Chapter 4: Comparing and Ordering Fractions
Learn Grade 5 mathematics from very beginner explanations through worked examples, practice, reasoning, and review.
Chapter Overview
This chapter teaches Comparing and Ordering Fractions with simple language, step-by-step reasoning, and original worked examples. Technical words are explained in plain language.
Key Technical Terms
- Comparing fractions with the same denominator (a Grade 5 idea used in this chapter)
- Comparing fractions with the same numerator (a Grade 5 idea used in this chapter)
- Using benchmarks such as one-half (a Grade 5 idea used in this chapter)
- Using number lines to compare fractions (a Grade 5 idea used in this chapter)
- Using equivalent fractions to compare (a Grade 5 idea used in this chapter)
- Comparing improper fractions (a Grade 5 idea used in this chapter)
- Comparing mixed numbers (a Grade 5 idea used in this chapter)
- Ordering several fractions (a Grade 5 idea used in this chapter)
- Fractions between two given fractions (a Grade 5 idea used in this chapter)
- Explaining fraction comparisons (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.
4.1 Comparing fractions with the same denominator
Comparing fractions with the same denominator focuses on equal parts of a whole or set. Learners use models, number lines, and equivalent forms so the fraction has a clear meaning before using a rule.
Beginner Note
Begin with meaning and a visual or concrete example. Work one step at a time and check the answer before moving on.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: In your own words, what is the main idea of Comparing fractions with the same denominator?
- 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: Comparing fractions with the same denominator focuses on equal parts of a whole or set. Learners use models, number lines, and equivalent forms so the fraction has a clear meaning before using a rule.
Answer: Comparing fractions with the same denominator focuses on equal parts of a whole or set. Learners use models, number lines, and equivalent forms so the fraction has a clear meaning before using a rule.
Worked Example 2
Problem: What should you identify first before solving a problem about Comparing fractions with the same denominator?
- Read the complete problem.
- Mark the information that is given.
- Identify exactly what the question asks you to find.
Very beginner explanation: Comparing fractions with the same denominator focuses on equal parts of a whole or set. Learners use models, number lines, and equivalent forms so the fraction has a clear meaning before using a rule.
Answer: Identify the known information and the unknown before choosing an operation or rule.
Worked Example 3
Problem: Choose a useful representation for Comparing fractions with the same denominator.
- 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: Comparing fractions with the same denominator focuses on equal parts of a whole or set. Learners use models, number lines, and equivalent forms so the fraction has a clear meaning before using a rule.
Answer: Use the representation that makes the important relationship easiest to see and check.
Worked Example 4
Problem: A learner gets an answer in a Comparing fractions with the same denominator 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: Comparing fractions with the same denominator focuses on equal parts of a whole or set. Learners use models, number lines, and equivalent forms so the fraction has a clear meaning before using a rule.
Answer: The answer should agree with an estimate, the problem conditions, and a second check.
Worked Example 5
Problem: Give a real-life use for Comparing fractions with the same denominator.
- 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: Comparing fractions with the same denominator focuses on equal parts of a whole or set. Learners use models, number lines, and equivalent forms so the fraction has a clear meaning before using a rule.
Answer: Comparing fractions with the same denominator can be used in a real situation where mathematics helps make a clear calculation, comparison, measurement, prediction, or decision.
Worked Example 6
Problem: Write 1/3 as a decimal.
- A fraction bar means division.
- Divide 1 by 3.
Very beginner explanation: Fractions and decimals can represent the same rational number.
Answer: 0.3333
Worked Example 7
Problem: Write 2/5 as a decimal.
- A fraction bar means division.
- Divide 2 by 5.
Very beginner explanation: Fractions and decimals can represent the same rational number.
Answer: 0.4
Worked Example 8
Problem: Write 3/8 as a decimal.
- A fraction bar means division.
- Divide 3 by 8.
Very beginner explanation: Fractions and decimals can represent the same rational number.
Answer: 0.375
Worked Example 9
Problem: Write 5/12 as a decimal.
- A fraction bar means division.
- Divide 5 by 12.
Very beginner explanation: Fractions and decimals can represent the same rational number.
Answer: 0.4167
Worked Example 10
Problem: Write 7/9 as a decimal.
- A fraction bar means division.
- Divide 7 by 9.
