Chapter 30: Pattern Rules, Predictions, and Missing Elements
Learn Grade 5 mathematics from very beginner explanations through worked examples, practice, reasoning, and review.
Chapter Overview
This chapter teaches Pattern Rules, Predictions, and Missing Elements with simple language, step-by-step reasoning, and original worked examples. Technical words are explained in plain language.
Key Technical Terms
- Finding a one-step pattern rule (a Grade 5 idea used in this chapter)
- Finding a two-step pattern rule (a Grade 5 idea used in this chapter)
- Extending growing patterns (a Grade 5 idea used in this chapter)
- Extending shrinking patterns (a Grade 5 idea used in this chapter)
- Finding missing terms (a Grade 5 idea used in this chapter)
- Making predictions (a Grade 5 idea used in this chapter)
- Justifying predictions (a Grade 5 idea used in this chapter)
- Testing a pattern rule (a Grade 5 idea used in this chapter)
- Finding errors in patterns (a Grade 5 idea used in this chapter)
- Writing a pattern rule in words (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.
30.1 Finding a one-step pattern rule
Finding a one-step pattern rule 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 Finding a one-step pattern rule?
- 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: Finding a one-step pattern rule introduces algebraic thinking by describing relationships, using symbols for unknown values, and checking whether mathematical statements are true.
Answer: Finding a one-step pattern rule 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 Finding a one-step pattern rule?
- Read the complete problem.
- Mark the information that is given.
- Identify exactly what the question asks you to find.
Very beginner explanation: Finding a one-step pattern rule 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 Finding a one-step pattern rule.
- 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: Finding a one-step pattern rule 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 Finding a one-step pattern rule 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: Finding a one-step pattern rule 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 Finding a one-step pattern rule.
- 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: Finding a one-step pattern rule introduces algebraic thinking by describing relationships, using symbols for unknown values, and checking whether mathematical statements are true.
Answer: Finding a one-step pattern rule can be used in a real situation where mathematics helps make a clear calculation, comparison, measurement, prediction, or decision.
Worked Example 6
Problem: Continue the pattern 2, 5, 8, ... for two more terms.
- The common difference is 3.
- Add the same difference each time.
Very beginner explanation: A constant-difference pattern changes by the same amount from one term to the next.
Answer: 11, 14
Worked Example 7
Problem: Continue the pattern 5, 9, 13, ... for two more terms.
- The common difference is 4.
- Add the same difference each time.
Very beginner explanation: A constant-difference pattern changes by the same amount from one term to the next.
Answer: 17, 21
Worked Example 8
Problem: Continue the pattern 10, 8, 6, ... for two more terms.
- The common difference is -2.
- Add the same difference each time.
Very beginner explanation: A constant-difference pattern changes by the same amount from one term to the next.
Answer: 4, 2
Worked Example 9
Problem: Continue the pattern 1.5, 2, 2.5, ... for two more terms.
- The common difference is 0.5.
- Add the same difference each time.
Very beginner explanation: A constant-difference pattern changes by the same amount from one term to the next.
Answer: 3, 3.5
Worked Example 10
Problem: Continue the pattern 20, 17.5, 15, ... for two more terms.
- The common difference is -2.5.
- Add the same difference each time.
Very beginner explanation: A constant-difference pattern changes by the same amount from one term to the next.
Answer: 12.5, 10
Practice Exercise
Create one new question about Finding a one-step pattern rule. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
30.2 Finding a two-step pattern rule
Finding a two-step pattern rule 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 Finding a two-step pattern rule?
- 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: Finding a two-step pattern rule introduces algebraic thinking by describing relationships, using symbols for unknown values, and checking whether mathematical statements are true.
Answer: Finding a two-step pattern rule 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 Finding a two-step pattern rule?
- Read the complete problem.
- Mark the information that is given.
- Identify exactly what the question asks you to find.
Very beginner explanation: Finding a two-step pattern rule 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 Finding a two-step pattern rule.
- 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: Finding a two-step pattern rule 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 Finding a two-step pattern rule 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: Finding a two-step pattern rule 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 Finding a two-step pattern rule.
- 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: Finding a two-step pattern rule introduces algebraic thinking by describing relationships, using symbols for unknown values, and checking whether mathematical statements are true.
Answer: Finding a two-step pattern rule can be used in a real situation where mathematics helps make a clear calculation, comparison, measurement, prediction, or decision.
Worked Example 6
Problem: Continue the pattern 2, 5, 8, ... for two more terms.
- The common difference is 3.
- Add the same difference each time.
