Chapter 27: Number Strand Review and Problem Solving
Learn Grade 5 mathematics from very beginner explanations through worked examples, practice, reasoning, and review.
Chapter Overview
This chapter teaches Number Strand Review and Problem Solving with simple language, step-by-step reasoning, and original worked examples. Technical words are explained in plain language.
Key Technical Terms
- Whole-number review (a Grade 5 idea used in this chapter)
- Fraction review (a Grade 5 idea used in this chapter)
- Decimal review (a Grade 5 idea used in this chapter)
- Percent review (a Grade 5 idea used in this chapter)
- Multiplication and division review (a Grade 5 idea used in this chapter)
- Fraction operations review (a Grade 5 idea used in this chapter)
- Ratios and rates review (a Grade 5 idea used in this chapter)
- Mental math review (a Grade 5 idea used in this chapter)
- Mixed word problems (a Grade 5 idea used in this chapter)
- Explaining mathematical reasoning (a Grade 5 idea used in this chapter)
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.
27.1 Whole-number review
Whole-number review develops spatial reasoning through shapes, position, movement, measurement, and labelled diagrams.
Beginner Note
Begin with meaning and a visual or concrete example. Work one step at a time and check the answer before moving on.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: In your own words, what is the main idea of Whole-number review?
- Read the topic name carefully.
- Identify the quantity, relationship, shape, or process it describes.
- Explain the idea without using a memorized sentence.
Very beginner explanation: Whole-number review develops spatial reasoning through shapes, position, movement, measurement, and labelled diagrams.
Answer: Whole-number review develops spatial reasoning through shapes, position, movement, measurement, and labelled diagrams.
Worked Example 2
Problem: What should you identify first before solving a problem about Whole-number review?
- Read the complete problem.
- Mark the information that is given.
- Identify exactly what the question asks you to find.
Very beginner explanation: Whole-number review develops spatial reasoning through shapes, position, movement, measurement, and labelled diagrams.
Answer: Identify the known information and the unknown before choosing an operation or rule.
Worked Example 3
Problem: Choose a useful representation for Whole-number review.
- Decide whether a model, number line, table, graph, diagram, equation, or labelled calculation best fits the idea.
- Label the representation clearly.
- Check that it matches the mathematical relationship.
Very beginner explanation: Whole-number review develops spatial reasoning through shapes, position, movement, measurement, and labelled diagrams.
Answer: Use the representation that makes the important relationship easiest to see and check.
Worked Example 4
Problem: A learner gets an answer in a Whole-number review problem. How can the learner check it?
- Estimate the expected size or direction of the answer.
- Check units, labels, signs, and place values.
- Use an inverse operation or a second method when possible.
Very beginner explanation: Whole-number review develops spatial reasoning through shapes, position, movement, measurement, and labelled diagrams.
Answer: The answer should agree with an estimate, the problem conditions, and a second check.
Worked Example 5
Problem: Give a real-life use for Whole-number review.
- Think about money, measurement, data, patterns, design, or everyday quantities.
- Choose a situation where the topic helps compare, calculate, measure, predict, or organize.
- Explain the connection in one sentence.
Very beginner explanation: Whole-number review develops spatial reasoning through shapes, position, movement, measurement, and labelled diagrams.
Answer: Whole-number review can be used in a real situation where mathematics helps make a clear calculation, comparison, measurement, prediction, or decision.
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.
27.2 Fraction review
Fraction review focuses on equal parts of a whole or set. Learners use models, number lines, and equivalent forms so the fraction has a clear meaning before using a rule.
Beginner Note
Begin with meaning and a visual or concrete example. Work one step at a time and check the answer before moving on.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: In your own words, what is the main idea of Fraction review?
- Read the topic name carefully.
- Identify the quantity, relationship, shape, or process it describes.
- Explain the idea without using a memorized sentence.
Very beginner explanation: Fraction review focuses on equal parts of a whole or set. Learners use models, number lines, and equivalent forms so the fraction has a clear meaning before using a rule.
Answer: Fraction review focuses on equal parts of a whole or set. Learners use models, number lines, and equivalent forms so the fraction has a clear meaning before using a rule.
Worked Example 2
Problem: What should you identify first before solving a problem about Fraction review?
- Read the complete problem.
- Mark the information that is given.
- Identify exactly what the question asks you to find.