Very beginner explanation: Fractions and decimals can represent the same rational number.
Answer: 0.7778
Practice Exercise
Create one new question about Comparing fractions with the same denominator. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
4.2 Comparing fractions with the same numerator
Comparing fractions with the same numerator focuses on equal parts of a whole or set. Learners use models, number lines, and equivalent forms so the fraction has a clear meaning before using a rule.
Beginner Note
Begin with meaning and a visual or concrete example. Work one step at a time and check the answer before moving on.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: In your own words, what is the main idea of Comparing fractions with the same numerator?
- 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: Comparing fractions with the same numerator focuses on equal parts of a whole or set. Learners use models, number lines, and equivalent forms so the fraction has a clear meaning before using a rule.
Answer: Comparing fractions with the same numerator focuses on equal parts of a whole or set. Learners use models, number lines, and equivalent forms so the fraction has a clear meaning before using a rule.
Worked Example 2
Problem: What should you identify first before solving a problem about Comparing fractions with the same numerator?
- Read the complete problem.
- Mark the information that is given.
- Identify exactly what the question asks you to find.
Very beginner explanation: Comparing fractions with the same numerator focuses on equal parts of a whole or set. Learners use models, number lines, and equivalent forms so the fraction has a clear meaning before using a rule.
Answer: Identify the known information and the unknown before choosing an operation or rule.
Worked Example 3
Problem: Choose a useful representation for Comparing fractions with the same numerator.
- 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: Comparing fractions with the same numerator focuses on equal parts of a whole or set. Learners use models, number lines, and equivalent forms so the fraction has a clear meaning before using a rule.
Answer: Use the representation that makes the important relationship easiest to see and check.
Worked Example 4
Problem: A learner gets an answer in a Comparing fractions with the same numerator 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: Comparing fractions with the same numerator focuses on equal parts of a whole or set. Learners use models, number lines, and equivalent forms so the fraction has a clear meaning before using a rule.
Answer: The answer should agree with an estimate, the problem conditions, and a second check.
Worked Example 5
Problem: Give a real-life use for Comparing fractions with the same numerator.
- 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: Comparing fractions with the same numerator focuses on equal parts of a whole or set. Learners use models, number lines, and equivalent forms so the fraction has a clear meaning before using a rule.
Answer: Comparing fractions with the same numerator can be used in a real situation where mathematics helps make a clear calculation, comparison, measurement, prediction, or decision.
Worked Example 6
Problem: Write 1/3 as a decimal.
- A fraction bar means division.
- Divide 1 by 3.
Very beginner explanation: Fractions and decimals can represent the same rational number.
Answer: 0.3333
Worked Example 7
Problem: Write 2/5 as a decimal.
- A fraction bar means division.
- Divide 2 by 5.
Very beginner explanation: Fractions and decimals can represent the same rational number.
Answer: 0.4
Worked Example 8
Problem: Write 3/8 as a decimal.
- A fraction bar means division.
- Divide 3 by 8.
Very beginner explanation: Fractions and decimals can represent the same rational number.
Answer: 0.375
Worked Example 9
Problem: Write 5/12 as a decimal.
- A fraction bar means division.
- Divide 5 by 12.
Very beginner explanation: Fractions and decimals can represent the same rational number.
Answer: 0.4167
Worked Example 10
Problem: Write 7/9 as a decimal.
- A fraction bar means division.
- Divide 7 by 9.
Very beginner explanation: Fractions and decimals can represent the same rational number.
Answer: 0.7778
Practice Exercise
Create one new question about Comparing fractions with the same numerator. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
4.3 Using benchmarks such as one-half
Using benchmarks such as one-half 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 Using benchmarks such as one-half?
- 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: Using benchmarks such as one-half 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: Using benchmarks such as one-half 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 Using benchmarks such as one-half?
- Read the complete problem.
- Mark the information that is given.
- Identify exactly what the question asks you to find.
Very beginner explanation: Using benchmarks such as one-half 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 Using benchmarks such as one-half.
- 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: Using benchmarks such as one-half 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 Using benchmarks such as one-half 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: Using benchmarks such as one-half 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 Using benchmarks such as one-half.
- 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: Using benchmarks such as one-half 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: Using benchmarks such as one-half can be used in a real situation where mathematics helps make a clear calculation, comparison, measurement, prediction, or decision.