Very beginner explanation: A constant-difference pattern changes by the same amount from one term to the next.
Answer: 11, 14
Worked Example 7
Problem: Continue the pattern 5, 9, 13, ... for two more terms.
- The common difference is 4.
- Add the same difference each time.
Very beginner explanation: A constant-difference pattern changes by the same amount from one term to the next.
Answer: 17, 21
Worked Example 8
Problem: Continue the pattern 10, 8, 6, ... for two more terms.
- The common difference is -2.
- Add the same difference each time.
Very beginner explanation: A constant-difference pattern changes by the same amount from one term to the next.
Answer: 4, 2
Worked Example 9
Problem: Continue the pattern 1.5, 2, 2.5, ... for two more terms.
- The common difference is 0.5.
- Add the same difference each time.
Very beginner explanation: A constant-difference pattern changes by the same amount from one term to the next.
Answer: 3, 3.5
Worked Example 10
Problem: Continue the pattern 20, 17.5, 15, ... for two more terms.
- The common difference is -2.5.
- Add the same difference each time.
Very beginner explanation: A constant-difference pattern changes by the same amount from one term to the next.
Answer: 12.5, 10
Practice Exercise
Create one new question about Finding a two-step pattern rule. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
30.3 Extending growing patterns
Extending growing patterns 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 Extending growing patterns?
- 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: Extending growing patterns introduces algebraic thinking by describing relationships, using symbols for unknown values, and checking whether mathematical statements are true.
Answer: Extending growing patterns 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 Extending growing patterns?
- Read the complete problem.
- Mark the information that is given.
- Identify exactly what the question asks you to find.
Very beginner explanation: Extending growing patterns 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 Extending growing patterns.
- 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: Extending growing patterns 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 Extending growing patterns 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: Extending growing patterns 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 Extending growing patterns.
- 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: Extending growing patterns introduces algebraic thinking by describing relationships, using symbols for unknown values, and checking whether mathematical statements are true.
Answer: Extending growing patterns can be used in a real situation where mathematics helps make a clear calculation, comparison, measurement, prediction, or decision.
Worked Example 6
Problem: Continue the pattern 2, 5, 8, ... for two more terms.
- The common difference is 3.
- Add the same difference each time.
Very beginner explanation: A constant-difference pattern changes by the same amount from one term to the next.
Answer: 11, 14
Worked Example 7
Problem: Continue the pattern 5, 9, 13, ... for two more terms.
- The common difference is 4.
- Add the same difference each time.
Very beginner explanation: A constant-difference pattern changes by the same amount from one term to the next.
Answer: 17, 21
Worked Example 8
Problem: Continue the pattern 10, 8, 6, ... for two more terms.
- The common difference is -2.
- Add the same difference each time.
Very beginner explanation: A constant-difference pattern changes by the same amount from one term to the next.
Answer: 4, 2
Worked Example 9
Problem: Continue the pattern 1.5, 2, 2.5, ... for two more terms.
- The common difference is 0.5.
- Add the same difference each time.
Very beginner explanation: A constant-difference pattern changes by the same amount from one term to the next.
Answer: 3, 3.5
Worked Example 10
Problem: Continue the pattern 20, 17.5, 15, ... for two more terms.
- The common difference is -2.5.
- Add the same difference each time.
Very beginner explanation: A constant-difference pattern changes by the same amount from one term to the next.
Answer: 12.5, 10
Practice Exercise
Create one new question about Extending growing patterns. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
30.4 Extending shrinking patterns
Extending shrinking patterns 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 Extending shrinking patterns?
- 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: Extending shrinking patterns introduces algebraic thinking by describing relationships, using symbols for unknown values, and checking whether mathematical statements are true.
Answer: Extending shrinking patterns 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 Extending shrinking patterns?
- Read the complete problem.
- Mark the information that is given.
- Identify exactly what the question asks you to find.
Very beginner explanation: Extending shrinking patterns 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 Extending shrinking patterns.
- 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: Extending shrinking patterns 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 Extending shrinking patterns 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: Extending shrinking patterns 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 Extending shrinking patterns.
- 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: Extending shrinking patterns introduces algebraic thinking by describing relationships, using symbols for unknown values, and checking whether mathematical statements are true.
Answer: Extending shrinking patterns can be used in a real situation where mathematics helps make a clear calculation, comparison, measurement, prediction, or decision.
Worked Example 6
Problem: Continue the pattern 2, 5, 8, ... for two more terms.
- The common difference is 3.
- Add the same difference each time.