Very beginner explanation: Fraction review focuses on equal parts of a whole or set. Learners use models, number lines, and equivalent forms so the fraction has a clear meaning before using a rule.
Answer: Identify the known information and the unknown before choosing an operation or rule.
Worked Example 3
Problem: Choose a useful representation for Fraction review.
- Decide whether a model, number line, table, graph, diagram, equation, or labelled calculation best fits the idea.
- Label the representation clearly.
- Check that it matches the mathematical relationship.
Very beginner explanation: Fraction review focuses on equal parts of a whole or set. Learners use models, number lines, and equivalent forms so the fraction has a clear meaning before using a rule.
Answer: Use the representation that makes the important relationship easiest to see and check.
Worked Example 4
Problem: A learner gets an answer in a Fraction review problem. How can the learner check it?
- Estimate the expected size or direction of the answer.
- Check units, labels, signs, and place values.
- Use an inverse operation or a second method when possible.
Very beginner explanation: Fraction review focuses on equal parts of a whole or set. Learners use models, number lines, and equivalent forms so the fraction has a clear meaning before using a rule.
Answer: The answer should agree with an estimate, the problem conditions, and a second check.
Worked Example 5
Problem: Give a real-life use for Fraction review.
- Think about money, measurement, data, patterns, design, or everyday quantities.
- Choose a situation where the topic helps compare, calculate, measure, predict, or organize.
- Explain the connection in one sentence.
Very beginner explanation: Fraction review focuses on equal parts of a whole or set. Learners use models, number lines, and equivalent forms so the fraction has a clear meaning before using a rule.
Answer: Fraction review can be used in a real situation where mathematics helps make a clear calculation, comparison, measurement, prediction, or decision.
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.
27.3 Decimal review
Decimal review uses place value to describe parts smaller than one. Keeping decimal places aligned makes comparisons and calculations easier to understand.
Beginner Note
Begin with meaning and a visual or concrete example. Work one step at a time and check the answer before moving on.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: In your own words, what is the main idea of Decimal review?
- Read the topic name carefully.
- Identify the quantity, relationship, shape, or process it describes.
- Explain the idea without using a memorized sentence.
Very beginner explanation: Decimal review uses place value to describe parts smaller than one. Keeping decimal places aligned makes comparisons and calculations easier to understand.
Answer: Decimal review uses place value to describe parts smaller than one. Keeping decimal places aligned makes comparisons and calculations easier to understand.
Worked Example 2
Problem: What should you identify first before solving a problem about Decimal review?
- Read the complete problem.
- Mark the information that is given.
- Identify exactly what the question asks you to find.
Very beginner explanation: Decimal review uses place value to describe parts smaller than one. Keeping decimal places aligned makes comparisons and calculations easier to understand.
Answer: Identify the known information and the unknown before choosing an operation or rule.
Worked Example 3
Problem: Choose a useful representation for Decimal review.
- Decide whether a model, number line, table, graph, diagram, equation, or labelled calculation best fits the idea.
- Label the representation clearly.
- Check that it matches the mathematical relationship.
Very beginner explanation: Decimal review uses place value to describe parts smaller than one. Keeping decimal places aligned makes comparisons and calculations easier to understand.
Answer: Use the representation that makes the important relationship easiest to see and check.
Worked Example 4
Problem: A learner gets an answer in a Decimal review problem. How can the learner check it?
- Estimate the expected size or direction of the answer.
- Check units, labels, signs, and place values.
- Use an inverse operation or a second method when possible.
Very beginner explanation: Decimal review uses place value to describe parts smaller than one. Keeping decimal places aligned makes comparisons and calculations easier to understand.
Answer: The answer should agree with an estimate, the problem conditions, and a second check.
Worked Example 5
Problem: Give a real-life use for Decimal review.
- Think about money, measurement, data, patterns, design, or everyday quantities.
- Choose a situation where the topic helps compare, calculate, measure, predict, or organize.
- Explain the connection in one sentence.
Very beginner explanation: Decimal review uses place value to describe parts smaller than one. Keeping decimal places aligned makes comparisons and calculations easier to understand.
Answer: Decimal review can be used in a real situation where mathematics helps make a clear calculation, comparison, measurement, prediction, or decision.
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.
27.4 Percent review
Percent review uses percent as a rate out of 100 and connects percent to familiar fractions and decimals.