Worked Example 6
Problem: Explain Using benchmarks such as one-half 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: Using benchmarks such as one-half 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 Using benchmarks such as one-half 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: Using benchmarks such as one-half 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 Using benchmarks such as one-half 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: Using benchmarks such as one-half 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 Using benchmarks such as one-half 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: Using benchmarks such as one-half 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 Using benchmarks such as one-half 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: Using benchmarks such as one-half 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 Using benchmarks such as one-half. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
4.4 Using number lines to compare fractions
Using number lines to compare fractions focuses on equal parts of a whole or set. Learners use models, number lines, and equivalent forms so the fraction has a clear meaning before using a rule.
Beginner Note
Begin with meaning and a visual or concrete example. Work one step at a time and check the answer before moving on.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: In your own words, what is the main idea of Using number lines to compare fractions?
- 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: Using number lines to compare fractions focuses on equal parts of a whole or set. Learners use models, number lines, and equivalent forms so the fraction has a clear meaning before using a rule.
Answer: Using number lines to compare fractions focuses on equal parts of a whole or set. Learners use models, number lines, and equivalent forms so the fraction has a clear meaning before using a rule.
Worked Example 2
Problem: What should you identify first before solving a problem about Using number lines to compare fractions?
- Read the complete problem.
- Mark the information that is given.
- Identify exactly what the question asks you to find.
Very beginner explanation: Using number lines to compare fractions focuses on equal parts of a whole or set. Learners use models, number lines, and equivalent forms so the fraction has a clear meaning before using a rule.
Answer: Identify the known information and the unknown before choosing an operation or rule.
Worked Example 3
Problem: Choose a useful representation for Using number lines to compare fractions.
- 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: Using number lines to compare fractions focuses on equal parts of a whole or set. Learners use models, number lines, and equivalent forms so the fraction has a clear meaning before using a rule.
Answer: Use the representation that makes the important relationship easiest to see and check.
Worked Example 4
Problem: A learner gets an answer in a Using number lines to compare fractions 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: Using number lines to compare fractions focuses on equal parts of a whole or set. Learners use models, number lines, and equivalent forms so the fraction has a clear meaning before using a rule.
Answer: The answer should agree with an estimate, the problem conditions, and a second check.
Worked Example 5
Problem: Give a real-life use for Using number lines to compare fractions.
- 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: Using number lines to compare fractions focuses on equal parts of a whole or set. Learners use models, number lines, and equivalent forms so the fraction has a clear meaning before using a rule.
Answer: Using number lines to compare fractions can be used in a real situation where mathematics helps make a clear calculation, comparison, measurement, prediction, or decision.
Worked Example 6
Problem: Which is farther right on a number line: 3,000 or 3,750?
- Numbers get larger as you move right.
- 3,750 is greater than 3,000.
Very beginner explanation: On a standard number line, the greater number is farther to the right.
Answer: 3,750
Worked Example 7
Problem: Which is farther right on a number line: 4,000 or 4,750?
- Numbers get larger as you move right.
- 4,750 is greater than 4,000.
Very beginner explanation: On a standard number line, the greater number is farther to the right.
Answer: 4,750
Worked Example 8
Problem: Which is farther right on a number line: 5,000 or 5,750?
- Numbers get larger as you move right.
- 5,750 is greater than 5,000.
Very beginner explanation: On a standard number line, the greater number is farther to the right.
Answer: 5,750
Worked Example 9
Problem: Which is farther right on a number line: 6,000 or 6,750?
- Numbers get larger as you move right.
- 6,750 is greater than 6,000.
Very beginner explanation: On a standard number line, the greater number is farther to the right.
Answer: 6,750
Worked Example 10
Problem: Which is farther right on a number line: 7,000 or 7,750?
- Numbers get larger as you move right.
- 7,750 is greater than 7,000.
Very beginner explanation: On a standard number line, the greater number is farther to the right.
Answer: 7,750
Practice Exercise
Create one new question about Using number lines to compare fractions. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
4.5 Using equivalent fractions to compare
Using equivalent fractions to compare focuses on equal parts of a whole or set. Learners use models, number lines, and equivalent forms so the fraction has a clear meaning before using a rule.
Beginner Note
Begin with meaning and a visual or concrete example. Work one step at a time and check the answer before moving on.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: In your own words, what is the main idea of Using equivalent fractions to compare?