Very beginner explanation: A constant-difference pattern changes by the same amount from one term to the next.
Answer: 11, 14
Worked Example 7
Problem: Continue the pattern 5, 9, 13, ... for two more terms.
- The common difference is 4.
- Add the same difference each time.
Very beginner explanation: A constant-difference pattern changes by the same amount from one term to the next.
Answer: 17, 21
Worked Example 8
Problem: Continue the pattern 10, 8, 6, ... for two more terms.
- The common difference is -2.
- Add the same difference each time.
Very beginner explanation: A constant-difference pattern changes by the same amount from one term to the next.
Answer: 4, 2
Worked Example 9
Problem: Continue the pattern 1.5, 2, 2.5, ... for two more terms.
- The common difference is 0.5.
- Add the same difference each time.
Very beginner explanation: A constant-difference pattern changes by the same amount from one term to the next.
Answer: 3, 3.5
Worked Example 10
Problem: Continue the pattern 20, 17.5, 15, ... for two more terms.
- The common difference is -2.5.
- Add the same difference each time.
Very beginner explanation: A constant-difference pattern changes by the same amount from one term to the next.
Answer: 12.5, 10
Practice Exercise
Create one new question about Extending shrinking patterns. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
30.5 Finding missing terms
Finding missing terms 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 Finding missing terms?
- 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: Finding missing terms 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: Finding missing terms 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 Finding missing terms?
- Read the complete problem.
- Mark the information that is given.
- Identify exactly what the question asks you to find.
Very beginner explanation: Finding missing terms 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 Finding missing terms.
- 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: Finding missing terms 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 Finding missing terms 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: Finding missing terms 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 Finding missing terms.
- 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: Finding missing terms 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: Finding missing terms can be used in a real situation where mathematics helps make a clear calculation, comparison, measurement, prediction, or decision.
Worked Example 6
Problem: Simplify 3x + 4x.
- Identify like terms or distribute first if parentheses are present.
- Combine coefficients carefully.
Very beginner explanation: Like terms have the same variable part. The distributive property multiplies every term inside parentheses.
Answer: 7x
Worked Example 7
Problem: Simplify 8y - 3y + 2.
- Identify like terms or distribute first if parentheses are present.
- Combine coefficients carefully.
Very beginner explanation: Like terms have the same variable part. The distributive property multiplies every term inside parentheses.
Answer: 5y + 2
Worked Example 8
Problem: Simplify 4(a + 2).
- Identify like terms or distribute first if parentheses are present.
- Combine coefficients carefully.
Very beginner explanation: Like terms have the same variable part. The distributive property multiplies every term inside parentheses.
Answer: 4a + 8
Worked Example 9
Problem: Simplify 2(3x - 5) + x.
- Identify like terms or distribute first if parentheses are present.
- Combine coefficients carefully.
Very beginner explanation: Like terms have the same variable part. The distributive property multiplies every term inside parentheses.
Answer: 7x - 10
Worked Example 10
Problem: Simplify 6m + 7 - 2m - 3.
- Identify like terms or distribute first if parentheses are present.
- Combine coefficients carefully.
Very beginner explanation: Like terms have the same variable part. The distributive property multiplies every term inside parentheses.
Answer: 4m + 4
Practice Exercise
Create one new question about Finding missing terms. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
30.6 Making predictions
Making predictions 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 Making predictions?
- 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: Making predictions 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: Making predictions 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 Making predictions?
- Read the complete problem.
- Mark the information that is given.
- Identify exactly what the question asks you to find.
Very beginner explanation: Making predictions 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 Making predictions.
- 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: Making predictions 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 Making predictions 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: Making predictions 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 Making predictions.
- 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: Making predictions 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: Making predictions can be used in a real situation where mathematics helps make a clear calculation, comparison, measurement, prediction, or decision.
Worked Example 6
Problem: Explain Making predictions 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: Making predictions 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 Making predictions 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: Making predictions 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 Making predictions 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: Making predictions 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 Making predictions 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: Making predictions 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 Making predictions 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: Making predictions 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 Making predictions. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
30.7 Justifying predictions
Justifying predictions 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 Justifying predictions?
- 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: Justifying predictions 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: Justifying predictions 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 Justifying predictions?
- Read the complete problem.
- Mark the information that is given.
- Identify exactly what the question asks you to find.
Very beginner explanation: Justifying predictions 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 Justifying predictions.
- 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: Justifying predictions 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 Justifying predictions 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: Justifying predictions 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 Justifying predictions.
- 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: Justifying predictions 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: Justifying predictions can be used in a real situation where mathematics helps make a clear calculation, comparison, measurement, prediction, or decision.
