Chapter 60: Comprehensive Grade 7 Review and Mathematical Processes
Learn Grade 7 mathematics from very beginner explanations through worked examples, practice, reasoning, and review.
Chapter Overview
This chapter teaches Comprehensive Grade 7 Review and Mathematical Processes with simple language, step-by-step reasoning, and original worked examples. Technical words are explained in plain language.
Key Technical Terms
- Rational Number (a number that can be written as a fraction of integers with a nonzero denominator)
- Integer (a positive whole number, negative whole number, or zero)
- Prime Number (a whole number greater than 1 with exactly two positive factors)
- Percent (a ratio out of 100)
- Ratio (a comparison of two quantities)
- Unit Rate (a rate per one unit)
- Proportion (an equation showing two equal ratios)
- Equation (a statement that two expressions are equal)
- Linear Relation (a relationship with a constant rate of change)
- Mathematical Model (a mathematical representation of a real situation)
How to Learn This Chapter
Read the explanation first, follow the examples one step at a time, cover the answer and retry the problem yourself, then use the practice and review questions to check understanding.
60.1 Whole-number review
Whole-number review is an important Grade 7 idea in Comprehensive Grade 7 Review and Mathematical Processes. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Beginner Note
Identify what is given, what must be found, choose the correct rule or representation, work one step at a time, and check whether the result makes sense.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: Number problem
- Estimate first, calculate, then check
- Use an inverse operation
- Explain why the strategy fits the problem.
Very beginner explanation: Whole-number review is an important Grade 7 idea in Comprehensive Grade 7 Review and Mathematical Processes. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: A complete solution includes reasoning, calculations, and a check.
Worked Example 2
Problem: Geometry problem
- Draw and label a diagram
- Check units
- Explain why the strategy fits the problem.
Very beginner explanation: Whole-number review is an important Grade 7 idea in Comprehensive Grade 7 Review and Mathematical Processes. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: A complete solution includes reasoning, calculations, and a check.
Worked Example 3
Problem: Data problem
- Choose a suitable graph or measure
- Explain what the result means
- Explain why the strategy fits the problem.
Very beginner explanation: Whole-number review is an important Grade 7 idea in Comprehensive Grade 7 Review and Mathematical Processes. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: A complete solution includes reasoning, calculations, and a check.
Worked Example 4
Problem: Financial problem
- Write amounts and rates clearly
- Compare options
- Explain why the strategy fits the problem.
Very beginner explanation: Whole-number review is an important Grade 7 idea in Comprehensive Grade 7 Review and Mathematical Processes. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: A complete solution includes reasoning, calculations, and a check.
Worked Example 5
Problem: Algebra problem
- Define the variable and write an equation
- Substitute to check
- Explain why the strategy fits the problem.
Very beginner explanation: Whole-number review is an important Grade 7 idea in Comprehensive Grade 7 Review and Mathematical Processes. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: A complete solution includes reasoning, calculations, and a check.
Worked Example 6
Problem: Explain Whole-number review in one simple sentence.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Whole-number review becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: State the meaning, then give one small example.
Worked Example 7
Problem: Give one example that does not satisfy Whole-number review and explain why.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Whole-number review becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: Identify the rule or condition that fails.
Worked Example 8
Problem: Show Whole-number review using a table, diagram, number line, graph, or equation.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Whole-number review becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: Choose the representation that makes the relationship easiest to see.
Worked Example 9
Problem: After solving a Whole-number review problem, how can you check the answer?
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Whole-number review becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: Use estimation, an inverse operation, substitution, or a second representation.
Worked Example 10
Problem: Give one real-life situation where Whole-number review could be useful.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Whole-number review becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: Connect the idea to money, measurement, data, design, or technology.
Practice Exercise
Create one new question about Whole-number review. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
60.2 Rational-number review
A rational number (a number that can be written as a fraction of integers with a nonzero denominator) includes integers, fractions, terminating decimals, and repeating decimals. This topic focuses on Rational-number review .
Beginner Note
Identify what is given, what must be found, choose the correct rule or representation, work one step at a time, and check whether the result makes sense.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: Show why 3/4 is rational.
- Write it as a fraction of integers.
Very beginner explanation: A rational number (a number that can be written as a fraction of integers with a nonzero denominator) includes integers, fractions, terminating decimals, and repeating decimals. This topic focuses on Rational-number review .
Answer: 3/4 = 0.75
Worked Example 2
Problem: Show why -2 is rational.
- Write it as a fraction of integers.
Very beginner explanation: A rational number (a number that can be written as a fraction of integers with a nonzero denominator) includes integers, fractions, terminating decimals, and repeating decimals. This topic focuses on Rational-number review .
Answer: -2 = -2/1
Worked Example 3
Problem: Show why 0.4 is rational.
- Write it as a fraction of integers.
Very beginner explanation: A rational number (a number that can be written as a fraction of integers with a nonzero denominator) includes integers, fractions, terminating decimals, and repeating decimals. This topic focuses on Rational-number review .
Answer: 0.4 = 2/5
Worked Example 4
Problem: Show why 0.333… is rational.
- Write it as a fraction of integers.
Very beginner explanation: A rational number (a number that can be written as a fraction of integers with a nonzero denominator) includes integers, fractions, terminating decimals, and repeating decimals. This topic focuses on Rational-number review .
Answer: 0.333… = 1/3
Worked Example 5
Problem: Show why 1.25 is rational.
- Write it as a fraction of integers.
Very beginner explanation: A rational number (a number that can be written as a fraction of integers with a nonzero denominator) includes integers, fractions, terminating decimals, and repeating decimals. This topic focuses on Rational-number review .
Answer: 1.25 = 5/4
Worked Example 6
Problem: Show why 3/5 is a rational number.
- A rational number can be written as a fraction of integers.
- Rewrite the given number as a fraction if needed.
Very beginner explanation: If a number can be written as a/b where a and b are integers and b is not zero, it is rational.
Answer: 3/5 = 0.6
Worked Example 7
Problem: Show why -7 is a rational number.
- A rational number can be written as a fraction of integers.
- Rewrite the given number as a fraction if needed.
Very beginner explanation: If a number can be written as a/b where a and b are integers and b is not zero, it is rational.
Answer: -7 = -7/1
Worked Example 8
Problem: Show why 0.25 is a rational number.
- A rational number can be written as a fraction of integers.
- Rewrite the given number as a fraction if needed.
Very beginner explanation: If a number can be written as a/b where a and b are integers and b is not zero, it is rational.
Answer: 0.25 = 1/4
Worked Example 9
Problem: Show why 0.666… is a rational number.
- A rational number can be written as a fraction of integers.
- Rewrite the given number as a fraction if needed.
Very beginner explanation: If a number can be written as a/b where a and b are integers and b is not zero, it is rational.
Answer: 0.666… = 2/3
Worked Example 10
Problem: Show why 1.4 is a rational number.
- A rational number can be written as a fraction of integers.
- Rewrite the given number as a fraction if needed.
Very beginner explanation: If a number can be written as a/b where a and b are integers and b is not zero, it is rational.
Answer: 1.4 = 7/5
Practice Exercise
Create one new question about Rational-number review. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
60.3 Integer review
An integer (a positive whole number, a negative whole number, or zero) is useful for temperatures, elevations, gains, and losses. This topic focuses on Integer review .
Beginner Note
Identify what is given, what must be found, choose the correct rule or representation, work one step at a time, and check whether the result makes sense.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: Place -7 on a number line.
- Start at 0.
- Move 7 units left.
Very beginner explanation: An integer (a positive whole number, a negative whole number, or zero) is useful for temperatures, elevations, gains, and losses. This topic focuses on Integer review .
Answer: -7 is negative.
Worked Example 2
Problem: Place 4 on a number line.
- Start at 0.
- Move 4 units right.
Very beginner explanation: An integer (a positive whole number, a negative whole number, or zero) is useful for temperatures, elevations, gains, and losses. This topic focuses on Integer review .
Answer: 4 is positive.
Worked Example 3
Problem: Place -6 on a number line.
- Start at 0.
- Move 6 units left.
Very beginner explanation: An integer (a positive whole number, a negative whole number, or zero) is useful for temperatures, elevations, gains, and losses. This topic focuses on Integer review .
Answer: -6 is negative.
Worked Example 4
Problem: Place 12 on a number line.
- Start at 0.
- Move 12 units right.
Very beginner explanation: An integer (a positive whole number, a negative whole number, or zero) is useful for temperatures, elevations, gains, and losses. This topic focuses on Integer review .
Answer: 12 is positive.
Worked Example 5
Problem: Place -15 on a number line.
- Start at 0.
- Move 15 units left.
Very beginner explanation: An integer (a positive whole number, a negative whole number, or zero) is useful for temperatures, elevations, gains, and losses. This topic focuses on Integer review .
Answer: -15 is negative.
Worked Example 6
Problem: Calculate -8 + (5).
- Use a number line or sign rules.
- If signs differ, subtract absolute values and keep the sign of the larger absolute value.
Very beginner explanation: Integer addition can be understood as movement left and right on a number line.
Answer: -3
Worked Example 7
Problem: Calculate 7 + (-12).
- Use a number line or sign rules.
- If signs differ, subtract absolute values and keep the sign of the larger absolute value.
Very beginner explanation: Integer addition can be understood as movement left and right on a number line.
Answer: -5
Worked Example 8
Problem: Calculate -4 + (-9).
- Use a number line or sign rules.
- If signs differ, subtract absolute values and keep the sign of the larger absolute value.
Very beginner explanation: Integer addition can be understood as movement left and right on a number line.
Answer: -13
Worked Example 9
Problem: Calculate 15 + (-6).
- Use a number line or sign rules.
- If signs differ, subtract absolute values and keep the sign of the larger absolute value.
Very beginner explanation: Integer addition can be understood as movement left and right on a number line.
Answer: 9
Worked Example 10
Problem: Calculate -20 + (13).
- Use a number line or sign rules.
- If signs differ, subtract absolute values and keep the sign of the larger absolute value.
Very beginner explanation: Integer addition can be understood as movement left and right on a number line.
