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Chapter 35: Mathematical Modelling Process and Design Project

Learn Grade 6 mathematics with very-beginner explanations, original worked examples, practice, reasoning, and review.

Grade 6Beginner Friendly10 Examples Per Topic30 Q&APractice + Review
Estimated reading time0% read
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Chapter Overview

This chapter teaches Mathematical Modelling Process and Design Project in original, simple language with step-by-step reasoning, worked examples, practice exercises, common mistakes, and review questions.

Key Technical Terms

  • Meaning of mathematical modelling (Mathematical modelling uses mathematics to represent, study, and improve a real-life situation.)
  • Identifying a real-life problem (Identifying a real-life problem is a Grade 6 mathematics idea in Mathematical Modelling Process and Design Project. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.)
  • Choosing important information (Choosing important information is a Grade 6 mathematics idea in Mathematical Modelling Process and Design Project. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.)
  • Making assumptions (Making assumptions is a Grade 6 mathematics idea in Mathematical Modelling Process and Design Project. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.)
  • Defining variables (A variable is a symbol used to represent a number that may change or may be unknown.)
  • Creating a model (The mode is the value that appears most often.)

How to Learn This Chapter

Read the explanation first, follow each worked example slowly, cover the answer and retry the problem yourself, then use the practice and review questions to check understanding.

35.1 Meaning of mathematical modelling

Mathematical modelling uses mathematics to represent, study, and improve a real-life situation. In this Grade 6 lesson, focus on meaning first, then use a small example, show the steps, and check whether the result fits the question.

Beginner Note

Identify what is given, what must be found, and which rule or representation fits this topic. Work one step at a time and explain why each step is valid.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: A simple cost model is C = 20 + 3n. Find the cost when n = 5.

  1. Substitute n = 5.
  2. Multiply 3 × 5.
  3. Add the fixed amount.

Very beginner explanation: A model is a simplified mathematical representation of a real situation.

Answer: $35

Worked Example 2

Problem: A simple cost model is C = 25 + 4n. Find the cost when n = 6.

  1. Substitute n = 6.
  2. Multiply 4 × 6.
  3. Add the fixed amount.

Very beginner explanation: A model is a simplified mathematical representation of a real situation.

Answer: $49

Worked Example 3

Problem: A simple cost model is C = 30 + 5n. Find the cost when n = 7.

  1. Substitute n = 7.
  2. Multiply 5 × 7.
  3. Add the fixed amount.

Very beginner explanation: A model is a simplified mathematical representation of a real situation.

Answer: $65

Worked Example 4

Problem: A simple cost model is C = 35 + 6n. Find the cost when n = 8.

  1. Substitute n = 8.
  2. Multiply 6 × 8.
  3. Add the fixed amount.

Very beginner explanation: A model is a simplified mathematical representation of a real situation.

Answer: $83

Worked Example 5

Problem: A simple cost model is C = 40 + 7n. Find the cost when n = 9.

  1. Substitute n = 9.
  2. Multiply 7 × 9.
  3. Add the fixed amount.

Very beginner explanation: A model is a simplified mathematical representation of a real situation.

Answer: $103

Worked Example 6

Problem: A simple cost model is C = 45 + 8n. Find the cost when n = 10.

  1. Substitute n = 10.
  2. Multiply 8 × 10.
  3. Add the fixed amount.

Very beginner explanation: A model is a simplified mathematical representation of a real situation.

Answer: $125

Worked Example 7

Problem: A simple cost model is C = 50 + 9n. Find the cost when n = 11.

  1. Substitute n = 11.
  2. Multiply 9 × 11.
  3. Add the fixed amount.

Very beginner explanation: A model is a simplified mathematical representation of a real situation.

Answer: $149

Worked Example 8

Problem: A simple cost model is C = 55 + 10n. Find the cost when n = 12.

  1. Substitute n = 12.
  2. Multiply 10 × 12.
  3. Add the fixed amount.

Very beginner explanation: A model is a simplified mathematical representation of a real situation.

Answer: $175

Worked Example 9

Problem: A simple cost model is C = 60 + 11n. Find the cost when n = 13.

  1. Substitute n = 13.
  2. Multiply 11 × 13.
  3. Add the fixed amount.

Very beginner explanation: A model is a simplified mathematical representation of a real situation.

Answer: $203

Worked Example 10

Problem: A simple cost model is C = 65 + 12n. Find the cost when n = 14.

  1. Substitute n = 14.
  2. Multiply 12 × 14.
  3. Add the fixed amount.

Very beginner explanation: A model is a simplified mathematical representation of a real situation.

Answer: $233

Practice Exercise

Create one new Grade 6 problem about Meaning of mathematical modelling. Solve it step by step, label any units, and explain how you checked your answer.

Common Mistake

A common mistake is starting a calculation before identifying what is given, what must be found, and which rule fits the problem.

35.2 Identifying a real-life problem

Identifying a real-life problem is a Grade 6 mathematics idea in Mathematical Modelling Process and Design Project. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable. In this Grade 6 lesson, focus on meaning first, then use a small example, show the steps, and check whether the result fits the question.

Beginner Note

Identify what is given, what must be found, and which rule or representation fits this topic. Work one step at a time and explain why each step is valid.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Use Identifying a real-life problem to analyze the values 10 and 13. What should be checked first?

  1. Identify what each value represents.
  2. Choose the rule, representation, or comparison that matches the topic.
  3. Show the reasoning before accepting the result.

Very beginner explanation: Identifying a real-life problem is a Grade 6 mathematics idea in Mathematical Modelling Process and Design Project. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

Answer: Check the meaning of the values and the rule required by the problem.

Worked Example 2

Problem: Use Identifying a real-life problem to analyze the values 10 and 16. What should be checked first?

  1. Identify what each value represents.
  2. Choose the rule, representation, or comparison that matches the topic.
  3. Show the reasoning before accepting the result.

Very beginner explanation: Identifying a real-life problem is a Grade 6 mathematics idea in Mathematical Modelling Process and Design Project. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

Answer: Check the meaning of the values and the rule required by the problem.

Worked Example 3

Problem: Use Identifying a real-life problem to analyze the values 30 and 19. What should be checked first?

  1. Identify what each value represents.
  2. Choose the rule, representation, or comparison that matches the topic.
  3. Show the reasoning before accepting the result.

Very beginner explanation: Identifying a real-life problem is a Grade 6 mathematics idea in Mathematical Modelling Process and Design Project. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

Answer: Check the meaning of the values and the rule required by the problem.

Worked Example 4

Problem: Use Identifying a real-life problem to analyze the values 15 and 17. What should be checked first?

  1. Identify what each value represents.
  2. Choose the rule, representation, or comparison that matches the topic.
  3. Show the reasoning before accepting the result.

Very beginner explanation: Identifying a real-life problem is a Grade 6 mathematics idea in Mathematical Modelling Process and Design Project. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

Answer: Check the meaning of the values and the rule required by the problem.

Worked Example 5

Problem: Use Identifying a real-life problem to analyze the values 21 and 18. What should be checked first?

  1. Identify what each value represents.
  2. Choose the rule, representation, or comparison that matches the topic.
  3. Show the reasoning before accepting the result.

Very beginner explanation: Identifying a real-life problem is a Grade 6 mathematics idea in Mathematical Modelling Process and Design Project. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

Answer: Check the meaning of the values and the rule required by the problem.

