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

Evaluating Expressions with Whole Numbers and Tenths

C2.2 A1About 35 minutes

Learning goals

I can substitute a whole number or a decimal tenth for a variable.
I can follow the order of operations to evaluate an expression.
I can evaluate expressions with brackets and with two variables.
I can use expressions to answer real-life questions.

Words to know

substitute
Replace a variable with a number.

evaluate
Find the value of an expression after substituting.

order of operations
The order for doing the steps: brackets first, then multiply and divide (left to right), then add and subtract (left to right).

decimal tenth
A decimal with one digit after the point, like 2.5 or 0.7.

Remember from Grade 5

In Grade 5 you evaluated expressions like 3 × n + 4 by substituting a whole number for n, and you used the order of operations: brackets first, then multiplying and dividing, then adding and subtracting.

In Grade 6, the numbers you substitute can also be decimal tenths, like 2.5 or 0.7, and you'll see a new way to write multiplication with brackets.

Let's start at the market

At the fall market, apples cost $3 per kilogram. The cost of a bag of apples that has a mass of w kilograms is:

3w (that is, 3 × w)

Aisha's bag has a mass of 2.5 kg. Substitute 2.5 for w: 3 × 2.5.

Think: 2.5 is 25 tenths. 3 × 25 tenths = 75 tenths = 7.5. The apples cost $7.50.

Wooden crates of red and green apples at an outdoor fall market, with a hanging scale and paper bags. — At a fall market, apples are weighed on a scale and the price depends on the mass. Substituting the mass for w in 3w tells you the cost. AI-generated image

How to evaluate an expression

Substitute: write the expression again with the number in place of the variable. Use brackets or × so it is clear: 3w becomes 3 × 2.5.
Follow the order of operations:

Brackets first
then × and ÷, from left to right
then + and −, from left to right

Check: is the answer reasonable? Estimate with whole numbers.

Evaluating 4n + 1.3 when n = 2.5: substitute, multiply before adding, and finish with 11.3 (about 9 to 13 is a sensible estimate).

Multiplying a tenth by a whole number	Think	Answer
4 × 2.5	4 × 25 tenths = 100 tenths	10
3 × 1.2	3 × 12 tenths = 36 tenths	3.6
5 × 0.8	5 × 8 tenths = 40 tenths	4
2 × 3.4	double 3 is 6, double 0.4 is 0.8	6.8

Two ways to multiply a decimal tenth by a whole number: count in tenths, or split the number.

A new way to write multiplication

3(x + 1.2) means 3 × (x + 1.2). When a number is right next to a bracket, it means multiply. Work out the bracket first, then multiply.

Worked example

Example 1: One variable

Evaluate 4n + 1.3 when n = 2.5.

Substitute: 4 × 2.5 + 1.3.
Multiply first: 4 × 2.5 = 10.
Then add: 10 + 1.3 = 11.3.
Estimate to check: n is about 2 or 3, so 4n is about 8 to 12, and the answer should be about 9 to 13 ✔.

11.3

Worked example

Example 2: Two variables

Evaluate 2a + b when a = 3.4 and b = 1.5.

Substitute: 2 × 3.4 + 1.5.
Multiply: 2 × 3.4 = 6.8.
Add: 6.8 + 1.5 = 8.3.

8.3

Worked example

Example 3: Brackets

Evaluate 3(x + 1.2) when x = 2.6.

Substitute: 3 × (2.6 + 1.2).
Brackets first: 2.6 + 1.2 = 3.8.
Multiply: 3 × 3.8 = 3 × 38 tenths = 114 tenths = 11.4.

11.4

Worked example

Example 4: Perimeter of a garden

The perimeter of a rectangle is 2l + 2w. A garden bed is l = 4.5 m long and w = 2.3 m wide. Find its perimeter.

Substitute: 2 × 4.5 + 2 × 2.3.
Multiply: 2 × 4.5 = 9 and 2 × 2.3 = 4.6.
Add: 9 + 4.6 = 13.6.

The perimeter is 13.6 m.

The garden bed from Example 4. Two lengths and two widths go all the way around: 13.6 m.

Worked example

Example 5: A table of values

Evaluate 5n − 0.5 for n = 1, n = 1.5 and n = 2.

n = 1: 5 × 1 − 0.5 = 5 − 0.5 = 4.5.
n = 1.5: 5 × 1.5 = 7.5, and 7.5 − 0.5 = 7.
n = 2: 5 × 2 − 0.5 = 10 − 0.5 = 9.5.