Answer: -7
Practice Exercise
Create one new question about Integer review. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
60.4 Fraction review
A fraction (a numerator over a denominator) can represent part of a whole, a quotient, or a ratio. This topic focuses on Fraction review .
Beginner Note
Identify what is given, what must be found, choose the correct rule or representation, work one step at a time, and check whether the result makes sense.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: Write 1/2 as a decimal.
- Divide 1 by 2.
Very beginner explanation: A fraction (a numerator over a denominator) can represent part of a whole, a quotient, or a ratio. This topic focuses on Fraction review .
Answer: 0.5
Worked Example 2
Problem: Write 2/3 as a decimal.
- Divide 2 by 3.
Very beginner explanation: A fraction (a numerator over a denominator) can represent part of a whole, a quotient, or a ratio. This topic focuses on Fraction review .
Answer: 0.6667
Worked Example 3
Problem: Write 3/4 as a decimal.
- Divide 3 by 4.
Very beginner explanation: A fraction (a numerator over a denominator) can represent part of a whole, a quotient, or a ratio. This topic focuses on Fraction review .
Answer: 0.75
Worked Example 4
Problem: Write 5/6 as a decimal.
- Divide 5 by 6.
Very beginner explanation: A fraction (a numerator over a denominator) can represent part of a whole, a quotient, or a ratio. This topic focuses on Fraction review .
Answer: 0.8333
Worked Example 5
Problem: Write 7/8 as a decimal.
- Divide 7 by 8.
Very beginner explanation: A fraction (a numerator over a denominator) can represent part of a whole, a quotient, or a ratio. This topic focuses on Fraction review .
Answer: 0.875
Worked Example 6
Problem: Write 1/3 as a decimal.
- A fraction bar means division.
- Divide 1 by 3.
Very beginner explanation: Fractions and decimals can represent the same rational number.
Answer: 0.3333
Worked Example 7
Problem: Write 2/5 as a decimal.
- A fraction bar means division.
- Divide 2 by 5.
Very beginner explanation: Fractions and decimals can represent the same rational number.
Answer: 0.4
Worked Example 8
Problem: Write 3/8 as a decimal.
- A fraction bar means division.
- Divide 3 by 8.
Very beginner explanation: Fractions and decimals can represent the same rational number.
Answer: 0.375
Worked Example 9
Problem: Write 5/12 as a decimal.
- A fraction bar means division.
- Divide 5 by 12.
Very beginner explanation: Fractions and decimals can represent the same rational number.
Answer: 0.4167
Worked Example 10
Problem: Write 7/9 as a decimal.
- A fraction bar means division.
- Divide 7 by 9.
Very beginner explanation: Fractions and decimals can represent the same rational number.
Answer: 0.7778
Practice Exercise
Create one new question about Fraction review. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
60.5 Decimal review
A decimal uses place value to represent parts of one, such as tenths, hundredths, and thousandths. This topic focuses on Decimal review .
Beginner Note
Identify what is given, what must be found, choose the correct rule or representation, work one step at a time, and check whether the result makes sense.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: Identify the tenths digit in 3.47.
- The tenths digit is the first digit to the right of the decimal point.
Very beginner explanation: A decimal uses place value to represent parts of one, such as tenths, hundredths, and thousandths. This topic focuses on Decimal review .
Answer: 4
Worked Example 2
Problem: Identify the tenths digit in 8.205.
- The tenths digit is the first digit to the right of the decimal point.
Very beginner explanation: A decimal uses place value to represent parts of one, such as tenths, hundredths, and thousandths. This topic focuses on Decimal review .
Answer: 2
Worked Example 3
Problem: Identify the tenths digit in 0.96.
- The tenths digit is the first digit to the right of the decimal point.
Very beginner explanation: A decimal uses place value to represent parts of one, such as tenths, hundredths, and thousandths. This topic focuses on Decimal review .
Answer: 9
Worked Example 4
Problem: Identify the tenths digit in 12.375.
- The tenths digit is the first digit to the right of the decimal point.
Very beginner explanation: A decimal uses place value to represent parts of one, such as tenths, hundredths, and thousandths. This topic focuses on Decimal review .
Answer: 3
Worked Example 5
Problem: Identify the tenths digit in 5.004.
- The tenths digit is the first digit to the right of the decimal point.
Very beginner explanation: A decimal uses place value to represent parts of one, such as tenths, hundredths, and thousandths. This topic focuses on Decimal review .
Answer: 0
Worked Example 6
Problem: Round 3.75 to the nearest tenth.
- Find the tenths digit.
- Look at the hundredths digit.
- Round up if the hundredths digit is 5 or more.
Very beginner explanation: Rounding uses the next digit to decide whether the target digit stays or increases.
Answer: 3.8
Worked Example 7
Problem: Round 8.4 to the nearest tenth.
- Find the tenths digit.
- Look at the hundredths digit.
- Round up if the hundredths digit is 5 or more.
Very beginner explanation: Rounding uses the next digit to decide whether the target digit stays or increases.
Answer: 8.4
Worked Example 8
Problem: Round 12.05 to the nearest tenth.
- Find the tenths digit.
- Look at the hundredths digit.
- Round up if the hundredths digit is 5 or more.
Very beginner explanation: Rounding uses the next digit to decide whether the target digit stays or increases.
Answer: 12.1
Worked Example 9
Problem: Round 0.96 to the nearest tenth.
- Find the tenths digit.
- Look at the hundredths digit.
- Round up if the hundredths digit is 5 or more.
Very beginner explanation: Rounding uses the next digit to decide whether the target digit stays or increases.
Answer: 1
Worked Example 10
Problem: Round 5.125 to the nearest tenth.
- Find the tenths digit.
- Look at the hundredths digit.
- Round up if the hundredths digit is 5 or more.
Very beginner explanation: Rounding uses the next digit to decide whether the target digit stays or increases.
Answer: 5.1
Practice Exercise
Create one new question about Decimal review. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
60.6 Percent review
Percent means per hundred, so 35% means 35 out of 100. This topic focuses on Percent review .
Beginner Note
Identify what is given, what must be found, choose the correct rule or representation, work one step at a time, and check whether the result makes sense.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: Find 10% of 80.
- 10% = 0.1.
- 0.1×80=8.
Very beginner explanation: Percent means per hundred, so 35% means 35 out of 100. This topic focuses on Percent review .
Answer: 8
Worked Example 2
Problem: Find 25% of 60.
- 25% = 0.25.
- 0.25×60=15.
Very beginner explanation: Percent means per hundred, so 35% means 35 out of 100. This topic focuses on Percent review .
Answer: 15
Worked Example 3
Problem: Find 15% of 120.
- 15% = 0.15.
- 0.15×120=18.
Very beginner explanation: Percent means per hundred, so 35% means 35 out of 100. This topic focuses on Percent review .
Answer: 18
Worked Example 4
Problem: Find 5% of 250.
- 5% = 0.05.
- 0.05×250=12.5.
Very beginner explanation: Percent means per hundred, so 35% means 35 out of 100. This topic focuses on Percent review .
Answer: 12.5
Worked Example 5
Problem: Find 13% of 75.
- 13% = 0.13.
- 0.13×75=9.75.
Very beginner explanation: Percent means per hundred, so 35% means 35 out of 100. This topic focuses on Percent review .
Answer: 9.75
Worked Example 6
Problem: Find 10% of 90.
- 10% = 0.1.
- 0.1 × 90 = 9.
Very beginner explanation: Percent means per hundred, so converting the percent to a decimal makes multiplication straightforward.
Answer: 9
Worked Example 7
Problem: Find 25% of 64.
- 25% = 0.25.
- 0.25 × 64 = 16.
Very beginner explanation: Percent means per hundred, so converting the percent to a decimal makes multiplication straightforward.
Answer: 16
Worked Example 8
Problem: Find 15% of 140.
- 15% = 0.15.
- 0.15 × 140 = 21.
Very beginner explanation: Percent means per hundred, so converting the percent to a decimal makes multiplication straightforward.
Answer: 21
Worked Example 9
Problem: Find 5% of 260.
- 5% = 0.05.
- 0.05 × 260 = 13.
Very beginner explanation: Percent means per hundred, so converting the percent to a decimal makes multiplication straightforward.
Answer: 13
Worked Example 10
Problem: Find 20% of 75.
- 20% = 0.2.
- 0.2 × 75 = 15.
Very beginner explanation: Percent means per hundred, so converting the percent to a decimal makes multiplication straightforward.
Answer: 15
Practice Exercise
Create one new question about Percent review. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
60.7 Ratio rate and proportion review
A ratio compares two quantities in a fixed order. This topic focuses on Ratio rate and proportion review .
Beginner Note
Identify what is given, what must be found, choose the correct rule or representation, work one step at a time, and check whether the result makes sense.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: Simplify 4:6.
- GCF(4,6)=2.
- Divide both terms by 2.
Very beginner explanation: A ratio compares two quantities in a fixed order. This topic focuses on Ratio rate and proportion review .
Answer: 2:3
Worked Example 2
Problem: Simplify 8:12.
- GCF(8,12)=4.
- Divide both terms by 4.
Very beginner explanation: A ratio compares two quantities in a fixed order. This topic focuses on Ratio rate and proportion review .
Answer: 2:3
Worked Example 3
Problem: Simplify 15:25.
- GCF(15,25)=5.
- Divide both terms by 5.
Very beginner explanation: A ratio compares two quantities in a fixed order. This topic focuses on Ratio rate and proportion review .
Answer: 3:5
Worked Example 4
Problem: Simplify 21:28.
- GCF(21,28)=7.
- Divide both terms by 7.
Very beginner explanation: A ratio compares two quantities in a fixed order. This topic focuses on Ratio rate and proportion review .
Answer: 3:4
Worked Example 5
Problem: Simplify 18:30.
- GCF(18,30)=6.
- Divide both terms by 6.
Very beginner explanation: A ratio compares two quantities in a fixed order. This topic focuses on Ratio rate and proportion review .
Answer: 3:5
Worked Example 6
Problem: 120 km are travelled in 2 hours. Find the unit rate.