Worked Example 6

Problem: Use Identifying a real-life problem to analyze the values 10 and 2. What should be checked first?

  1. Identify what each value represents.
  2. Choose the rule, representation, or comparison that matches the topic.
  3. Show the reasoning before accepting the result.

Very beginner explanation: Identifying a real-life problem is a Grade 6 mathematics idea in Mathematical Modelling Process and Design Project. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

Answer: Check the meaning of the values and the rule required by the problem.

Worked Example 7

Problem: Use Identifying a real-life problem to analyze the values 25 and 19. What should be checked first?

  1. Identify what each value represents.
  2. Choose the rule, representation, or comparison that matches the topic.
  3. Show the reasoning before accepting the result.

Very beginner explanation: Identifying a real-life problem is a Grade 6 mathematics idea in Mathematical Modelling Process and Design Project. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

Answer: Check the meaning of the values and the rule required by the problem.

Worked Example 8

Problem: Use Identifying a real-life problem to analyze the values 8 and 3. What should be checked first?

  1. Identify what each value represents.
  2. Choose the rule, representation, or comparison that matches the topic.
  3. Show the reasoning before accepting the result.

Very beginner explanation: Identifying a real-life problem is a Grade 6 mathematics idea in Mathematical Modelling Process and Design Project. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

Answer: Check the meaning of the values and the rule required by the problem.

Worked Example 9

Problem: Use Identifying a real-life problem to analyze the values 5 and 9. What should be checked first?

  1. Identify what each value represents.
  2. Choose the rule, representation, or comparison that matches the topic.
  3. Show the reasoning before accepting the result.

Very beginner explanation: Identifying a real-life problem is a Grade 6 mathematics idea in Mathematical Modelling Process and Design Project. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

Answer: Check the meaning of the values and the rule required by the problem.

Worked Example 10

Problem: Use Identifying a real-life problem to analyze the values 17 and 11. What should be checked first?

  1. Identify what each value represents.
  2. Choose the rule, representation, or comparison that matches the topic.
  3. Show the reasoning before accepting the result.

Very beginner explanation: Identifying a real-life problem is a Grade 6 mathematics idea in Mathematical Modelling Process and Design Project. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

Answer: Check the meaning of the values and the rule required by the problem.

Practice Exercise

Create one new Grade 6 problem about Identifying a real-life problem. Solve it step by step, label any units, and explain how you checked your answer.

Common Mistake

A common mistake is starting a calculation before identifying what is given, what must be found, and which rule fits the problem.

35.3 Choosing important information

Choosing important information is a Grade 6 mathematics idea in Mathematical Modelling Process and Design Project. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable. In this Grade 6 lesson, focus on meaning first, then use a small example, show the steps, and check whether the result fits the question.

Beginner Note

Identify what is given, what must be found, and which rule or representation fits this topic. Work one step at a time and explain why each step is valid.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Use Choosing important information to analyze the values 6 and 7. What should be checked first?

  1. Identify what each value represents.
  2. Choose the rule, representation, or comparison that matches the topic.
  3. Show the reasoning before accepting the result.

Very beginner explanation: Choosing important information is a Grade 6 mathematics idea in Mathematical Modelling Process and Design Project. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

Answer: Check the meaning of the values and the rule required by the problem.

Worked Example 2

Problem: Use Choosing important information to analyze the values 5 and 14. What should be checked first?

  1. Identify what each value represents.
  2. Choose the rule, representation, or comparison that matches the topic.
  3. Show the reasoning before accepting the result.

Very beginner explanation: Choosing important information is a Grade 6 mathematics idea in Mathematical Modelling Process and Design Project. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

Answer: Check the meaning of the values and the rule required by the problem.

Worked Example 3

Problem: Use Choosing important information to analyze the values 4 and 2. What should be checked first?

  1. Identify what each value represents.
  2. Choose the rule, representation, or comparison that matches the topic.
  3. Show the reasoning before accepting the result.

Very beginner explanation: Choosing important information is a Grade 6 mathematics idea in Mathematical Modelling Process and Design Project. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

Answer: Check the meaning of the values and the rule required by the problem.

Worked Example 4

Problem: Use Choosing important information to analyze the values 7 and 9. What should be checked first?

  1. Identify what each value represents.
  2. Choose the rule, representation, or comparison that matches the topic.
  3. Show the reasoning before accepting the result.

Very beginner explanation: Choosing important information is a Grade 6 mathematics idea in Mathematical Modelling Process and Design Project. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

Answer: Check the meaning of the values and the rule required by the problem.

Worked Example 5

Problem: Use Choosing important information to analyze the values 30 and 18. What should be checked first?

  1. Identify what each value represents.
  2. Choose the rule, representation, or comparison that matches the topic.
  3. Show the reasoning before accepting the result.

Very beginner explanation: Choosing important information is a Grade 6 mathematics idea in Mathematical Modelling Process and Design Project. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

Answer: Check the meaning of the values and the rule required by the problem.

Worked Example 6

Problem: Use Choosing important information to analyze the values 9 and 12. What should be checked first?

  1. Identify what each value represents.
  2. Choose the rule, representation, or comparison that matches the topic.
  3. Show the reasoning before accepting the result.

Very beginner explanation: Choosing important information is a Grade 6 mathematics idea in Mathematical Modelling Process and Design Project. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

Answer: Check the meaning of the values and the rule required by the problem.

Worked Example 7

Problem: Use Choosing important information to analyze the values 8 and 13. What should be checked first?

  1. Identify what each value represents.
  2. Choose the rule, representation, or comparison that matches the topic.
  3. Show the reasoning before accepting the result.

Very beginner explanation: Choosing important information is a Grade 6 mathematics idea in Mathematical Modelling Process and Design Project. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

Answer: Check the meaning of the values and the rule required by the problem.

Worked Example 8

Problem: Use Choosing important information to analyze the values 3 and 18. What should be checked first?

  1. Identify what each value represents.
  2. Choose the rule, representation, or comparison that matches the topic.
  3. Show the reasoning before accepting the result.

Very beginner explanation: Choosing important information is a Grade 6 mathematics idea in Mathematical Modelling Process and Design Project. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

Answer: Check the meaning of the values and the rule required by the problem.

Worked Example 9

Problem: Use Choosing important information to analyze the values 5 and 7. What should be checked first?

  1. Identify what each value represents.
  2. Choose the rule, representation, or comparison that matches the topic.
  3. Show the reasoning before accepting the result.

Very beginner explanation: Choosing important information is a Grade 6 mathematics idea in Mathematical Modelling Process and Design Project. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

Answer: Check the meaning of the values and the rule required by the problem.

Worked Example 10

Problem: Use Choosing important information to analyze the values 26 and 7. What should be checked first?

  1. Identify what each value represents.
  2. Choose the rule, representation, or comparison that matches the topic.
  3. Show the reasoning before accepting the result.

Very beginner explanation: Choosing important information is a Grade 6 mathematics idea in Mathematical Modelling Process and Design Project. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

Answer: Check the meaning of the values and the rule required by the problem.

Practice Exercise

Create one new Grade 6 problem about Choosing important information. Solve it step by step, label any units, and explain how you checked your answer.

