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Algebraic expressions and substitution
Replacing letters with positive and negative values.
In Lesson 5.1 a red ? stood for a number you did not know yet. In algebra we use a letter instead. In this lesson you will learn how to:
- use letters for numbers, and write algebra the short way
- turn words into an expression
- substitute: swap each letter for its number and work out the answer
- move terms around without changing the answer
- substitute negative numbers
1. Letters stand for numbers
A bag and 3 more sweets
A bag holds n sweets. You cannot see inside. One bag and 3 more sweets is n + 3.
Three bags
Three bags is n + n + n. The short way to write it is 3n, which means 3 × n.
Algebra the short way
| The long way | The short way | Remember |
|---|---|---|
| 3 × n or n + n + n | 3n | Leave out the × sign |
| n × 3 | 3n | The number goes in front |
| 1 × n | n | Nobody writes the 1 |
| n × n | n2 | Say “n squared”, as in Lesson 1.7 |
| a × b | ab | Two letters side by side are multiplied |
| n ÷ 2 | n ÷ 2 or n2 | A division can be written as a fraction |
Long way, short way
- Tap a card on the left, then tap the card on the right that means the same.
- Watch out for twins that look alike, such as 2n and n2.
2. Writing an expression
Example: think of a number n. Multiply it by 4, then take away 1.
- Start with the number:n
- Multiply it by 4:4n
- Then take away 1:4n − 1
5 less than n
Start with n and take 5 away: n − 5. Test it with a number. If n = 10, then 5 less than 10 is 5, and 10 − 5 = 5. Writing 5 − n would give 5 − 10, which is below zero.
Add 2, then multiply by 3
The adding comes first, so it goes in brackets: 3(n + 2). Without them, 3n + 2 would multiply first (BIDMAS, Lesson 2.4). If n = 10, 3(n + 2) is 36 but 3n + 2 is 32.
A story: cinema tickets
A cinema ticket costs £t. Four tickets cost 4 × t pounds, so the expression is 4t. You can pick any letter. A sensible letter helps you remember what it stands for: t for ticket.
Expression builder
- Drag tiles into the box, or tap a tile to add it to the end. Drag a tile along the box to move it, or tap it to send it back.
- Two with no slips moves you up a level. Level 1 has one step (add or take away), level 2 multiplies or divides, and level 3 has two steps, sometimes with brackets.
3. Substitution
Example: work out 3n + 2 when n = 5
- Swap n for 5, and put back the × sign:3 × 5 + 2
- Multiply first:15 + 2
- Then add:17
Two letters: a = 4 and b = 3
ab = 4 × 3 = 12
2a + b = 2 × 4 + 3 = 8 + 3 = 11
A square: n = 6
n2 = 6 × 6 = 36
Squaring is not doubling: 2 × 6 is 12, which is 2n.
A formula: the taxi ride
A formula is a rule for working something out. A taxi costs C = 2m + 3 pounds, where m is the number of miles. For a 6 mile ride, m = 6: C = 2 × 6 + 3 = 12 + 3 = 15. The ride costs £15.
Substitution machine
- Drag a number tile onto a red letter, or tap a tile to drop it in. The working shows you every step.
- Try the numbers in order: 1, 2, 3 … How does the answer change each time? The table keeps a record.
4. Moving terms around
Example: the same expression, in a new order
Written: 3n − 4 + 10
Written: 10 + 3n − 4
Check with n = 2: 3 × 2 − 4 + 10 = 12, and 10 + 3 × 2 − 4 = 12. The same answer, because every term kept its own sign.
Swapping is fine for + and ×
n + 5 is the same as 5 + n, and 3 × n is the same as n × 3, whatever n is.
Not for − and ÷
n − 5 and 5 − n are different. If n = 8, they give 3 and −3. In the same way, n ÷ 2 and 2 ÷ n are different.
Term shuffle
- Drag a term to a new place, or tap one term and then another to swap them. Can you make the order asked for?
- Then press “Leave the signs behind” and move the terms again. What happens to the answer?
Same or different?
- Drag numbers onto the n = box (or tap them) to test both expressions with the same number.
- When you are sure, pick a verdict. To show they are not always the same, find a number that gives two different answers.
5. Substituting negative numbers
Example: work out 2n + 7 when n = −3
- Swap n for −3. Brackets keep the minus sign safe:2 × (−3) + 7
- 2 lots of −3: two jumps of 3 down from 0:−6 + 7
- Then jump 7 up:1
n + 5 when n = −2
Start at −2 and jump 5 up: −2 + 5 = 3.
n − 4 when n = −3
Start at −3 and jump 4 down: −3 − 4 = −7. Going down from a negative number takes you further below zero.
Jump along the line
- Drag a number onto the red letter, or tap it. The jumps for the letter are red, and the jumps for the numbers are blue.
- Before the jumps finish, say where you think you will land.
Substitution workshop
- Step 1: pick the line with the numbers put in. Step 2: work it out and type the answer.
- Two with no slips moves you up a level. Level 1 has one step, level 2 has two steps, squares and brackets, level 3 has two letters, level 4 puts in negative numbers, and level 5 has negative numbers with two steps. The keypad gets a minus key on levels 4 and 5.
6. Practice: pick your level
Start with Foundation if you are in Year 4 or just starting. Move up when you are getting most of them right.
The mistakes that cost marks
- Writing the number next to the other number. Saying 3n is 35 when n = 5. 3n means 3 × n, so it is 3 × 5 = 15.
- Doubling instead of squaring. Saying n2 is 12 when n = 6. n2 means n × n: 6 × 6 = 36.
- Turning “less than” round. Writing “5 less than n” as 5 − n. start with n and take 5 away: n − 5. Test it with a number.
- Leaving out the brackets. Writing “add 2, then multiply by 3” as 3n + 2. the adding comes first, so it goes in brackets: 3(n + 2).
- Adding before multiplying. Working out 2n + 7 when n = 4 as 2 × 11 = 22. multiply first: 2 × 4 = 8, then 8 + 7 = 15.
- Leaving a sign behind. Rewriting 10 − n + 3 as 10 + n − 3. each term takes the sign in front of it: 10 + 3 − n.
- Losing the minus sign. Saying 2n is 6 when n = −3. two jumps of 3 down from 0 land on −6.
- Testing only one number. Saying 2n and n2 are the same because both give 4 when n = 2. try another number: when n = 3 they give 6 and 9.
Quick checklist
Tick each one when you can do it without help.
All ticked. Brilliant work! You have finished Lesson 5.3.
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