Beginner Note
Begin with meaning and a visual or concrete example. Work one step at a time and check the answer before moving on.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: In your own words, what is the main idea of Percent review?
- Read the topic name carefully.
- Identify the quantity, relationship, shape, or process it describes.
- Explain the idea without using a memorized sentence.
Very beginner explanation: Percent review uses percent as a rate out of 100 and connects percent to familiar fractions and decimals.
Answer: Percent review uses percent as a rate out of 100 and connects percent to familiar fractions and decimals.
Worked Example 2
Problem: What should you identify first before solving a problem about Percent review?
- Read the complete problem.
- Mark the information that is given.
- Identify exactly what the question asks you to find.
Very beginner explanation: Percent review uses percent as a rate out of 100 and connects percent to familiar fractions and decimals.
Answer: Identify the known information and the unknown before choosing an operation or rule.
Worked Example 3
Problem: Choose a useful representation for Percent review.
- Decide whether a model, number line, table, graph, diagram, equation, or labelled calculation best fits the idea.
- Label the representation clearly.
- Check that it matches the mathematical relationship.
Very beginner explanation: Percent review uses percent as a rate out of 100 and connects percent to familiar fractions and decimals.
Answer: Use the representation that makes the important relationship easiest to see and check.
Worked Example 4
Problem: A learner gets an answer in a Percent review problem. How can the learner check it?
- Estimate the expected size or direction of the answer.
- Check units, labels, signs, and place values.
- Use an inverse operation or a second method when possible.
Very beginner explanation: Percent review uses percent as a rate out of 100 and connects percent to familiar fractions and decimals.
Answer: The answer should agree with an estimate, the problem conditions, and a second check.
Worked Example 5
Problem: Give a real-life use for Percent review.
- Think about money, measurement, data, patterns, design, or everyday quantities.
- Choose a situation where the topic helps compare, calculate, measure, predict, or organize.
- Explain the connection in one sentence.
Very beginner explanation: Percent review uses percent as a rate out of 100 and connects percent to familiar fractions and decimals.
Answer: Percent review can be used in a real situation where mathematics helps make a clear calculation, comparison, measurement, prediction, or decision.
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.
27.5 Multiplication and division review
Multiplication and division review develops accurate calculation and problem solving. Learners choose a strategy, show the steps, estimate when useful, and check the result with an inverse operation or another method.
Beginner Note
Begin with meaning and a visual or concrete example. Work one step at a time and check the answer before moving on.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: In your own words, what is the main idea of Multiplication and division review?
- Read the topic name carefully.
- Identify the quantity, relationship, shape, or process it describes.
- Explain the idea without using a memorized sentence.
Very beginner explanation: Multiplication and division review develops accurate calculation and problem solving. Learners choose a strategy, show the steps, estimate when useful, and check the result with an inverse operation or another method.
Answer: Multiplication and division review develops accurate calculation and problem solving. Learners choose a strategy, show the steps, estimate when useful, and check the result with an inverse operation or another method.
Worked Example 2
Problem: What should you identify first before solving a problem about Multiplication and division review?
- Read the complete problem.
- Mark the information that is given.
- Identify exactly what the question asks you to find.
Very beginner explanation: Multiplication and division review develops accurate calculation and problem solving. Learners choose a strategy, show the steps, estimate when useful, and check the result with an inverse operation or another method.
Answer: Identify the known information and the unknown before choosing an operation or rule.
Worked Example 3
Problem: Choose a useful representation for Multiplication and division review.
- Decide whether a model, number line, table, graph, diagram, equation, or labelled calculation best fits the idea.
- Label the representation clearly.
- Check that it matches the mathematical relationship.
Very beginner explanation: Multiplication and division review develops accurate calculation and problem solving. Learners choose a strategy, show the steps, estimate when useful, and check the result with an inverse operation or another method.
Answer: Use the representation that makes the important relationship easiest to see and check.
Worked Example 4
Problem: A learner gets an answer in a Multiplication and division review problem. How can the learner check it?
- Estimate the expected size or direction of the answer.
- Check units, labels, signs, and place values.
- Use an inverse operation or a second method when possible.
Very beginner explanation: Multiplication and division review develops accurate calculation and problem solving. Learners choose a strategy, show the steps, estimate when useful, and check the result with an inverse operation or another method.