- 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: Using equivalent fractions to compare focuses on equal parts of a whole or set. Learners use models, number lines, and equivalent forms so the fraction has a clear meaning before using a rule.
Answer: Using equivalent fractions to compare focuses on equal parts of a whole or set. Learners use models, number lines, and equivalent forms so the fraction has a clear meaning before using a rule.
Worked Example 2
Problem: What should you identify first before solving a problem about Using equivalent fractions to compare?
- Read the complete problem.
- Mark the information that is given.
- Identify exactly what the question asks you to find.
Very beginner explanation: Using equivalent fractions to compare focuses on equal parts of a whole or set. Learners use models, number lines, and equivalent forms so the fraction has a clear meaning before using a rule.
Answer: Identify the known information and the unknown before choosing an operation or rule.
Worked Example 3
Problem: Choose a useful representation for Using equivalent fractions to compare.
- 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: Using equivalent fractions to compare focuses on equal parts of a whole or set. Learners use models, number lines, and equivalent forms so the fraction has a clear meaning before using a rule.
Answer: Use the representation that makes the important relationship easiest to see and check.
Worked Example 4
Problem: A learner gets an answer in a Using equivalent fractions to compare 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: Using equivalent fractions to compare focuses on equal parts of a whole or set. Learners use models, number lines, and equivalent forms so the fraction has a clear meaning before using a rule.
Answer: The answer should agree with an estimate, the problem conditions, and a second check.
Worked Example 5
Problem: Give a real-life use for Using equivalent fractions to compare.
- 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: Using equivalent fractions to compare focuses on equal parts of a whole or set. Learners use models, number lines, and equivalent forms so the fraction has a clear meaning before using a rule.
Answer: Using equivalent fractions to compare can be used in a real situation where mathematics helps make a clear calculation, comparison, measurement, prediction, or decision.
Worked Example 6
Problem: Write 1/3 as a decimal.
- A fraction bar means division.
- Divide 1 by 3.
Very beginner explanation: Fractions and decimals can represent the same rational number.
Answer: 0.3333
Worked Example 7
Problem: Write 2/5 as a decimal.
- A fraction bar means division.
- Divide 2 by 5.
Very beginner explanation: Fractions and decimals can represent the same rational number.
Answer: 0.4
Worked Example 8
Problem: Write 3/8 as a decimal.
- A fraction bar means division.
- Divide 3 by 8.
Very beginner explanation: Fractions and decimals can represent the same rational number.
Answer: 0.375
Worked Example 9
Problem: Write 5/12 as a decimal.
- A fraction bar means division.
- Divide 5 by 12.
Very beginner explanation: Fractions and decimals can represent the same rational number.
Answer: 0.4167
Worked Example 10
Problem: Write 7/9 as a decimal.
- A fraction bar means division.
- Divide 7 by 9.
Very beginner explanation: Fractions and decimals can represent the same rational number.
Answer: 0.7778
Practice Exercise
Create one new question about Using equivalent fractions to compare. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
4.6 Comparing improper fractions
Comparing improper fractions focuses on equal parts of a whole or set. Learners use models, number lines, and equivalent forms so the fraction has a clear meaning before using a rule.
Beginner Note
Begin with meaning and a visual or concrete example. Work one step at a time and check the answer before moving on.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: In your own words, what is the main idea of Comparing improper fractions?
- 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: Comparing improper fractions focuses on equal parts of a whole or set. Learners use models, number lines, and equivalent forms so the fraction has a clear meaning before using a rule.
Answer: Comparing improper fractions focuses on equal parts of a whole or set. Learners use models, number lines, and equivalent forms so the fraction has a clear meaning before using a rule.
Worked Example 2
Problem: What should you identify first before solving a problem about Comparing improper fractions?
- Read the complete problem.
- Mark the information that is given.
- Identify exactly what the question asks you to find.
Very beginner explanation: Comparing improper fractions focuses on equal parts of a whole or set. Learners use models, number lines, and equivalent forms so the fraction has a clear meaning before using a rule.
Answer: Identify the known information and the unknown before choosing an operation or rule.
Worked Example 3
Problem: Choose a useful representation for Comparing improper fractions.
- 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: Comparing improper fractions focuses on equal parts of a whole or set. Learners use models, number lines, and equivalent forms so the fraction has a clear meaning before using a rule.
