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Chapter 34: Mathematical Modelling Process

Learn Grade 7 mathematics from very beginner explanations through worked examples, practice, reasoning, and review.

Grade 7Beginner Friendly10 Examples Per Topic30 Q&APractice + Review
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Chapter Overview

This chapter teaches Mathematical Modelling Process with simple language, step-by-step reasoning, and original worked examples. Technical words are explained in plain language.

Key Technical Terms

  • Variable (a letter or symbol representing a number)
  • Equation (a statement that two expressions are equal)
  • Mathematical Model (a mathematical representation of a real situation)
  • Circle Graph (a graph using sectors of a circle to show parts of a whole)
  • Estimate (a close approximation used to check whether an answer is reasonable)
  • Solution (a value or result that satisfies the problem)
  • Representation (a way to show mathematics using symbols, diagrams, tables, graphs, or words)
  • Reasonableness (whether an answer makes sense in the context of the problem)

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.

34.1 What is mathematical modelling?

Mathematical modelling is the process of representing a real situation with mathematics, testing the model, and improving it when needed. This topic focuses on What is mathematical modelling? .

Beginner Note

Identify what is given, what must be found, choose the correct rule or representation, work one step at a time, and check whether the result makes sense.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Paint a room

  1. Model wall area and paint coverage
  2. State assumptions.
  3. Calculate using the model.
  4. Check whether the result is realistic.
  5. Revise if needed.

Very beginner explanation: Mathematical modelling is the process of representing a real situation with mathematics, testing the model, and improving it when needed. This topic focuses on What is mathematical modelling? .

Answer: Estimate litres and cost

Worked Example 2

Problem: Tile a floor

  1. Model floor area and tile area
  2. State assumptions.
  3. Calculate using the model.
  4. Check whether the result is realistic.
  5. Revise if needed.

Very beginner explanation: Mathematical modelling is the process of representing a real situation with mathematics, testing the model, and improving it when needed. This topic focuses on What is mathematical modelling? .

Answer: Estimate tile count plus waste

Worked Example 3

Problem: Plan a school event

  1. Model attendance and cost per person
  2. State assumptions.
  3. Calculate using the model.
  4. Check whether the result is realistic.
  5. Revise if needed.

Very beginner explanation: Mathematical modelling is the process of representing a real situation with mathematics, testing the model, and improving it when needed. This topic focuses on What is mathematical modelling? .

Answer: Estimate total cost

Worked Example 4

Problem: Compare transportation

  1. Model fixed and per-kilometre costs
  2. State assumptions.
  3. Calculate using the model.
  4. Check whether the result is realistic.
  5. Revise if needed.

Very beginner explanation: Mathematical modelling is the process of representing a real situation with mathematics, testing the model, and improving it when needed. This topic focuses on What is mathematical modelling? .

Answer: Find a break-even distance

Worked Example 5

Problem: Plan monthly savings

  1. Model starting savings plus monthly deposits
  2. State assumptions.
  3. Calculate using the model.
  4. Check whether the result is realistic.
  5. Revise if needed.

Very beginner explanation: Mathematical modelling is the process of representing a real situation with mathematics, testing the model, and improving it when needed. This topic focuses on What is mathematical modelling? .

Answer: Predict when a goal is reached

Worked Example 6

Problem: Tile a floor

  1. Write a simple rule: area = length × width.
  2. State any assumptions.
  3. Use the rule on sample values.
  4. Check whether the result is realistic.

Very beginner explanation: Coding and modelling organize a real situation into clear steps, variables, rules, and checks.

Answer: Estimate tile quantity and include a small waste allowance.

Worked Example 7

Problem: Plan monthly savings

  1. Write a simple rule: total = starting amount + monthly deposit × months.
  2. State any assumptions.
  3. Use the rule on sample values.
  4. Check whether the result is realistic.

Very beginner explanation: Coding and modelling organize a real situation into clear steps, variables, rules, and checks.

Answer: Predict when the savings goal is reached.

Worked Example 8

Problem: Simulate a die

  1. Write a simple rule: generate a random whole number from 1 to 6.
  2. State any assumptions.
  3. Use the rule on sample values.
  4. Check whether the result is realistic.

Very beginner explanation: Coding and modelling organize a real situation into clear steps, variables, rules, and checks.

Answer: Repeat many times and compare frequencies.