- Use rate = distance ÷ time.
- 120 ÷ 2 = 60.
Very beginner explanation: A unit rate compares a quantity with exactly one unit of another quantity.
Answer: 60 km/h
Worked Example 7
Problem: 180 km are travelled in 3 hours. Find the unit rate.
- Use rate = distance ÷ time.
- 180 ÷ 3 = 60.
Very beginner explanation: A unit rate compares a quantity with exactly one unit of another quantity.
Answer: 60 km/h
Worked Example 8
Problem: 240 km are travelled in 4 hours. Find the unit rate.
- Use rate = distance ÷ time.
- 240 ÷ 4 = 60.
Very beginner explanation: A unit rate compares a quantity with exactly one unit of another quantity.
Answer: 60 km/h
Worked Example 9
Problem: 300 km are travelled in 5 hours. Find the unit rate.
- Use rate = distance ÷ time.
- 300 ÷ 5 = 60.
Very beginner explanation: A unit rate compares a quantity with exactly one unit of another quantity.
Answer: 60 km/h
Worked Example 10
Problem: 360 km are travelled in 6 hours. Find the unit rate.
- Use rate = distance ÷ time.
- 360 ÷ 6 = 60.
Very beginner explanation: A unit rate compares a quantity with exactly one unit of another quantity.
Answer: 60 km/h
Practice Exercise
Create one new question about Ratio rate and proportion review. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
60.8 Algebra review
Algebra review is an important Grade 7 idea in Comprehensive Grade 7 Review and Mathematical Processes. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Beginner Note
Identify what is given, what must be found, choose the correct rule or representation, work one step at a time, and check whether the result makes sense.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: Number problem
- Estimate first, calculate, then check
- Use an inverse operation
- Explain why the strategy fits the problem.
Very beginner explanation: Algebra review is an important Grade 7 idea in Comprehensive Grade 7 Review and Mathematical Processes. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: A complete solution includes reasoning, calculations, and a check.
Worked Example 2
Problem: Geometry problem
- Draw and label a diagram
- Check units
- Explain why the strategy fits the problem.
Very beginner explanation: Algebra review is an important Grade 7 idea in Comprehensive Grade 7 Review and Mathematical Processes. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: A complete solution includes reasoning, calculations, and a check.
Worked Example 3
Problem: Data problem
- Choose a suitable graph or measure
- Explain what the result means
- Explain why the strategy fits the problem.
Very beginner explanation: Algebra review is an important Grade 7 idea in Comprehensive Grade 7 Review and Mathematical Processes. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: A complete solution includes reasoning, calculations, and a check.
Worked Example 4
Problem: Financial problem
- Write amounts and rates clearly
- Compare options
- Explain why the strategy fits the problem.
Very beginner explanation: Algebra review is an important Grade 7 idea in Comprehensive Grade 7 Review and Mathematical Processes. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: A complete solution includes reasoning, calculations, and a check.
Worked Example 5
Problem: Algebra problem
- Define the variable and write an equation
- Substitute to check
- Explain why the strategy fits the problem.
Very beginner explanation: Algebra review is an important Grade 7 idea in Comprehensive Grade 7 Review and Mathematical Processes. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: A complete solution includes reasoning, calculations, and a check.
Worked Example 6
Problem: Explain Algebra review in one simple sentence.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Algebra review becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: State the meaning, then give one small example.
Worked Example 7
Problem: Give one example that does not satisfy Algebra review and explain why.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Algebra review becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: Identify the rule or condition that fails.
Worked Example 8
Problem: Show Algebra review using a table, diagram, number line, graph, or equation.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Algebra review becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: Choose the representation that makes the relationship easiest to see.
Worked Example 9
Problem: After solving a Algebra review problem, how can you check the answer?
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Algebra review becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: Use estimation, an inverse operation, substitution, or a second representation.
Worked Example 10
Problem: Give one real-life situation where Algebra review could be useful.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Algebra review becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: Connect the idea to money, measurement, data, design, or technology.
Practice Exercise
Create one new question about Algebra review. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
60.9 Multi-term equation review
A term is one part of an expression separated by addition or subtraction signs. This topic focuses on Multi-term equation review .
Beginner Note
Identify what is given, what must be found, choose the correct rule or representation, work one step at a time, and check whether the result makes sense.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: Simplify 3x+2x.
- Identify like terms or use distribution.
- Combine carefully.
Very beginner explanation: A term is one part of an expression separated by addition or subtraction signs. This topic focuses on Multi-term equation review .
Answer: 5x
Worked Example 2
Problem: Simplify 4(a+3).
- Identify like terms or use distribution.
- Combine carefully.
Very beginner explanation: A term is one part of an expression separated by addition or subtraction signs. This topic focuses on Multi-term equation review .
Answer: 4a+12
Worked Example 3
Problem: Simplify 7y-2y+5.
- Identify like terms or use distribution.
- Combine carefully.
Very beginner explanation: A term is one part of an expression separated by addition or subtraction signs. This topic focuses on Multi-term equation review .
Answer: 5y+5
Worked Example 4
Problem: Simplify 2(3x-4).
- Identify like terms or use distribution.
- Combine carefully.
Very beginner explanation: A term is one part of an expression separated by addition or subtraction signs. This topic focuses on Multi-term equation review .
Answer: 6x-8
Worked Example 5
Problem: Simplify 5m+3-2m.
- Identify like terms or use distribution.
- Combine carefully.
Very beginner explanation: A term is one part of an expression separated by addition or subtraction signs. This topic focuses on Multi-term equation review .
Answer: 3m+3
Worked Example 6
Problem: Simplify 3x + 4x.
- Identify like terms or distribute first if parentheses are present.
- Combine coefficients carefully.
Very beginner explanation: Like terms have the same variable part. The distributive property multiplies every term inside parentheses.
Answer: 7x
Worked Example 7
Problem: Simplify 8y - 3y + 2.
- Identify like terms or distribute first if parentheses are present.
- Combine coefficients carefully.
Very beginner explanation: Like terms have the same variable part. The distributive property multiplies every term inside parentheses.
Answer: 5y + 2
Worked Example 8
Problem: Simplify 4(a + 2).
- Identify like terms or distribute first if parentheses are present.
- Combine coefficients carefully.
Very beginner explanation: Like terms have the same variable part. The distributive property multiplies every term inside parentheses.
Answer: 4a + 8
Worked Example 9
Problem: Simplify 2(3x - 5) + x.
- Identify like terms or distribute first if parentheses are present.
- Combine coefficients carefully.
Very beginner explanation: Like terms have the same variable part. The distributive property multiplies every term inside parentheses.
Answer: 7x - 10
Worked Example 10
Problem: Simplify 6m + 7 - 2m - 3.
- Identify like terms or distribute first if parentheses are present.
- Combine coefficients carefully.
Very beginner explanation: Like terms have the same variable part. The distributive property multiplies every term inside parentheses.
Answer: 4m + 4
Practice Exercise
Create one new question about Multi-term equation review. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
60.10 Pattern and relation review
A pattern follows a rule that can be used to predict missing or future terms. This topic focuses on Pattern and relation review .
Beginner Note
Identify what is given, what must be found, choose the correct rule or representation, work one step at a time, and check whether the result makes sense.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: Extend the pattern 2, 5, 8, ... two more terms.
- Common difference = 3.
- Keep adding the same difference.
Very beginner explanation: A pattern follows a rule that can be used to predict missing or future terms. This topic focuses on Pattern and relation review .
Answer: 11, 14
Worked Example 2
Problem: Extend the pattern 5, 9, 13, ... two more terms.
- Common difference = 4.
- Keep adding the same difference.
Very beginner explanation: A pattern follows a rule that can be used to predict missing or future terms. This topic focuses on Pattern and relation review .
Answer: 17, 21
Worked Example 3
Problem: Extend the pattern 10, 8, 6, ... two more terms.
- Common difference = -2.
- Keep adding the same difference.
Very beginner explanation: A pattern follows a rule that can be used to predict missing or future terms. This topic focuses on Pattern and relation review .
Answer: 4, 2
Worked Example 4
Problem: Extend the pattern 1.5, 2, 2.5, ... two more terms.
- Common difference = 0.5.
- Keep adding the same difference.
Very beginner explanation: A pattern follows a rule that can be used to predict missing or future terms. This topic focuses on Pattern and relation review .
Answer: 3, 3.5
Worked Example 5
Problem: Extend the pattern 20, 17.5, 15, ... two more terms.
- Common difference = -2.5.
- Keep adding the same difference.
Very beginner explanation: A pattern follows a rule that can be used to predict missing or future terms. This topic focuses on Pattern and relation review .
Answer: 12.5, 10
Worked Example 6
Problem: For y = 2x + (1), find y when x = 3.
- Substitute x = 3.
- y = 2(3) + (1).
- Multiply first, then add.
Very beginner explanation: A relation pairs inputs with outputs. In a linear relation, the rate of change is constant.
Answer: 7
Worked Example 7
Problem: For y = -1x + (4), find y when x = 3.
- Substitute x = 3.
- y = -1(3) + (4).
- Multiply first, then add.
Very beginner explanation: A relation pairs inputs with outputs. In a linear relation, the rate of change is constant.
Answer: 1
Worked Example 8
Problem: For y = 0.5x + (-2), find y when x = 3.
- Substitute x = 3.
- y = 0.5(3) + (-2).
- Multiply first, then add.
Very beginner explanation: A relation pairs inputs with outputs. In a linear relation, the rate of change is constant.
Answer: -0.5
Worked Example 9
Problem: For y = 3x + (0), find y when x = 3.
- Substitute x = 3.
- y = 3(3) + (0).
- Multiply first, then add.
Very beginner explanation: A relation pairs inputs with outputs. In a linear relation, the rate of change is constant.
Answer: 9
Worked Example 10
Problem: For y = -2x + (5), find y when x = 3.
- Substitute x = 3.
- y = -2(3) + (5).
- Multiply first, then add.
Very beginner explanation: A relation pairs inputs with outputs. In a linear relation, the rate of change is constant.
