Grade by Grade
B. Number · Number: Number Sets, Powers and Rational Numbers in Action

Number: Number Sets, Powers and Rational Numbers in Action: Unit Review

A1 A2 B1.1 B1.2 B1.3 B2.1 B2.2 B3.1 B3.2 B3.3 B3.4 B3.5About 40 minutes
Learning goals
  • I can classify real numbers, explain number sets and tell the story of a number concept.
  • I can evaluate powers, use scientific notation and simplify expressions with the exponent laws.
  • I can use integers, unit fractions and signed fractions and decimals in real contexts.
  • I can solve problems with rates, proportions and percentages.

Words to know

real numbers ℝ
All rational and irrational numbers: every point on the number line.
rational number
A number that can be written as a/b with integers a and b, b ≠ 0.
exponent laws
Rules for multiplying, dividing and raising powers, such as aᵐ × aⁿ = aᵐ⁺ⁿ.
scientific notation
a × 10ⁿ, with 1 ≤ a < 10 (ignoring sign).
rate of change
Change in one quantity ÷ change in another.
proportion
Two equal ratios, like 3/5 = 24/40.
multiplier
The decimal that applies a percent change, like 1.13 for adding 13% HST.

Let's review!

In this unit you told the story of zero and place value, built the real number system with set notation, proved facts about repeating decimals, density, infinity and limits, explored powers, scientific notation and the exponent laws, and used integers, unit fractions, signed fractions, rates, proportions and percentages to solve real problems.

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

Real numbers ℝRational numbers ℚIntegers ℤWhole numbers 𝕎Natural ℕ17√162500−3−122/3−0.250.333…5 ½Irrational numbers√2π−√70.1010010001…
The real number system from this unit. Every number you used, from −4 1/2 to π to 2.1 × 10⁶, has a place in this diagram.
aᵐ × aⁿ = aᵐ⁺ⁿx³ · x⁴ = x⁷aᵐ ÷ aⁿ = aᵐ⁻ⁿx⁹ ÷ x² = x⁷(aᵐ)ⁿ = aᵐⁿ(x²)⁵ = x¹⁰(ab)ⁿ = aⁿbⁿ(3x)² = 9x²a⁰ = 1(−4)⁰ = 1a⁻ⁿ = 1/aⁿ2⁻³ = 1/8
All the exponent laws on one card, each with an example. Every law comes from writing powers in expanded form (a ≠ 0, b ≠ 0 where needed).

The Story of Numbers: Zero, Place Value and Number Systems Around the World

1

Write 2026 in base 20 and describe it in Maya digits.

2

Convert 1 h 5 min 20 s to seconds.

3

Convert 100110₂ to base 10.

4

Name one culture's contribution to the story of zero and one way it matters today.

Number Sets: How the Real Number System Is Built

1

List every set −√16 belongs to.

2

Write {x ∈ ℤ | −2 < x ≤ 3} in roster form.

3

Give a counterexample showing ℕ is not closed under division.

Rational or Irrational? Decimals, Fractions and Repeating Patterns

1

Will 9/40 terminate or repeat? Explain without dividing.

2

Write 0.272727… as a fraction in lowest terms.

3

Write 0.4666… as a fraction.

4

Is 5 − √3 rational or irrational?

Density, Infinity and Limits: How Number Sets Behave

1

Find a rational number between 5/7 and 3/4.

2

What is the limit of 0.3, 0.33, 0.333, …?

3

Why is ℤ not dense?

Powers: How the Sign and Size of the Exponent Change the Value

1

Evaluate (−2)⁶ and −2⁶.

2

Evaluate (3/5)⁻².

3

Order 4⁻¹, (−4)¹, 4⁰, (1/4)² from least to greatest.

Scientific Notation: Calculating with Very Large and Very Small Numbers

1

Write 0.000 506 in scientific notation.

2

Calculate (7 × 10⁻³)(3 × 10⁸).

3

Calculate (1.2 × 10⁴) ÷ (4 × 10⁻²).

Exponent Laws: Multiplying and Dividing Powers

1

Simplify (−5m³n)(4m²n⁵).

2

Simplify (42a⁹b⁴) ÷ (−7a³b⁴).

3

Evaluate 2⁷ × 2⁻⁴ ÷ 2².

Exponent Laws: Powers of Powers, Products and Quotients, and Simplifying Expressions

1

Simplify (−3x⁴)³.

2

Simplify (4a²b³)² ÷ (8ab⁷).

3

Simplify (x⁻²y)⁻³ with positive exponents.

4

Evaluate (2x²)³ ÷ (4x⁵) when x = −3 by simplifying first.

Integers in Context: Location, Direction, Amount and Change

1

Find the change from −17 °C to −4 °C.

2

A diver at −9 m descends 14 m. Where is she?