Answer: The answer should agree with an estimate, the problem conditions, and a second check.
Worked Example 5
Problem: Give a real-life use for Multiplication and division review.
- Think about money, measurement, data, patterns, design, or everyday quantities.
- Choose a situation where the topic helps compare, calculate, measure, predict, or organize.
- Explain the connection in one sentence.
Very beginner explanation: Multiplication and division review develops accurate calculation and problem solving. Learners choose a strategy, show the steps, estimate when useful, and check the result with an inverse operation or another method.
Answer: Multiplication and division review can be used in a real situation where mathematics helps make a clear calculation, comparison, measurement, prediction, or decision.
Worked Example 6
Problem: Find 10% of $90.
- Convert 10% to 0.1.
- Multiply by 90.
Very beginner explanation: Percent of an amount is found by multiplying the amount by the percent written as a decimal.
Answer: $9.00
Worked Example 7
Problem: Find 25% of $64.
- Convert 25% to 0.25.
- Multiply by 64.
Very beginner explanation: Percent of an amount is found by multiplying the amount by the percent written as a decimal.
Answer: $16.00
Worked Example 8
Problem: Find 15% of $140.
- Convert 15% to 0.15.
- Multiply by 140.
Very beginner explanation: Percent of an amount is found by multiplying the amount by the percent written as a decimal.
Answer: $21.00
Worked Example 9
Problem: Find 5% of $260.
- Convert 5% to 0.05.
- Multiply by 260.
Very beginner explanation: Percent of an amount is found by multiplying the amount by the percent written as a decimal.
Answer: $13.00
Worked Example 10
Problem: Find 20% of $75.
- Convert 20% to 0.2.
- Multiply by 75.
Very beginner explanation: Percent of an amount is found by multiplying the amount by the percent written as a decimal.
Answer: $15.00
Practice Exercise
Create one new question about Multiplication and division review. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
27.6 Fraction operations review
Fraction operations review focuses on equal parts of a whole or set. Learners use models, number lines, and equivalent forms so the fraction has a clear meaning before using a rule.
Beginner Note
Begin with meaning and a visual or concrete example. Work one step at a time and check the answer before moving on.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: In your own words, what is the main idea of Fraction operations review?
- Read the topic name carefully.
- Identify the quantity, relationship, shape, or process it describes.
- Explain the idea without using a memorized sentence.
Very beginner explanation: Fraction operations review focuses on equal parts of a whole or set. Learners use models, number lines, and equivalent forms so the fraction has a clear meaning before using a rule.
Answer: Fraction operations review focuses on equal parts of a whole or set. Learners use models, number lines, and equivalent forms so the fraction has a clear meaning before using a rule.
Worked Example 2
Problem: What should you identify first before solving a problem about Fraction operations review?
- Read the complete problem.
- Mark the information that is given.
- Identify exactly what the question asks you to find.
Very beginner explanation: Fraction operations review focuses on equal parts of a whole or set. Learners use models, number lines, and equivalent forms so the fraction has a clear meaning before using a rule.
Answer: Identify the known information and the unknown before choosing an operation or rule.
Worked Example 3
Problem: Choose a useful representation for Fraction operations review.
- Decide whether a model, number line, table, graph, diagram, equation, or labelled calculation best fits the idea.
- Label the representation clearly.
- Check that it matches the mathematical relationship.
Very beginner explanation: Fraction operations review focuses on equal parts of a whole or set. Learners use models, number lines, and equivalent forms so the fraction has a clear meaning before using a rule.
Answer: Use the representation that makes the important relationship easiest to see and check.
Worked Example 4
Problem: A learner gets an answer in a Fraction operations review problem. How can the learner check it?
- Estimate the expected size or direction of the answer.
- Check units, labels, signs, and place values.
- Use an inverse operation or a second method when possible.
Very beginner explanation: Fraction operations review focuses on equal parts of a whole or set. Learners use models, number lines, and equivalent forms so the fraction has a clear meaning before using a rule.
Answer: The answer should agree with an estimate, the problem conditions, and a second check.
Worked Example 5
Problem: Give a real-life use for Fraction operations review.
- Think about money, measurement, data, patterns, design, or everyday quantities.
- Choose a situation where the topic helps compare, calculate, measure, predict, or organize.