Answer: -1
Practice Exercise
Create one new question about Pattern and relation review. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
60.11 Coding review
Coding review is an important Grade 7 idea in Comprehensive Grade 7 Review and Mathematical Processes. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Beginner Note
Identify what is given, what must be found, choose the correct rule or representation, work one step at a time, and check whether the result makes sense.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: Number problem
- Estimate first, calculate, then check
- Use an inverse operation
- Explain why the strategy fits the problem.
Very beginner explanation: Coding review is an important Grade 7 idea in Comprehensive Grade 7 Review and Mathematical Processes. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: A complete solution includes reasoning, calculations, and a check.
Worked Example 2
Problem: Geometry problem
- Draw and label a diagram
- Check units
- Explain why the strategy fits the problem.
Very beginner explanation: Coding review is an important Grade 7 idea in Comprehensive Grade 7 Review and Mathematical Processes. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: A complete solution includes reasoning, calculations, and a check.
Worked Example 3
Problem: Data problem
- Choose a suitable graph or measure
- Explain what the result means
- Explain why the strategy fits the problem.
Very beginner explanation: Coding review is an important Grade 7 idea in Comprehensive Grade 7 Review and Mathematical Processes. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: A complete solution includes reasoning, calculations, and a check.
Worked Example 4
Problem: Financial problem
- Write amounts and rates clearly
- Compare options
- Explain why the strategy fits the problem.
Very beginner explanation: Coding review is an important Grade 7 idea in Comprehensive Grade 7 Review and Mathematical Processes. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: A complete solution includes reasoning, calculations, and a check.
Worked Example 5
Problem: Algebra problem
- Define the variable and write an equation
- Substitute to check
- Explain why the strategy fits the problem.
Very beginner explanation: Coding review is an important Grade 7 idea in Comprehensive Grade 7 Review and Mathematical Processes. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: A complete solution includes reasoning, calculations, and a check.
Worked Example 6
Problem: Tile a floor
- Write a simple rule: area = length × width.
- State any assumptions.
- Use the rule on sample values.
- Check whether the result is realistic.
Very beginner explanation: Coding and modelling organize a real situation into clear steps, variables, rules, and checks.
Answer: Estimate tile quantity and include a small waste allowance.
Worked Example 7
Problem: Plan monthly savings
- Write a simple rule: total = starting amount + monthly deposit × months.
- State any assumptions.
- Use the rule on sample values.
- Check whether the result is realistic.
Very beginner explanation: Coding and modelling organize a real situation into clear steps, variables, rules, and checks.
Answer: Predict when the savings goal is reached.
Worked Example 8
Problem: Simulate a die
- Write a simple rule: generate a random whole number from 1 to 6.
- State any assumptions.
- Use the rule on sample values.
- Check whether the result is realistic.
Very beginner explanation: Coding and modelling organize a real situation into clear steps, variables, rules, and checks.
Answer: Repeat many times and compare frequencies.
Worked Example 9
Problem: Translate a point
- Write a simple rule: new x = x + 3; new y = y - 2.
- State any assumptions.
- Use the rule on sample values.
- Check whether the result is realistic.
Very beginner explanation: Coding and modelling organize a real situation into clear steps, variables, rules, and checks.
Answer: Apply the same rule to every point.
Worked Example 10
Problem: Estimate paint needed
- Write a simple rule: paint = wall area ÷ coverage per can.
- State any assumptions.
- Use the rule on sample values.
- Check whether the result is realistic.
Very beginner explanation: Coding and modelling organize a real situation into clear steps, variables, rules, and checks.
Answer: Round up because a partial can may not be enough.
Practice Exercise
Create one new question about Coding review. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
60.12 Mathematical modelling review
Mathematical modelling is the process of representing a real situation with mathematics, testing the model, and improving it when needed. This topic focuses on Mathematical modelling review .
Beginner Note
Identify what is given, what must be found, choose the correct rule or representation, work one step at a time, and check whether the result makes sense.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: Paint a room
- Model wall area and paint coverage
- State assumptions.
- Calculate using the model.
- Check whether the result is realistic.
- Revise if needed.
Very beginner explanation: Mathematical modelling is the process of representing a real situation with mathematics, testing the model, and improving it when needed. This topic focuses on Mathematical modelling review .
Answer: Estimate litres and cost
Worked Example 2
Problem: Tile a floor
- Model floor area and tile area
- State assumptions.
- Calculate using the model.
- Check whether the result is realistic.
- Revise if needed.
Very beginner explanation: Mathematical modelling is the process of representing a real situation with mathematics, testing the model, and improving it when needed. This topic focuses on Mathematical modelling review .
Answer: Estimate tile count plus waste
Worked Example 3
Problem: Plan a school event
- Model attendance and cost per person
- State assumptions.
- Calculate using the model.
- Check whether the result is realistic.
- Revise if needed.
Very beginner explanation: Mathematical modelling is the process of representing a real situation with mathematics, testing the model, and improving it when needed. This topic focuses on Mathematical modelling review .
Answer: Estimate total cost
Worked Example 4
Problem: Compare transportation
- Model fixed and per-kilometre costs
- State assumptions.
- Calculate using the model.
- Check whether the result is realistic.
- Revise if needed.
Very beginner explanation: Mathematical modelling is the process of representing a real situation with mathematics, testing the model, and improving it when needed. This topic focuses on Mathematical modelling review .
Answer: Find a break-even distance
Worked Example 5
Problem: Plan monthly savings
- Model starting savings plus monthly deposits
- State assumptions.
- Calculate using the model.
- Check whether the result is realistic.
- Revise if needed.
Very beginner explanation: Mathematical modelling is the process of representing a real situation with mathematics, testing the model, and improving it when needed. This topic focuses on Mathematical modelling review .
Answer: Predict when a goal is reached
Worked Example 6
Problem: Tile a floor
- Write a simple rule: area = length × width.
- State any assumptions.
- Use the rule on sample values.
- Check whether the result is realistic.
Very beginner explanation: Coding and modelling organize a real situation into clear steps, variables, rules, and checks.
Answer: Estimate tile quantity and include a small waste allowance.
Worked Example 7
Problem: Plan monthly savings
- Write a simple rule: total = starting amount + monthly deposit × months.
- State any assumptions.
- Use the rule on sample values.
- Check whether the result is realistic.
Very beginner explanation: Coding and modelling organize a real situation into clear steps, variables, rules, and checks.
Answer: Predict when the savings goal is reached.
Worked Example 8
Problem: Simulate a die
- Write a simple rule: generate a random whole number from 1 to 6.
- State any assumptions.
- Use the rule on sample values.
- Check whether the result is realistic.
Very beginner explanation: Coding and modelling organize a real situation into clear steps, variables, rules, and checks.
Answer: Repeat many times and compare frequencies.
Worked Example 9
Problem: Translate a point
- Write a simple rule: new x = x + 3; new y = y - 2.
- State any assumptions.
- Use the rule on sample values.
- Check whether the result is realistic.
Very beginner explanation: Coding and modelling organize a real situation into clear steps, variables, rules, and checks.
Answer: Apply the same rule to every point.
Worked Example 10
Problem: Estimate paint needed
- Write a simple rule: paint = wall area ÷ coverage per can.
- State any assumptions.
- Use the rule on sample values.
- Check whether the result is realistic.
Very beginner explanation: Coding and modelling organize a real situation into clear steps, variables, rules, and checks.
Answer: Round up because a partial can may not be enough.
Practice Exercise
Create one new question about Mathematical modelling review. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
60.13 Data and circle-graph review
Data are observations, measurements, or categories collected to answer questions. This topic focuses on Data and circle-graph review .
Beginner Note
Identify what is given, what must be found, choose the correct rule or representation, work one step at a time, and check whether the result makes sense.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: Favourite school subject
- Classify it as categorical/qualitative data.
- Choose an appropriate collection/organization method.
Very beginner explanation: Data are observations, measurements, or categories collected to answer questions. This topic focuses on Data and circle-graph review .
Answer: Use a frequency table
Worked Example 2
Problem: Student heights
- Classify it as quantitative continuous data.
- Choose an appropriate collection/organization method.
Very beginner explanation: Data are observations, measurements, or categories collected to answer questions. This topic focuses on Data and circle-graph review .
Answer: Measure in centimetres
Worked Example 3
Problem: Number of siblings
- Classify it as quantitative discrete data.
- Choose an appropriate collection/organization method.
Very beginner explanation: Data are observations, measurements, or categories collected to answer questions. This topic focuses on Data and circle-graph review .
Answer: Count whole numbers
Worked Example 4
Problem: Survey every 10th student
- Classify it as systematic sample.
- Choose an appropriate collection/organization method.
Very beginner explanation: Data are observations, measurements, or categories collected to answer questions. This topic focuses on Data and circle-graph review .
Answer: Check that the list order does not create bias
Worked Example 5
Problem: Survey only basketball team members about school sports
- Classify it as biased sample.
- Choose an appropriate collection/organization method.
Very beginner explanation: Data are observations, measurements, or categories collected to answer questions. This topic focuses on Data and circle-graph review .
Answer: Broaden the sample
Worked Example 6
Problem: Classify the data [4, 6, 8].
- These values are numerical.
- Decide whether they represent counts or measurements.
Very beginner explanation: Quantitative data are numerical values used for calculations or comparisons.
Answer: Quantitative data
Worked Example 7
Problem: Classify the data [5, 9, 10, 12].
- These values are numerical.
- Decide whether they represent counts or measurements.
Very beginner explanation: Quantitative data are numerical values used for calculations or comparisons.
Answer: Quantitative data
Worked Example 8
Problem: Classify the data [3, 7, 7, 11].
- These values are numerical.
- Decide whether they represent counts or measurements.
Very beginner explanation: Quantitative data are numerical values used for calculations or comparisons.
Answer: Quantitative data
Worked Example 9
Problem: Classify the data [20, 25, 30].
- These values are numerical.
- Decide whether they represent counts or measurements.
Very beginner explanation: Quantitative data are numerical values used for calculations or comparisons.
Answer: Quantitative data
Worked Example 10
Problem: Classify the data [6, 8, 9, 12, 15].
