Chapter 9: Properties of Operations and Multi-Step Problems

Learn Grade 5 mathematics from very beginner explanations through worked examples, practice, reasoning, and review.

Grade 5Beginner Friendly10 Examples Per Topic30 Q&APractice + Review

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Chapter Overview

This chapter teaches Properties of Operations and Multi-Step Problems with simple language, step-by-step reasoning, and original worked examples. Technical words are explained in plain language.

Key Technical Terms

Commutative property of addition (a Grade 5 idea used in this chapter)
Commutative property of multiplication (a Grade 5 idea used in this chapter)
Associative property of addition (a Grade 5 idea used in this chapter)
Associative property of multiplication (a Grade 5 idea used in this chapter)
Distributive property (a Grade 5 idea used in this chapter)
Identity properties (a Grade 5 idea used in this chapter)
Relationships between inverse operations (a Grade 5 idea used in this chapter)
Order of operations with grouping (a Grade 5 idea used in this chapter)
Multi-operation whole-number problems (a Grade 5 idea used in this chapter)
Checking multi-step calculations (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.

9.1 Commutative property of addition

Commutative property of addition 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 Commutative property of addition?

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: Commutative property of addition 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: Commutative property of addition 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 Commutative property of addition?

Read the complete problem.
Mark the information that is given.
Identify exactly what the question asks you to find.

Answer: Identify the known information and the unknown before choosing an operation or rule.

Worked Example 3

Problem: Choose a useful representation for Commutative property of addition.

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.

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 Commutative property of addition 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.

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 Commutative property of addition.

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.

Answer: Commutative property of addition can be used in a real situation where mathematics helps make a clear calculation, comparison, measurement, prediction, or decision.

Worked Example 6

Problem: Explain Commutative property of addition 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: Commutative property of addition 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 Commutative property of addition 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: Commutative property of addition 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 Commutative property of addition 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: Commutative property of addition 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 Commutative property of addition 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: Commutative property of addition 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 Commutative property of addition 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: Commutative property of addition 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 Commutative property of addition. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.

9.2 Commutative property of multiplication

Commutative property of multiplication 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 Commutative property of multiplication?

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: Commutative property of multiplication 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: Commutative property of multiplication 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 Commutative property of multiplication?

Read the complete problem.
Mark the information that is given.
Identify exactly what the question asks you to find.

Answer: Identify the known information and the unknown before choosing an operation or rule.

Worked Example 3

Problem: Choose a useful representation for Commutative property of multiplication.

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.

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 Commutative property of multiplication 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.

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 Commutative property of multiplication.

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.

Answer: Commutative property of multiplication 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 Commutative property of multiplication. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.

9.3 Associative property of addition

Associative property of addition 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 Associative property of addition?

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: Associative property of addition 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: Associative property of addition 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 Associative property of addition?

Read the complete problem.
Mark the information that is given.
Identify exactly what the question asks you to find.

Answer: Identify the known information and the unknown before choosing an operation or rule.

Worked Example 3

Problem: Choose a useful representation for Associative property of addition.

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.

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 Associative property of addition 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.

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 Associative property of addition.

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.

Answer: Associative property of addition can be used in a real situation where mathematics helps make a clear calculation, comparison, measurement, prediction, or decision.

Worked Example 6

Problem: Explain Associative property of addition 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: Associative property of addition 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 Associative property of addition 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: Associative property of addition 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 Associative property of addition 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: Associative property of addition 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 Associative property of addition 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: Associative property of addition 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 Associative property of addition 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: Associative property of addition 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 Associative property of addition. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.

9.4 Associative property of multiplication

Associative property of multiplication 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 Associative property of multiplication?

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: Associative property of multiplication 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: Associative property of multiplication 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 Associative property of multiplication?

Read the complete problem.
Mark the information that is given.
Identify exactly what the question asks you to find.

Answer: Identify the known information and the unknown before choosing an operation or rule.

Worked Example 3

Problem: Choose a useful representation for Associative property of multiplication.

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.

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 Associative property of multiplication 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.