- Explain the connection in one sentence.
Very beginner explanation: Fraction operations review focuses on equal parts of a whole or set. Learners use models, number lines, and equivalent forms so the fraction has a clear meaning before using a rule.
Answer: Fraction operations review can be used in a real situation where mathematics helps make a clear calculation, comparison, measurement, prediction, or decision.
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 operations review. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
27.7 Ratios and rates review
Ratios and rates review compares quantities. Equivalent ratios and unit rates help learners scale situations and compare choices fairly.
Beginner Note
Begin with meaning and a visual or concrete example. Work one step at a time and check the answer before moving on.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: In your own words, what is the main idea of Ratios and rates review?
- Read the topic name carefully.
- Identify the quantity, relationship, shape, or process it describes.
- Explain the idea without using a memorized sentence.
Very beginner explanation: Ratios and rates review compares quantities. Equivalent ratios and unit rates help learners scale situations and compare choices fairly.
Answer: Ratios and rates review compares quantities. Equivalent ratios and unit rates help learners scale situations and compare choices fairly.
Worked Example 2
Problem: What should you identify first before solving a problem about Ratios and rates review?
- Read the complete problem.
- Mark the information that is given.
- Identify exactly what the question asks you to find.
Very beginner explanation: Ratios and rates review compares quantities. Equivalent ratios and unit rates help learners scale situations and compare choices fairly.
Answer: Identify the known information and the unknown before choosing an operation or rule.
Worked Example 3
Problem: Choose a useful representation for Ratios and rates review.
- Decide whether a model, number line, table, graph, diagram, equation, or labelled calculation best fits the idea.
- Label the representation clearly.
- Check that it matches the mathematical relationship.
Very beginner explanation: Ratios and rates review compares quantities. Equivalent ratios and unit rates help learners scale situations and compare choices fairly.
Answer: Use the representation that makes the important relationship easiest to see and check.
Worked Example 4
Problem: A learner gets an answer in a Ratios and rates review problem. How can the learner check it?
- Estimate the expected size or direction of the answer.
- Check units, labels, signs, and place values.
- Use an inverse operation or a second method when possible.
Very beginner explanation: Ratios and rates review compares quantities. Equivalent ratios and unit rates help learners scale situations and compare choices fairly.
Answer: The answer should agree with an estimate, the problem conditions, and a second check.
Worked Example 5
Problem: Give a real-life use for Ratios and rates review.
- Think about money, measurement, data, patterns, design, or everyday quantities.
- Choose a situation where the topic helps compare, calculate, measure, predict, or organize.
- Explain the connection in one sentence.
Very beginner explanation: Ratios and rates review compares quantities. Equivalent ratios and unit rates help learners scale situations and compare choices fairly.
Answer: Ratios and rates review can be used in a real situation where mathematics helps make a clear calculation, comparison, measurement, prediction, or decision.
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 Ratios and rates review. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
27.8 Mental math review
Mental math review develops accurate calculation and problem solving. Learners choose a strategy, show the steps, estimate when useful, and check the result with an inverse operation or another method.
Beginner Note
Begin with meaning and a visual or concrete example. Work one step at a time and check the answer before moving on.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: In your own words, what is the main idea of Mental math review?
- Read the topic name carefully.
- Identify the quantity, relationship, shape, or process it describes.
- Explain the idea without using a memorized sentence.
Very beginner explanation: Mental math review develops accurate calculation and problem solving. Learners choose a strategy, show the steps, estimate when useful, and check the result with an inverse operation or another method.
Answer: Mental math review develops accurate calculation and problem solving. Learners choose a strategy, show the steps, estimate when useful, and check the result with an inverse operation or another method.
Worked Example 2
Problem: What should you identify first before solving a problem about Mental math review?
- Read the complete problem.
- Mark the information that is given.
- Identify exactly what the question asks you to find.
Very beginner explanation: Mental math review develops accurate calculation and problem solving. Learners choose a strategy, show the steps, estimate when useful, and check the result with an inverse operation or another method.
Answer: Identify the known information and the unknown before choosing an operation or rule.
Worked Example 3
Problem: Choose a useful representation for Mental math review.
- Decide whether a model, number line, table, graph, diagram, equation, or labelled calculation best fits the idea.
- Label the representation clearly.
- Check that it matches the mathematical relationship.