- These values are numerical.
- Decide whether they represent counts or measurements.
Very beginner explanation: Quantitative data are numerical values used for calculations or comparisons.
Answer: Quantitative data
Practice Exercise
Create one new question about Data and circle-graph review. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
60.14 Probability review
Probability measures likelihood from 0 to 1, or from 0% to 100%. This topic focuses on Probability review .
Beginner Note
Identify what is given, what must be found, choose the correct rule or representation, work one step at a time, and check whether the result makes sense.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: Roll an even number on a fair die
- Favourable outcomes=3.
- Total equally likely outcomes=6.
- P=3/6.
Very beginner explanation: Probability measures likelihood from 0 to 1, or from 0% to 100%. This topic focuses on Probability review .
Answer: 3/6
Worked Example 2
Problem: Flip heads on a fair coin
- Favourable outcomes=1.
- Total equally likely outcomes=2.
- P=1/2.
Very beginner explanation: Probability measures likelihood from 0 to 1, or from 0% to 100%. This topic focuses on Probability review .
Answer: 1/2
Worked Example 3
Problem: Draw red from 5 red and 3 blue marbles
- Favourable outcomes=5.
- Total equally likely outcomes=8.
- P=5/8.
Very beginner explanation: Probability measures likelihood from 0 to 1, or from 0% to 100%. This topic focuses on Probability review .
Answer: 5/8
Worked Example 4
Problem: Choose a vowel from A,B,C,E
- Favourable outcomes=2.
- Total equally likely outcomes=4.
- P=2/4.
Very beginner explanation: Probability measures likelihood from 0 to 1, or from 0% to 100%. This topic focuses on Probability review .
Answer: 2/4
Worked Example 5
Problem: Spin 1 on a 4-equal-section spinner
- Favourable outcomes=1.
- Total equally likely outcomes=4.
- P=1/4.
Very beginner explanation: Probability measures likelihood from 0 to 1, or from 0% to 100%. This topic focuses on Probability review .
Answer: 1/4
Worked Example 6
Problem: Find the probability to roll an even number on a fair die.
- Favourable outcomes = 3.
- Total equally likely outcomes = 6.
- Probability = 3/6.
- Simplify if possible.
Very beginner explanation: Probability compares favourable outcomes with all equally likely outcomes.
Answer: 1/2
Worked Example 7
Problem: Find the probability to flip heads on a fair coin.
- Favourable outcomes = 1.
- Total equally likely outcomes = 2.
- Probability = 1/2.
- Simplify if possible.
Very beginner explanation: Probability compares favourable outcomes with all equally likely outcomes.
Answer: 1/2
Worked Example 8
Problem: Find the probability to draw red from 5 red and 3 blue counters.
- Favourable outcomes = 5.
- Total equally likely outcomes = 8.
- Probability = 5/8.
- Simplify if possible.
Very beginner explanation: Probability compares favourable outcomes with all equally likely outcomes.
Answer: 5/8
Worked Example 9
Problem: Find the probability to choose a vowel from A, B, C, E.
- Favourable outcomes = 2.
- Total equally likely outcomes = 4.
- Probability = 2/4.
- Simplify if possible.
Very beginner explanation: Probability compares favourable outcomes with all equally likely outcomes.
Answer: 1/2
Worked Example 10
Problem: Find the probability to spin section 1 on four equal sections.
- Favourable outcomes = 1.
- Total equally likely outcomes = 4.
- Probability = 1/4.
- Simplify if possible.
Very beginner explanation: Probability compares favourable outcomes with all equally likely outcomes.
Answer: 1/4
Practice Exercise
Create one new question about Probability review. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
60.15 Circle and cylinder review
A cylinder is a 3D object with two congruent circular bases connected by a curved surface. This topic focuses on Circle and cylinder review .
Beginner Note
Identify what is given, what must be found, choose the correct rule or representation, work one step at a time, and check whether the result makes sense.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: Circle radius 3 cm. Find circumference.
- Use C=2πr.
- C≈18.85.
Very beginner explanation: A cylinder is a 3D object with two congruent circular bases connected by a curved surface. This topic focuses on Circle and cylinder review .
Answer: 18.85 cm
Worked Example 2
Problem: Circle radius 4 cm. Find circumference.
- Use C=2πr.
- C≈25.13.
Very beginner explanation: A cylinder is a 3D object with two congruent circular bases connected by a curved surface. This topic focuses on Circle and cylinder review .
Answer: 25.13 cm
Worked Example 3
Problem: Circle radius 5 cm. Find circumference.
- Use C=2πr.
- C≈31.42.
Very beginner explanation: A cylinder is a 3D object with two congruent circular bases connected by a curved surface. This topic focuses on Circle and cylinder review .
Answer: 31.42 cm
Worked Example 4
Problem: Circle radius 6 cm. Find circumference.
- Use C=2πr.
- C≈37.70.
Very beginner explanation: A cylinder is a 3D object with two congruent circular bases connected by a curved surface. This topic focuses on Circle and cylinder review .
Answer: 37.70 cm
Worked Example 5
Problem: Circle radius 7 cm. Find circumference.
- Use C=2πr.
- C≈43.98.
Very beginner explanation: A cylinder is a 3D object with two congruent circular bases connected by a curved surface. This topic focuses on Circle and cylinder review .
Answer: 43.98 cm
Worked Example 6
Problem: Find the volume of a cylinder with radius 2 cm and height 5 cm.
- Use V = πr²h.
- V = π(2)²(5).
Very beginner explanation: Cylinder volume equals the area of the circular base multiplied by height.
Answer: 62.83 cm³
Worked Example 7
Problem: Find the volume of a cylinder with radius 3 cm and height 6 cm.
- Use V = πr²h.
- V = π(3)²(6).
Very beginner explanation: Cylinder volume equals the area of the circular base multiplied by height.
Answer: 169.65 cm³
Worked Example 8
Problem: Find the volume of a cylinder with radius 4 cm and height 7 cm.
- Use V = πr²h.
- V = π(4)²(7).
Very beginner explanation: Cylinder volume equals the area of the circular base multiplied by height.
Answer: 351.86 cm³
Worked Example 9
Problem: Find the volume of a cylinder with radius 5 cm and height 8 cm.
- Use V = πr²h.
- V = π(5)²(8).
Very beginner explanation: Cylinder volume equals the area of the circular base multiplied by height.
Answer: 628.32 cm³
Worked Example 10
Problem: Find the volume of a cylinder with radius 6 cm and height 9 cm.
- Use V = πr²h.
- V = π(6)²(9).
Very beginner explanation: Cylinder volume equals the area of the circular base multiplied by height.
Answer: 1017.88 cm³
Practice Exercise
Create one new question about Circle and cylinder review. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
60.16 Dilation and transformation review
A dilation enlarges or reduces a figure using a scale factor while preserving shape. This topic focuses on Dilation and transformation review .
Beginner Note
Identify what is given, what must be found, choose the correct rule or representation, work one step at a time, and check whether the result makes sense.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: Dilate (2,3) from the origin by scale factor 2.
- Multiply both coordinates by 2.
Very beginner explanation: A dilation enlarges or reduces a figure using a scale factor while preserving shape. This topic focuses on Dilation and transformation review .
Answer: (4,6)
Worked Example 2
Problem: Dilate (-1,4) from the origin by scale factor 2.
- Multiply both coordinates by 2.
Very beginner explanation: A dilation enlarges or reduces a figure using a scale factor while preserving shape. This topic focuses on Dilation and transformation review .
Answer: (-2,8)
Worked Example 3
Problem: Dilate (5,-2) from the origin by scale factor 2.
- Multiply both coordinates by 2.
Very beginner explanation: A dilation enlarges or reduces a figure using a scale factor while preserving shape. This topic focuses on Dilation and transformation review .
Answer: (10,-4)
Worked Example 4
Problem: Dilate (0,6) from the origin by scale factor 2.
- Multiply both coordinates by 2.
Very beginner explanation: A dilation enlarges or reduces a figure using a scale factor while preserving shape. This topic focuses on Dilation and transformation review .
Answer: (0,12)
Worked Example 5
Problem: Dilate (-3,-5) from the origin by scale factor 2.
- Multiply both coordinates by 2.
Very beginner explanation: A dilation enlarges or reduces a figure using a scale factor while preserving shape. This topic focuses on Dilation and transformation review .
Answer: (-6,-10)
Worked Example 6
Problem: Dilate (2,3) from the origin by scale factor 2.
- Multiply both coordinates by the scale factor.
Very beginner explanation: A dilation from the origin multiplies x and y by the same scale factor.
Answer: (4, 6)
Worked Example 7
Problem: Dilate (-1,4) from the origin by scale factor 0.5.
- Multiply both coordinates by the scale factor.
Very beginner explanation: A dilation from the origin multiplies x and y by the same scale factor.
Answer: (-0.5, 2)
Worked Example 8
Problem: Dilate (5,-2) from the origin by scale factor 3.
- Multiply both coordinates by the scale factor.
Very beginner explanation: A dilation from the origin multiplies x and y by the same scale factor.
Answer: (15, -6)
Worked Example 9
Problem: Dilate (0,6) from the origin by scale factor 1.5.
- Multiply both coordinates by the scale factor.
Very beginner explanation: A dilation from the origin multiplies x and y by the same scale factor.
Answer: (0, 9)
Worked Example 10
Problem: Dilate (-3,-5) from the origin by scale factor 0.25.
- Multiply both coordinates by the scale factor.
Very beginner explanation: A dilation from the origin multiplies x and y by the same scale factor.
Answer: (-0.75, -1.25)
Practice Exercise
Create one new question about Dilation and transformation review. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
60.17 Financial-literacy review
Financial-literacy review is an important Grade 7 idea in Comprehensive Grade 7 Review and Mathematical Processes. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Beginner Note
Identify what is given, what must be found, choose the correct rule or representation, work one step at a time, and check whether the result makes sense.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: Number problem
- Estimate first, calculate, then check
- Use an inverse operation
- Explain why the strategy fits the problem.