Common Mistake

A common mistake is starting a calculation before identifying what is given, what must be found, and which rule fits the problem.

35.4 Making assumptions

Making assumptions is a Grade 6 mathematics idea in Mathematical Modelling Process and Design Project. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable. In this Grade 6 lesson, focus on meaning first, then use a small example, show the steps, and check whether the result fits the question.

Beginner Note

Identify what is given, what must be found, and which rule or representation fits this topic. Work one step at a time and explain why each step is valid.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Use Making assumptions to analyze the values 5 and 9. What should be checked first?

  1. Identify what each value represents.
  2. Choose the rule, representation, or comparison that matches the topic.
  3. Show the reasoning before accepting the result.

Very beginner explanation: Making assumptions is a Grade 6 mathematics idea in Mathematical Modelling Process and Design Project. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

Answer: Check the meaning of the values and the rule required by the problem.

Worked Example 2

Problem: Use Making assumptions to analyze the values 26 and 11. What should be checked first?

  1. Identify what each value represents.
  2. Choose the rule, representation, or comparison that matches the topic.
  3. Show the reasoning before accepting the result.

Very beginner explanation: Making assumptions is a Grade 6 mathematics idea in Mathematical Modelling Process and Design Project. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

Answer: Check the meaning of the values and the rule required by the problem.

Worked Example 3

Problem: Use Making assumptions to analyze the values 12 and 19. What should be checked first?

  1. Identify what each value represents.
  2. Choose the rule, representation, or comparison that matches the topic.
  3. Show the reasoning before accepting the result.

Very beginner explanation: Making assumptions is a Grade 6 mathematics idea in Mathematical Modelling Process and Design Project. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

Answer: Check the meaning of the values and the rule required by the problem.

Worked Example 4

Problem: Use Making assumptions to analyze the values 12 and 7. What should be checked first?

  1. Identify what each value represents.
  2. Choose the rule, representation, or comparison that matches the topic.
  3. Show the reasoning before accepting the result.

Very beginner explanation: Making assumptions is a Grade 6 mathematics idea in Mathematical Modelling Process and Design Project. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

Answer: Check the meaning of the values and the rule required by the problem.

Worked Example 5

Problem: Use Making assumptions to analyze the values 16 and 17. What should be checked first?

  1. Identify what each value represents.
  2. Choose the rule, representation, or comparison that matches the topic.
  3. Show the reasoning before accepting the result.

Very beginner explanation: Making assumptions is a Grade 6 mathematics idea in Mathematical Modelling Process and Design Project. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

Answer: Check the meaning of the values and the rule required by the problem.

Worked Example 6

Problem: Use Making assumptions to analyze the values 9 and 16. What should be checked first?

  1. Identify what each value represents.
  2. Choose the rule, representation, or comparison that matches the topic.
  3. Show the reasoning before accepting the result.

Very beginner explanation: Making assumptions is a Grade 6 mathematics idea in Mathematical Modelling Process and Design Project. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

Answer: Check the meaning of the values and the rule required by the problem.

Worked Example 7

Problem: Use Making assumptions to analyze the values 12 and 18. What should be checked first?

  1. Identify what each value represents.
  2. Choose the rule, representation, or comparison that matches the topic.
  3. Show the reasoning before accepting the result.

Very beginner explanation: Making assumptions is a Grade 6 mathematics idea in Mathematical Modelling Process and Design Project. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

Answer: Check the meaning of the values and the rule required by the problem.

Worked Example 8

Problem: Use Making assumptions to analyze the values 27 and 19. What should be checked first?

  1. Identify what each value represents.
  2. Choose the rule, representation, or comparison that matches the topic.
  3. Show the reasoning before accepting the result.

Very beginner explanation: Making assumptions is a Grade 6 mathematics idea in Mathematical Modelling Process and Design Project. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

Answer: Check the meaning of the values and the rule required by the problem.

Worked Example 9

Problem: Use Making assumptions to analyze the values 28 and 15. What should be checked first?

  1. Identify what each value represents.
  2. Choose the rule, representation, or comparison that matches the topic.
  3. Show the reasoning before accepting the result.

Very beginner explanation: Making assumptions is a Grade 6 mathematics idea in Mathematical Modelling Process and Design Project. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

Answer: Check the meaning of the values and the rule required by the problem.

Worked Example 10

Problem: Use Making assumptions to analyze the values 9 and 10. What should be checked first?

  1. Identify what each value represents.
  2. Choose the rule, representation, or comparison that matches the topic.
  3. Show the reasoning before accepting the result.

Very beginner explanation: Making assumptions is a Grade 6 mathematics idea in Mathematical Modelling Process and Design Project. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

Answer: Check the meaning of the values and the rule required by the problem.

Practice Exercise

Create one new Grade 6 problem about Making assumptions. Solve it step by step, label any units, and explain how you checked your answer.

Common Mistake

A common mistake is starting a calculation before identifying what is given, what must be found, and which rule fits the problem.

35.5 Defining variables

A variable is a symbol used to represent a number that may change or may be unknown. In this Grade 6 lesson, focus on meaning first, then use a small example, show the steps, and check whether the result fits the question.

Beginner Note

Identify what is given, what must be found, and which rule or representation fits this topic. Work one step at a time and explain why each step is valid.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Evaluate 3x + 4 when x = 5.

  1. Replace x with 5.
  2. Follow order of operations.
  3. Calculate carefully.

Very beginner explanation: Substitution replaces a variable with a known value before calculating.

Answer: 19

Worked Example 2

Problem: Evaluate 2n + 7 when n = 6.

  1. Replace n with 6.
  2. Follow order of operations.
  3. Calculate carefully.

Very beginner explanation: Substitution replaces a variable with a known value before calculating.

Answer: 19

Worked Example 3

Problem: Evaluate 5a - 3 when a = 4.

  1. Replace a with 4.
  2. Follow order of operations.
  3. Calculate carefully.

Very beginner explanation: Substitution replaces a variable with a known value before calculating.

Answer: 17

Worked Example 4

Problem: Evaluate 4p + 1 when p = 8.

  1. Replace p with 8.
  2. Follow order of operations.
  3. Calculate carefully.

Very beginner explanation: Substitution replaces a variable with a known value before calculating.

Answer: 33

Worked Example 5

Problem: Evaluate 6m - 5 when m = 3.

  1. Replace m with 3.
  2. Follow order of operations.
  3. Calculate carefully.

Very beginner explanation: Substitution replaces a variable with a known value before calculating.

Answer: 13

Worked Example 6

Problem: Evaluate 2.5x + 1 when x = 4.

  1. Replace x with 4.
  2. Follow order of operations.
  3. Calculate carefully.

Very beginner explanation: Substitution replaces a variable with a known value before calculating.

Answer: 11

Worked Example 7

Problem: Evaluate 7q + 2 when q = 2.

  1. Replace q with 2.
  2. Follow order of operations.
  3. Calculate carefully.

Very beginner explanation: Substitution replaces a variable with a known value before calculating.

Answer: 16

Worked Example 8

Problem: Evaluate 3r - 1 when r = 9.

  1. Replace r with 9.
  2. Follow order of operations.
  3. Calculate carefully.

Very beginner explanation: Substitution replaces a variable with a known value before calculating.

Answer: 26

Worked Example 9

Problem: Evaluate 4y + 6 when y = 5.