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 Associative property of multiplication.

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.

Answer: Associative property of multiplication 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 Associative property of multiplication. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.

9.5 Distributive property

Distributive property 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 Distributive property?

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: Distributive property 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: Distributive property 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 Distributive property?

Read the complete problem.
Mark the information that is given.
Identify exactly what the question asks you to find.

Very beginner explanation: Distributive property 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 Distributive property.

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: Distributive property 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 Distributive property 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: Distributive property 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 Distributive property.

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: Distributive property 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: Distributive property can be used in a real situation where mathematics helps make a clear calculation, comparison, measurement, prediction, or decision.

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 Distributive property. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.

9.6 Identity properties

Identity properties 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 Identity properties?

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: Identity properties 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: Identity properties 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 Identity properties?

Read the complete problem.
Mark the information that is given.
Identify exactly what the question asks you to find.

Very beginner explanation: Identity properties 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 Identity properties.

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: Identity properties 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 Identity properties 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: Identity properties 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 Identity properties.

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: Identity properties 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: Identity properties can be used in a real situation where mathematics helps make a clear calculation, comparison, measurement, prediction, or decision.

Worked Example 6

Problem: Solve 2x + 5 = 17.

Simplify each side if needed.
Use inverse operations to collect variable terms and constants.
Keep both sides balanced.
Check by substitution.

Very beginner explanation: An equation stays balanced when equivalent operations are applied correctly.

Answer: x = 6

Worked Example 7

Problem: Solve 3x - 4 = 11.

Simplify each side if needed.
Use inverse operations to collect variable terms and constants.
Keep both sides balanced.
Check by substitution.

Very beginner explanation: An equation stays balanced when equivalent operations are applied correctly.

Answer: x = 5

Worked Example 8

Problem: Solve 5x + 7 = 2x + 22.

Simplify each side if needed.
Use inverse operations to collect variable terms and constants.
Keep both sides balanced.
Check by substitution.

Very beginner explanation: An equation stays balanced when equivalent operations are applied correctly.

Answer: x = 5

Worked Example 9

Problem: Solve 4(x + 2) = 24.

Simplify each side if needed.
Use inverse operations to collect variable terms and constants.
Keep both sides balanced.
Check by substitution.

Very beginner explanation: An equation stays balanced when equivalent operations are applied correctly.

Answer: x = 4

Worked Example 10

Problem: Solve 7x - 3 = 4x + 12.

Simplify each side if needed.
Use inverse operations to collect variable terms and constants.
Keep both sides balanced.
Check by substitution.

Very beginner explanation: An equation stays balanced when equivalent operations are applied correctly.

Answer: x = 5

Practice Exercise

Create one new question about Identity properties. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.

9.7 Relationships between inverse operations

Relationships between inverse operations 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 Relationships between inverse operations?

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: Relationships between inverse operations compares quantities. Equivalent ratios and unit rates help learners scale situations and compare choices fairly.

Answer: Relationships between inverse operations 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 Relationships between inverse operations?

Read the complete problem.
Mark the information that is given.
Identify exactly what the question asks you to find.

Very beginner explanation: Relationships between inverse operations 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 Relationships between inverse operations.

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: Relationships between inverse operations 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 Relationships between inverse operations 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: Relationships between inverse operations 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 Relationships between inverse operations.

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: Relationships between inverse operations compares quantities. Equivalent ratios and unit rates help learners scale situations and compare choices fairly.

Answer: Relationships between inverse operations can be used in a real situation where mathematics helps make a clear calculation, comparison, measurement, prediction, or decision.

Worked Example 6

Problem: Simplify the ratio 4:6.

Find the greatest common factor, 2.
Divide both terms by 2.

Very beginner explanation: Simplifying a ratio is like simplifying a fraction: divide both quantities by the same common factor.

Answer: 2:3

Worked Example 7

Problem: Simplify the ratio 8:12.

Find the greatest common factor, 4.
Divide both terms by 4.

Very beginner explanation: Simplifying a ratio is like simplifying a fraction: divide both quantities by the same common factor.