Very beginner explanation: Financial-literacy review is an important Grade 7 idea in Comprehensive Grade 7 Review and Mathematical Processes. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: A complete solution includes reasoning, calculations, and a check.
Worked Example 2
Problem: Geometry problem
- Draw and label a diagram
- Check units
- Explain why the strategy fits the problem.
Very beginner explanation: Financial-literacy review is an important Grade 7 idea in Comprehensive Grade 7 Review and Mathematical Processes. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: A complete solution includes reasoning, calculations, and a check.
Worked Example 3
Problem: Data problem
- Choose a suitable graph or measure
- Explain what the result means
- Explain why the strategy fits the problem.
Very beginner explanation: Financial-literacy review is an important Grade 7 idea in Comprehensive Grade 7 Review and Mathematical Processes. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: A complete solution includes reasoning, calculations, and a check.
Worked Example 4
Problem: Financial problem
- Write amounts and rates clearly
- Compare options
- Explain why the strategy fits the problem.
Very beginner explanation: Financial-literacy review is an important Grade 7 idea in Comprehensive Grade 7 Review and Mathematical Processes. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: A complete solution includes reasoning, calculations, and a check.
Worked Example 5
Problem: Algebra problem
- Define the variable and write an equation
- Substitute to check
- Explain why the strategy fits the problem.
Very beginner explanation: Financial-literacy review is an important Grade 7 idea in Comprehensive Grade 7 Review and Mathematical Processes. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: A complete solution includes reasoning, calculations, and a check.
Worked Example 6
Problem: Explain Financial-literacy review in one simple sentence.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Financial-literacy review becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: State the meaning, then give one small example.
Worked Example 7
Problem: Give one example that does not satisfy Financial-literacy review and explain why.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Financial-literacy review becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: Identify the rule or condition that fails.
Worked Example 8
Problem: Show Financial-literacy review using a table, diagram, number line, graph, or equation.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Financial-literacy review becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: Choose the representation that makes the relationship easiest to see.
Worked Example 9
Problem: After solving a Financial-literacy review problem, how can you check the answer?
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Financial-literacy review becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: Use estimation, an inverse operation, substitution, or a second representation.
Worked Example 10
Problem: Give one real-life situation where Financial-literacy review could be useful.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Financial-literacy review becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: Connect the idea to money, measurement, data, design, or technology.
Practice Exercise
Create one new question about Financial-literacy review. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
60.18 Problem solving
Problem solving is an important Grade 7 idea in Comprehensive Grade 7 Review and Mathematical Processes. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Beginner Note
Identify what is given, what must be found, choose the correct rule or representation, work one step at a time, and check whether the result makes sense.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: Number problem
- Estimate first, calculate, then check
- Use an inverse operation
- Explain why the strategy fits the problem.
Very beginner explanation: Problem solving is an important Grade 7 idea in Comprehensive Grade 7 Review and Mathematical Processes. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: A complete solution includes reasoning, calculations, and a check.
Worked Example 2
Problem: Geometry problem
- Draw and label a diagram
- Check units
- Explain why the strategy fits the problem.
Very beginner explanation: Problem solving is an important Grade 7 idea in Comprehensive Grade 7 Review and Mathematical Processes. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: A complete solution includes reasoning, calculations, and a check.
Worked Example 3
Problem: Data problem
- Choose a suitable graph or measure
- Explain what the result means
- Explain why the strategy fits the problem.
Very beginner explanation: Problem solving is an important Grade 7 idea in Comprehensive Grade 7 Review and Mathematical Processes. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: A complete solution includes reasoning, calculations, and a check.
Worked Example 4
Problem: Financial problem
- Write amounts and rates clearly
- Compare options
- Explain why the strategy fits the problem.
Very beginner explanation: Problem solving is an important Grade 7 idea in Comprehensive Grade 7 Review and Mathematical Processes. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: A complete solution includes reasoning, calculations, and a check.
Worked Example 5
Problem: Algebra problem
- Define the variable and write an equation
- Substitute to check
- Explain why the strategy fits the problem.
Very beginner explanation: Problem solving is an important Grade 7 idea in Comprehensive Grade 7 Review and Mathematical Processes. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: A complete solution includes reasoning, calculations, and a check.
Worked Example 6
Problem: Explain Problem solving in one simple sentence.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Problem solving becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: State the meaning, then give one small example.
Worked Example 7
Problem: Give one example that does not satisfy Problem solving and explain why.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Problem solving becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: Identify the rule or condition that fails.
Worked Example 8
Problem: Show Problem solving using a table, diagram, number line, graph, or equation.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Problem solving becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: Choose the representation that makes the relationship easiest to see.
Worked Example 9
Problem: After solving a Problem solving problem, how can you check the answer?
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Problem solving becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: Use estimation, an inverse operation, substitution, or a second representation.
Worked Example 10
Problem: Give one real-life situation where Problem solving could be useful.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Problem solving becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: Connect the idea to money, measurement, data, design, or technology.
Practice Exercise
Create one new question about Problem solving. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
60.19 Reasoning and proving
Mathematical reasoning means using facts, definitions, patterns, and logical steps to justify a conclusion. This topic focuses on Reasoning and proving .
Beginner Note
Identify what is given, what must be found, choose the correct rule or representation, work one step at a time, and check whether the result makes sense.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: Number problem
- Estimate first, calculate, then check
- Use an inverse operation
- Explain why the strategy fits the problem.
Very beginner explanation: Mathematical reasoning means using facts, definitions, patterns, and logical steps to justify a conclusion. This topic focuses on Reasoning and proving .
Answer: A complete solution includes reasoning, calculations, and a check.
Worked Example 2
Problem: Geometry problem
- Draw and label a diagram
- Check units
- Explain why the strategy fits the problem.
Very beginner explanation: Mathematical reasoning means using facts, definitions, patterns, and logical steps to justify a conclusion. This topic focuses on Reasoning and proving .
Answer: A complete solution includes reasoning, calculations, and a check.
Worked Example 3
Problem: Data problem
- Choose a suitable graph or measure
- Explain what the result means
- Explain why the strategy fits the problem.
Very beginner explanation: Mathematical reasoning means using facts, definitions, patterns, and logical steps to justify a conclusion. This topic focuses on Reasoning and proving .
Answer: A complete solution includes reasoning, calculations, and a check.
Worked Example 4
Problem: Financial problem
- Write amounts and rates clearly
- Compare options
- Explain why the strategy fits the problem.
Very beginner explanation: Mathematical reasoning means using facts, definitions, patterns, and logical steps to justify a conclusion. This topic focuses on Reasoning and proving .
Answer: A complete solution includes reasoning, calculations, and a check.
Worked Example 5
Problem: Algebra problem
- Define the variable and write an equation
- Substitute to check
- Explain why the strategy fits the problem.
Very beginner explanation: Mathematical reasoning means using facts, definitions, patterns, and logical steps to justify a conclusion. This topic focuses on Reasoning and proving .
Answer: A complete solution includes reasoning, calculations, and a check.
Worked Example 6
Problem: Explain Reasoning and proving in one simple sentence.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Reasoning and proving becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: State the meaning, then give one small example.
Worked Example 7
Problem: Give one example that does not satisfy Reasoning and proving and explain why.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Reasoning and proving becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: Identify the rule or condition that fails.
Worked Example 8
Problem: Show Reasoning and proving using a table, diagram, number line, graph, or equation.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Reasoning and proving becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: Choose the representation that makes the relationship easiest to see.
Worked Example 9
Problem: After solving a Reasoning and proving problem, how can you check the answer?
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Reasoning and proving becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: Use estimation, an inverse operation, substitution, or a second representation.
Worked Example 10
Problem: Give one real-life situation where Reasoning and proving could be useful.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Reasoning and proving becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: Connect the idea to money, measurement, data, design, or technology.
Practice Exercise
Create one new question about Reasoning and proving. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
60.20 Reflecting
Reflecting is an important Grade 7 idea in Comprehensive Grade 7 Review and Mathematical Processes. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Beginner Note
Identify what is given, what must be found, choose the correct rule or representation, work one step at a time, and check whether the result makes sense.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: Number problem
- Estimate first, calculate, then check
- Use an inverse operation
- Explain why the strategy fits the problem.
Very beginner explanation: Reflecting is an important Grade 7 idea in Comprehensive Grade 7 Review and Mathematical Processes. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: A complete solution includes reasoning, calculations, and a check.
Worked Example 2
Problem: Geometry problem
- Draw and label a diagram
- Check units
- Explain why the strategy fits the problem.
Very beginner explanation: Reflecting is an important Grade 7 idea in Comprehensive Grade 7 Review and Mathematical Processes. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: A complete solution includes reasoning, calculations, and a check.
Worked Example 3
Problem: Data problem
- Choose a suitable graph or measure
- Explain what the result means
- Explain why the strategy fits the problem.
Very beginner explanation: Reflecting is an important Grade 7 idea in Comprehensive Grade 7 Review and Mathematical Processes. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: A complete solution includes reasoning, calculations, and a check.
Worked Example 4
Problem: Financial problem
- Write amounts and rates clearly
- Compare options
- Explain why the strategy fits the problem.
Very beginner explanation: Reflecting is an important Grade 7 idea in Comprehensive Grade 7 Review and Mathematical Processes. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: A complete solution includes reasoning, calculations, and a check.
Worked Example 5
Problem: Algebra problem
- Define the variable and write an equation
- Substitute to check
- Explain why the strategy fits the problem.
Very beginner explanation: Reflecting is an important Grade 7 idea in Comprehensive Grade 7 Review and Mathematical Processes. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: A complete solution includes reasoning, calculations, and a check.
Worked Example 6
Problem: Explain Reflecting in one simple sentence.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Reflecting becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: State the meaning, then give one small example.
Worked Example 7
Problem: Give one example that does not satisfy Reflecting and explain why.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Reflecting becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: Identify the rule or condition that fails.
Worked Example 8
Problem: Show Reflecting using a table, diagram, number line, graph, or equation.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Reflecting becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: Choose the representation that makes the relationship easiest to see.