  1. Replace y with 5.
  2. Follow order of operations.
  3. Calculate carefully.

Very beginner explanation: Substitution replaces a variable with a known value before calculating.

Answer: 26

Worked Example 10

Problem: Evaluate 8k - 4 when k = 3.

  1. Replace k with 3.
  2. Follow order of operations.
  3. Calculate carefully.

Very beginner explanation: Substitution replaces a variable with a known value before calculating.

Answer: 20

Practice Exercise

Create one new Grade 6 problem about Defining variables. Solve it step by step, label any units, and explain how you checked your answer.

Common Mistake

A common mistake is starting a calculation before identifying what is given, what must be found, and which rule fits the problem.

35.6 Creating a model

The mode is the value that appears most often. In this Grade 6 lesson, focus on meaning first, then use a small example, show the steps, and check whether the result fits the question.

Beginner Note

Identify what is given, what must be found, and which rule or representation fits this topic. Work one step at a time and explain why each step is valid.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: A simple cost model is C = 20 + 3n. Find the cost when n = 5.

  1. Substitute n = 5.
  2. Multiply 3 × 5.
  3. Add the fixed amount.

Very beginner explanation: A model is a simplified mathematical representation of a real situation.

Answer: $35

Worked Example 2

Problem: A simple cost model is C = 25 + 4n. Find the cost when n = 6.

  1. Substitute n = 6.
  2. Multiply 4 × 6.
  3. Add the fixed amount.

Very beginner explanation: A model is a simplified mathematical representation of a real situation.

Answer: $49

Worked Example 3

Problem: A simple cost model is C = 30 + 5n. Find the cost when n = 7.

  1. Substitute n = 7.
  2. Multiply 5 × 7.
  3. Add the fixed amount.

Very beginner explanation: A model is a simplified mathematical representation of a real situation.

Answer: $65

Worked Example 4

Problem: A simple cost model is C = 35 + 6n. Find the cost when n = 8.

  1. Substitute n = 8.
  2. Multiply 6 × 8.
  3. Add the fixed amount.

Very beginner explanation: A model is a simplified mathematical representation of a real situation.

Answer: $83

Worked Example 5

Problem: A simple cost model is C = 40 + 7n. Find the cost when n = 9.

  1. Substitute n = 9.
  2. Multiply 7 × 9.
  3. Add the fixed amount.

Very beginner explanation: A model is a simplified mathematical representation of a real situation.

Answer: $103

Worked Example 6

Problem: A simple cost model is C = 45 + 8n. Find the cost when n = 10.

  1. Substitute n = 10.
  2. Multiply 8 × 10.
  3. Add the fixed amount.

Very beginner explanation: A model is a simplified mathematical representation of a real situation.

Answer: $125

Worked Example 7

Problem: A simple cost model is C = 50 + 9n. Find the cost when n = 11.

  1. Substitute n = 11.
  2. Multiply 9 × 11.
  3. Add the fixed amount.

Very beginner explanation: A model is a simplified mathematical representation of a real situation.

Answer: $149

Worked Example 8

Problem: A simple cost model is C = 55 + 10n. Find the cost when n = 12.

  1. Substitute n = 12.
  2. Multiply 10 × 12.
  3. Add the fixed amount.

Very beginner explanation: A model is a simplified mathematical representation of a real situation.

Answer: $175

Worked Example 9

Problem: A simple cost model is C = 60 + 11n. Find the cost when n = 13.

  1. Substitute n = 13.
  2. Multiply 11 × 13.
  3. Add the fixed amount.

Very beginner explanation: A model is a simplified mathematical representation of a real situation.

Answer: $203

Worked Example 10

Problem: A simple cost model is C = 65 + 12n. Find the cost when n = 14.

  1. Substitute n = 14.
  2. Multiply 12 × 14.
  3. Add the fixed amount.

Very beginner explanation: A model is a simplified mathematical representation of a real situation.

Answer: $233

Practice Exercise

Create one new Grade 6 problem about Creating a model. Solve it step by step, label any units, and explain how you checked your answer.

Common Mistake

A common mistake is starting a calculation before identifying what is given, what must be found, and which rule fits the problem.

35.7 Using tables in a model

The mode is the value that appears most often. In this Grade 6 lesson, focus on meaning first, then use a small example, show the steps, and check whether the result fits the question.

Beginner Note

Identify what is given, what must be found, and which rule or representation fits this topic. Work one step at a time and explain why each step is valid.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: A simple cost model is C = 20 + 3n. Find the cost when n = 5.

  1. Substitute n = 5.
  2. Multiply 3 × 5.
  3. Add the fixed amount.

Very beginner explanation: A model is a simplified mathematical representation of a real situation.

Answer: $35

Worked Example 2

Problem: A simple cost model is C = 25 + 4n. Find the cost when n = 6.

  1. Substitute n = 6.
  2. Multiply 4 × 6.
  3. Add the fixed amount.

Very beginner explanation: A model is a simplified mathematical representation of a real situation.

Answer: $49

Worked Example 3

Problem: A simple cost model is C = 30 + 5n. Find the cost when n = 7.

  1. Substitute n = 7.
  2. Multiply 5 × 7.
  3. Add the fixed amount.

Very beginner explanation: A model is a simplified mathematical representation of a real situation.

Answer: $65

Worked Example 4

Problem: A simple cost model is C = 35 + 6n. Find the cost when n = 8.

  1. Substitute n = 8.
  2. Multiply 6 × 8.
  3. Add the fixed amount.

Very beginner explanation: A model is a simplified mathematical representation of a real situation.

Answer: $83

Worked Example 5

Problem: A simple cost model is C = 40 + 7n. Find the cost when n = 9.

  1. Substitute n = 9.
  2. Multiply 7 × 9.
  3. Add the fixed amount.

Very beginner explanation: A model is a simplified mathematical representation of a real situation.

Answer: $103

Worked Example 6

Problem: A simple cost model is C = 45 + 8n. Find the cost when n = 10.

  1. Substitute n = 10.
  2. Multiply 8 × 10.
  3. Add the fixed amount.

Very beginner explanation: A model is a simplified mathematical representation of a real situation.

Answer: $125

Worked Example 7

Problem: A simple cost model is C = 50 + 9n. Find the cost when n = 11.

  1. Substitute n = 11.
  2. Multiply 9 × 11.
  3. Add the fixed amount.

Very beginner explanation: A model is a simplified mathematical representation of a real situation.

Answer: $149

Worked Example 8

Problem: A simple cost model is C = 55 + 10n. Find the cost when n = 12.

  1. Substitute n = 12.
  2. Multiply 10 × 12.
  3. Add the fixed amount.

Very beginner explanation: A model is a simplified mathematical representation of a real situation.

Answer: $175

Worked Example 9

Problem: A simple cost model is C = 60 + 11n. Find the cost when n = 13.

  1. Substitute n = 13.
  2. Multiply 11 × 13.
  3. Add the fixed amount.

Very beginner explanation: A model is a simplified mathematical representation of a real situation.

Answer: $203

Worked Example 10

Problem: A simple cost model is C = 65 + 12n. Find the cost when n = 14.

  1. Substitute n = 14.
  2. Multiply 12 × 14.
  3. Add the fixed amount.

Very beginner explanation: A model is a simplified mathematical representation of a real situation.