3

A point moves from (−6, −1) to (2, −8). Find Δx and Δy.

Unit Fractions: Building Fractions and Measuring with Tools

1

Order 1/7, 1/3, 1/12, 1/5 from greatest to least.

2

A tape measure reads 5 in and 10 sixteenths. Write it in simplest form.

3

Use the greedy method to write 9/10 as an Egyptian fraction.

Positive and Negative Signs: Fractions, Decimals, Ratios and Rates

1

Order −4/5, −0.85, −7/9 from least to greatest.

2

Calculate (−0.6) ÷ (−0.15).

3

A battery goes from 92% to 68% in 3 h. Find the rate of change.

Adding and Subtracting Positive and Negative Fractions and Mixed Numbers

1

Calculate −4/9 + 1/6.

2

Calculate 1 3/4 − (−2 5/6).

3

Calculate −3 1/5 − 1 1/2.

Multiplying and Dividing Positive and Negative Fractions: Formulas and Linear Relations

1

Calculate (−1 2/3)(−3/10).

2

Calculate −4/7 ÷ 2 2/7.

3

Use F = 9/5 C + 32 to convert −25 °C.

Rates and Proportions: Unit Rates, Scales, Similar Figures and Direct Variation

1

Solve 9/12 = x/40.

2

Convert 54 km/h to m/s.

3

A 1.2 m child casts a 1.5 m shadow; a tree casts a 20 m shadow. How tall is the tree?

Percentages: Taxes, Discounts, Percent Change and Interest

1

Find the total for an $85 item at 20% off with 13% HST.

2

A total including HST is $203.40. Find the pre-tax price.

3

Find the percent change from $2.50 to $3.10.

4

Find the simple interest on $2400 at 3% for 18 months.

Just for fun

Why do exponent laws make great friends?

They always know how to simplify your problems.

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 1 | 0 | 7 in base 20 (top to bottom) as a base-10 number?

2.

Which number is rational but not an integer?

3.

What is 0.151515… as a fraction?

4.

What is (−5)⁻²?

5.

What is (4 × 10⁻⁵)(6 × 10⁹) in scientific notation?

6.

Simplify (2a³b)⁴ ÷ (8a⁵b²).

7.

The temperature changes from 4 °C to −6 °C in 5 h. What is the rate of change?

8.

What is −2 1/4 + 3 2/3?

9.

Using F = 9/5 C + 32, what is 15 °C?

10.

A price rises 20% and then falls 20%. Overall it:

For parents and teachers

How to use the review results:

  • Mixing up place values in other bases, or not knowing why zero matters → revisit The Story of Numbers: Zero, Place Value and Number Systems Around the World.
  • Missing a set a number belongs to, or trouble with set notation and closure → revisit Number Sets: How the Real Number System Is Built.
  • Wrong fractions for repeating decimals, or calling long calculator decimals irrational → revisit Rational or Irrational? Decimals, Fractions and Repeating Patterns.
  • Trouble finding a number between two fractions, or doubting 0.999… = 1 → revisit Density, Infinity and Limits: How Number Sets Behave.
  • Writing 2⁻³ as −8, or mixing up (−3)² and −3² → revisit Powers: How the Sign and Size of the Exponent Change the Value.
  • Exponent sign errors in scientific notation, or coefficients outside 1 to 10 → revisit Scientific Notation: Calculating with Very Large and Very Small Numbers.
  • Multiplying bases or exponents instead of adding/subtracting exponents → revisit Exponent Laws: Multiplying and Dividing Powers.
  • Forgetting to raise the coefficient, or leaving negative exponents → revisit Exponent Laws: Powers of Powers, Products and Quotients, and Simplifying Expressions.
  • Subtracting initial − final, or confusing distance with displacement → revisit Integers in Context: Location, Direction, Amount and Change.
  • Thinking 1/8 > 1/4, or misreading a ruler → revisit Unit Fractions: Building Fractions and Measuring with Tools.
  • Ordering negative fractions wrongly, or sign errors in rates → revisit Positive and Negative Signs: Fractions, Decimals, Ratios and Rates.
  • Adding denominators, or treating −2 1/3 as −2 + 1/3 → revisit Adding and Subtracting Positive and Negative Fractions and Mixed Numbers.
  • Multiplying mixed numbers part by part, or errors in formulas → revisit Multiplying and Dividing Positive and Negative Fractions: Formulas and Linear Relations.
  • Mismatched ratios in a proportion, or using proportions where there is a starting fee → revisit Rates and Proportions: Unit Rates, Scales, Similar Figures and Direct Variation.
  • Removing tax by subtracting 13%, or adding successive percents → revisit Percentages: Taxes, Discounts, Percent Change and Interest.

Keep practice short and frequent. Ask your teen to explain one answer out loud each day: explaining and justifying is part of the Grade 9 mathematical processes (A1).