Worked Example 9
Problem: After solving a Reflecting problem, how can you check the answer?
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Reflecting becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: Use estimation, an inverse operation, substitution, or a second representation.
Worked Example 10
Problem: Give one real-life situation where Reflecting could be useful.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Reflecting becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: Connect the idea to money, measurement, data, design, or technology.
Practice Exercise
Create one new question about Reflecting. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
60.21 Selecting tools and strategies
A rate compares two quantities measured in different units. This topic focuses on Selecting tools and strategies .
Beginner Note
Identify what is given, what must be found, choose the correct rule or representation, work one step at a time, and check whether the result makes sense.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: 120 km in 2 h. Find the unit rate.
- Use rate = distance ÷ time.
- 120÷2=60.
Very beginner explanation: A rate compares two quantities measured in different units. This topic focuses on Selecting tools and strategies .
Answer: 60 km/h
Worked Example 2
Problem: 180 km in 3 h. Find the unit rate.
- Use rate = distance ÷ time.
- 180÷3=60.
Very beginner explanation: A rate compares two quantities measured in different units. This topic focuses on Selecting tools and strategies .
Answer: 60 km/h
Worked Example 3
Problem: 250 km in 5 h. Find the unit rate.
- Use rate = distance ÷ time.
- 250÷5=50.
Very beginner explanation: A rate compares two quantities measured in different units. This topic focuses on Selecting tools and strategies .
Answer: 50 km/h
Worked Example 4
Problem: 72 km in 1.5 h. Find the unit rate.
- Use rate = distance ÷ time.
- 72÷1.5=48.
Very beginner explanation: A rate compares two quantities measured in different units. This topic focuses on Selecting tools and strategies .
Answer: 48 km/h
Worked Example 5
Problem: 315 km in 4.5 h. Find the unit rate.
- Use rate = distance ÷ time.
- 315÷4.5=70.
Very beginner explanation: A rate compares two quantities measured in different units. This topic focuses on Selecting tools and strategies .
Answer: 70 km/h
Worked Example 6
Problem: 120 km are travelled in 2 hours. Find the unit rate.
- Use rate = distance ÷ time.
- 120 ÷ 2 = 60.
Very beginner explanation: A unit rate compares a quantity with exactly one unit of another quantity.
Answer: 60 km/h
Worked Example 7
Problem: 180 km are travelled in 3 hours. Find the unit rate.
- Use rate = distance ÷ time.
- 180 ÷ 3 = 60.
Very beginner explanation: A unit rate compares a quantity with exactly one unit of another quantity.
Answer: 60 km/h
Worked Example 8
Problem: 240 km are travelled in 4 hours. Find the unit rate.
- Use rate = distance ÷ time.
- 240 ÷ 4 = 60.
Very beginner explanation: A unit rate compares a quantity with exactly one unit of another quantity.
Answer: 60 km/h
Worked Example 9
Problem: 300 km are travelled in 5 hours. Find the unit rate.
- Use rate = distance ÷ time.
- 300 ÷ 5 = 60.
Very beginner explanation: A unit rate compares a quantity with exactly one unit of another quantity.
Answer: 60 km/h
Worked Example 10
Problem: 360 km are travelled in 6 hours. Find the unit rate.
- Use rate = distance ÷ time.
- 360 ÷ 6 = 60.
Very beginner explanation: A unit rate compares a quantity with exactly one unit of another quantity.
Answer: 60 km/h
Practice Exercise
Create one new question about Selecting tools and strategies. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
60.22 Connecting
Connecting is an important Grade 7 idea in Comprehensive Grade 7 Review and Mathematical Processes. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Beginner Note
Identify what is given, what must be found, choose the correct rule or representation, work one step at a time, and check whether the result makes sense.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: Number problem
- Estimate first, calculate, then check
- Use an inverse operation
- Explain why the strategy fits the problem.
Very beginner explanation: Connecting is an important Grade 7 idea in Comprehensive Grade 7 Review and Mathematical Processes. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: A complete solution includes reasoning, calculations, and a check.
Worked Example 2
Problem: Geometry problem
- Draw and label a diagram
- Check units
- Explain why the strategy fits the problem.
Very beginner explanation: Connecting is an important Grade 7 idea in Comprehensive Grade 7 Review and Mathematical Processes. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: A complete solution includes reasoning, calculations, and a check.
Worked Example 3
Problem: Data problem
- Choose a suitable graph or measure
- Explain what the result means
- Explain why the strategy fits the problem.
Very beginner explanation: Connecting is an important Grade 7 idea in Comprehensive Grade 7 Review and Mathematical Processes. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: A complete solution includes reasoning, calculations, and a check.
Worked Example 4
Problem: Financial problem
- Write amounts and rates clearly
- Compare options
- Explain why the strategy fits the problem.
Very beginner explanation: Connecting is an important Grade 7 idea in Comprehensive Grade 7 Review and Mathematical Processes. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: A complete solution includes reasoning, calculations, and a check.
Worked Example 5
Problem: Algebra problem
- Define the variable and write an equation
- Substitute to check
- Explain why the strategy fits the problem.
Very beginner explanation: Connecting is an important Grade 7 idea in Comprehensive Grade 7 Review and Mathematical Processes. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: A complete solution includes reasoning, calculations, and a check.
Worked Example 6
Problem: Explain Connecting in one simple sentence.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Connecting becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: State the meaning, then give one small example.
Worked Example 7
Problem: Give one example that does not satisfy Connecting and explain why.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Connecting becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: Identify the rule or condition that fails.
Worked Example 8
Problem: Show Connecting using a table, diagram, number line, graph, or equation.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Connecting becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: Choose the representation that makes the relationship easiest to see.
Worked Example 9
Problem: After solving a Connecting problem, how can you check the answer?
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Connecting becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: Use estimation, an inverse operation, substitution, or a second representation.
Worked Example 10
Problem: Give one real-life situation where Connecting could be useful.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Connecting becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: Connect the idea to money, measurement, data, design, or technology.
Practice Exercise
Create one new question about Connecting. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
60.23 Representing
Representing is an important Grade 7 idea in Comprehensive Grade 7 Review and Mathematical Processes. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Beginner Note
Identify what is given, what must be found, choose the correct rule or representation, work one step at a time, and check whether the result makes sense.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: Paint a room
- Model wall area and paint coverage
- State assumptions.
- Calculate using the model.
- Check whether the result is realistic.
- Revise if needed.
Very beginner explanation: Representing is an important Grade 7 idea in Comprehensive Grade 7 Review and Mathematical Processes. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: Estimate litres and cost
Worked Example 2
Problem: Tile a floor
- Model floor area and tile area
- State assumptions.
- Calculate using the model.
- Check whether the result is realistic.
- Revise if needed.
Very beginner explanation: Representing is an important Grade 7 idea in Comprehensive Grade 7 Review and Mathematical Processes. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: Estimate tile count plus waste
Worked Example 3
Problem: Plan a school event
- Model attendance and cost per person
- State assumptions.
- Calculate using the model.
- Check whether the result is realistic.
- Revise if needed.
Very beginner explanation: Representing is an important Grade 7 idea in Comprehensive Grade 7 Review and Mathematical Processes. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: Estimate total cost
Worked Example 4
Problem: Compare transportation
- Model fixed and per-kilometre costs
- State assumptions.
- Calculate using the model.
- Check whether the result is realistic.
- Revise if needed.
Very beginner explanation: Representing is an important Grade 7 idea in Comprehensive Grade 7 Review and Mathematical Processes. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: Find a break-even distance
Worked Example 5
Problem: Plan monthly savings
- Model starting savings plus monthly deposits
- State assumptions.
- Calculate using the model.
- Check whether the result is realistic.
- Revise if needed.
Very beginner explanation: Representing is an important Grade 7 idea in Comprehensive Grade 7 Review and Mathematical Processes. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: Predict when a goal is reached
Worked Example 6
Problem: Explain Representing in one simple sentence.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Representing becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: State the meaning, then give one small example.
Worked Example 7
Problem: Give one example that does not satisfy Representing and explain why.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Representing becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: Identify the rule or condition that fails.
Worked Example 8
Problem: Show Representing using a table, diagram, number line, graph, or equation.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Representing becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: Choose the representation that makes the relationship easiest to see.
Worked Example 9
Problem: After solving a Representing problem, how can you check the answer?
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Representing becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: Use estimation, an inverse operation, substitution, or a second representation.
Worked Example 10
Problem: Give one real-life situation where Representing could be useful.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Representing becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: Connect the idea to money, measurement, data, design, or technology.
Practice Exercise
Create one new question about Representing. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
60.24 Communicating
Mathematical communication means clearly explaining ideas using words, numbers, diagrams, symbols, tables, or graphs. This topic focuses on Communicating .
Beginner Note
Identify what is given, what must be found, choose the correct rule or representation, work one step at a time, and check whether the result makes sense.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: Number problem
- Estimate first, calculate, then check
- Use an inverse operation
- Explain why the strategy fits the problem.
Very beginner explanation: Mathematical communication means clearly explaining ideas using words, numbers, diagrams, symbols, tables, or graphs. This topic focuses on Communicating .
Answer: A complete solution includes reasoning, calculations, and a check.
Worked Example 2
Problem: Geometry problem
- Draw and label a diagram
- Check units
- Explain why the strategy fits the problem.
Very beginner explanation: Mathematical communication means clearly explaining ideas using words, numbers, diagrams, symbols, tables, or graphs. This topic focuses on Communicating .
Answer: A complete solution includes reasoning, calculations, and a check.
Worked Example 3
Problem: Data problem
- Choose a suitable graph or measure
- Explain what the result means
- Explain why the strategy fits the problem.
Very beginner explanation: Mathematical communication means clearly explaining ideas using words, numbers, diagrams, symbols, tables, or graphs. This topic focuses on Communicating .
Answer: A complete solution includes reasoning, calculations, and a check.
Worked Example 4
Problem: Financial problem
- Write amounts and rates clearly
- Compare options
- Explain why the strategy fits the problem.