Answer: 2:3

Worked Example 8

Problem: Simplify the ratio 15:25.

Find the greatest common factor, 5.
Divide both terms by 5.

Very beginner explanation: Simplifying a ratio is like simplifying a fraction: divide both quantities by the same common factor.

Answer: 3:5

Worked Example 9

Problem: Simplify the ratio 18:30.

Find the greatest common factor, 6.
Divide both terms by 6.

Very beginner explanation: Simplifying a ratio is like simplifying a fraction: divide both quantities by the same common factor.

Answer: 3:5

Worked Example 10

Problem: Simplify the ratio 21:28.

Find the greatest common factor, 7.
Divide both terms by 7.

Very beginner explanation: Simplifying a ratio is like simplifying a fraction: divide both quantities by the same common factor.

Answer: 3:4

Practice Exercise

Create one new question about Relationships between inverse operations. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.

9.8 Order of operations with grouping

Order of operations with grouping 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 Order of operations with grouping?

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: Order of operations with grouping compares quantities. Equivalent ratios and unit rates help learners scale situations and compare choices fairly.

Answer: Order of operations with grouping 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 Order of operations with grouping?

Read the complete problem.
Mark the information that is given.
Identify exactly what the question asks you to find.

Very beginner explanation: Order of operations with grouping 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 Order of operations with grouping.

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: Order of operations with grouping 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 Order of operations with grouping 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: Order of operations with grouping 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 Order of operations with grouping.

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: Order of operations with grouping compares quantities. Equivalent ratios and unit rates help learners scale situations and compare choices fairly.

Answer: Order of operations with grouping can be used in a real situation where mathematics helps make a clear calculation, comparison, measurement, prediction, or decision.

Worked Example 6

Problem: Simplify the ratio 4:6.

Find the greatest common factor, 2.
Divide both terms by 2.

Very beginner explanation: Simplifying a ratio is like simplifying a fraction: divide both quantities by the same common factor.

Answer: 2:3

Worked Example 7

Problem: Simplify the ratio 8:12.

Find the greatest common factor, 4.
Divide both terms by 4.

Very beginner explanation: Simplifying a ratio is like simplifying a fraction: divide both quantities by the same common factor.

Answer: 2:3

Worked Example 8

Problem: Simplify the ratio 15:25.

Find the greatest common factor, 5.
Divide both terms by 5.

Very beginner explanation: Simplifying a ratio is like simplifying a fraction: divide both quantities by the same common factor.

Answer: 3:5

Worked Example 9

Problem: Simplify the ratio 18:30.

Find the greatest common factor, 6.
Divide both terms by 6.

Very beginner explanation: Simplifying a ratio is like simplifying a fraction: divide both quantities by the same common factor.

Answer: 3:5

Worked Example 10

Problem: Simplify the ratio 21:28.

Find the greatest common factor, 7.
Divide both terms by 7.

Very beginner explanation: Simplifying a ratio is like simplifying a fraction: divide both quantities by the same common factor.

Answer: 3:4

Practice Exercise

Create one new question about Order of operations with grouping. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.

9.9 Multi-operation whole-number problems

Multi-operation whole-number problems 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 Multi-operation whole-number 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: Multi-operation whole-number problems compares quantities. Equivalent ratios and unit rates help learners scale situations and compare choices fairly.

Answer: Multi-operation whole-number problems 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 Multi-operation whole-number problems?

Read the complete problem.
Mark the information that is given.
Identify exactly what the question asks you to find.

Very beginner explanation: Multi-operation whole-number problems 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 Multi-operation whole-number 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: Multi-operation whole-number problems 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 Multi-operation whole-number 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: Multi-operation whole-number problems 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 Multi-operation whole-number 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: Multi-operation whole-number problems compares quantities. Equivalent ratios and unit rates help learners scale situations and compare choices fairly.

Answer: Multi-operation whole-number problems can be used in a real situation where mathematics helps make a clear calculation, comparison, measurement, prediction, or decision.

Worked Example 6

Problem: Simplify the ratio 4:6.