Worked Example 6
Problem: Explain Justifying predictions 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: Justifying predictions 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 Justifying predictions 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: Justifying predictions 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 Justifying predictions 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: Justifying predictions 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 Justifying predictions 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: Justifying predictions 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 Justifying predictions 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: Justifying predictions 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 Justifying predictions. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
30.8 Testing a pattern rule
Testing a pattern rule 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 Testing a pattern rule?
- 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: Testing a pattern rule introduces algebraic thinking by describing relationships, using symbols for unknown values, and checking whether mathematical statements are true.
Answer: Testing a pattern rule 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 Testing a pattern rule?
- Read the complete problem.
- Mark the information that is given.
- Identify exactly what the question asks you to find.
Very beginner explanation: Testing a pattern rule 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 Testing a pattern rule.
- 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: Testing a pattern rule 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 Testing a pattern rule 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: Testing a pattern rule 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 Testing a pattern rule.
- 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: Testing a pattern rule introduces algebraic thinking by describing relationships, using symbols for unknown values, and checking whether mathematical statements are true.
Answer: Testing a pattern rule can be used in a real situation where mathematics helps make a clear calculation, comparison, measurement, prediction, or decision.
Worked Example 6
Problem: Continue the pattern 2, 5, 8, ... for two more terms.
- The common difference is 3.
- Add the same difference each time.
Very beginner explanation: A constant-difference pattern changes by the same amount from one term to the next.
Answer: 11, 14
Worked Example 7
Problem: Continue the pattern 5, 9, 13, ... for two more terms.
- The common difference is 4.
- Add the same difference each time.
Very beginner explanation: A constant-difference pattern changes by the same amount from one term to the next.
Answer: 17, 21
Worked Example 8
Problem: Continue the pattern 10, 8, 6, ... for two more terms.
- The common difference is -2.
- Add the same difference each time.
Very beginner explanation: A constant-difference pattern changes by the same amount from one term to the next.
Answer: 4, 2
Worked Example 9
Problem: Continue the pattern 1.5, 2, 2.5, ... for two more terms.
- The common difference is 0.5.
- Add the same difference each time.
Very beginner explanation: A constant-difference pattern changes by the same amount from one term to the next.
Answer: 3, 3.5
Worked Example 10
Problem: Continue the pattern 20, 17.5, 15, ... for two more terms.
- The common difference is -2.5.
- Add the same difference each time.
Very beginner explanation: A constant-difference pattern changes by the same amount from one term to the next.
Answer: 12.5, 10
Practice Exercise
Create one new question about Testing a pattern rule. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
30.9 Finding errors in patterns
Finding errors in patterns 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 Finding errors in patterns?
- 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: Finding errors in patterns introduces algebraic thinking by describing relationships, using symbols for unknown values, and checking whether mathematical statements are true.
Answer: Finding errors in patterns 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 Finding errors in patterns?
- Read the complete problem.
- Mark the information that is given.
- Identify exactly what the question asks you to find.
Very beginner explanation: Finding errors in patterns 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 Finding errors in patterns.
- 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: Finding errors in patterns 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 Finding errors in patterns 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: Finding errors in patterns 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 Finding errors in patterns.
- 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: Finding errors in patterns introduces algebraic thinking by describing relationships, using symbols for unknown values, and checking whether mathematical statements are true.
Answer: Finding errors in patterns can be used in a real situation where mathematics helps make a clear calculation, comparison, measurement, prediction, or decision.
Worked Example 6
Problem: Continue the pattern 2, 5, 8, ... for two more terms.
- The common difference is 3.
- Add the same difference each time.
Very beginner explanation: A constant-difference pattern changes by the same amount from one term to the next.
Answer: 11, 14
Worked Example 7
Problem: Continue the pattern 5, 9, 13, ... for two more terms.
- The common difference is 4.
- Add the same difference each time.
Very beginner explanation: A constant-difference pattern changes by the same amount from one term to the next.
Answer: 17, 21
Worked Example 8
Problem: Continue the pattern 10, 8, 6, ... for two more terms.
- The common difference is -2.
- Add the same difference each time.
Very beginner explanation: A constant-difference pattern changes by the same amount from one term to the next.
Answer: 4, 2
Worked Example 9
Problem: Continue the pattern 1.5, 2, 2.5, ... for two more terms.
- The common difference is 0.5.
- Add the same difference each time.
Very beginner explanation: A constant-difference pattern changes by the same amount from one term to the next.
