C. Algebra · Patterns, Equations, Inequalities, Coding and Modelling

Solving Equations with Multiplying and Dividing

C2.2 A1About 40 minutes

Learning goals

I can solve multiplication and division equations with numbers up to 50.
I can use multiplication and division facts to solve equations.
I can write an equation for a story problem and solve it.
I can verify my solution and explain my thinking.

Words to know

factor
A number you multiply. In 6 × 7 = 42, 6 and 7 are factors.

product
The answer when you multiply.

quotient
The answer when you divide.

fact family
Facts that use the same three numbers, like 6 × 7 = 42, 7 × 6 = 42, 42 ÷ 6 = 7 and 42 ÷ 7 = 6.

Remember

Multiplication and division are partners: they undo each other. If you know 6 × 7 = 42, you also know 42 ÷ 6 = 7 and 42 ÷ 7 = 6. That fact family is the key to solving today's equations.

The fact family for 6, 7 and 42. Cover any one number in the triangle: the other two tell you how to find it.

Let's start with a story

The class is going to the sugar bush to see how maple syrup is made. Each van holds 6 students. There are 42 students. How many vans are needed?

Let v be the number of vans. Then 6 × v = 42.

This array shows 4 × 6 = 24: 4 rows (vans) with 6 dots (students) in each row. For 42 students we would need more rows.

Ask: 6 times what is 42? You know 6 × 7 = 42. So v = 7. They need 7 vans.

Verify: 6 × 7 = 42 ✔

Two ways to solve multiplication equations

Think of a fact: 6 × ☐ = 42. I know 6 × 7 = 42.
Use the opposite operation: undo multiplying by 6 by dividing by 6: 42 ÷ 6 = 7.

If the equation is like ...	Undo it with ...	Example
4 × n = 28	divide by 4	n = 28 ÷ 4 = 7
n × 5 = 45	divide by 5	n = 45 ÷ 5 = 9
n ÷ 3 = 8	multiply by 3	n = 8 × 3 = 24
36 ÷ n = 4	think: 36 shared into how many groups gives 4?	n = 36 ÷ 4 = 9

Multiplying and dividing undo each other.

Worked example

Example 1: A multiplication equation

Solve 8 × n = 48.

Think: 8 times what is 48?
Skip count by 8: 8, 16, 24, 32, 40, 48. That is 6 eights.
Or divide: 48 ÷ 8 = 6.
Verify: 8 × 6 = 48 ✔

n = 6

Worked example

Example 2: A division equation

Solve n ÷ 4 = 9.

Some number, shared into 4 equal groups, gives 9 in each group.
Undo dividing by 4 by multiplying by 4: 9 × 4 = 36.
Verify: 36 ÷ 4 = 9 ✔

n = 36

Worked example

Example 3: The variable is the divisor

Solve 35 ÷ k = 5.

35 shared into k groups gives 5 in each group. How many groups?
Think of the fact family: 5 × 7 = 35, so 35 ÷ 7 = 5.
Verify: 35 ÷ 7 = 5 ✔

k = 7

Worked example

Example 4: A story problem

Ava has $40. Each bag of birdseed costs $5. How many bags can she buy? Write an equation, solve it and verify.

Let b be the number of bags. Each costs $5, so b bags cost 5 × b.
Equation: 5 × b = 40.
Think: 5 times what is 40? 5 × 8 = 40. Or 40 ÷ 5 = 8.
Verify: 5 × 8 = 40 ✔

b = 8. Ava can buy 8 bags of birdseed.

Worked example

Example 5: Which operation?

Some hockey pucks are shared equally among 3 teams. Each team gets 12 pucks. How many pucks were there?

Let p be the number of pucks. They were shared into 3 groups: p ÷ 3.
Equation: p ÷ 3 = 12.
Undo dividing by multiplying: 12 × 3 = 36.
Verify: 36 ÷ 3 = 12 ✔

p = 36 pucks.

Story clues

Equal groups, each, every, per → think multiplication or division.
If you know the number in each group and the number of groups, multiply to find the total.
If you know the total, divide to find the other number.
Always write what your variable means: b = number of bags.

A sweet-but-true fact

It takes about 40 litres of maple sap to make just 1 litre of maple syrup. So for 3 litres of syrup, a sugar bush needs about 40 × 3 = 120 litres of sap. That's a lot of sap for one pancake breakfast!

Hands-on activity

Equal groups detective

You need: 48 small items (dry pasta, Lego pieces or paper squares), 8 cups or bowls, paper and a pencil. (Grown-up note: keep small items away from babies and toddlers.)

Your grown-up says a story: I put the same number of pasta pieces in each of 4 cups. There are 28 pieces in all.
Write an equation: 4 × p = 28.
Solve it on paper first. Then share 28 pieces into 4 cups to verify.
Try a division story: I shared some pieces into 6 cups. Each cup has 5. Write n ÷ 6 = 5, solve, and build it.
Make up 2 stories of your own for your grown-up to solve.

What to look for: your child writes an equation that matches the story and checks the answer with the objects.

Your turn

Solve 5 × n = 30.

Solve n × 7 = 49.

Solve k ÷ 2 = 15.

Solve 24 ÷ m = 3.

Solve 9 × a = 36.

Each mitten pair has 2 mittens. A box has 18 mittens. Write and solve an equation for the number of pairs.

Is n = 5 the solution to n ÷ 5 = 5? Verify.

More practice: warm-up

Solve 2 × n = 10.

Solve 10 × n = 40.

Solve n ÷ 2 = 4.

Solve 3 × n = 12.

What operation undoes multiplying?

More practice: on your own

Solve 6 × t = 42.

Solve h ÷ 4 = 12.

Solve 45 ÷ w = 9.

A spider has 8 legs. Some spiders have 32 legs altogether. Write and solve an equation.

Some muffins are shared equally on 4 plates. Each plate has 6 muffins. Write and solve an equation for the number of muffins.

Challenge

Solve 3 × n + 5 = 32. (Hint: first find what 3 × n must be.)

Find every whole number pair that works for a × b = 24 (a and b both less than 25).

Student tickets for the school play cost $4 each. Adult tickets cost $8 each. A family buys 1 adult ticket and some student tickets. They pay $28 in all. How many student tickets did they buy? Write an equation and solve it.

Check your understanding

Quiz

What is the solution to 7 × n = 42?

What is the solution to n ÷ 5 = 8?

What is the solution to 36 ÷ k = 6?

Pencils come in boxes of 10. Ms. Roy has 50 pencils. Which equation tells how many boxes, b?

Which number makes 4 × n = 32 true?

Tom says the solution to m ÷ 3 = 9 is 3. What is correct?

For parents and teachers

Big idea: This lesson continues Ontario C2.2: solve equations that involve whole numbers up to 50 in various contexts, and verify solutions, now with multiplication and division. Knowing multiplication facts and fact families makes these equations much easier.

Common difficulties:

Using the wrong opposite (n ÷ 4 = 9 solved as 9 ÷ 4). Ask: "What was done to n? What undoes dividing?"
Not knowing facts quickly. Skip counting is a fine strategy while facts are being learned.
Mixing up groups and items in stories. Have your child say what the variable means before writing the equation.
Two-step challenge questions. These are an extension; help your child do one step at a time.

How to help: Ask questions at the store or in the kitchen: "Juice boxes come in packs of 6. We need 30. How many packs?" Have your child write the equation, solve it, and check with real items (Strand A: connecting math to real life).