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

Evaluating Expressions with Whole Numbers and Decimals

C2.2 A1About 45 minutes

Learning goals

I can substitute whole numbers and decimals for variables in an expression.
I can follow the order of operations to evaluate an expression.
I can evaluate expressions with more than one variable.
I can use formulas from real life by evaluating them.

Words to know

evaluate
Find the value of an expression by replacing each variable with a number and calculating.

substitute
Replace a variable with a given number.

formula
An equation that describes a rule used in real life, like P = 2l + 2w for the perimeter of a rectangle.

order of operations
Brackets first, then × and ÷ from left to right, then + and − from left to right.

expression
Numbers, variables and operations with no equal sign, like 4.5a + 2b.

Remember from Grade 6

In Grade 6 you evaluated expressions with whole numbers and decimal tenths, like 3n + 2 when n = 1.5. You used the order of operations and wrote the multiplication sign back in: 3n means 3 × n.

In Grade 7 the decimals can have hundredths (and more), expressions can have two variables, and you'll use real formulas.

Let's start at the sugar shack

A maple farm sells syrup for $12.75 a bottle and charges $4.50 for shipping an order. The cost of an order of b bottles is:

12.75b + 4.50

How much do 4 bottles cost? Substitute b = 4: 12.75 × 4 + 4.50 = 51.00 + 4.50 = $55.50.

That's evaluating an expression. (These prices are made up for practice.)

Bottles of amber maple syrup lined up on a wooden shelf in a sugar shack, with a steaming evaporator pan behind them. — Maple farms boil sap in evaporators like this to make syrup. The expression 12.75b + 4.50 gives the cost of any number of bottles, b. AI-generated image

How to evaluate

Write the expression.
Substitute: replace each variable with its value. Put the multiplication sign back in: 4.5a becomes 4.5 × a. Use brackets around the number if it helps you see it.
Calculate using the order of operations: brackets first, then × and ÷ from left to right, then + and − from left to right.
Check that the answer makes sense. Estimating helps.

An expression is like a machine. Put in n = 4. Multiply by 2.5 to get 10, then add 1 to get 11.

n	Substitute into 2.5n + 1	Value
0	2.5 × 0 + 1	1
1.2	2.5 × 1.2 + 1	4
4	2.5 × 4 + 1	11
6.4	2.5 × 6.4 + 1	17

One expression, different values for n. Each line multiplies first, then adds.

Estimate first

Before evaluating 12.75 × 4 + 4.50, estimate: about 13 × 4 + 5 = 57. The exact answer, $55.50, is close, so it's reasonable. A misplaced decimal point (like $514.50, from 127.5 × 4 + 4.50) would stand out right away.

Worked example

Example 1: Two variables

Evaluate 4.5a + 2b when a = 3 and b = 1.25.

Substitute: 4.5 × 3 + 2 × 1.25.
Multiply: 4.5 × 3 = 13.5 and 2 × 1.25 = 2.5.
Add: 13.5 + 2.5 = 16.

Worked example

Example 2: A formula

The perimeter of a rectangle is P = 2l + 2w. Find P when l = 4.25 m and w = 2.5 m.

Substitute: P = 2 × 4.25 + 2 × 2.5.
Multiply: 8.5 + 5.
Add: 13.5.

P = 13.5 m

Substitute each length into its spot in P = 2l + 2w. Multiply first, then add: two lengths of 4.25 m and two widths of 2.5 m make 13.5 m all the way around.

Worked example

Example 3: Brackets

Evaluate 3(n + 2.5) when n = 1.5.

3(n + 2.5) means 3 × (n + 2.5).
Substitute: 3 × (1.5 + 2.5).
Brackets first: 1.5 + 2.5 = 4. Then 3 × 4 = 12.

Worked example

Example 4: Division and subtraction

Evaluate m ÷ 4 − 0.75 when m = 10, and 7.2 − 2k when k = 1.35.

m ÷ 4 − 0.75: 10 ÷ 4 = 2.5. Then 2.5 − 0.75 = 1.75.
7.2 − 2k: multiply first, 2 × 1.35 = 2.7. Then 7.2 − 2.7 = 4.5.