Very beginner explanation: Mathematical communication means clearly explaining ideas using words, numbers, diagrams, symbols, tables, or graphs. This topic focuses on Communicating .
Answer: A complete solution includes reasoning, calculations, and a check.
Worked Example 5
Problem: Algebra problem
- Define the variable and write an equation
- Substitute to check
- Explain why the strategy fits the problem.
Very beginner explanation: Mathematical communication means clearly explaining ideas using words, numbers, diagrams, symbols, tables, or graphs. This topic focuses on Communicating .
Answer: A complete solution includes reasoning, calculations, and a check.
Worked Example 6
Problem: Explain Communicating in one simple sentence.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Communicating becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: State the meaning, then give one small example.
Worked Example 7
Problem: Give one example that does not satisfy Communicating and explain why.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Communicating becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: Identify the rule or condition that fails.
Worked Example 8
Problem: Show Communicating using a table, diagram, number line, graph, or equation.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Communicating becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: Choose the representation that makes the relationship easiest to see.
Worked Example 9
Problem: After solving a Communicating problem, how can you check the answer?
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Communicating becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: Use estimation, an inverse operation, substitution, or a second representation.
Worked Example 10
Problem: Give one real-life situation where Communicating could be useful.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Communicating becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: Connect the idea to money, measurement, data, design, or technology.
Practice Exercise
Create one new question about Communicating. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
60.25 Final mixed practice
Final mixed practice is an important Grade 7 idea in Comprehensive Grade 7 Review and Mathematical Processes. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Beginner Note
Identify what is given, what must be found, choose the correct rule or representation, work one step at a time, and check whether the result makes sense.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: Number problem
- Estimate first, calculate, then check
- Use an inverse operation
- Explain why the strategy fits the problem.
Very beginner explanation: Final mixed practice is an important Grade 7 idea in Comprehensive Grade 7 Review and Mathematical Processes. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: A complete solution includes reasoning, calculations, and a check.
Worked Example 2
Problem: Geometry problem
- Draw and label a diagram
- Check units
- Explain why the strategy fits the problem.
Very beginner explanation: Final mixed practice is an important Grade 7 idea in Comprehensive Grade 7 Review and Mathematical Processes. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: A complete solution includes reasoning, calculations, and a check.
Worked Example 3
Problem: Data problem
- Choose a suitable graph or measure
- Explain what the result means
- Explain why the strategy fits the problem.
Very beginner explanation: Final mixed practice is an important Grade 7 idea in Comprehensive Grade 7 Review and Mathematical Processes. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: A complete solution includes reasoning, calculations, and a check.
Worked Example 4
Problem: Financial problem
- Write amounts and rates clearly
- Compare options
- Explain why the strategy fits the problem.
Very beginner explanation: Final mixed practice is an important Grade 7 idea in Comprehensive Grade 7 Review and Mathematical Processes. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: A complete solution includes reasoning, calculations, and a check.
Worked Example 5
Problem: Algebra problem
- Define the variable and write an equation
- Substitute to check
- Explain why the strategy fits the problem.
Very beginner explanation: Final mixed practice is an important Grade 7 idea in Comprehensive Grade 7 Review and Mathematical Processes. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: A complete solution includes reasoning, calculations, and a check.
Worked Example 6
Problem: Explain Final mixed practice in one simple sentence.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Final mixed practice becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: State the meaning, then give one small example.
Worked Example 7
Problem: Give one example that does not satisfy Final mixed practice and explain why.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Final mixed practice becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: Identify the rule or condition that fails.
Worked Example 8
Problem: Show Final mixed practice using a table, diagram, number line, graph, or equation.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Final mixed practice becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: Choose the representation that makes the relationship easiest to see.
Worked Example 9
Problem: After solving a Final mixed practice problem, how can you check the answer?
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Final mixed practice becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: Use estimation, an inverse operation, substitution, or a second representation.
Worked Example 10
Problem: Give one real-life situation where Final mixed practice could be useful.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Final mixed practice becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: Connect the idea to money, measurement, data, design, or technology.
Practice Exercise
Create one new question about Final mixed practice. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
Chapter Notes, Practice, and Common Mistakes
Chapter Notes
- Read the complete question before choosing an operation, formula, graph, or model.
- Write technical words together with their meaning until you are comfortable using them.
- Show all important steps so another student can follow your reasoning.
- Keep units, labels, signs, axes, variables, and mathematical symbols clear.
- Estimate before or after calculating when an estimate can help check reasonableness.
- For real-life problems, explain what the final number means in the situation.
Extra Practice
- Create and solve a new question about Whole-number review . Show your reasoning and check your answer.
- Create and solve a new question about Rational-number review . Show your reasoning and check your answer.
- Create and solve a new question about Integer review . Show your reasoning and check your answer.
- Create and solve a new question about Fraction review . Show your reasoning and check your answer.
- Create and solve a new question about Decimal review . Show your reasoning and check your answer.
- Create and solve a new question about Percent review . Show your reasoning and check your answer.
- Create and solve a new question about Ratio rate and proportion review . Show your reasoning and check your answer.
- Create and solve a new question about Algebra review . Show your reasoning and check your answer.
- Create and solve a new question about Multi-term equation review . Show your reasoning and check your answer.
- Create and solve a new question about Pattern and relation review . Show your reasoning and check your answer.
- Create and solve a new question about Coding review . Show your reasoning and check your answer.
- Create and solve a new question about Mathematical modelling review . Show your reasoning and check your answer.
Common Mistakes
- Reversing an exchange-rate calculation.
- Comparing financial options without including fees.
- Treating interest rate as a whole number instead of a decimal in calculations.
- Ignoring time when comparing savings or borrowing.
- Giving a number without explaining what it means.
30 Review Questions and Answers
Q1. What is important to remember about Whole-number review?
Answer: Whole-number review is an important Grade 7 idea in Comprehensive Grade 7 Review and Mathematical Processes. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Q2. What is important to remember about Rational-number review?
Answer: A rational number (a number that can be written as a fraction of integers with a nonzero denominator) includes integers, fractions, terminating decimals, and repeating decimals. This topic focuses on Rational-number review.
Q3. What is important to remember about Integer review?
Answer: An integer (a positive whole number, a negative whole number, or zero) is useful for temperatures, elevations, gains, and losses. This topic focuses on Integer review.
Q4. What is important to remember about Fraction review?
Answer: A fraction (a numerator over a denominator) can represent part of a whole, a quotient, or a ratio. This topic focuses on Fraction review.
Q5. What is important to remember about Decimal review?
Answer: A decimal uses place value to represent parts of one, such as tenths, hundredths, and thousandths. This topic focuses on Decimal review.
Q6. What is important to remember about Percent review?
Answer: Percent means per hundred, so 35% means 35 out of 100. This topic focuses on Percent review.
Q7. What is important to remember about Ratio rate and proportion review?
Answer: A ratio compares two quantities in a fixed order. This topic focuses on Ratio rate and proportion review.
Q8. What is important to remember about Algebra review?
Answer: Algebra review is an important Grade 7 idea in Comprehensive Grade 7 Review and Mathematical Processes. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Q9. What is important to remember about Multi-term equation review?
Answer: A term is one part of an expression separated by addition or subtraction signs. This topic focuses on Multi-term equation review.
Q10. What is important to remember about Pattern and relation review?
Answer: A pattern follows a rule that can be used to predict missing or future terms. This topic focuses on Pattern and relation review.
Q11. What is important to remember about Coding review?
Answer: Coding review is an important Grade 7 idea in Comprehensive Grade 7 Review and Mathematical Processes. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Q12. What is important to remember about Mathematical modelling review?
Answer: Mathematical modelling is the process of representing a real situation with mathematics, testing the model, and improving it when needed. This topic focuses on Mathematical modelling review.
Q13. What is important to remember about Data and circle-graph review?
Answer: Data are observations, measurements, or categories collected to answer questions. This topic focuses on Data and circle-graph review.
Q14. What is important to remember about Probability review?
Answer: Probability measures likelihood from 0 to 1, or from 0% to 100%. This topic focuses on Probability review.
Q15. What is important to remember about Circle and cylinder review?
Answer: A cylinder is a 3D object with two congruent circular bases connected by a curved surface. This topic focuses on Circle and cylinder review.
Q16. What is important to remember about Dilation and transformation review?
Answer: A dilation enlarges or reduces a figure using a scale factor while preserving shape. This topic focuses on Dilation and transformation review.
Q17. What is important to remember about Financial-literacy review?
Answer: Financial-literacy review is an important Grade 7 idea in Comprehensive Grade 7 Review and Mathematical Processes. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Q18. What is important to remember about Problem solving?
Answer: Problem solving is an important Grade 7 idea in Comprehensive Grade 7 Review and Mathematical Processes. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Q19. Why should you show your steps?
Answer: Showing steps makes reasoning easier to verify and helps locate mistakes.
Q20. Why is estimation useful?
Answer: Estimation helps you judge whether a final answer is reasonable before accepting it.
Q21. Why do units matter?
Answer: Units tell what a value measures and prevent incompatible quantities from being mixed.
Q22. When should you round?
Answer: Usually round near the end of a calculation unless the question specifically asks for earlier rounding.
Q23. How can you check an answer?
Answer: Use an inverse operation, substitution, estimation, a graph, or a second method.
Q24. What should you identify first in a word problem?
Answer: Identify what is known, what is unknown, and the relationship connecting them.
Q25. What makes a final answer complete?
Answer: Give the value, correct units when needed, and a brief interpretation in context.
Q26. What should you do if an answer seems unreasonable?
Answer: Re-read the question, check copied values, signs, units, formulas, and calculations.
Q27. Why are diagrams useful?
Answer: A labelled diagram can reveal relationships between lengths, angles, areas, and other quantities.
Q28. Why are tables useful?
Answer: Tables organize values and help reveal patterns, relationships, rates, and missing information.
Q29. Why should you explain your reasoning?
Answer: Reasoning shows why a method works, not just what answer you obtained.
Q30. How do mistakes help learning?
Answer: Analyzing a mistake identifies the misunderstood rule or step and helps prevent the same error later.