Answer: Use the representation that makes the important relationship easiest to see and check.
Worked Example 4
Problem: A learner gets an answer in a Comparing improper fractions 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: Comparing improper fractions focuses on equal parts of a whole or set. Learners use models, number lines, and equivalent forms so the fraction has a clear meaning before using a rule.
Answer: The answer should agree with an estimate, the problem conditions, and a second check.
Worked Example 5
Problem: Give a real-life use for Comparing improper fractions.
- 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: Comparing improper fractions focuses on equal parts of a whole or set. Learners use models, number lines, and equivalent forms so the fraction has a clear meaning before using a rule.
Answer: Comparing improper fractions can be used in a real situation where mathematics helps make a clear calculation, comparison, measurement, prediction, or decision.
Worked Example 6
Problem: Write 1/3 as a decimal.
- A fraction bar means division.
- Divide 1 by 3.
Very beginner explanation: Fractions and decimals can represent the same rational number.
Answer: 0.3333
Worked Example 7
Problem: Write 2/5 as a decimal.
- A fraction bar means division.
- Divide 2 by 5.
Very beginner explanation: Fractions and decimals can represent the same rational number.
Answer: 0.4
Worked Example 8
Problem: Write 3/8 as a decimal.
- A fraction bar means division.
- Divide 3 by 8.
Very beginner explanation: Fractions and decimals can represent the same rational number.
Answer: 0.375
Worked Example 9
Problem: Write 5/12 as a decimal.
- A fraction bar means division.
- Divide 5 by 12.
Very beginner explanation: Fractions and decimals can represent the same rational number.
Answer: 0.4167
Worked Example 10
Problem: Write 7/9 as a decimal.
- A fraction bar means division.
- Divide 7 by 9.
Very beginner explanation: Fractions and decimals can represent the same rational number.
Answer: 0.7778
Practice Exercise
Create one new question about Comparing improper fractions. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
4.7 Comparing mixed numbers
Comparing mixed numbers 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 Comparing mixed numbers?
- 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: Comparing mixed numbers 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: Comparing mixed numbers 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 Comparing mixed numbers?
- Read the complete problem.
- Mark the information that is given.
- Identify exactly what the question asks you to find.
Very beginner explanation: Comparing mixed numbers 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 Comparing mixed numbers.
- 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: Comparing mixed numbers 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 Comparing mixed numbers 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: Comparing mixed numbers 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 Comparing mixed numbers.
- 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: Comparing mixed numbers 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: Comparing mixed numbers can be used in a real situation where mathematics helps make a clear calculation, comparison, measurement, prediction, or decision.
Worked Example 6
Problem: Write 1/3 as a decimal.
- A fraction bar means division.
- Divide 1 by 3.
Very beginner explanation: Fractions and decimals can represent the same rational number.
Answer: 0.3333
Worked Example 7
Problem: Write 2/5 as a decimal.
- A fraction bar means division.
- Divide 2 by 5.
Very beginner explanation: Fractions and decimals can represent the same rational number.
Answer: 0.4
Worked Example 8
Problem: Write 3/8 as a decimal.
- A fraction bar means division.
- Divide 3 by 8.
Very beginner explanation: Fractions and decimals can represent the same rational number.
Answer: 0.375
Worked Example 9
Problem: Write 5/12 as a decimal.
- A fraction bar means division.
- Divide 5 by 12.
Very beginner explanation: Fractions and decimals can represent the same rational number.
Answer: 0.4167
Worked Example 10
Problem: Write 7/9 as a decimal.
- A fraction bar means division.
- Divide 7 by 9.
Very beginner explanation: Fractions and decimals can represent the same rational number.
Answer: 0.7778
Practice Exercise
Create one new question about Comparing mixed numbers. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
4.8 Ordering several fractions
Ordering several fractions focuses on equal parts of a whole or set. Learners use models, number lines, and equivalent forms so the fraction has a clear meaning before using a rule.
Beginner Note
Begin with meaning and a visual or concrete example. Work one step at a time and check the answer before moving on.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: In your own words, what is the main idea of Ordering several fractions?
- 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: Ordering several fractions focuses on equal parts of a whole or set. Learners use models, number lines, and equivalent forms so the fraction has a clear meaning before using a rule.
