Grade by Grade
C. Algebra · Patterns, Equations, Coding and Modelling

Variables: Letters That Stand for Numbers

C2.1 A1About 35 minutes
Learning goals
  • I can explain what a variable is.
  • I can use a variable for an unknown number and find its value.
  • I can use a variable for an amount that changes, like the number of bikes.
  • I can write a number sentence with a variable to match a story.

Words to know

variable
A letter or symbol that stands for a number. It can be a number we don't know yet, or an amount that can change.
unknown
A number we do not know yet and need to find.
equation
A number sentence with an equal sign, like n + 4 = 11.
value
The number that a variable stands for. In n + 4 = 11, the value of n is 7.
expression
Numbers, variables and operation signs without an equal sign, like 2 × b or 15 − 6.

Remember from Grade 2

In Grade 2 you used symbols like ☐, ? and the letter n to stand for a mystery number, like 8 + ☐ = 12.

These symbols have a special name in math: variables. In Grade 3 you will learn the two big jobs a variable can do.

Job 1: A variable can stand for an unknown number

A grown-up puts some counters in a bag. You can't see how many. Then 4 more counters are put next to the bag. Altogether there are 11 counters. How many are in the bag?

We can call the number in the bag n.

n+=
A bag with n counters, plus 4 counters, makes 11 counters. n + 4 = 11.

We write: n + 4 = 11. Think: what number plus 4 makes 11? 7 + 4 = 11, so n = 7. We found the value of the variable.

In this kind of problem, the variable stands for one number that we don't know yet.

Job 2: A variable can stand for an amount that changes

A bike has 2 wheels. How many wheels are on b bikes?

The number of bikes can change. It could be 1 bike, 5 bikes or 20 bikes. So we use a variable, b, for the number of bikes.

The number of wheels is 2 × b (2 times b). When b changes, the number of wheels changes too. That is why it is called a variable: it can vary (change).

Number of bikes (b)Number of wheels (2 × b)
12 × 1 = 2
22 × 2 = 4
32 × 3 = 6
42 × 4 = 8
52 × 5 = 10
When b changes, 2 × b changes too.
b = 12 × 1 = 2 wheelsb = 22 × 2 = 4 wheelsb = 32 × 3 = 6 wheels
The number of bikes, b, can change. Each bike adds 2 wheels, so there are always 2 × b wheels.

Choosing a letter

  • You can use any letter for a variable. It helps to pick one that reminds you of the story: c for cookies, b for bikes, m for marbles.
  • In one problem, the same letter always stands for the same number. In n + n = 10, both n's are 5.
  • Write 2 × b with a times sign so you don't mix it up with a number.
Worked example

Example 1: Find the unknown

Find the value of m in m + 9 = 21.

  1. Ask: what number plus 9 makes 21?
  2. Count up from 9 to 21: 9 + 1 = 10, then 10 + 11 = 21. That is 1 + 11 = 12.
  3. Check: 12 + 9 = 21. ✔

m = 12

Worked example

Example 2: A story with a variable

Ana had some hockey cards. Her cousin gave her 15 more. Now she has 40. How many did she have at the start?

  1. We don't know how many she had at the start. Call that number c (for cards).
  2. Write the equation: c + 15 = 40.
  3. What plus 15 makes 40? 25 + 15 = 40.
  4. Check: 25 + 15 = 40. ✔

c = 25. Ana had 25 hockey cards at the start.

Worked example

Example 3: Multiplying with a variable

Find p in 4 × p = 20.

  1. 4 × p means 4 groups of p.
  2. 4 groups of what number make 20? Skip count by 5s: 5, 10, 15, 20. That is 4 groups of 5.
  3. Check: 4 × 5 = 20. ✔

p = 5

Worked example

Example 4: The same letter twice

Find n in n + n = 16.

  1. Both n's stand for the same number.
  2. What number added to itself makes 16? That is a double: 8 + 8 = 16.

n = 8

Just for fun

Why was the letter n so relaxed in math class?

It knew it could stand for any number it wanted!

Worked example

Example 5: A changing amount

Tickets to the school play cost $3 each. Write an expression for the cost of t tickets. What is the cost when t = 6?

  1. Each ticket costs $3, so t tickets cost 3 × t dollars.
  2. When t = 6: 3 × 6 = 18.

The cost is 3 × t dollars. For 6 tickets, it is $18.

Your turn

1

What is a variable?

2

Find n: n + 6 = 14

3

Find k: 20 − k = 12

4

Find a: 3 × a = 12

5

Find y: y + y = 18

6

A cat has 4 legs. Write an expression for the number of legs on c cats.

7

Use your expression: how many legs do 3 cats have?

Hands-on activity

Mystery cup

You need: a plastic cup (that you can't see through) and about 20 small things (dry beans, buttons or coins).

  1. Close your eyes. A grown-up hides some beans under the cup and puts some beans beside it.
  2. The grown-up tells you the total. For example: "There are 13 beans in all."
  3. Count the beans beside the cup. Write an equation with a variable, like b + 5 = 13.
  4. Find the value of b. Then lift the cup to check!
  5. Switch jobs and play 5 rounds.

Math talk: "The variable b stands for ___."

Grown-up note: keep small items away from younger children who might put them in their mouths.

More practice: warm-up

1

Find n: n + 3 = 10

2

Find x: x − 4 = 6

3

Find w: 2 × w = 10

4

In b + 5 = 9, what is the variable?

5

If t = 4, what is t + 10?

More practice: on your own

1

Write an equation with a variable, then solve: Jo had some stickers. She gave away 7. She has 18 left.

2

Find m: 45 + m = 100

3

Each box holds 5 muffins. Write an expression for the muffins in b boxes. How many muffins are in 7 boxes?

4

Find r: r ÷ 2 = 6

5

Max says that in n + n = 12, the first n could be 5 and the second could be 7. Is he right? Explain.

Challenge

1

Find n: n + n + n = 15

2

A spider has 8 legs. Make a table for the number of legs on s spiders, for s = 1, 2, 3 and 4. What expression tells the legs?

3

In a + b = 10, can you find just one answer for a and b? Explain.

Check your understanding

Quiz

1.

What is a variable?

2.

Find n: n + 7 = 15

3.

Find p: 5 × p = 30

4.

A tricycle has 3 wheels. Which expression shows the wheels on t tricycles?

5.

Find y: y + y = 14

6.

Lia had some marbles. She found 12 more. Now she has 30. Which equation matches?

For parents and teachers

Big idea: A variable is a letter or symbol that stands for a number (C2.1). In Grade 3, children see two uses: an unknown (n + 4 = 11, so n = 7) and an amount that changes (2 × b wheels for b bikes). This is an important early step toward algebra.

Common difficulties:

  • Thinking a letter has a fixed value ("a is 1 because it is the first letter"). Explain that we choose the letter; it can stand for any number.
  • Using different values for the same letter in one equation. In n + n = 12, both are 6.
  • Missing-start problems like s − 7 = 18. Children often subtract. Ask: "Is the variable the whole or a part?"

How to help: Play "mystery cup" with snacks. When shopping, ask: "Apples cost 50¢ each. How much for a apples? What if a = 4?"