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Chapter 32: Variables and Algebraic Expressions

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 Variables and Algebraic Expressions with simple language, step-by-step reasoning, and original worked examples. Technical words are explained in plain language.

Key Technical Terms

  • What a variable means (a Grade 5 idea used in this chapter)
  • Words to algebraic expressions (a Grade 5 idea used in this chapter)
  • Pictures to algebraic expressions (a Grade 5 idea used in this chapter)
  • Addition expressions (a Grade 5 idea used in this chapter)
  • Subtraction expressions (a Grade 5 idea used in this chapter)
  • Multiplication expressions (a Grade 5 idea used in this chapter)
  • Division expressions (a Grade 5 idea used in this chapter)
  • Equivalent relationships (a Grade 5 idea used in this chapter)
  • Matching expressions to situations (a Grade 5 idea used in this chapter)
  • Writing expressions from word problems (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.

32.1 What a variable means

What a variable means introduces algebraic thinking by describing relationships, using symbols for unknown values, and checking whether mathematical statements are true.

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 What a variable means?

  1. Read the topic name carefully.
  2. Identify the quantity, relationship, shape, or process it describes.
  3. Explain the idea without using a memorized sentence.

Very beginner explanation: What a variable means introduces algebraic thinking by describing relationships, using symbols for unknown values, and checking whether mathematical statements are true.

Answer: What a variable means introduces algebraic thinking by describing relationships, using symbols for unknown values, and checking whether mathematical statements are true.

Worked Example 2

Problem: What should you identify first before solving a problem about What a variable means?

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

Very beginner explanation: What a variable means introduces algebraic thinking by describing relationships, using symbols for unknown values, and checking whether mathematical statements are true.

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

Worked Example 3

Problem: Choose a useful representation for What a variable means.

  1. Decide whether a model, number line, table, graph, diagram, equation, or labelled calculation best fits the idea.
  2. Label the representation clearly.
  3. Check that it matches the mathematical relationship.

Very beginner explanation: What a variable means introduces algebraic thinking by describing relationships, using symbols for unknown values, and checking whether mathematical statements are true.

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 What a variable means problem. How can the learner check it?

  1. Estimate the expected size or direction of the answer.
  2. Check units, labels, signs, and place values.
  3. Use an inverse operation or a second method when possible.

Very beginner explanation: What a variable means introduces algebraic thinking by describing relationships, using symbols for unknown values, and checking whether mathematical statements are true.

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 What a variable means.

  1. Think about money, measurement, data, patterns, design, or everyday quantities.
  2. Choose a situation where the topic helps compare, calculate, measure, predict, or organize.
  3. Explain the connection in one sentence.

Very beginner explanation: What a variable means introduces algebraic thinking by describing relationships, using symbols for unknown values, and checking whether mathematical statements are true.

Answer: What a variable means 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.

  1. Identify like terms or distribute first if parentheses are present.
  2. 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.

  1. Identify like terms or distribute first if parentheses are present.
  2. 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).

  1. Identify like terms or distribute first if parentheses are present.
  2. 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.

  1. Identify like terms or distribute first if parentheses are present.
  2. 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.

  1. Identify like terms or distribute first if parentheses are present.
  2. 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 What a variable means. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.

32.2 Words to algebraic expressions

Words to algebraic expressions introduces algebraic thinking by describing relationships, using symbols for unknown values, and checking whether mathematical statements are true.

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 Words to algebraic expressions?

  1. Read the topic name carefully.
  2. Identify the quantity, relationship, shape, or process it describes.
  3. Explain the idea without using a memorized sentence.

Very beginner explanation: Words to algebraic expressions introduces algebraic thinking by describing relationships, using symbols for unknown values, and checking whether mathematical statements are true.

Answer: Words to algebraic expressions introduces algebraic thinking by describing relationships, using symbols for unknown values, and checking whether mathematical statements are true.

Worked Example 2

Problem: What should you identify first before solving a problem about Words to algebraic expressions?