Answer: Ordering several fractions focuses on equal parts of a whole or set. Learners use models, number lines, and equivalent forms so the fraction has a clear meaning before using a rule.
Worked Example 2
Problem: What should you identify first before solving a problem about Ordering several fractions?
- Read the complete problem.
- Mark the information that is given.
- Identify exactly what the question asks you to find.
Very beginner explanation: Ordering several fractions focuses on equal parts of a whole or set. Learners use models, number lines, and equivalent forms so the fraction has a clear meaning before using a rule.
Answer: Identify the known information and the unknown before choosing an operation or rule.
Worked Example 3
Problem: Choose a useful representation for Ordering several fractions.
- 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: Ordering several fractions focuses on equal parts of a whole or set. Learners use models, number lines, and equivalent forms so the fraction has a clear meaning before using a rule.
Answer: Use the representation that makes the important relationship easiest to see and check.
Worked Example 4
Problem: A learner gets an answer in a Ordering several fractions 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: Ordering several fractions focuses on equal parts of a whole or set. Learners use models, number lines, and equivalent forms so the fraction has a clear meaning before using a rule.
Answer: The answer should agree with an estimate, the problem conditions, and a second check.
Worked Example 5
Problem: Give a real-life use for Ordering several fractions.
- 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: Ordering several fractions focuses on equal parts of a whole or set. Learners use models, number lines, and equivalent forms so the fraction has a clear meaning before using a rule.
Answer: Ordering several fractions can be used in a real situation where mathematics helps make a clear calculation, comparison, measurement, prediction, or decision.
Worked Example 6
Problem: Write 1/3 as a decimal.
- A fraction bar means division.
- Divide 1 by 3.
Very beginner explanation: Fractions and decimals can represent the same rational number.
Answer: 0.3333
Worked Example 7
Problem: Write 2/5 as a decimal.
- A fraction bar means division.
- Divide 2 by 5.
Very beginner explanation: Fractions and decimals can represent the same rational number.
Answer: 0.4
Worked Example 8
Problem: Write 3/8 as a decimal.
- A fraction bar means division.
- Divide 3 by 8.
Very beginner explanation: Fractions and decimals can represent the same rational number.
Answer: 0.375
Worked Example 9
Problem: Write 5/12 as a decimal.
- A fraction bar means division.
- Divide 5 by 12.
Very beginner explanation: Fractions and decimals can represent the same rational number.
Answer: 0.4167
Worked Example 10
Problem: Write 7/9 as a decimal.
- A fraction bar means division.
- Divide 7 by 9.
Very beginner explanation: Fractions and decimals can represent the same rational number.
Answer: 0.7778
Practice Exercise
Create one new question about Ordering several fractions. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
4.9 Fractions between two given fractions
Fractions between two given fractions focuses on equal parts of a whole or set. Learners use models, number lines, and equivalent forms so the fraction has a clear meaning before using a rule.
Beginner Note
Begin with meaning and a visual or concrete example. Work one step at a time and check the answer before moving on.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: In your own words, what is the main idea of Fractions between two given fractions?
- 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: Fractions between two given fractions focuses on equal parts of a whole or set. Learners use models, number lines, and equivalent forms so the fraction has a clear meaning before using a rule.
Answer: Fractions between two given fractions focuses on equal parts of a whole or set. Learners use models, number lines, and equivalent forms so the fraction has a clear meaning before using a rule.
Worked Example 2
Problem: What should you identify first before solving a problem about Fractions between two given fractions?
- Read the complete problem.
- Mark the information that is given.
- Identify exactly what the question asks you to find.
Very beginner explanation: Fractions between two given fractions focuses on equal parts of a whole or set. Learners use models, number lines, and equivalent forms so the fraction has a clear meaning before using a rule.
Answer: Identify the known information and the unknown before choosing an operation or rule.
Worked Example 3
Problem: Choose a useful representation for Fractions between two given fractions.
- 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: Fractions between two given fractions focuses on equal parts of a whole or set. Learners use models, number lines, and equivalent forms so the fraction has a clear meaning before using a rule.
Answer: Use the representation that makes the important relationship easiest to see and check.
Worked Example 4
Problem: A learner gets an answer in a Fractions between two given fractions 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: Fractions between two given fractions focuses on equal parts of a whole or set. Learners use models, number lines, and equivalent forms so the fraction has a clear meaning before using a rule.