Worked Example 9

Problem: Translate a point

  1. Write a simple rule: new x = x + 3; new y = y - 2.
  2. State any assumptions.
  3. Use the rule on sample values.
  4. Check whether the result is realistic.

Very beginner explanation: Coding and modelling organize a real situation into clear steps, variables, rules, and checks.

Answer: Apply the same rule to every point.

Worked Example 10

Problem: Estimate paint needed

  1. Write a simple rule: paint = wall area ÷ coverage per can.
  2. State any assumptions.
  3. Use the rule on sample values.
  4. Check whether the result is realistic.

Very beginner explanation: Coding and modelling organize a real situation into clear steps, variables, rules, and checks.

Answer: Round up because a partial can may not be enough.

Practice Exercise

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

34.2 Identifying a real-life problem

Identifying a real-life problem is an important Grade 7 idea in Mathematical Modelling Process. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Beginner Note

Identify what is given, what must be found, choose the correct rule or representation, work one step at a time, and check whether the result makes sense.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Paint a room

  1. Model wall area and paint coverage
  2. State assumptions.
  3. Calculate using the model.
  4. Check whether the result is realistic.
  5. Revise if needed.

Very beginner explanation: Identifying a real-life problem is an important Grade 7 idea in Mathematical Modelling Process. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Answer: Estimate litres and cost

Worked Example 2

Problem: Tile a floor

  1. Model floor area and tile area
  2. State assumptions.
  3. Calculate using the model.
  4. Check whether the result is realistic.
  5. Revise if needed.

Very beginner explanation: Identifying a real-life problem is an important Grade 7 idea in Mathematical Modelling Process. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Answer: Estimate tile count plus waste

Worked Example 3

Problem: Plan a school event

  1. Model attendance and cost per person
  2. State assumptions.
  3. Calculate using the model.
  4. Check whether the result is realistic.
  5. Revise if needed.

Very beginner explanation: Identifying a real-life problem is an important Grade 7 idea in Mathematical Modelling Process. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Answer: Estimate total cost

Worked Example 4

Problem: Compare transportation

  1. Model fixed and per-kilometre costs
  2. State assumptions.
  3. Calculate using the model.
  4. Check whether the result is realistic.
  5. Revise if needed.

Very beginner explanation: Identifying a real-life problem is an important Grade 7 idea in Mathematical Modelling Process. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Answer: Find a break-even distance

Worked Example 5

Problem: Plan monthly savings

  1. Model starting savings plus monthly deposits
  2. State assumptions.
  3. Calculate using the model.
  4. Check whether the result is realistic.
  5. Revise if needed.

Very beginner explanation: Identifying a real-life problem is an important Grade 7 idea in Mathematical Modelling Process. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Answer: Predict when a goal is reached

Worked Example 6

Problem: Explain Identifying a real-life problem in one simple sentence.

  1. Read the definition of the topic.
  2. Use the definition to build the response.
  3. Check that the response matches every important condition.

Very beginner explanation: Identifying a real-life problem 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 Identifying a real-life problem and explain why.

  1. Read the definition of the topic.
  2. Use the definition to build the response.
  3. Check that the response matches every important condition.

Very beginner explanation: Identifying a real-life problem 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 Identifying a real-life problem using a table, diagram, number line, graph, or equation.

  1. Read the definition of the topic.
  2. Use the definition to build the response.
  3. Check that the response matches every important condition.

Very beginner explanation: Identifying a real-life problem 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 Identifying a real-life problem problem, how can you check the answer?

  1. Read the definition of the topic.
  2. Use the definition to build the response.
  3. Check that the response matches every important condition.

Very beginner explanation: Identifying a real-life problem 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 Identifying a real-life problem could be useful.

  1. Read the definition of the topic.
  2. Use the definition to build the response.
  3. Check that the response matches every important condition.

Very beginner explanation: Identifying a real-life problem 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 Identifying a real-life problem. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.

34.3 Choosing important information

Choosing important information is an important Grade 7 idea in Mathematical Modelling Process. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Beginner Note

Identify what is given, what must be found, choose the correct rule or representation, work one step at a time, and check whether the result makes sense.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Choosing important information: Identify a correct example.

  1. State the definition and show why the example fits.
  2. Show all important reasoning.

Very beginner explanation: Choosing important information is an important Grade 7 idea in Mathematical Modelling Process. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Answer: A correct response must satisfy the definition or rule and include a check.

Worked Example 2

Problem: Choosing important information: Identify a non-example.