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

Very beginner explanation: Words to algebraic expressions introduces algebraic thinking by describing relationships, using symbols for unknown values, and checking whether mathematical statements are true.

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

Worked Example 3

Problem: Choose a useful representation for Words to algebraic expressions.

  1. Decide whether a model, number line, table, graph, diagram, equation, or labelled calculation best fits the idea.
  2. Label the representation clearly.
  3. Check that it matches the mathematical relationship.

Very beginner explanation: Words to algebraic expressions introduces algebraic thinking by describing relationships, using symbols for unknown values, and checking whether mathematical statements are true.

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 Words to algebraic expressions problem. How can the learner check it?

  1. Estimate the expected size or direction of the answer.
  2. Check units, labels, signs, and place values.
  3. Use an inverse operation or a second method when possible.

Very beginner explanation: Words to algebraic expressions introduces algebraic thinking by describing relationships, using symbols for unknown values, and checking whether mathematical statements are true.

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 Words to algebraic expressions.

  1. Think about money, measurement, data, patterns, design, or everyday quantities.
  2. Choose a situation where the topic helps compare, calculate, measure, predict, or organize.
  3. Explain the connection in one sentence.

Very beginner explanation: Words to algebraic expressions introduces algebraic thinking by describing relationships, using symbols for unknown values, and checking whether mathematical statements are true.

Answer: Words to algebraic expressions 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.

  1. Identify like terms or distribute first if parentheses are present.
  2. 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.

  1. Identify like terms or distribute first if parentheses are present.
  2. 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).

  1. Identify like terms or distribute first if parentheses are present.
  2. 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.

  1. Identify like terms or distribute first if parentheses are present.
  2. 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.

  1. Identify like terms or distribute first if parentheses are present.
  2. 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 Words to algebraic expressions. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.

32.3 Pictures to algebraic expressions

Pictures to algebraic expressions introduces algebraic thinking by describing relationships, using symbols for unknown values, and checking whether mathematical statements are true.

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 Pictures to algebraic expressions?

  1. Read the topic name carefully.
  2. Identify the quantity, relationship, shape, or process it describes.
  3. Explain the idea without using a memorized sentence.

Very beginner explanation: Pictures to algebraic expressions introduces algebraic thinking by describing relationships, using symbols for unknown values, and checking whether mathematical statements are true.

Answer: Pictures to algebraic expressions introduces algebraic thinking by describing relationships, using symbols for unknown values, and checking whether mathematical statements are true.

Worked Example 2

Problem: What should you identify first before solving a problem about Pictures to algebraic expressions?

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

Very beginner explanation: Pictures to algebraic expressions introduces algebraic thinking by describing relationships, using symbols for unknown values, and checking whether mathematical statements are true.

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

Worked Example 3

Problem: Choose a useful representation for Pictures to algebraic expressions.

  1. Decide whether a model, number line, table, graph, diagram, equation, or labelled calculation best fits the idea.
  2. Label the representation clearly.
  3. Check that it matches the mathematical relationship.

Very beginner explanation: Pictures to algebraic expressions introduces algebraic thinking by describing relationships, using symbols for unknown values, and checking whether mathematical statements are true.

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 Pictures to algebraic expressions problem. How can the learner check it?

  1. Estimate the expected size or direction of the answer.
  2. Check units, labels, signs, and place values.
  3. Use an inverse operation or a second method when possible.

Very beginner explanation: Pictures to algebraic expressions introduces algebraic thinking by describing relationships, using symbols for unknown values, and checking whether mathematical statements are true.

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 Pictures to algebraic expressions.

  1. Think about money, measurement, data, patterns, design, or everyday quantities.
  2. Choose a situation where the topic helps compare, calculate, measure, predict, or organize.
  3. Explain the connection in one sentence.

Very beginner explanation: Pictures to algebraic expressions introduces algebraic thinking by describing relationships, using symbols for unknown values, and checking whether mathematical statements are true.

Answer: Pictures to algebraic expressions 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.

  1. Identify like terms or distribute first if parentheses are present.
  2. 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.

  1. Identify like terms or distribute first if parentheses are present.
  2. 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).