Answer: The answer should agree with an estimate, the problem conditions, and a second check.
Worked Example 5
Problem: Give a real-life use for Fractions between two given fractions.
- 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: Fractions between two given fractions focuses on equal parts of a whole or set. Learners use models, number lines, and equivalent forms so the fraction has a clear meaning before using a rule.
Answer: Fractions between two given fractions can be used in a real situation where mathematics helps make a clear calculation, comparison, measurement, prediction, or decision.
Worked Example 6
Problem: Write 1/3 as a decimal.
- A fraction bar means division.
- Divide 1 by 3.
Very beginner explanation: Fractions and decimals can represent the same rational number.
Answer: 0.3333
Worked Example 7
Problem: Write 2/5 as a decimal.
- A fraction bar means division.
- Divide 2 by 5.
Very beginner explanation: Fractions and decimals can represent the same rational number.
Answer: 0.4
Worked Example 8
Problem: Write 3/8 as a decimal.
- A fraction bar means division.
- Divide 3 by 8.
Very beginner explanation: Fractions and decimals can represent the same rational number.
Answer: 0.375
Worked Example 9
Problem: Write 5/12 as a decimal.
- A fraction bar means division.
- Divide 5 by 12.
Very beginner explanation: Fractions and decimals can represent the same rational number.
Answer: 0.4167
Worked Example 10
Problem: Write 7/9 as a decimal.
- A fraction bar means division.
- Divide 7 by 9.
Very beginner explanation: Fractions and decimals can represent the same rational number.
Answer: 0.7778
Practice Exercise
Create one new question about Fractions between two given fractions. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
4.10 Explaining fraction comparisons
Explaining fraction comparisons focuses on equal parts of a whole or set. Learners use models, number lines, and equivalent forms so the fraction has a clear meaning before using a rule.
Beginner Note
Begin with meaning and a visual or concrete example. Work one step at a time and check the answer before moving on.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: In your own words, what is the main idea of Explaining fraction comparisons?
- Read the topic name carefully.
- Identify the quantity, relationship, shape, or process it describes.
- Explain the idea without using a memorized sentence.
Very beginner explanation: Explaining fraction comparisons focuses on equal parts of a whole or set. Learners use models, number lines, and equivalent forms so the fraction has a clear meaning before using a rule.
Answer: Explaining fraction comparisons focuses on equal parts of a whole or set. Learners use models, number lines, and equivalent forms so the fraction has a clear meaning before using a rule.
Worked Example 2
Problem: What should you identify first before solving a problem about Explaining fraction comparisons?
- Read the complete problem.
- Mark the information that is given.
- Identify exactly what the question asks you to find.
Very beginner explanation: Explaining fraction comparisons focuses on equal parts of a whole or set. Learners use models, number lines, and equivalent forms so the fraction has a clear meaning before using a rule.
Answer: Identify the known information and the unknown before choosing an operation or rule.
Worked Example 3
Problem: Choose a useful representation for Explaining fraction comparisons.
- Decide whether a model, number line, table, graph, diagram, equation, or labelled calculation best fits the idea.
- Label the representation clearly.
- Check that it matches the mathematical relationship.
Very beginner explanation: Explaining fraction comparisons focuses on equal parts of a whole or set. Learners use models, number lines, and equivalent forms so the fraction has a clear meaning before using a rule.
Answer: Use the representation that makes the important relationship easiest to see and check.
Worked Example 4
Problem: A learner gets an answer in a Explaining fraction comparisons problem. How can the learner check it?
- Estimate the expected size or direction of the answer.
- Check units, labels, signs, and place values.
- Use an inverse operation or a second method when possible.
Very beginner explanation: Explaining fraction comparisons focuses on equal parts of a whole or set. Learners use models, number lines, and equivalent forms so the fraction has a clear meaning before using a rule.
Answer: The answer should agree with an estimate, the problem conditions, and a second check.
Worked Example 5
Problem: Give a real-life use for Explaining fraction comparisons.
- Think about money, measurement, data, patterns, design, or everyday quantities.
- Choose a situation where the topic helps compare, calculate, measure, predict, or organize.
- Explain the connection in one sentence.
Very beginner explanation: Explaining fraction comparisons focuses on equal parts of a whole or set. Learners use models, number lines, and equivalent forms so the fraction has a clear meaning before using a rule.