Find the greatest common factor, 2.
Divide both terms by 2.

Very beginner explanation: Simplifying a ratio is like simplifying a fraction: divide both quantities by the same common factor.

Answer: 2:3

Worked Example 7

Problem: Simplify the ratio 8:12.

Find the greatest common factor, 4.
Divide both terms by 4.

Very beginner explanation: Simplifying a ratio is like simplifying a fraction: divide both quantities by the same common factor.

Answer: 2:3

Worked Example 8

Problem: Simplify the ratio 15:25.

Find the greatest common factor, 5.
Divide both terms by 5.

Very beginner explanation: Simplifying a ratio is like simplifying a fraction: divide both quantities by the same common factor.

Answer: 3:5

Worked Example 9

Problem: Simplify the ratio 18:30.

Find the greatest common factor, 6.
Divide both terms by 6.

Very beginner explanation: Simplifying a ratio is like simplifying a fraction: divide both quantities by the same common factor.

Answer: 3:5

Worked Example 10

Problem: Simplify the ratio 21:28.

Find the greatest common factor, 7.
Divide both terms by 7.

Very beginner explanation: Simplifying a ratio is like simplifying a fraction: divide both quantities by the same common factor.

Answer: 3:4

Practice Exercise

Create one new question about Multi-operation whole-number problems. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.

9.10 Checking multi-step calculations

Checking multi-step calculations 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 Checking multi-step calculations?

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: Checking multi-step calculations 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: Checking multi-step calculations 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 Checking multi-step calculations?

Read the complete problem.
Mark the information that is given.
Identify exactly what the question asks you to find.

Answer: Identify the known information and the unknown before choosing an operation or rule.

Worked Example 3

Problem: Choose a useful representation for Checking multi-step calculations.

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.

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 Checking multi-step calculations 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.

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 Checking multi-step calculations.

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.

Answer: Checking multi-step calculations can be used in a real situation where mathematics helps make a clear calculation, comparison, measurement, prediction, or decision.

Worked Example 6

Problem: Explain Checking multi-step calculations 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: Checking multi-step calculations 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 Checking multi-step calculations 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: Checking multi-step calculations 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 Checking multi-step calculations 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: Checking multi-step calculations 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 Checking multi-step calculations 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: Checking multi-step calculations 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 Checking multi-step calculations 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: Checking multi-step calculations 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 Checking multi-step calculations. 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 Commutative property of addition.
Create and solve one original problem about Commutative property of multiplication.
Create and solve one original problem about Associative property of addition.
Create and solve one original problem about Associative property of multiplication.
Create and solve one original problem about Distributive property.
Create and solve one original problem about Identity properties.

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.

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30 Review Questions and Answers

Q1. What is the key idea in Commutative property of addition?

Q2. What is the key idea in Commutative property of multiplication?

Q3. What is the key idea in Associative property of addition?

Q4. What is the key idea in Associative property of multiplication?

Q5. What is the key idea in Distributive property?

Answer: Distributive property 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.

Q6. What is the key idea in Identity properties?

Answer: Identity properties 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.

Q7. What is the key idea in Relationships between inverse operations?

Answer: Relationships between inverse operations compares quantities. Equivalent ratios and unit rates help learners scale situations and compare choices fairly.

Q8. What is the key idea in Order of operations with grouping?

Answer: Order of operations with grouping compares quantities. Equivalent ratios and unit rates help learners scale situations and compare choices fairly.

Q9. What is the key idea in Multi-operation whole-number problems?

Answer: Multi-operation whole-number problems compares quantities. Equivalent ratios and unit rates help learners scale situations and compare choices fairly.

Q10. What is the key idea in Checking multi-step calculations?

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 Properties of Operations and Multi-Step Problems 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 Properties of Operations and Multi-Step Problems 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 Properties of Operations and Multi-Step Problems 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 Properties of Operations and Multi-Step Problems 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 Properties of Operations and Multi-Step Problems effectively?

Answer: Work a small example, explain every step, check the answer, then try a similar problem without looking at the first solution.