  1. Identify like terms or distribute first if parentheses are present.
  2. 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.

  1. Identify like terms or distribute first if parentheses are present.
  2. 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.

  1. Identify like terms or distribute first if parentheses are present.
  2. 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 Pictures to algebraic expressions. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.

32.4 Addition expressions

Addition expressions 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 Addition expressions?

  1. Read the topic name carefully.
  2. Identify the quantity, relationship, shape, or process it describes.
  3. Explain the idea without using a memorized sentence.

Very beginner explanation: Addition expressions 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: Addition expressions 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 Addition expressions?

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

Very beginner explanation: Addition expressions 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 Addition expressions.

  1. Decide whether a model, number line, table, graph, diagram, equation, or labelled calculation best fits the idea.
  2. Label the representation clearly.
  3. Check that it matches the mathematical relationship.

Very beginner explanation: Addition expressions 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 Addition expressions problem. How can the learner check it?

  1. Estimate the expected size or direction of the answer.
  2. Check units, labels, signs, and place values.
  3. Use an inverse operation or a second method when possible.

Very beginner explanation: Addition expressions 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 Addition expressions.

  1. Think about money, measurement, data, patterns, design, or everyday quantities.
  2. Choose a situation where the topic helps compare, calculate, measure, predict, or organize.
  3. Explain the connection in one sentence.

Very beginner explanation: Addition expressions 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: Addition expressions 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.

  1. Identify like terms or distribute first if parentheses are present.
  2. 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.

  1. Identify like terms or distribute first if parentheses are present.
  2. 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).

  1. Identify like terms or distribute first if parentheses are present.
  2. 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.

  1. Identify like terms or distribute first if parentheses are present.
  2. 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.

  1. Identify like terms or distribute first if parentheses are present.
  2. 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 Addition expressions. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.

32.5 Subtraction expressions

Subtraction expressions 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 Subtraction expressions?

  1. Read the topic name carefully.
  2. Identify the quantity, relationship, shape, or process it describes.
  3. Explain the idea without using a memorized sentence.

Very beginner explanation: Subtraction expressions 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: Subtraction expressions 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 Subtraction expressions?

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

Very beginner explanation: Subtraction expressions 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 Subtraction expressions.

  1. Decide whether a model, number line, table, graph, diagram, equation, or labelled calculation best fits the idea.
  2. Label the representation clearly.
  3. Check that it matches the mathematical relationship.

Very beginner explanation: Subtraction expressions 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 Subtraction expressions problem. How can the learner check it?

  1. Estimate the expected size or direction of the answer.
  2. Check units, labels, signs, and place values.
  3. Use an inverse operation or a second method when possible.

Very beginner explanation: Subtraction expressions 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 Subtraction expressions.

  1. Think about money, measurement, data, patterns, design, or everyday quantities.
  2. Choose a situation where the topic helps compare, calculate, measure, predict, or organize.
  3. Explain the connection in one sentence.

Very beginner explanation: Subtraction expressions 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: Subtraction expressions 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.

  1. Identify like terms or distribute first if parentheses are present.
  2. 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.

  1. Identify like terms or distribute first if parentheses are present.
  2. 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).

  1. Identify like terms or distribute first if parentheses are present.
  2. 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.

  1. Identify like terms or distribute first if parentheses are present.
  2. 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.

  1. Identify like terms or distribute first if parentheses are present.
  2. 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 Subtraction expressions. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.

32.6 Multiplication expressions

Multiplication expressions 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 expressions?

  1. Read the topic name carefully.
  2. Identify the quantity, relationship, shape, or process it describes.
  3. Explain the idea without using a memorized sentence.

Very beginner explanation: Multiplication expressions 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 expressions 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 expressions?

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

Very beginner explanation: Multiplication expressions 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 expressions.

  1. Decide whether a model, number line, table, graph, diagram, equation, or labelled calculation best fits the idea.
  2. Label the representation clearly.
  3. Check that it matches the mathematical relationship.

Very beginner explanation: Multiplication expressions 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 expressions problem. How can the learner check it?

