.
A machine multiplies by 5. What comes out when 9 goes in?
Answer: 45
A machine doubles a number, then adds 5. What comes out when 12 goes in?
Answer: 29
I think of a number, multiply it by 6, then subtract 8. I get 40. What was my number?
Answer: 8
Tom and Asha have 46 stickers altogether. Tom has 8 more than Asha. How many stickers does Tom have?
Answer: 27
A square (■) and a circle (●) stand for numbers. ■ + ● = 15 and ■ + ■ + ● = 22. What is ● + ● + ■?
Answer: 23
Lots of children make these slips. Spot them early and you will win marks.
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.
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.
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.
Five questions, easy to hard. Have a go on paper, then open the answer.
A machine subtracts 9. What comes out when 35 goes in?
26
35 − 9 = 26.
I think of a number, triple it and add 4. I get 55. What was my number?
17
Undo + 4: 55 − 4 = 51. Undo × 3: 51 ÷ 3 = 17. Check: 17 × 3 + 4 = 55.
The sequence is 3, 10, 17, 24, ... What is the 8th term?
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.)
C = 5n + 3. Find C when n = 7.
38
5 × 7 = 35, then 35 + 3 = 38.
Two numbers add up to 74. They differ by 8. What are they?
41 and 33
(74 + 8) ÷ 2 = 41. Then 41 − 8 = 33. Check: 41 + 33 = 74.