Answer: $233

Practice Exercise

Create one new Grade 6 problem about Using tables in a model. Solve it step by step, label any units, and explain how you checked your answer.

Common Mistake

A common mistake is starting a calculation before identifying what is given, what must be found, and which rule fits the problem.

35.8 Using equations in a model

An equation states that two mathematical expressions have the same value. In this Grade 6 lesson, focus on meaning first, then use a small example, show the steps, and check whether the result fits the question.

Beginner Note

Identify what is given, what must be found, and which rule or representation fits this topic. Work one step at a time and explain why each step is valid.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Solve x + 7 = 18.

  1. Simplify each side if needed.
  2. Use inverse operations to isolate the variable.
  3. Keep the relationship balanced.
  4. Substitute the answer to check.

Very beginner explanation: An equation is balanced when both sides have equal value.

Answer: x = 11

Worked Example 2

Problem: Solve x - 9 = 14.

  1. Simplify each side if needed.
  2. Use inverse operations to isolate the variable.
  3. Keep the relationship balanced.
  4. Substitute the answer to check.

Very beginner explanation: An equation is balanced when both sides have equal value.

Answer: x = 23

Worked Example 3

Problem: Solve 4x = 28.

  1. Simplify each side if needed.
  2. Use inverse operations to isolate the variable.
  3. Keep the relationship balanced.
  4. Substitute the answer to check.

Very beginner explanation: An equation is balanced when both sides have equal value.

Answer: x = 7

Worked Example 4

Problem: Solve x ÷ 5 = 6.

  1. Simplify each side if needed.
  2. Use inverse operations to isolate the variable.
  3. Keep the relationship balanced.
  4. Substitute the answer to check.

Very beginner explanation: An equation is balanced when both sides have equal value.

Answer: x = 30

Worked Example 5

Problem: Solve 2x + 3x = 25.

  1. Simplify each side if needed.
  2. Use inverse operations to isolate the variable.
  3. Keep the relationship balanced.
  4. Substitute the answer to check.

Very beginner explanation: An equation is balanced when both sides have equal value.

Answer: x = 5

Worked Example 6

Problem: Solve 3x + 4 = 19.

  1. Simplify each side if needed.
  2. Use inverse operations to isolate the variable.
  3. Keep the relationship balanced.
  4. Substitute the answer to check.

Very beginner explanation: An equation is balanced when both sides have equal value.

Answer: x = 5

Worked Example 7

Problem: Solve 5x - 2 = 23.

  1. Simplify each side if needed.
  2. Use inverse operations to isolate the variable.
  3. Keep the relationship balanced.
  4. Substitute the answer to check.

Very beginner explanation: An equation is balanced when both sides have equal value.

Answer: x = 5

Worked Example 8

Problem: Solve 0.5x + 2 = 6.

  1. Simplify each side if needed.
  2. Use inverse operations to isolate the variable.
  3. Keep the relationship balanced.
  4. Substitute the answer to check.

Very beginner explanation: An equation is balanced when both sides have equal value.

Answer: x = 8

Worked Example 9

Problem: Solve 4x + 1 = 2x + 13.

  1. Simplify each side if needed.
  2. Use inverse operations to isolate the variable.
  3. Keep the relationship balanced.
  4. Substitute the answer to check.

Very beginner explanation: An equation is balanced when both sides have equal value.

Answer: x = 6

Worked Example 10

Problem: Solve 6x - 3 = 3x + 12.

  1. Simplify each side if needed.
  2. Use inverse operations to isolate the variable.
  3. Keep the relationship balanced.
  4. Substitute the answer to check.

Very beginner explanation: An equation is balanced when both sides have equal value.

Answer: x = 5

Practice Exercise

Create one new Grade 6 problem about Using equations in a model. Solve it step by step, label any units, and explain how you checked your answer.

Common Mistake

A common mistake is changing only one side of a relationship or forgetting to check the solution.

35.9 Using graphs in a model

The mode is the value that appears most often. In this Grade 6 lesson, focus on meaning first, then use a small example, show the steps, and check whether the result fits the question.

Beginner Note

Identify what is given, what must be found, and which rule or representation fits this topic. Work one step at a time and explain why each step is valid.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: A simple cost model is C = 20 + 3n. Find the cost when n = 5.

  1. Substitute n = 5.
  2. Multiply 3 × 5.
  3. Add the fixed amount.

Very beginner explanation: A model is a simplified mathematical representation of a real situation.

Answer: $35

Worked Example 2

Problem: A simple cost model is C = 25 + 4n. Find the cost when n = 6.

  1. Substitute n = 6.
  2. Multiply 4 × 6.
  3. Add the fixed amount.

Very beginner explanation: A model is a simplified mathematical representation of a real situation.

Answer: $49

Worked Example 3

Problem: A simple cost model is C = 30 + 5n. Find the cost when n = 7.

  1. Substitute n = 7.
  2. Multiply 5 × 7.
  3. Add the fixed amount.

Very beginner explanation: A model is a simplified mathematical representation of a real situation.

Answer: $65

Worked Example 4

Problem: A simple cost model is C = 35 + 6n. Find the cost when n = 8.

  1. Substitute n = 8.
  2. Multiply 6 × 8.
  3. Add the fixed amount.

Very beginner explanation: A model is a simplified mathematical representation of a real situation.

Answer: $83

Worked Example 5

Problem: A simple cost model is C = 40 + 7n. Find the cost when n = 9.

  1. Substitute n = 9.
  2. Multiply 7 × 9.
  3. Add the fixed amount.

Very beginner explanation: A model is a simplified mathematical representation of a real situation.

Answer: $103

Worked Example 6

Problem: A simple cost model is C = 45 + 8n. Find the cost when n = 10.

  1. Substitute n = 10.
  2. Multiply 8 × 10.
  3. Add the fixed amount.

Very beginner explanation: A model is a simplified mathematical representation of a real situation.

Answer: $125

Worked Example 7

Problem: A simple cost model is C = 50 + 9n. Find the cost when n = 11.

  1. Substitute n = 11.
  2. Multiply 9 × 11.
  3. Add the fixed amount.

Very beginner explanation: A model is a simplified mathematical representation of a real situation.

Answer: $149

Worked Example 8

Problem: A simple cost model is C = 55 + 10n. Find the cost when n = 12.

  1. Substitute n = 12.
  2. Multiply 10 × 12.
  3. Add the fixed amount.

Very beginner explanation: A model is a simplified mathematical representation of a real situation.

Answer: $175

Worked Example 9

Problem: A simple cost model is C = 60 + 11n. Find the cost when n = 13.

  1. Substitute n = 13.
  2. Multiply 11 × 13.
  3. Add the fixed amount.

Very beginner explanation: A model is a simplified mathematical representation of a real situation.

Answer: $203

Worked Example 10

Problem: A simple cost model is C = 65 + 12n. Find the cost when n = 14.

  1. Substitute n = 14.
  2. Multiply 12 × 14.
  3. Add the fixed amount.

Very beginner explanation: A model is a simplified mathematical representation of a real situation.

Answer: $233

Practice Exercise

Create one new Grade 6 problem about Using graphs in a model. Solve it step by step, label any units, and explain how you checked your answer.

Common Mistake

A common mistake is ignoring the scale, labels, units, or the type of data shown.