Very beginner explanation: Mental math review develops accurate calculation and problem solving. Learners choose a strategy, show the steps, estimate when useful, and check the result with an inverse operation or another method.
Answer: Use the representation that makes the important relationship easiest to see and check.
Worked Example 4
Problem: A learner gets an answer in a Mental math review problem. How can the learner check it?
- Estimate the expected size or direction of the answer.
- Check units, labels, signs, and place values.
- Use an inverse operation or a second method when possible.
Very beginner explanation: Mental math review develops accurate calculation and problem solving. Learners choose a strategy, show the steps, estimate when useful, and check the result with an inverse operation or another method.
Answer: The answer should agree with an estimate, the problem conditions, and a second check.
Worked Example 5
Problem: Give a real-life use for Mental math review.
- Think about money, measurement, data, patterns, design, or everyday quantities.
- Choose a situation where the topic helps compare, calculate, measure, predict, or organize.
- Explain the connection in one sentence.
Very beginner explanation: Mental math review develops accurate calculation and problem solving. Learners choose a strategy, show the steps, estimate when useful, and check the result with an inverse operation or another method.
Answer: Mental math review can be used in a real situation where mathematics helps make a clear calculation, comparison, measurement, prediction, or decision.
Worked Example 6
Problem: Explain Mental math 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: Mental math 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 Mental math 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: Mental math 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 Mental math 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: Mental math 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 Mental math 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: Mental math 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 Mental math 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: Mental math 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 Mental math review. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
27.9 Mixed word problems
Mixed word problems is an important Grade 5 mathematics idea. Learn its meaning first, then connect it to a representation, a worked example, and a real-life use.
Beginner Note
Begin with meaning and a visual or concrete example. Work one step at a time and check the answer before moving on.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: In your own words, what is the main idea of Mixed word problems?
- Read the topic name carefully.
- Identify the quantity, relationship, shape, or process it describes.
- Explain the idea without using a memorized sentence.
Very beginner explanation: Mixed word problems is an important Grade 5 mathematics idea. Learn its meaning first, then connect it to a representation, a worked example, and a real-life use.
Answer: Mixed word problems is an important Grade 5 mathematics idea. Learn its meaning first, then connect it to a representation, a worked example, and a real-life use.
Worked Example 2
Problem: What should you identify first before solving a problem about Mixed word problems?
- Read the complete problem.
- Mark the information that is given.
- Identify exactly what the question asks you to find.
Very beginner explanation: Mixed word problems is an important Grade 5 mathematics idea. Learn its meaning first, then connect it to a representation, a worked example, and a real-life use.
Answer: Identify the known information and the unknown before choosing an operation or rule.
Worked Example 3
Problem: Choose a useful representation for Mixed word problems.
- Decide whether a model, number line, table, graph, diagram, equation, or labelled calculation best fits the idea.
- Label the representation clearly.
- Check that it matches the mathematical relationship.
Very beginner explanation: Mixed word problems is an important Grade 5 mathematics idea. Learn its meaning first, then connect it to a representation, a worked example, and a real-life use.
Answer: Use the representation that makes the important relationship easiest to see and check.
Worked Example 4
Problem: A learner gets an answer in a Mixed word problems problem. How can the learner check it?
- Estimate the expected size or direction of the answer.
- Check units, labels, signs, and place values.
- Use an inverse operation or a second method when possible.
Very beginner explanation: Mixed word problems is an important Grade 5 mathematics idea. Learn its meaning first, then connect it to a representation, a worked example, and a real-life use.
Answer: The answer should agree with an estimate, the problem conditions, and a second check.
Worked Example 5
Problem: Give a real-life use for Mixed word problems.
- Think about money, measurement, data, patterns, design, or everyday quantities.
- Choose a situation where the topic helps compare, calculate, measure, predict, or organize.
- Explain the connection in one sentence.
Very beginner explanation: Mixed word problems is an important Grade 5 mathematics idea. Learn its meaning first, then connect it to a representation, a worked example, and a real-life use.
Answer: Mixed word problems can be used in a real situation where mathematics helps make a clear calculation, comparison, measurement, prediction, or decision.