1.75 and 4.5

Worked example

Example 5: Compare two expressions

Which is greater when x = 0.8: 5x + 2 or 3x + 4?

5x + 2: 5 × 0.8 + 2 = 4 + 2 = 6.
3x + 4: 3 × 0.8 + 4 = 2.4 + 4 = 6.4.
6.4 > 6.

3x + 4 is greater when x = 0.8. (When x is bigger, like 2, 5x + 2 wins: 12 compared with 10.)

Just for fun

Sugar shack fact: it takes about 40 litres of maple sap to make just 1 litre of maple syrup. That's a lot of boiling for one pancake breakfast!

Hands-on activity

Grocery flyer expressions

You need: a grocery flyer (paper or online, with a grown-up), paper, a pencil and a calculator for checking.

Choose two items, like apples per kilogram and a bag of milk. Write an expression for the cost of buying some of each, for example 3.49a + 6.29m, where a is kilograms of apples and m is bags of milk. (Use the prices in your flyer.)
Evaluate it for a = 2.5 and m = 2. Estimate first!
Evaluate it for two other amounts your family might really buy.
Check one answer with a calculator.

Math talk: "Which part did you calculate first? Why?"

Your turn

Evaluate 5x + 3 when x = 2.4.

Evaluate 3a + 2b when a = 1.5 and b = 4.

Evaluate 7.2 − 2m when m = 1.35.

Evaluate 4(k + 0.6) when k = 2.

Evaluate p ÷ 5 + 1.2 when p = 8.

Use P = 2l + 2w to find the perimeter when l = 6.5 cm and w = 3.25 cm.

A skating party costs 2.25n + 3.5 dollars per family, where n is the number of people. Find the cost for 6 people.

More practice: warm-up

Evaluate x + 4 when x = 2.5.

Evaluate 3y when y = 1.2.

Evaluate 10 − n when n = 3.7.

Evaluate 2a + 1 when a = 0.5.

Evaluate 4b when b = 0.25.

More practice: on your own

The area of a triangle is A = b × h ÷ 2. Find A when b = 7.4 cm and h = 5 cm.

Evaluate 6.5x − 2y when x = 4 and y = 3.75.

A hiker walks at 4.5 km an hour. The distance is d = 4.5t. How far does she walk in 2.5 hours?

Evaluate (a + b) ÷ 2 when a = 3.75 and b = 6.25. What does this expression find?

Evaluate 0.25n + 1.75 for n = 0, 1, 2 and 3. What kind of pattern do the values make?

Challenge

Find a value of x that makes 3x + 1.5 and 2x + 4 equal. (Try some values.)

Lena evaluates 2 + 3n when n = 1.5 and gets 7.5. What did she do wrong? What is the right answer?

A made-up taxi fare is 4.25 + 1.75k dollars for k kilometres. Find the fare for 8.4 km. Is $20 enough for a 9 km trip?

Check your understanding

Quiz

Evaluate 4n + 1.5 when n = 2.5.

Evaluate 2a + 3b when a = 1.25 and b = 2.

Evaluate 5(x − 0.4) when x = 1.

What is the first step to evaluate 9 − 2y when y = 3.5?

Use P = 2l + 2w with l = 5.5 m and w = 3 m.

Syrup costs 12.75b + 4.50 dollars for b bottles. What do 2 bottles cost?

For parents and teachers

Big idea: Grade 7 students evaluate algebraic expressions that involve whole numbers and decimal numbers (C2.2). They substitute values for variables and use the order of operations. Grade 6 used decimal tenths; Grade 7 uses any decimals, more than one variable and real formulas.

Common difficulties:

Reading 3n as 3 followed by n (n = 1.5 makes "31.5"). Remind your child that 3n means 3 × n.
Adding before multiplying (2 + 3n). Write the multiplication sign back in and underline it.
Decimal placement. Estimating first catches most errors.

How to help: Use flyers, recipes and receipts to make expressions with real prices. Ask for an estimate before every calculation. This lesson builds A1 skills: making connections to real life and checking for reasonableness.