35.10 Testing a model

The mode is the value that appears most often. In this Grade 6 lesson, focus on meaning first, then use a small example, show the steps, and check whether the result fits the question.

Beginner Note

Identify what is given, what must be found, and which rule or representation fits this topic. Work one step at a time and explain why each step is valid.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: A simple cost model is C = 20 + 3n. Find the cost when n = 5.

  1. Substitute n = 5.
  2. Multiply 3 × 5.
  3. Add the fixed amount.

Very beginner explanation: A model is a simplified mathematical representation of a real situation.

Answer: $35

Worked Example 2

Problem: A simple cost model is C = 25 + 4n. Find the cost when n = 6.

  1. Substitute n = 6.
  2. Multiply 4 × 6.
  3. Add the fixed amount.

Very beginner explanation: A model is a simplified mathematical representation of a real situation.

Answer: $49

Worked Example 3

Problem: A simple cost model is C = 30 + 5n. Find the cost when n = 7.

  1. Substitute n = 7.
  2. Multiply 5 × 7.
  3. Add the fixed amount.

Very beginner explanation: A model is a simplified mathematical representation of a real situation.

Answer: $65

Worked Example 4

Problem: A simple cost model is C = 35 + 6n. Find the cost when n = 8.

  1. Substitute n = 8.
  2. Multiply 6 × 8.
  3. Add the fixed amount.

Very beginner explanation: A model is a simplified mathematical representation of a real situation.

Answer: $83

Worked Example 5

Problem: A simple cost model is C = 40 + 7n. Find the cost when n = 9.

  1. Substitute n = 9.
  2. Multiply 7 × 9.
  3. Add the fixed amount.

Very beginner explanation: A model is a simplified mathematical representation of a real situation.

Answer: $103

Worked Example 6

Problem: A simple cost model is C = 45 + 8n. Find the cost when n = 10.

  1. Substitute n = 10.
  2. Multiply 8 × 10.
  3. Add the fixed amount.

Very beginner explanation: A model is a simplified mathematical representation of a real situation.

Answer: $125

Worked Example 7

Problem: A simple cost model is C = 50 + 9n. Find the cost when n = 11.

  1. Substitute n = 11.
  2. Multiply 9 × 11.
  3. Add the fixed amount.

Very beginner explanation: A model is a simplified mathematical representation of a real situation.

Answer: $149

Worked Example 8

Problem: A simple cost model is C = 55 + 10n. Find the cost when n = 12.

  1. Substitute n = 12.
  2. Multiply 10 × 12.
  3. Add the fixed amount.

Very beginner explanation: A model is a simplified mathematical representation of a real situation.

Answer: $175

Worked Example 9

Problem: A simple cost model is C = 60 + 11n. Find the cost when n = 13.

  1. Substitute n = 13.
  2. Multiply 11 × 13.
  3. Add the fixed amount.

Very beginner explanation: A model is a simplified mathematical representation of a real situation.

Answer: $203

Worked Example 10

Problem: A simple cost model is C = 65 + 12n. Find the cost when n = 14.

  1. Substitute n = 14.
  2. Multiply 12 × 14.
  3. Add the fixed amount.

Very beginner explanation: A model is a simplified mathematical representation of a real situation.

Answer: $233

Practice Exercise

Create one new Grade 6 problem about Testing a model. Solve it step by step, label any units, and explain how you checked your answer.

Common Mistake

A common mistake is starting a calculation before identifying what is given, what must be found, and which rule fits the problem.

35.11 Revising a model

The mode is the value that appears most often. In this Grade 6 lesson, focus on meaning first, then use a small example, show the steps, and check whether the result fits the question.

Beginner Note

Identify what is given, what must be found, and which rule or representation fits this topic. Work one step at a time and explain why each step is valid.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: A simple cost model is C = 20 + 3n. Find the cost when n = 5.

  1. Substitute n = 5.
  2. Multiply 3 × 5.
  3. Add the fixed amount.

Very beginner explanation: A model is a simplified mathematical representation of a real situation.

Answer: $35

Worked Example 2

Problem: A simple cost model is C = 25 + 4n. Find the cost when n = 6.

  1. Substitute n = 6.
  2. Multiply 4 × 6.
  3. Add the fixed amount.

Very beginner explanation: A model is a simplified mathematical representation of a real situation.

Answer: $49

Worked Example 3

Problem: A simple cost model is C = 30 + 5n. Find the cost when n = 7.

  1. Substitute n = 7.
  2. Multiply 5 × 7.
  3. Add the fixed amount.

Very beginner explanation: A model is a simplified mathematical representation of a real situation.

Answer: $65

Worked Example 4

Problem: A simple cost model is C = 35 + 6n. Find the cost when n = 8.

  1. Substitute n = 8.
  2. Multiply 6 × 8.
  3. Add the fixed amount.

Very beginner explanation: A model is a simplified mathematical representation of a real situation.

Answer: $83

Worked Example 5

Problem: A simple cost model is C = 40 + 7n. Find the cost when n = 9.

  1. Substitute n = 9.
  2. Multiply 7 × 9.
  3. Add the fixed amount.

Very beginner explanation: A model is a simplified mathematical representation of a real situation.

Answer: $103

Worked Example 6

Problem: A simple cost model is C = 45 + 8n. Find the cost when n = 10.

  1. Substitute n = 10.
  2. Multiply 8 × 10.
  3. Add the fixed amount.

Very beginner explanation: A model is a simplified mathematical representation of a real situation.

Answer: $125

Worked Example 7

Problem: A simple cost model is C = 50 + 9n. Find the cost when n = 11.

  1. Substitute n = 11.
  2. Multiply 9 × 11.
  3. Add the fixed amount.

Very beginner explanation: A model is a simplified mathematical representation of a real situation.

Answer: $149

Worked Example 8

Problem: A simple cost model is C = 55 + 10n. Find the cost when n = 12.

  1. Substitute n = 12.
  2. Multiply 10 × 12.
  3. Add the fixed amount.

Very beginner explanation: A model is a simplified mathematical representation of a real situation.

Answer: $175

Worked Example 9

Problem: A simple cost model is C = 60 + 11n. Find the cost when n = 13.

  1. Substitute n = 13.
  2. Multiply 11 × 13.
  3. Add the fixed amount.

Very beginner explanation: A model is a simplified mathematical representation of a real situation.

Answer: $203

Worked Example 10

Problem: A simple cost model is C = 65 + 12n. Find the cost when n = 14.

  1. Substitute n = 14.
  2. Multiply 12 × 14.
  3. Add the fixed amount.

Very beginner explanation: A model is a simplified mathematical representation of a real situation.

Answer: $233

Practice Exercise

Create one new Grade 6 problem about Revising a model. Solve it step by step, label any units, and explain how you checked your answer.

Common Mistake

A common mistake is starting a calculation before identifying what is given, what must be found, and which rule fits the problem.

35.12 Play-area design project

Area measures the amount of two-dimensional space inside a shape. In this Grade 6 lesson, focus on meaning first, then use a small example, show the steps, and check whether the result fits the question.

Beginner Note

Identify what is given, what must be found, and which rule or representation fits this topic. Work one step at a time and explain why each step is valid.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Find the area of a rectangle 5 cm by 2 cm.