Worked Example 6
Problem: Explain Mixed word problems 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: Mixed word problems 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 Mixed word problems 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: Mixed word problems 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 Mixed word problems 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: Mixed word problems 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 Mixed word problems 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: Mixed word problems 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 Mixed word problems 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: Mixed word problems 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 Mixed word problems. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
27.10 Explaining mathematical reasoning
Explaining mathematical reasoning is an important Grade 5 mathematics idea. Learn its meaning first, then connect it to a representation, a worked example, and a real-life use.
Beginner Note
Begin with meaning and a visual or concrete example. Work one step at a time and check the answer before moving on.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: In your own words, what is the main idea of Explaining mathematical reasoning?
- Read the topic name carefully.
- Identify the quantity, relationship, shape, or process it describes.
- Explain the idea without using a memorized sentence.
Very beginner explanation: Explaining mathematical reasoning is an important Grade 5 mathematics idea. Learn its meaning first, then connect it to a representation, a worked example, and a real-life use.
Answer: Explaining mathematical reasoning is an important Grade 5 mathematics idea. Learn its meaning first, then connect it to a representation, a worked example, and a real-life use.
Worked Example 2
Problem: What should you identify first before solving a problem about Explaining mathematical reasoning?
- Read the complete problem.
- Mark the information that is given.
- Identify exactly what the question asks you to find.
Very beginner explanation: Explaining mathematical reasoning is an important Grade 5 mathematics idea. Learn its meaning first, then connect it to a representation, a worked example, and a real-life use.
Answer: Identify the known information and the unknown before choosing an operation or rule.
Worked Example 3
Problem: Choose a useful representation for Explaining mathematical reasoning.
- Decide whether a model, number line, table, graph, diagram, equation, or labelled calculation best fits the idea.
- Label the representation clearly.
- Check that it matches the mathematical relationship.
Very beginner explanation: Explaining mathematical reasoning is an important Grade 5 mathematics idea. Learn its meaning first, then connect it to a representation, a worked example, and a real-life use.
Answer: Use the representation that makes the important relationship easiest to see and check.
Worked Example 4
Problem: A learner gets an answer in a Explaining mathematical reasoning problem. How can the learner check it?
- Estimate the expected size or direction of the answer.
- Check units, labels, signs, and place values.
- Use an inverse operation or a second method when possible.
Very beginner explanation: Explaining mathematical reasoning is an important Grade 5 mathematics idea. Learn its meaning first, then connect it to a representation, a worked example, and a real-life use.
Answer: The answer should agree with an estimate, the problem conditions, and a second check.
Worked Example 5
Problem: Give a real-life use for Explaining mathematical reasoning.
- Think about money, measurement, data, patterns, design, or everyday quantities.
- Choose a situation where the topic helps compare, calculate, measure, predict, or organize.
- Explain the connection in one sentence.
Very beginner explanation: Explaining mathematical reasoning is an important Grade 5 mathematics idea. Learn its meaning first, then connect it to a representation, a worked example, and a real-life use.
Answer: Explaining mathematical reasoning can be used in a real situation where mathematics helps make a clear calculation, comparison, measurement, prediction, or decision.
Worked Example 6
Problem: Explain Explaining mathematical reasoning 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: Explaining mathematical reasoning 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 Explaining mathematical reasoning 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: Explaining mathematical reasoning 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 Explaining mathematical reasoning 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: Explaining mathematical reasoning 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 Explaining mathematical reasoning 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: Explaining mathematical reasoning 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 Explaining mathematical reasoning 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: Explaining mathematical reasoning 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 Explaining mathematical reasoning. 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 question before calculating.
- Represent the mathematics with a model, table, graph, diagram, number line, or equation when useful.
- Show the reasoning and check the final answer with a second method when possible.
Extra Practice
- Create and solve one original problem about Whole-number review.
- Create and solve one original problem about Fraction review.
- Create and solve one original problem about Decimal review.
- Create and solve one original problem about Percent review.
- Create and solve one original problem about Multiplication and division review.
- Create and solve one original problem about Fraction operations review.
Common Mistakes
- Skipping the meaning and trying to memorize a rule only.
- Using the wrong operation because the question was not read completely.
- Ignoring units, labels, place values, or the context of the problem.
- Not estimating or checking whether the final answer is reasonable.
30 Review Questions and Answers
Q1. What is the key idea in Whole-number review?
Answer: Whole-number review develops spatial reasoning through shapes, position, movement, measurement, and labelled diagrams.