Answer: Explaining fraction comparisons can be used in a real situation where mathematics helps make a clear calculation, comparison, measurement, prediction, or decision.
Worked Example 6
Problem: Write 1/3 as a decimal.
- A fraction bar means division.
- Divide 1 by 3.
Very beginner explanation: Fractions and decimals can represent the same rational number.
Answer: 0.3333
Worked Example 7
Problem: Write 2/5 as a decimal.
- A fraction bar means division.
- Divide 2 by 5.
Very beginner explanation: Fractions and decimals can represent the same rational number.
Answer: 0.4
Worked Example 8
Problem: Write 3/8 as a decimal.
- A fraction bar means division.
- Divide 3 by 8.
Very beginner explanation: Fractions and decimals can represent the same rational number.
Answer: 0.375
Worked Example 9
Problem: Write 5/12 as a decimal.
- A fraction bar means division.
- Divide 5 by 12.
Very beginner explanation: Fractions and decimals can represent the same rational number.
Answer: 0.4167
Worked Example 10
Problem: Write 7/9 as a decimal.
- A fraction bar means division.
- Divide 7 by 9.
Very beginner explanation: Fractions and decimals can represent the same rational number.
Answer: 0.7778
Practice Exercise
Create one new question about Explaining fraction comparisons. 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 Comparing fractions with the same denominator.
- Create and solve one original problem about Comparing fractions with the same numerator.
- Create and solve one original problem about Using benchmarks such as one-half.
- Create and solve one original problem about Using number lines to compare fractions.
- Create and solve one original problem about Using equivalent fractions to compare.
- Create and solve one original problem about Comparing improper fractions.
Common Mistakes
- Skipping the meaning and trying to memorize a rule only.
- Using the wrong operation because the question was not read completely.
- Ignoring units, labels, place values, or the context of the problem.
- Not estimating or checking whether the final answer is reasonable.
30 Review Questions and Answers
Q1. What is the key idea in Comparing fractions with the same denominator?
Answer: Comparing fractions with the same denominator focuses on equal parts of a whole or set. Learners use models, number lines, and equivalent forms so the fraction has a clear meaning before using a rule.
Q2. What is the key idea in Comparing fractions with the same numerator?
Answer: Comparing fractions with the same numerator focuses on equal parts of a whole or set. Learners use models, number lines, and equivalent forms so the fraction has a clear meaning before using a rule.
Q3. What is the key idea in Using benchmarks such as one-half?
Answer: Using benchmarks such as one-half 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.
Q4. What is the key idea in Using number lines to compare fractions?
Answer: Using number lines to compare fractions focuses on equal parts of a whole or set. Learners use models, number lines, and equivalent forms so the fraction has a clear meaning before using a rule.
Q5. What is the key idea in Using equivalent fractions to compare?
Answer: Using equivalent fractions to compare focuses on equal parts of a whole or set. Learners use models, number lines, and equivalent forms so the fraction has a clear meaning before using a rule.
Q6. What is the key idea in Comparing improper fractions?
Answer: Comparing improper fractions focuses on equal parts of a whole or set. Learners use models, number lines, and equivalent forms so the fraction has a clear meaning before using a rule.
Q7. What is the key idea in Comparing mixed numbers?
Answer: Comparing mixed numbers 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.
Q8. What is the key idea in Ordering several fractions?
Answer: Ordering several fractions focuses on equal parts of a whole or set. Learners use models, number lines, and equivalent forms so the fraction has a clear meaning before using a rule.
Q9. What is the key idea in Fractions between two given fractions?
Answer: Fractions between two given fractions focuses on equal parts of a whole or set. Learners use models, number lines, and equivalent forms so the fraction has a clear meaning before using a rule.
Q10. What is the key idea in Explaining fraction comparisons?
Answer: Explaining fraction comparisons focuses on equal parts of a whole or set. Learners use models, number lines, and equivalent forms so the fraction has a clear meaning before using a rule.
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 Comparing and Ordering Fractions 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 Comparing and Ordering Fractions 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 Comparing and Ordering Fractions 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 Comparing and Ordering Fractions 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 Comparing and Ordering Fractions effectively?
Answer: Work a small example, explain every step, check the answer, then try a similar problem without looking at the first solution.