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C. Algebra · Algebra: Patterns, Rational Numbers, Equations, Inequalities, Coding and Modelling

Evaluating Algebraic Expressions with Rational Numbers

C2.2 A1About 45 minutes
Learning goals
  • I can substitute integers, fractions and decimals, positive or negative, into an algebraic expression.
  • I can use brackets and the order of operations to evaluate expressions correctly.
  • I can evaluate formulas, such as temperature conversions, in real situations.

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 number.
variable
A letter or symbol that stands for a number that can change or is unknown.
formula
An equation that shows how to calculate one quantity from others, like F = 1.8C + 32.
order of operations
The order for doing calculations: Brackets, Exponents, Division and Multiplication (left to right), Addition and Subtraction (left to right).

Remember from Grade 7

In Grade 7 you evaluated expressions with whole numbers and decimals, including expressions with more than one variable, like 2a + 3b when a = 1.5 and b = 4. You also used exponents and BEDMAS.

In Grade 8 the values can be any rational numbers: negative numbers and fractions too. That means you'll multiply and divide with negative numbers. Quick reminder (from the Operations unit): when you multiply or divide two numbers, same signs give a positive answer and different signs give a negative answer.

Let's start with a cold fact

Canada measures temperature in degrees Celsius, but some apps and recipes use degrees Fahrenheit. This formula changes Celsius to Fahrenheit:

F = 1.8C + 32 (1.8 is the same as 9/5)

What is −15°C in Fahrenheit? Substitute C = −15:

F = 1.8(−15) + 32 = −27 + 32 = 5°F.

And here's a famous fact you can check: at −40, both scales show the same number. F = 1.8(−40) + 32 = −72 + 32 = −40 ✓.

−3× 2−6+ 5−12x + 5 when x = −3: 2(−3) + 5 = −6 + 5 = −1
An expression is like a machine. For 2x + 5, the input x = −3 is multiplied by 2 (giving −6), then 5 is added, giving −1.

How to evaluate

  1. Write the expression.
  2. Substitute each value in brackets, especially negatives and fractions: 2(−3) + 5.
  3. Follow the order of operations (BEDMAS).
  4. Ask: Does the answer make sense?
ExpressionValue of the variableSubstituteValue
4x − 7x = −24(−2) − 7−15
x² + 1x = −3(−3)² + 110
(1/2)n − 3n = −5(1/2)(−5) − 3−5.5
5 − 2pp = 3/45 − 2(3/4)3.5
Brackets keep the sign of the number together with the number.

Why brackets matter

If x = −3, then x² means (−3)² = (−3) × (−3) = 9. Writing −3² without brackets means −(3²) = −9. Always put the value you substitute in brackets.

With bracketssquare the whole −3(−3)²= (−3) × (−3)= 9Without bracketssquare 3, then take the opposite−3²= −(3 × 3)= −9
Brackets change the answer. (−3)² squares −3 and gives 9. −3² squares 3 first and then takes the opposite, giving −9.
Worked example

Example 1: Two variables with negatives and decimals

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

  1. Substitute: 3(−4) − 2(1.5).
  2. Multiply: −12 − 3.
  3. Subtract: −15.

−15

Worked example

Example 2: A temperature formula

Use F = 1.8C + 32 to change 22.5°C (a warm room) to Fahrenheit.

  1. Substitute: F = 1.8(22.5) + 32.
  2. 1.8 × 22.5 = 40.5.
  3. 40.5 + 32 = 72.5.

72.5°F

Worked example

Example 3: Fractions

Evaluate (2/3)x + 1/4 when x = −3/4.

  1. Substitute: (2/3)(−3/4) + 1/4.
  2. Multiply: (2 × −3)/(3 × 4) = −6/12 = −1/2.
  3. Add: −1/2 + 1/4 = −2/4 + 1/4 = −1/4.

−1/4

Worked example

Example 4: An exponent and a negative

Evaluate x² − 3x when x = −2.

  1. Substitute with brackets: (−2)² − 3(−2).
  2. Exponent first: (−2)² = 4.
  3. Multiply: 3(−2) = −6, so we have 4 − (−6).
  4. Subtracting a negative: 4 + 6 = 10.