  1. Use A = length × width.
  2. A = 5 × 2.

Very beginner explanation: Area covers a two-dimensional region, so it uses square units.

Answer: 10 cm²

Worked Example 2

Problem: Find the area of a rectangle 6 cm by 3 cm.

  1. Use A = length × width.
  2. A = 6 × 3.

Very beginner explanation: Area covers a two-dimensional region, so it uses square units.

Answer: 18 cm²

Worked Example 3

Problem: Find the area of a rectangle 7 cm by 4 cm.

  1. Use A = length × width.
  2. A = 7 × 4.

Very beginner explanation: Area covers a two-dimensional region, so it uses square units.

Answer: 28 cm²

Worked Example 4

Problem: Find the area of a rectangle 8 cm by 5 cm.

  1. Use A = length × width.
  2. A = 8 × 5.

Very beginner explanation: Area covers a two-dimensional region, so it uses square units.

Answer: 40 cm²

Worked Example 5

Problem: Find the area of a rectangle 9 cm by 6 cm.

  1. Use A = length × width.
  2. A = 9 × 6.

Very beginner explanation: Area covers a two-dimensional region, so it uses square units.

Answer: 54 cm²

Worked Example 6

Problem: Find the area of a rectangle 10 cm by 7 cm.

  1. Use A = length × width.
  2. A = 10 × 7.

Very beginner explanation: Area covers a two-dimensional region, so it uses square units.

Answer: 70 cm²

Worked Example 7

Problem: Find the area of a rectangle 11 cm by 8 cm.

  1. Use A = length × width.
  2. A = 11 × 8.

Very beginner explanation: Area covers a two-dimensional region, so it uses square units.

Answer: 88 cm²

Worked Example 8

Problem: Find the area of a rectangle 12 cm by 9 cm.

  1. Use A = length × width.
  2. A = 12 × 9.

Very beginner explanation: Area covers a two-dimensional region, so it uses square units.

Answer: 108 cm²

Worked Example 9

Problem: Find the area of a rectangle 13 cm by 10 cm.

  1. Use A = length × width.
  2. A = 13 × 10.

Very beginner explanation: Area covers a two-dimensional region, so it uses square units.

Answer: 130 cm²

Worked Example 10

Problem: Find the area of a rectangle 14 cm by 11 cm.

  1. Use A = length × width.
  2. A = 14 × 11.

Very beginner explanation: Area covers a two-dimensional region, so it uses square units.

Answer: 154 cm²

Practice Exercise

Create one new Grade 6 problem about Play-area design project. Solve it step by step, label any units, and explain how you checked your answer.

Common Mistake

A common mistake is using the wrong formula or forgetting square units for area.

35.13 Cost comparison in a model

The mode is the value that appears most often. In this Grade 6 lesson, focus on meaning first, then use a small example, show the steps, and check whether the result fits the question.

Beginner Note

Identify what is given, what must be found, and which rule or representation fits this topic. Work one step at a time and explain why each step is valid.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: A simple cost model is C = 20 + 3n. Find the cost when n = 5.

  1. Substitute n = 5.
  2. Multiply 3 × 5.
  3. Add the fixed amount.

Very beginner explanation: A model is a simplified mathematical representation of a real situation.

Answer: $35

Worked Example 2

Problem: A simple cost model is C = 25 + 4n. Find the cost when n = 6.

  1. Substitute n = 6.
  2. Multiply 4 × 6.
  3. Add the fixed amount.

Very beginner explanation: A model is a simplified mathematical representation of a real situation.

Answer: $49

Worked Example 3

Problem: A simple cost model is C = 30 + 5n. Find the cost when n = 7.

  1. Substitute n = 7.
  2. Multiply 5 × 7.
  3. Add the fixed amount.

Very beginner explanation: A model is a simplified mathematical representation of a real situation.

Answer: $65

Worked Example 4

Problem: A simple cost model is C = 35 + 6n. Find the cost when n = 8.

  1. Substitute n = 8.
  2. Multiply 6 × 8.
  3. Add the fixed amount.

Very beginner explanation: A model is a simplified mathematical representation of a real situation.

Answer: $83

Worked Example 5

Problem: A simple cost model is C = 40 + 7n. Find the cost when n = 9.

  1. Substitute n = 9.
  2. Multiply 7 × 9.
  3. Add the fixed amount.

Very beginner explanation: A model is a simplified mathematical representation of a real situation.

Answer: $103

Worked Example 6

Problem: A simple cost model is C = 45 + 8n. Find the cost when n = 10.

  1. Substitute n = 10.
  2. Multiply 8 × 10.
  3. Add the fixed amount.

Very beginner explanation: A model is a simplified mathematical representation of a real situation.

Answer: $125

Worked Example 7

Problem: A simple cost model is C = 50 + 9n. Find the cost when n = 11.

  1. Substitute n = 11.
  2. Multiply 9 × 11.
  3. Add the fixed amount.

Very beginner explanation: A model is a simplified mathematical representation of a real situation.

Answer: $149

Worked Example 8

Problem: A simple cost model is C = 55 + 10n. Find the cost when n = 12.

  1. Substitute n = 12.
  2. Multiply 10 × 12.
  3. Add the fixed amount.

Very beginner explanation: A model is a simplified mathematical representation of a real situation.

Answer: $175

Worked Example 9

Problem: A simple cost model is C = 60 + 11n. Find the cost when n = 13.

  1. Substitute n = 13.
  2. Multiply 11 × 13.
  3. Add the fixed amount.

Very beginner explanation: A model is a simplified mathematical representation of a real situation.

Answer: $203

Worked Example 10

Problem: A simple cost model is C = 65 + 12n. Find the cost when n = 14.

  1. Substitute n = 14.
  2. Multiply 12 × 14.
  3. Add the fixed amount.

Very beginner explanation: A model is a simplified mathematical representation of a real situation.

Answer: $233

Practice Exercise

Create one new Grade 6 problem about Cost comparison in a model. Solve it step by step, label any units, and explain how you checked your answer.

Common Mistake

A common mistake is starting a calculation before identifying what is given, what must be found, and which rule fits the problem.

35.14 Communicating conclusions

Communicating conclusions is a Grade 6 mathematics idea in Mathematical Modelling Process and Design Project. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable. In this Grade 6 lesson, focus on meaning first, then use a small example, show the steps, and check whether the result fits the question.

Beginner Note

Identify what is given, what must be found, and which rule or representation fits this topic. Work one step at a time and explain why each step is valid.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Use Communicating conclusions to analyze the values 18 and 19. What should be checked first?

  1. Identify what each value represents.
  2. Choose the rule, representation, or comparison that matches the topic.
  3. Show the reasoning before accepting the result.

Very beginner explanation: Communicating conclusions is a Grade 6 mathematics idea in Mathematical Modelling Process and Design Project. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

Answer: Check the meaning of the values and the rule required by the problem.

Worked Example 2

Problem: Use Communicating conclusions to analyze the values 16 and 19. What should be checked first?

  1. Identify what each value represents.
  2. Choose the rule, representation, or comparison that matches the topic.
  3. Show the reasoning before accepting the result.

Very beginner explanation: Communicating conclusions is a Grade 6 mathematics idea in Mathematical Modelling Process and Design Project. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

Answer: Check the meaning of the values and the rule required by the problem.

