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

Evaluating Algebraic Expressions

C2.2 A1About 35 minutes

Learning goals

I can substitute a whole number for a variable.
I can evaluate an expression in the right order: brackets, then multiply or divide, then add or subtract.
I can evaluate expressions with more than one variable.
I can use an expression to make a table of values.

Words to know

evaluate
Find the value of an expression.

substitute
Put a number in place of a variable.

value
The number an expression is worth after you substitute and calculate.

order of operations
The order to do the steps: brackets first, then multiplying and dividing, then adding and subtracting.

formula
An expression or equation that is always used for the same job, like perimeter = 2 × l + 2 × w.

Remember

In Grade 4 you found the value of an expression like 4 × b when b = 6: 4 × 6 = 24. In the last lesson you wrote expressions like 3 × n + 4 and 2 × (n + 3).

Today you will evaluate expressions with more than one step, and expressions with more than one variable.

Let's start with a story

The community centre rents skates. It costs $4 per hour, plus a $3 helmet fee. The cost for h hours is:

4 × h + 3

How much for 2 hours? Substitute 2 for h: 4 × 2 + 3 = 8 + 3 = $11.

How much for 5 hours? 4 × 5 + 3 = 20 + 3 = $23.

One expression can answer many questions. Just change the number you substitute!

Rows of rental skates on wooden shelves, with helmets on the top shelf, beside an indoor ice rink. — Skates and helmets waiting at a rental counter. The rink can use one expression, 4 × h + 3, to find the cost for any number of hours. AI-generated image

Think of 4 × h + 3 as a machine: put in the hours, multiply by 4, then add 3. Putting in 2 gives $11, and putting in 5 gives $23.

Do the steps in the right order

What is 4 × 2 + 3? If you add first, you get 4 × 5 = 20. If you multiply first, you get 8 + 3 = 11. Only one can be right! Mathematicians agree on this order:

Brackets first.
Then multiply and divide, from left to right.
Then add and subtract, from left to right.

So 4 × 2 + 3 = 8 + 3 = 11, which matches the skate story: 2 hours cost $8, then add the $3 helmet fee.

Two ways to work out 4 × 2 + 3. Only multiplying first follows the agreed order, and it gives 11, the real cost of 2 hours of skating.

Expression	n = 3	n = 10
n + 15	3 + 15 = 18	10 + 15 = 25
6 × n	6 × 3 = 18	6 × 10 = 60
5 × n − 4	15 − 4 = 11	50 − 4 = 46
n × n	3 × 3 = 9	10 × 10 = 100
2 × (n + 4)	2 × 7 = 14	2 × 14 = 28

Evaluating expressions for different values of n.

Worked example

Example 1: Two steps

Evaluate 7 × n − 5 when n = 8.

Substitute: 7 × 8 − 5.
Multiply first: 7 × 8 = 56.
Then subtract: 56 − 5 = 51.

Worked example

Example 2: Brackets first

Evaluate 3 × (k + 9) when k = 6.

Substitute: 3 × (6 + 9).
Brackets first: 6 + 9 = 15.
Then multiply: 3 × 15 = 45.

Worked example

Example 3: Two variables

The perimeter of a rectangle is 2 × l + 2 × w. Find the perimeter when l = 12 cm and w = 5 cm.

Substitute both: 2 × 12 + 2 × 5.
Multiply first: 2 × 12 = 24 and 2 × 5 = 10.
Then add: 24 + 10 = 34.

The perimeter is 34 cm. Check by adding all four sides: 12 + 5 + 12 + 5 = 34 ✔

Worked example

Example 4: Make a table of values from an expression

Make a table for 4 × n + 1 for n = 1, 2, 3, 4, 5. What do you notice?

n = 1: 4 + 1 = 5. n = 2: 8 + 1 = 9. n = 3: 12 + 1 = 13.
n = 4: 16 + 1 = 17. n = 5: 20 + 1 = 21.