10

Worked example

Example 5: The mean of three temperatures

The expression (a + b + c) ÷ 3 gives the mean of three numbers. Find the mean of the daily highs −8°C, −3.5°C and 2.5°C.

  1. Substitute: (−8 + (−3.5) + 2.5) ÷ 3.
  2. Brackets first: −8 − 3.5 + 2.5 = −9.
  3. Divide: −9 ÷ 3 = −3.

−3°C

Hands-on activity

Convert temperatures at home

You need: a weather app, an outdoor thermometer, or a weather report; a recipe or oven dial with °F; pencil and paper.

  1. Write down today's high and low in °C. Use F = 1.8C + 32 to change them to °F. If your app can switch to °F, check your answers.
  2. Find an oven temperature in °F (for example 350°F). Change it to °C with C = (F − 32) ÷ 1.8.
  3. Make a table of 5 temperatures from −30°C to 30°C, in steps of 15, and their °F values. Is the pattern linear?

What to look for: 350°F → (350 − 32) ÷ 1.8 = 318 ÷ 1.8 ≈ 177°C. The table: −30 → −22, −15 → 5, 0 → 32, 15 → 59, 30 → 86. Each step of 15°C adds 27°F, so it is linear.

Just for fun: a seriously cold fact

Canada's coldest recorded temperature was −63°C, at Snag, Yukon, in February 1947. In Fahrenheit: F = 1.8(−63) + 32 = −81.4°F. Either way, it's a two-toque day.

Your turn

1

Evaluate 4x − 7 when x = −2.

2

Evaluate −3y + 10 when y = −4.

3

Evaluate 2a + b when a = 1.5 and b = −6.

4

Evaluate (1/2)n − 3 when n = −5.

5

Evaluate x² + 1 when x = −3.

6

Use F = 1.8C + 32 to change −10°C to °F.

7

Evaluate 5 − 2p when p = 3/4.

More practice: warm-up

1

Evaluate 3x when x = −2.

2

Evaluate x + 8 when x = −8.

3

Evaluate 10 − m when m = 2.5.

4

Evaluate 2k when k = 1/4.

5

Evaluate −x when x = −7.

More practice: on your own

1

Evaluate 3m − 2n when m = −1 and n = −2.5.

2

Evaluate (x − 4) ÷ 2 when x = −6.

3

Evaluate (2/5)t + 1/2 when t = 5/4.

4

Evaluate a² − b² when a = −3 and b = 2.

5

Use C = (F − 32) ÷ 1.8 to change 5°F to °C.

Challenge

1

Evaluate −x² and (−x)² when x = 4. Explain why the answers are different.

2

A rectangle has length l = 3 1/4 m and width w = 1 5/8 m. Evaluate P = 2(l + w).

3

Evaluate (a − b) ÷ (a + b) when a = −2 and b = 0.5. Give your answer as a fraction.

Check your understanding

Quiz

1.

Evaluate 2x + 5 when x = −3.

2.

Evaluate x² when x = −5.

3.

Evaluate (1/2)y − 1 when y = −4.

4.

Evaluate a − b when a = −2 and b = −6.

5.

Use F = 1.8C + 32 to change 25°C to °F.

6.

Why do we write (−3)² when substituting x = −3 into x²?

For parents and teachers

Big idea: C2.2 asks Grade 8 students to evaluate algebraic expressions that involve rational numbers: negative integers, fractions and decimals. This uses integer multiplication and fraction skills from the Number strand.

Common difficulties:

  • Dropping brackets when substituting negatives, which leads to sign errors (especially with exponents).
  • Subtracting a negative (3 − (−5)). Remind your child it means adding the opposite.
  • Order of operations. Multiply before adding, and do brackets first.
  • Fraction multiplication. Multiply numerators and denominators, then simplify.

How to help: Convert real temperatures together using the formula and check with a weather app. Ask "Does that answer make sense?" (−15°C should be a cold number in °F too.) This lesson builds A1 skills: checking work and persevering through multi-step calculations.