Worked Example 3

Problem: Use Communicating conclusions to analyze the values 2 and 11. What should be checked first?

  1. Identify what each value represents.
  2. Choose the rule, representation, or comparison that matches the topic.
  3. Show the reasoning before accepting the result.

Very beginner explanation: Communicating conclusions is a Grade 6 mathematics idea in Mathematical Modelling Process and Design Project. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

Answer: Check the meaning of the values and the rule required by the problem.

Worked Example 4

Problem: Use Communicating conclusions to analyze the values 20 and 5. What should be checked first?

  1. Identify what each value represents.
  2. Choose the rule, representation, or comparison that matches the topic.
  3. Show the reasoning before accepting the result.

Very beginner explanation: Communicating conclusions is a Grade 6 mathematics idea in Mathematical Modelling Process and Design Project. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

Answer: Check the meaning of the values and the rule required by the problem.

Worked Example 5

Problem: Use Communicating conclusions to analyze the values 2 and 6. What should be checked first?

  1. Identify what each value represents.
  2. Choose the rule, representation, or comparison that matches the topic.
  3. Show the reasoning before accepting the result.

Very beginner explanation: Communicating conclusions is a Grade 6 mathematics idea in Mathematical Modelling Process and Design Project. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

Answer: Check the meaning of the values and the rule required by the problem.

Worked Example 6

Problem: Use Communicating conclusions to analyze the values 13 and 17. What should be checked first?

  1. Identify what each value represents.
  2. Choose the rule, representation, or comparison that matches the topic.
  3. Show the reasoning before accepting the result.

Very beginner explanation: Communicating conclusions is a Grade 6 mathematics idea in Mathematical Modelling Process and Design Project. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

Answer: Check the meaning of the values and the rule required by the problem.

Worked Example 7

Problem: Use Communicating conclusions to analyze the values 30 and 13. What should be checked first?

  1. Identify what each value represents.
  2. Choose the rule, representation, or comparison that matches the topic.
  3. Show the reasoning before accepting the result.

Very beginner explanation: Communicating conclusions is a Grade 6 mathematics idea in Mathematical Modelling Process and Design Project. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

Answer: Check the meaning of the values and the rule required by the problem.

Worked Example 8

Problem: Use Communicating conclusions to analyze the values 21 and 20. What should be checked first?

  1. Identify what each value represents.
  2. Choose the rule, representation, or comparison that matches the topic.
  3. Show the reasoning before accepting the result.

Very beginner explanation: Communicating conclusions is a Grade 6 mathematics idea in Mathematical Modelling Process and Design Project. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

Answer: Check the meaning of the values and the rule required by the problem.

Worked Example 9

Problem: Use Communicating conclusions to analyze the values 5 and 19. What should be checked first?

  1. Identify what each value represents.
  2. Choose the rule, representation, or comparison that matches the topic.
  3. Show the reasoning before accepting the result.

Very beginner explanation: Communicating conclusions is a Grade 6 mathematics idea in Mathematical Modelling Process and Design Project. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

Answer: Check the meaning of the values and the rule required by the problem.

Worked Example 10

Problem: Use Communicating conclusions to analyze the values 15 and 16. What should be checked first?

  1. Identify what each value represents.
  2. Choose the rule, representation, or comparison that matches the topic.
  3. Show the reasoning before accepting the result.

Very beginner explanation: Communicating conclusions is a Grade 6 mathematics idea in Mathematical Modelling Process and Design Project. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

Answer: Check the meaning of the values and the rule required by the problem.

Practice Exercise

Create one new Grade 6 problem about Communicating conclusions. Solve it step by step, label any units, and explain how you checked your answer.

Common Mistake

A common mistake is starting a calculation before identifying what is given, what must be found, and which rule fits the problem.

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30 Review Questions and Answers

Q1. What is important to understand about Meaning of mathematical modelling?

Answer: Mathematical modelling uses mathematics to represent, study, and improve a real-life situation.

Q2. What is important to understand about Identifying a real-life problem?

Answer: Identifying a real-life problem is a Grade 6 mathematics idea in Mathematical Modelling Process and Design Project. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

Q3. What is important to understand about Choosing important information?

Answer: Choosing important information is a Grade 6 mathematics idea in Mathematical Modelling Process and Design Project. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

Q4. What is important to understand about Making assumptions?

Answer: Making assumptions is a Grade 6 mathematics idea in Mathematical Modelling Process and Design Project. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

Q5. What is important to understand about Defining variables?

Answer: A variable is a symbol used to represent a number that may change or may be unknown.

Q6. What is important to understand about Creating a model?

Answer: The mode is the value that appears most often.

Q7. What is important to understand about Using tables in a model?

Answer: The mode is the value that appears most often.

Q8. What is important to understand about Using equations in a model?

Answer: An equation states that two mathematical expressions have the same value.

Q9. What is important to understand about Using graphs in a model?

Answer: The mode is the value that appears most often.

Q10. What is important to understand about Testing a model?

Answer: The mode is the value that appears most often.

Q11. What is important to understand about Revising a model?

Answer: The mode is the value that appears most often.

Q12. What is important to understand about Play-area design project?

Answer: Area measures the amount of two-dimensional space inside a shape.

Q13. What is important to understand about Cost comparison in a model?

Answer: The mode is the value that appears most often.

Q14. What is important to understand about Communicating conclusions?

Answer: Communicating conclusions is a Grade 6 mathematics idea in Mathematical Modelling Process and Design Project. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

Q15. Why should you show your steps?

Answer: Showing steps makes reasoning easier to verify and helps locate mistakes.

Q16. Why is estimation useful?

Answer: Estimation helps you judge whether a final answer is reasonable before accepting it.

Q17. Why do units matter?

Answer: Units tell what a value measures and help prevent incompatible quantities from being mixed.

Q18. When should you round?

Answer: Usually round near the end unless the question specifically asks for earlier rounding.

Q19. How can you check an answer?

Answer: Use an inverse operation, substitution, estimation, a graph, a table, or a second method.

Q20. What should you identify first in a word problem?

Answer: Identify what is known, what is unknown, and the relationship connecting them.

Q21. What makes a final answer complete?

Answer: Give the value, correct units when needed, and a brief interpretation in context.

Q22. What should you do if an answer seems unreasonable?

Answer: Re-read the question and check copied values, operations, signs, units, rules, and calculations.

Q23. Why are diagrams useful?

Answer: A labelled diagram can reveal relationships that are harder to see in words alone.

Q24. Why are tables useful?

Answer: Tables organize values so patterns, rates, comparisons, and missing information are easier to see.

Q25. Why are graphs useful?

Answer: Graphs make relationships, changes, trends, and comparisons visible.

Q26. Why should you explain your reasoning?

Answer: An explanation shows why a method works instead of only giving a final number.

Q27. Why can more than one strategy be correct?

Answer: Different correct strategies can represent the same mathematics in different but equivalent ways.

Q28. What should you do before using a formula or rule?

Answer: Check what each quantity means and whether the rule matches the situation.

Q29. How does practice help in mathematics?

Answer: Practice builds accuracy and fluency while helping you recognize which strategies fit different problems.

Q30. What is a good way to learn from a mistake?

Answer: Find the exact step where the reasoning changed from correct to incorrect, then solve a similar problem again.