  1. Estimate the expected size or direction of the answer.
  2. Check units, labels, signs, and place values.
  3. Use an inverse operation or a second method when possible.

Very beginner explanation: Multiplication expressions 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 expressions.

  1. Think about money, measurement, data, patterns, design, or everyday quantities.
  2. Choose a situation where the topic helps compare, calculate, measure, predict, or organize.
  3. Explain the connection in one sentence.

Very beginner explanation: Multiplication expressions 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 expressions 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.

  1. Convert 10% to 0.1.
  2. 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.

  1. Convert 25% to 0.25.
  2. 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.

  1. Convert 15% to 0.15.
  2. 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.

  1. Convert 5% to 0.05.
  2. 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.

  1. Convert 20% to 0.2.
  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 expressions. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.

32.7 Division expressions

Division expressions 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 Division expressions?

  1. Read the topic name carefully.
  2. Identify the quantity, relationship, shape, or process it describes.
  3. Explain the idea without using a memorized sentence.

Very beginner explanation: Division expressions 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: Division expressions 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 Division expressions?

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

Very beginner explanation: Division expressions 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 Division expressions.

  1. Decide whether a model, number line, table, graph, diagram, equation, or labelled calculation best fits the idea.
  2. Label the representation clearly.
  3. Check that it matches the mathematical relationship.

Very beginner explanation: Division expressions 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 Division expressions problem. How can the learner check it?

  1. Estimate the expected size or direction of the answer.
  2. Check units, labels, signs, and place values.
  3. Use an inverse operation or a second method when possible.

Very beginner explanation: Division expressions 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 Division expressions.

  1. Think about money, measurement, data, patterns, design, or everyday quantities.
  2. Choose a situation where the topic helps compare, calculate, measure, predict, or organize.
  3. Explain the connection in one sentence.

Very beginner explanation: Division expressions 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: Division expressions 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.

  1. Identify like terms or distribute first if parentheses are present.
  2. 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.

  1. Identify like terms or distribute first if parentheses are present.
  2. 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).

  1. Identify like terms or distribute first if parentheses are present.
  2. 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.

  1. Identify like terms or distribute first if parentheses are present.
  2. 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.

  1. Identify like terms or distribute first if parentheses are present.
  2. 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 Division expressions. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.

32.8 Equivalent relationships

Equivalent relationships 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 Equivalent relationships?

  1. Read the topic name carefully.
  2. Identify the quantity, relationship, shape, or process it describes.
  3. Explain the idea without using a memorized sentence.

Very beginner explanation: Equivalent relationships 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: Equivalent relationships 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 Equivalent relationships?

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

Very beginner explanation: Equivalent relationships 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 Equivalent relationships.

  1. Decide whether a model, number line, table, graph, diagram, equation, or labelled calculation best fits the idea.
  2. Label the representation clearly.
  3. Check that it matches the mathematical relationship.

Very beginner explanation: Equivalent relationships 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 Equivalent relationships problem. How can the learner check it?

  1. Estimate the expected size or direction of the answer.
  2. Check units, labels, signs, and place values.
  3. Use an inverse operation or a second method when possible.

Very beginner explanation: Equivalent relationships 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 Equivalent relationships.

  1. Think about money, measurement, data, patterns, design, or everyday quantities.
  2. Choose a situation where the topic helps compare, calculate, measure, predict, or organize.
  3. Explain the connection in one sentence.

Very beginner explanation: Equivalent relationships 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: Equivalent relationships can be used in a real situation where mathematics helps make a clear calculation, comparison, measurement, prediction, or decision.

Worked Example 6

Problem: For y = 2x + (1), find y when x = 3.

  1. Substitute x = 3.
  2. y = 2(3) + (1).
  3. Multiply first, then add.

Very beginner explanation: A relation pairs inputs with outputs. In a linear relation, the rate of change is constant.

Answer: 7

Worked Example 7

Problem: For y = -1x + (4), find y when x = 3.

