Grade by Grade
B. Number · Operations: Rational Numbers, Integers, Fractions and Proportional Reasoning

Operations: Rational Numbers, Integers, Fractions and Proportional Reasoning: Unit Review

A1 B2.1 B2.2 B2.3 B2.4 B2.5 B2.6 B2.7 B2.8About 40 minutes
Learning goals
  • I can use properties, relationships between operations and the order of operations with rational numbers.
  • I can recall square numbers and square roots, and multiply and divide by powers of ten in my head.
  • I can add, subtract, multiply and divide integers, and add, subtract, multiply and divide fractions and mixed numbers.
  • I can compare proportional situations and solve proportional problems, including percents.

Words to know

rational number
Any number that can be written as a fraction of two integers.
distributive property
a × (b + c) = a × b + a × c.
reciprocal
The number you multiply by to get 1: the reciprocal of 2/3 is 3/2.
square root
A number that, multiplied by itself, gives the original number: √81 = 9.
power of ten
10, 100, 1 000, ... written as 10¹, 10², 10³, ...
unit rate
A rate for 1 unit, like $1.55 per litre.
proportion
Two equal ratios, like 3/5 = 24/40.

Let's review!

In this unit you used properties and the order of operations with rational numbers, recalled squares and square roots, multiplied and divided by powers of ten in your head, added, subtracted, multiplied and divided integers, worked with fractions and mixed numbers in all four operations, and used proportional reasoning to compare situations and to solve rate, ratio and percent problems.

This review has a few questions from every lesson. Try each one, then press Show answer to check. If a part feels tricky, go back to that lesson for more practice.

second numberpositivesecond numbernegativefirst numberpositive+(+6) ÷ (+2) = +3−(+6) × (−2) = −12first numbernegative−(−6) ÷ (+2) = −3+(−6) × (−2) = +12Same signs → positive. Different signs → negative.
One chart for both multiplying and dividing integers: same signs give a positive answer and different signs give a negative answer.
Mixed numbers? Write them asimproper fractions first.Add or subtractFind a commondenominator, then add orsubtract the numerators.1/2 + 1/3 = 5/6MultiplySimplify common factors,then multiply tops andmultiply bottoms.2/3 × 3/4 = 1/2DivideMultiply by thereciprocal of thedivisor (flip it).3 ÷ 3/4 = 3 × 4/3 = 4
Which fraction method? Change mixed numbers to improper fractions, then choose by operation. (For + and −, you can also work with the wholes and the parts separately.)

Properties of Operations with Rational Numbers

1

Find 0.25 × 9.6 × 4 in your head.

2

Find 5 × 7.98 using the distributive property.

3

Find 36 ÷ 0.25. Which relationship did you use?

The Order of Operations with Fractions, Decimals and Exponents

1

7 − 0.5 × 6 = ?

2

(1/2)³ + 3/8 = ?

3

(3.5 + 0.5²) / 0.25 = ?

Square Numbers and Square Roots to Know by Heart

1

14² = ?

2

√0.49 = ?

3

A square rug has an area of 225 dm². How long is each side?

Multiplying and Dividing by Powers of Ten in Your Head

1

0.063 × 1 000 = ?

2

4.5 ÷ 100 = ?

3

Change 2 750 mL to litres.

Adding and Subtracting Integers: Choosing Strategies

1

(−11) + (+6) − (−3) = ?

2

(−4) − (+9) = ?

3

The temperature goes from −17°C to −5°C. What is the change?

Multiplying Integers

1

(−8) × (+7) = ?

2

(−3) × (−4) × (−5) = ?

3

Find (−4)² and −4².

Dividing Integers and the Order of Operations with Integers

1

(−64) ÷ (−8) = ?

2

(+27) ÷ (−3) − (−2) = ?

3

Find the mean of −9, −4, +3 and −2.

Adding and Subtracting Fractions, Mixed Numbers and Negative Fractions

1

3/4 + 1/6 = ?