1 → 4.5, 1.5 → 7, 2 → 9.5. Each time n goes up by 0.5, the value goes up by 2.5.

Worked example

Example 6: Dividing

Evaluate k ÷ 2 + 0.4 when k = 7.

Substitute: 7 ÷ 2 + 0.4.
Divide first: 7 ÷ 2 = 3.5.
Add: 3.5 + 0.4 = 3.9.

3.9

Avoid these slips

In 2a, multiply! 2a with a = 3.4 is 6.8, not 23.4.
Do × and ÷ before + and −, unless there are brackets.
Line up the decimal points when you add or subtract: 10.0 − 1.8 = 8.2.
Stuck? Try the same question with a whole number first, then use the same steps with the decimal.

Just for fun

Joke: What did the variable say after it was substituted?

"I feel like a whole new number!"

Hands-on activity

Grocery expressions

You need: a few fruits or vegetables, a kitchen scale (if you have one), a grocery flyer or receipt, paper and a pencil.

Find a price per kilogram on a flyer, receipt or store sign (or pretend: apples $4 per kg).
Write an expression for the cost of w kilograms, like 4w.
Weigh some fruit (or pretend it has a mass of 1.5 kg, 2.5 kg and 0.5 kg). Round the mass to the nearest tenth of a kilogram.
Evaluate your expression for each mass and make a table.
Add a bag fee: write 4w + 0.5 and evaluate it again.

What to look for: with $4 per kg, 1.5 kg costs $6, 2.5 kg costs $10 and 0.5 kg costs $2. With the bag fee: $6.50, $10.50 and $2.50.

Your turn

Evaluate 5x when x = 1.2.

Evaluate y + 4.7 when y = 3.6.

Evaluate 2n + 3 when n = 4.5.

Evaluate 10m − 2.5 when m = 0.7.

Evaluate 3(a + 0.4) when a = 1.6.

Evaluate p + q + 1.1 when p = 2.3 and q = 0.6.

Hot chocolate costs $2.50 a cup, so c cups cost 2.5c dollars. What do 6 cups cost?

More practice: warm-up

Evaluate n + 0.5 when n = 3.

Evaluate 2x when x = 0.4.

Evaluate 10 − y when y = 0.3.

What is 6 × 0.5?

In 2 + 3 × 1.5, which operation do you do first?

More practice: on your own

Evaluate 6r − 1.8 when r = 2.5.

Evaluate 2(w + 3.5) when w = 1.5.

Evaluate 4a + 2b when a = 1.5 and b = 0.8.

Make a table for 3n + 0.2 for n = 0.5, 1, 1.5 and 2. What pattern do you see?

A cyclist rides at 12.5 km each hour. The distance after h hours is 12.5h. How far in 3 hours?

Challenge

Find a value of n with one decimal place that makes 2n + 1.4 equal to 5.

Evaluate 3(x + 2) and 3x + 2 when x = 1.5. Why are they different?

For 4n − 0.6, find the values when n = 0.5, 1.5 and 2.5. Then find which n gives 13.4.

Check your understanding

Quiz

Evaluate 3n when n = 2.4.

Evaluate x + 1.6 when x = 2.7.

Evaluate 2(y + 0.5) when y = 3.

Evaluate 5a − 1.5 when a = 2.

What do you do first in 4 + 2 × 1.5?

The perimeter of a rectangle is 2l + 2w. What is it when l = 3.5 m and w = 1.5 m?

For parents and teachers

Big idea: Grade 6 students evaluate algebraic expressions involving whole numbers and decimal tenths (C2.2). They substitute a value like 2.5 for the variable and follow the order of operations.

Common difficulties:

Treating 2a as digits side by side (a = 3.4 gives "23.4"). 2a always means 2 × a.
Ignoring the order of operations (doing 4 + 2 first in 4 + 2 × 1.5).
Decimal multiplication slips. Counting in tenths helps: 3 × 2.4 is 3 × 24 tenths = 72 tenths = 7.2.

How to help: Use prices per kilogram or per litre at the grocery store, or distances and speeds on a car or bike trip. Ask your child to write the expression first, then substitute. Encourage estimating before calculating, so they can catch an answer that's way off. Mathematical processes (A1): reasoning and checking are as important as the final number.