  1. Substitute x = 3.
  2. y = -1(3) + (4).
  3. Multiply first, then add.

Very beginner explanation: A relation pairs inputs with outputs. In a linear relation, the rate of change is constant.

Answer: 1

Worked Example 8

Problem: For y = 0.5x + (-2), find y when x = 3.

  1. Substitute x = 3.
  2. y = 0.5(3) + (-2).
  3. Multiply first, then add.

Very beginner explanation: A relation pairs inputs with outputs. In a linear relation, the rate of change is constant.

Answer: -0.5

Worked Example 9

Problem: For y = 3x + (0), find y when x = 3.

  1. Substitute x = 3.
  2. y = 3(3) + (0).
  3. Multiply first, then add.

Very beginner explanation: A relation pairs inputs with outputs. In a linear relation, the rate of change is constant.

Answer: 9

Worked Example 10

Problem: For y = -2x + (5), find y when x = 3.

  1. Substitute x = 3.
  2. y = -2(3) + (5).
  3. Multiply first, then add.

Very beginner explanation: A relation pairs inputs with outputs. In a linear relation, the rate of change is constant.

Answer: -1

Practice Exercise

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

32.9 Matching expressions to situations

Matching expressions to situations introduces algebraic thinking by describing relationships, using symbols for unknown values, and checking whether mathematical statements are true.

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 Matching expressions to situations?

  1. Read the topic name carefully.
  2. Identify the quantity, relationship, shape, or process it describes.
  3. Explain the idea without using a memorized sentence.

Very beginner explanation: Matching expressions to situations introduces algebraic thinking by describing relationships, using symbols for unknown values, and checking whether mathematical statements are true.

Answer: Matching expressions to situations introduces algebraic thinking by describing relationships, using symbols for unknown values, and checking whether mathematical statements are true.

Worked Example 2

Problem: What should you identify first before solving a problem about Matching expressions to situations?

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

Very beginner explanation: Matching expressions to situations introduces algebraic thinking by describing relationships, using symbols for unknown values, and checking whether mathematical statements are true.

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

Worked Example 3

Problem: Choose a useful representation for Matching expressions to situations.

  1. Decide whether a model, number line, table, graph, diagram, equation, or labelled calculation best fits the idea.
  2. Label the representation clearly.
  3. Check that it matches the mathematical relationship.

Very beginner explanation: Matching expressions to situations introduces algebraic thinking by describing relationships, using symbols for unknown values, and checking whether mathematical statements are true.

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 Matching expressions to situations problem. How can the learner check it?

  1. Estimate the expected size or direction of the answer.
  2. Check units, labels, signs, and place values.
  3. Use an inverse operation or a second method when possible.

Very beginner explanation: Matching expressions to situations introduces algebraic thinking by describing relationships, using symbols for unknown values, and checking whether mathematical statements are true.

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 Matching expressions to situations.

  1. Think about money, measurement, data, patterns, design, or everyday quantities.
  2. Choose a situation where the topic helps compare, calculate, measure, predict, or organize.
  3. Explain the connection in one sentence.

Very beginner explanation: Matching expressions to situations introduces algebraic thinking by describing relationships, using symbols for unknown values, and checking whether mathematical statements are true.

Answer: Matching expressions to situations 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.

  1. Identify like terms or distribute first if parentheses are present.
  2. 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.

  1. Identify like terms or distribute first if parentheses are present.
  2. 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).

  1. Identify like terms or distribute first if parentheses are present.
  2. 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.

  1. Identify like terms or distribute first if parentheses are present.
  2. 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.

  1. Identify like terms or distribute first if parentheses are present.
  2. 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 Matching expressions to situations. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.

32.10 Writing expressions from word problems

Writing expressions from word problems introduces algebraic thinking by describing relationships, using symbols for unknown values, and checking whether mathematical statements are true.

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 Writing expressions from word problems?

  1. Read the topic name carefully.
  2. Identify the quantity, relationship, shape, or process it describes.
  3. Explain the idea without using a memorized sentence.

Very beginner explanation: Writing expressions from word problems introduces algebraic thinking by describing relationships, using symbols for unknown values, and checking whether mathematical statements are true.