Q2. What is the key idea in Fraction review?
Answer: Fraction review focuses on equal parts of a whole or set. Learners use models, number lines, and equivalent forms so the fraction has a clear meaning before using a rule.
Q3. What is the key idea in Decimal review?
Answer: Decimal review uses place value to describe parts smaller than one. Keeping decimal places aligned makes comparisons and calculations easier to understand.
Q4. What is the key idea in Percent review?
Answer: Percent review uses percent as a rate out of 100 and connects percent to familiar fractions and decimals.
Q5. What is the key idea in Multiplication and division review?
Answer: Multiplication and division review develops accurate calculation and problem solving. Learners choose a strategy, show the steps, estimate when useful, and check the result with an inverse operation or another method.
Q6. What is the key idea in Fraction operations review?
Answer: Fraction operations review focuses on equal parts of a whole or set. Learners use models, number lines, and equivalent forms so the fraction has a clear meaning before using a rule.
Q7. What is the key idea in Ratios and rates review?
Answer: Ratios and rates review compares quantities. Equivalent ratios and unit rates help learners scale situations and compare choices fairly.
Q8. What is the key idea in Mental math review?
Answer: Mental math review develops accurate calculation and problem solving. Learners choose a strategy, show the steps, estimate when useful, and check the result with an inverse operation or another method.
Q9. What is the key idea in Mixed word problems?
Answer: Mixed word problems is an important Grade 5 mathematics idea. Learn its meaning first, then connect it to a representation, a worked example, and a real-life use.
Q10. What is the key idea in Explaining mathematical reasoning?
Answer: Explaining mathematical reasoning is an important Grade 5 mathematics idea. Learn its meaning first, then connect it to a representation, a worked example, and a real-life use.
Q11. Why should you estimate before or after a calculation?
Answer: Estimation gives a reasonable range and can reveal place-value or operation mistakes.
Q12. Why are labels and units important?
Answer: They show what a number represents and help prevent mixing unlike quantities.
Q13. How can a diagram help solve a problem?
Answer: A diagram makes quantities and relationships visible before calculation.
Q14. How can inverse operations check an answer?
Answer: An inverse operation reverses the original operation and should recover the starting value when the work is correct.
Q15. Why should you show steps?
Answer: Showing steps makes reasoning clear and helps find where an error happened.
Q16. What should you do after making a mistake?
Answer: Find the step where the reasoning changed, correct it, and try a similar problem again.
Q17. How can a number line support reasoning?
Answer: It shows order, distance, relative size, fractions, decimals, and inequality solutions visually.
Q18. When is a table useful?
Answer: A table organizes related values so patterns and comparisons are easier to see.
Q19. When is a graph useful?
Answer: A graph makes trends, comparisons, locations, or data patterns visible.
Q20. How do you decide which operation to use?
Answer: Use the meaning of the situation: combine, compare, find a difference, form equal groups, or find how many groups fit.
Q21. Why should you check place value?
Answer: A digit or decimal has a different value depending on its position.
Q22. What makes an answer reasonable?
Answer: It fits the context, has the expected size and units, and agrees with an estimate or second method.
Q23. How can you explain mathematical reasoning clearly?
Answer: Name the information, the rule or operation, the steps, and why the final answer fits the problem.
Q24. Why can more than one strategy be correct?
Answer: Different valid strategies can represent the same mathematical relationship and lead to the same result.
Q25. How should you approach a difficult Grade 5 problem?
Answer: Break it into smaller parts, solve one part at a time, and check each result before continuing.
Q26. How can you practise Number Strand Review and Problem Solving effectively?
Answer: Work a small example, explain every step, check the answer, then try a similar problem without looking at the first solution.
Q27. How can you practise Number Strand Review and Problem Solving effectively?
Answer: Work a small example, explain every step, check the answer, then try a similar problem without looking at the first solution.
Q28. How can you practise Number Strand Review and Problem Solving effectively?
Answer: Work a small example, explain every step, check the answer, then try a similar problem without looking at the first solution.
Q29. How can you practise Number Strand Review and Problem Solving effectively?
Answer: Work a small example, explain every step, check the answer, then try a similar problem without looking at the first solution.
Q30. How can you practise Number Strand Review and Problem Solving effectively?
Answer: Work a small example, explain every step, check the answer, then try a similar problem without looking at the first solution.