The values are 5, 9, 13, 17, 21. They grow by 4 each time. It's the same pattern as in the Pattern Rules lesson: term number × 4 + 1! A term-number rule is an expression.

Check your work

Write the expression with the number in place of the letter before you calculate.
Underline what you will do first.
Ask: does my answer make sense? If the skates cost $4 an hour, 5 hours should cost more than $20.
Mistakes are part of learning. If your answer doesn't make sense, look for the step where the order went wrong.

Just for fun

What does the skate rental cost for 0 hours? 4 × 0 + 3 = $3. That's someone who just wanted to rent a helmet and wear it to the snack bar!

Hands-on activity

Roll and evaluate

You need: one die (or slips of paper numbered 1 to 6 in a cup), paper and a pencil.

Copy these expressions onto a paper: n + 17, 9 × n, 6 × n − 4, 2 × (n + 5), 30 − 3 × n, n × n + 1.
Roll the die. That number is n for this round.
Evaluate every expression. Your grown-up checks one of them.
Circle the expression that gave the biggest value this round.
Play 4 rounds. Does the same expression always win? Talk about why.

What to look for: when n = 4, the values are 21, 36, 20, 18, 18 and 17. The winner changes! For n = 1 or 2, 30 − 3 × n is biggest (27 and 24). For n = 3 to 6, 9 × n is biggest, because it grows by 9 every time n goes up by 1. Watch for children who add before multiplying in 6 × n − 4 or 30 − 3 × n.

Your turn

Evaluate n + 28 when n = 45.

Evaluate 9 × a when a = 7.

Evaluate 6 × n + 5 when n = 4.

Evaluate 50 − 4 × t when t = 8.

Evaluate 5 × (m + 3) when m = 7.

Evaluate x + y when x = 36 and y = 27.

Skate rental costs 4 × h + 3 dollars. What is the cost for 3 hours?

More practice: warm-up

Evaluate n + 9 when n = 11.

Evaluate 3 × n when n = 6.

Evaluate k − 4 when k = 30.

Evaluate p ÷ 2 when p = 18.

What do you do first in 2 × 5 + 1?

More practice: on your own

Evaluate 8 × n − 7 when n = 9.

Evaluate 48 ÷ d + 6 when d = 8.

Evaluate 2 × l + 2 × w when l = 15 and w = 9.

Make a table for 3 × n + 2 for n = 1 to 5.

Owen says 2 × (6 + 4) = 16. Is he right?

Challenge

A bake sale sells muffins for $2 and cookies for $1. The money made is 2 × m + c. How much money for 23 muffins and 38 cookies?

Evaluate n × n − n when n = 9.

Which value of n from 1, 2, 3, 4, 5, 6 makes 10 × n − 15 equal to 45?

Check your understanding

Quiz

What is 5 × n + 2 when n = 6?

What is 4 × (n + 1) when n = 5?

What is a + b when a = 19 and b = 42?

What is 30 − 2 × n when n = 7?

In 8 + 3 × 4, what do you do first?

Skates cost 4 × h + 3 dollars for h hours. What do 6 hours cost?

For parents and teachers

Big idea: Grade 5 students evaluate algebraic expressions that involve whole numbers (C2.2): substitute a value for each variable and calculate. To get one correct answer, everyone follows the same order: brackets, then multiplication and division, then addition and subtraction.

Common difficulties:

Working left to right no matter what. In 8 + 3 × 4 many children add first. Connect it to a story (8 loose cookies plus 3 bags of 4).
Writing 4 × 5 as 45 after substituting. Encourage writing the × sign every time when substituting.
Forgetting one variable when there are two (2 × l + 2 × w).

How to help: Use real formulas: the cost of tickets plus a fee, the perimeter of a table, points in a game. Ask your child to write the expression, substitute, then check that the answer makes sense. Social-emotional learning (A1): model calm checking ("Let's see which step went wrong") when an answer doesn't make sense.