Answer: Writing expressions from word problems introduces algebraic thinking by describing relationships, using symbols for unknown values, and checking whether mathematical statements are true.

Worked Example 2

Problem: What should you identify first before solving a problem about Writing expressions from word problems?

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

Very beginner explanation: Writing expressions from word problems introduces algebraic thinking by describing relationships, using symbols for unknown values, and checking whether mathematical statements are true.

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

Worked Example 3

Problem: Choose a useful representation for Writing expressions from word problems.

  1. Decide whether a model, number line, table, graph, diagram, equation, or labelled calculation best fits the idea.
  2. Label the representation clearly.
  3. Check that it matches the mathematical relationship.

Very beginner explanation: Writing expressions from word problems introduces algebraic thinking by describing relationships, using symbols for unknown values, and checking whether mathematical statements are true.

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 Writing expressions from word problems problem. How can the learner check it?

  1. Estimate the expected size or direction of the answer.
  2. Check units, labels, signs, and place values.
  3. Use an inverse operation or a second method when possible.

Very beginner explanation: Writing expressions from word problems introduces algebraic thinking by describing relationships, using symbols for unknown values, and checking whether mathematical statements are true.

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 Writing expressions from word problems.

  1. Think about money, measurement, data, patterns, design, or everyday quantities.
  2. Choose a situation where the topic helps compare, calculate, measure, predict, or organize.
  3. Explain the connection in one sentence.

Very beginner explanation: Writing expressions from word problems introduces algebraic thinking by describing relationships, using symbols for unknown values, and checking whether mathematical statements are true.

Answer: Writing expressions from 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: Simplify 3x + 4x.

  1. Identify like terms or distribute first if parentheses are present.
  2. 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.

  1. Identify like terms or distribute first if parentheses are present.
  2. 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).

  1. Identify like terms or distribute first if parentheses are present.
  2. 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.

  1. Identify like terms or distribute first if parentheses are present.
  2. 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.

  1. Identify like terms or distribute first if parentheses are present.
  2. 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 Writing expressions from word problems. 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

  1. Create and solve one original problem about What a variable means.
  2. Create and solve one original problem about Words to algebraic expressions.
  3. Create and solve one original problem about Pictures to algebraic expressions.
  4. Create and solve one original problem about Addition expressions.
  5. Create and solve one original problem about Subtraction expressions.
  6. Create and solve one original problem about Multiplication expressions.

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 What a variable means?

Answer: What a variable means introduces algebraic thinking by describing relationships, using symbols for unknown values, and checking whether mathematical statements are true.

Q2. What is the key idea in Words to algebraic expressions?

Answer: Words to algebraic expressions introduces algebraic thinking by describing relationships, using symbols for unknown values, and checking whether mathematical statements are true.

Q3. What is the key idea in Pictures to algebraic expressions?

Answer: Pictures to algebraic expressions introduces algebraic thinking by describing relationships, using symbols for unknown values, and checking whether mathematical statements are true.

Q4. What is the key idea in Addition expressions?

Answer: Addition expressions 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.

Q5. What is the key idea in Subtraction expressions?

Answer: Subtraction expressions 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 Multiplication expressions?

Answer: Multiplication expressions 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.

Q7. What is the key idea in Division expressions?

Answer: Division expressions 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.

Q8. What is the key idea in Equivalent relationships?

Answer: Equivalent relationships 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.

Q9. What is the key idea in Matching expressions to situations?

Answer: Matching expressions to situations introduces algebraic thinking by describing relationships, using symbols for unknown values, and checking whether mathematical statements are true.

Q10. What is the key idea in Writing expressions from word problems?

Answer: Writing expressions from word problems introduces algebraic thinking by describing relationships, using symbols for unknown values, and checking whether mathematical statements are true.

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 Variables and Algebraic Expressions 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 Variables and Algebraic Expressions 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 Variables and Algebraic Expressions 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 Variables and Algebraic Expressions 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 Variables and Algebraic Expressions effectively?

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