2

5 1/3 − 2 3/4 = ?

3

(−1/3) − 1/2 = ?

Multiplying Fractions by Whole Numbers and Mixed Numbers

1

8 × 3/4 = ?

2

2 1/2 × 1 3/5 = ?

3

A recipe uses 1 1/4 cups of milk. How much for 2 1/2 batches?

Dividing with Fractions, Whole Numbers and Mixed Numbers

1

4 ÷ 2/3 = ?

2

3/5 ÷ 3 = ?

3

How many 1 1/2 m pieces can be cut from a 7 1/2 m board?

Comparing Rates and Ratios: Which Is Better?

1

Juice: 1.75 L for $3.50 or 2.5 L for $4.75. Which is the better buy?

2

Mo reads 84 pages in 3 h. Lu reads 110 pages in 4 h. Who reads faster?

3

Which is stronger: 2 scoops to 7 cups of water or 3 scoops to 10 cups?

Solving Proportional Problems: Missing Values, Percents and Multi-Step Problems

1

Solve 5/6 = x/42.

2

A $90 coat is 20% off, then 13% HST is added. What is the final price?

3

12 is 15% of what number?

Just for fun

Riddle: I am a perfect square, and I am my own square root. I am my own reciprocal. Multiply any number by me and nothing changes. What am I?

1

How to use this review

  • Work through one section at a time and write your steps, not just the answers.
  • Put a check mark beside each question you got right on your own. Put a star beside any you needed help with.
  • If you starred two or more questions in a section, go back to that lesson and redo its "More practice" questions.
  • Finish with the quiz. For every quiz mistake, read the explanation and find the lesson it came from.
  • Mistakes are useful: they show exactly what to practise next.
Check your understanding

Quiz

1.

What is 6 × 4.99?

2.

What is 3 + 0.5 × 2³?

3.

What is √1.44?

4.

What is 0.47 ÷ 0.01?

5.

What is (−6) − (−10) + (−3)?

6.

What is (−5) × (−4) ÷ (−2)?

7.

What is 2 1/3 − 3/4?

8.

What is 1 1/2 × 2 2/3?

9.

What is 2 1/4 ÷ 3/4?

10.

A $40 item has 13% HST added. What is the total?

For parents and teachers

How to use the review results:

  • Mistakes with shortcuts like 6 × 4.99 or 0.25 × 9.6 × 4 → revisit Properties of Operations with Rational Numbers.
  • Wrong order of steps, or 0.5² written as 2.5 or 0.5 → revisit The Order of Operations with Fractions, Decimals and Exponents.
  • Slow or wrong square facts, or decimal places in square roots → revisit Square Numbers and Square Roots to Know by Heart and keep using the flashcards.
  • Digits moved the wrong way, or missing placeholder zeros → revisit Multiplying and Dividing by Powers of Ten in Your Head.
  • Sign errors when adding or subtracting integers → revisit Adding and Subtracting Integers: Choosing Strategies.
  • Sign errors when multiplying, dividing, or with (−4)² versus −4² → revisit Multiplying Integers and Dividing Integers and the Order of Operations with Integers.
  • Adding denominators, or trouble regrouping mixed numbers → revisit Adding and Subtracting Fractions, Mixed Numbers and Negative Fractions.
  • Multiplying wholes and fractions separately → revisit Multiplying Fractions by Whole Numbers and Mixed Numbers.
  • Flipping the wrong fraction, or not changing mixed numbers first → revisit Dividing with Fractions, Whole Numbers and Mixed Numbers.
  • Comparing amounts that are not "per the same thing" → revisit Comparing Rates and Ratios: Which Is Better?
  • Trouble with missing values, discounts, HST or percent change → revisit Solving Proportional Problems.

Keep practice short and frequent. Ask your child to explain one answer out loud each day: explaining is part of the Grade 8 math processes (A1).