Writing Algebraic Expressions
Learn to use letters to represent unknown numbers and write expressions
1. Letters for Unknown Numbers
The Big Idea
In algebra, we use a letter (like n, x, or y) to stand for an unknown number. We use a letter when we don't know the exact value of something.
Example: Mystery Cupcakes
There are some cupcakes in a box, but we don't know how many. We can use the letter n to represent this unknown number.
Key Point: We can use ANY letter to represent an unknown number. Common choices are n, x, y, p, q, and m.
Why Use Letters?
If there are n cupcakes in the box and 3 cupcakes on a plate, how many cupcakes are there altogether?
| Cupcakes in box | Total cupcakes |
|---|---|
| 5 | 5 + 3 = 8 |
| 6 | 6 + 3 = 9 |
| 10 | 10 + 3 = 13 |
| n | n + 3 |
(n + 3) is an algebraic expression!
2. Addition Expressions
When we add, we write the sum with a + sign, for example n + 3 or 7 + y. The order does not matter in addition.
Example 1: Raju's Cookies
Raju had 7 cookies at first. His mother gave him y more cookies. How many cookies does Raju have now?
Answer: Raju has (7 + y) cookies.
Example 2: Alan's Oranges
After eating k oranges, Alan has 3 oranges left. How many oranges did he have at first? Express your answer in terms of k.
Answer: He had (k + 3) oranges at first.
Reasoning: eaten + left = total, so k + 3 = total
Practice: Addition Expression
Hassan is 12 years old now. In p years' time, how old will Hassan be? Express your answer in terms of p.
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Solution: Hassan will be (12 + p) years old.
Current age + years to pass = Future age
3. Subtraction Expressions
When we subtract, the ORDER matters!
- p - 2 means "start with p, take away 2"
- 12 - j means "start with 12, take away j"
Example 1: Buns in a Bag
There are p buns in a bag. Leila takes out 2 buns. How many buns are left in the bag?
Answer: There are (p - 2) buns left in the bag.
Example 2: Mary's Apples
Mary had 12 apples at first. She gave away j apples. How many apples did she have left? Express your answer in terms of j.
Answer: She had (12 - j) apples left.
Notice: This is NOT (j - 12)! Order matters in subtraction.
Common Subtraction Phrases
Watch out! "15 less than d" means d - 15, NOT 15 - d
Practice: Subtraction Expression
Hassan is 12 years old now. r years ago, how old was Hassan? Express your answer in terms of r.
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Solution: Hassan was (12 - r) years old.
4. Multiplication Expressions
Special Rule for Multiplication
In algebra, we write m × 8 as 8m (number before letter, no × sign!)
- 5 × q = 5q
- n × 3 = 3n
- 2 × y × 4 = 8y
Example 1: Shirts and Buttons
Each shirt has 8 buttons. If Jiahao has m shirts, how many buttons are there altogether?
| Number of shirts | Number of buttons |
|---|---|
| 1 | 1 × 8 = 8 |
| 2 | 2 × 8 = 16 |
| 5 | 5 × 8 = 40 |
| m | m × 8 = 8m |
Answer: There are 8m buttons altogether.
Example 2: Leila and Siti's Stickers
Leila has q stickers. Siti has 5 times as many stickers as Leila. How many stickers does Siti have?
q + q + q + q + q = 5 × q = 5q
Answer: Siti has 5q stickers.
3 groups of x
Practice: Multiplication Expression
A notebook costs $w. Amy buys 6 notebooks. How much does she pay? Express your answer in terms of w.
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Solution: Amy pays $6w or $(6w).
5. Division Expressions
Writing Division
We write division as a fraction:
- y ÷ 6 =
- p ÷ 3 =
Example 1: Sharing Money
$p is shared equally among 3 children. How much does each child get?
Answer: Each child gets $.
Let's check with p = 120:
$120 ÷ 3 = $40 for each child
Example 2: Simon and Lucas
Simon has 5 times as much money as Lucas. They have a total of $q. Express the amount of money Lucas has in terms of q.
Step 1: Count total units: 5 + 1 = 6 units
Step 2: 6 units = $q
Step 3: 1 unit = $q ÷ 6 = $
Answer: Lucas has $.
Divide y by 6
Practice: Addition and Multiplication Together
Mr Low rented a bus for p students. He was charged $30 for the bus rental and $6 for each student. Express the total amount Mr Low paid in terms of p.
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Answer: Mr Low paid $(30 + 6p).
Summary: Ways to Write Expressions
| Statement | Expression |
|---|---|
| Add 10 to a | a + 10 |
| 12 more than b | b + 12 |
| Subtract c from 20 | 20 - c |
| 15 less than d | d - 15 |
| 3 groups of x | 3x |
| Divide y by 6 |
Key Takeaways
- ✓ A letter can stand for an unknown number
- ✓ Addition: n + 3 means "n plus 3"
- ✓ Subtraction: Be careful with order! "5 less than m" = m - 5
- ✓ Multiplication: Write 4 × n as 4n (number first, no × sign)
- ✓ Division: Write n ÷ 5 as (fraction form)
- ✓ Always read word problems carefully to identify the operation!