Answer: 3, 3.5
Worked Example 10
Problem: Continue the pattern 20, 17.5, 15, ... for two more terms.
- The common difference is -2.5.
- Add the same difference each time.
Very beginner explanation: A constant-difference pattern changes by the same amount from one term to the next.
Answer: 12.5, 10
Practice Exercise
Create one new question about Finding errors in patterns. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
30.10 Writing a pattern rule in words
Writing a pattern rule in words 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 a pattern rule in words?
- 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: Writing a pattern rule in words introduces algebraic thinking by describing relationships, using symbols for unknown values, and checking whether mathematical statements are true.
Answer: Writing a pattern rule in words 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 a pattern rule in words?
- Read the complete problem.
- Mark the information that is given.
- Identify exactly what the question asks you to find.
Very beginner explanation: Writing a pattern rule in words 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 a pattern rule in words.
- 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: Writing a pattern rule in words 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 a pattern rule in words 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: Writing a pattern rule in words 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 a pattern rule in words.
- 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: Writing a pattern rule in words introduces algebraic thinking by describing relationships, using symbols for unknown values, and checking whether mathematical statements are true.
Answer: Writing a pattern rule in words can be used in a real situation where mathematics helps make a clear calculation, comparison, measurement, prediction, or decision.
Worked Example 6
Problem: Continue the pattern 2, 5, 8, ... for two more terms.
- The common difference is 3.
- Add the same difference each time.
Very beginner explanation: A constant-difference pattern changes by the same amount from one term to the next.
Answer: 11, 14
Worked Example 7
Problem: Continue the pattern 5, 9, 13, ... for two more terms.
- The common difference is 4.
- Add the same difference each time.
Very beginner explanation: A constant-difference pattern changes by the same amount from one term to the next.
Answer: 17, 21
Worked Example 8
Problem: Continue the pattern 10, 8, 6, ... for two more terms.
- The common difference is -2.
- Add the same difference each time.
Very beginner explanation: A constant-difference pattern changes by the same amount from one term to the next.
Answer: 4, 2
Worked Example 9
Problem: Continue the pattern 1.5, 2, 2.5, ... for two more terms.
- The common difference is 0.5.
- Add the same difference each time.
Very beginner explanation: A constant-difference pattern changes by the same amount from one term to the next.
Answer: 3, 3.5
Worked Example 10
Problem: Continue the pattern 20, 17.5, 15, ... for two more terms.
- The common difference is -2.5.
- Add the same difference each time.
Very beginner explanation: A constant-difference pattern changes by the same amount from one term to the next.
Answer: 12.5, 10
Practice Exercise
Create one new question about Writing a pattern rule in words. 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 Finding a one-step pattern rule.
- Create and solve one original problem about Finding a two-step pattern rule.
- Create and solve one original problem about Extending growing patterns.
- Create and solve one original problem about Extending shrinking patterns.
- Create and solve one original problem about Finding missing terms.
- Create and solve one original problem about Making predictions.
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 Finding a one-step pattern rule?
Answer: Finding a one-step pattern rule 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 Finding a two-step pattern rule?
Answer: Finding a two-step pattern rule 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 Extending growing patterns?
Answer: Extending growing patterns 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 Extending shrinking patterns?
Answer: Extending shrinking patterns introduces algebraic thinking by describing relationships, using symbols for unknown values, and checking whether mathematical statements are true.
Q5. What is the key idea in Finding missing terms?
Answer: Finding missing terms is an important Grade 5 mathematics idea. Learn its meaning first, then connect it to a representation, a worked example, and a real-life use.
Q6. What is the key idea in Making predictions?
Answer: Making predictions is an important Grade 5 mathematics idea. Learn its meaning first, then connect it to a representation, a worked example, and a real-life use.
Q7. What is the key idea in Justifying predictions?
Answer: Justifying predictions 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 Testing a pattern rule?
Answer: Testing a pattern rule introduces algebraic thinking by describing relationships, using symbols for unknown values, and checking whether mathematical statements are true.
Q9. What is the key idea in Finding errors in patterns?
Answer: Finding errors in patterns 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 a pattern rule in words?
Answer: Writing a pattern rule in words 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 Pattern Rules, Predictions, and Missing Elements 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 Pattern Rules, Predictions, and Missing Elements 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 Pattern Rules, Predictions, and Missing Elements 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 Pattern Rules, Predictions, and Missing Elements 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 Pattern Rules, Predictions, and Missing Elements effectively?
Answer: Work a small example, explain every step, check the answer, then try a similar problem without looking at the first solution.