  1. Explain which requirement is missing.
  2. Show all important reasoning.

Very beginner explanation: Choosing important information is an important Grade 7 idea in Mathematical Modelling Process. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Answer: A correct response must satisfy the definition or rule and include a check.

Worked Example 3

Problem: Choosing important information: Compare two cases.

  1. State one similarity and one difference.
  2. Show all important reasoning.

Very beginner explanation: Choosing important information is an important Grade 7 idea in Mathematical Modelling Process. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Answer: A correct response must satisfy the definition or rule and include a check.

Worked Example 4

Problem: Choosing important information: Apply the idea in a real situation.

  1. Translate the situation into mathematical language.
  2. Show all important reasoning.

Very beginner explanation: Choosing important information is an important Grade 7 idea in Mathematical Modelling Process. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Answer: A correct response must satisfy the definition or rule and include a check.

Worked Example 5

Problem: Choosing important information: Create and check your own example.

  1. Use the definition, then verify each condition.
  2. Show all important reasoning.

Very beginner explanation: Choosing important information is an important Grade 7 idea in Mathematical Modelling Process. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Answer: A correct response must satisfy the definition or rule and include a check.

Worked Example 6

Problem: Explain Choosing important information in one simple sentence.

  1. Read the definition of the topic.
  2. Use the definition to build the response.
  3. Check that the response matches every important condition.

Very beginner explanation: Choosing important information 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 Choosing important information and explain why.

  1. Read the definition of the topic.
  2. Use the definition to build the response.
  3. Check that the response matches every important condition.

Very beginner explanation: Choosing important information 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 Choosing important information using a table, diagram, number line, graph, or equation.

  1. Read the definition of the topic.
  2. Use the definition to build the response.
  3. Check that the response matches every important condition.

Very beginner explanation: Choosing important information 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 Choosing important information problem, how can you check the answer?

  1. Read the definition of the topic.
  2. Use the definition to build the response.
  3. Check that the response matches every important condition.

Very beginner explanation: Choosing important information 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 Choosing important information could be useful.

  1. Read the definition of the topic.
  2. Use the definition to build the response.
  3. Check that the response matches every important condition.

Very beginner explanation: Choosing important information 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 Choosing important information. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.

34.4 Making assumptions

An assumption is a reasonable statement accepted for the purpose of building a model. This topic focuses on Making assumptions .

Beginner Note

Identify what is given, what must be found, choose the correct rule or representation, work one step at a time, and check whether the result makes sense.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Paint a room

  1. Model wall area and paint coverage
  2. State assumptions.
  3. Calculate using the model.
  4. Check whether the result is realistic.
  5. Revise if needed.

Very beginner explanation: An assumption is a reasonable statement accepted for the purpose of building a model. This topic focuses on Making assumptions .

Answer: Estimate litres and cost

Worked Example 2

Problem: Tile a floor

  1. Model floor area and tile area
  2. State assumptions.
  3. Calculate using the model.
  4. Check whether the result is realistic.
  5. Revise if needed.

Very beginner explanation: An assumption is a reasonable statement accepted for the purpose of building a model. This topic focuses on Making assumptions .

Answer: Estimate tile count plus waste

Worked Example 3

Problem: Plan a school event

  1. Model attendance and cost per person
  2. State assumptions.
  3. Calculate using the model.
  4. Check whether the result is realistic.
  5. Revise if needed.

Very beginner explanation: An assumption is a reasonable statement accepted for the purpose of building a model. This topic focuses on Making assumptions .

Answer: Estimate total cost

Worked Example 4

Problem: Compare transportation

  1. Model fixed and per-kilometre costs
  2. State assumptions.
  3. Calculate using the model.
  4. Check whether the result is realistic.
  5. Revise if needed.

Very beginner explanation: An assumption is a reasonable statement accepted for the purpose of building a model. This topic focuses on Making assumptions .

Answer: Find a break-even distance

Worked Example 5

Problem: Plan monthly savings

  1. Model starting savings plus monthly deposits
  2. State assumptions.
  3. Calculate using the model.
  4. Check whether the result is realistic.
  5. Revise if needed.

Very beginner explanation: An assumption is a reasonable statement accepted for the purpose of building a model. This topic focuses on Making assumptions .

Answer: Predict when a goal is reached

Worked Example 6

Problem: Tile a floor

  1. Write a simple rule: area = length × width.
  2. State any assumptions.
  3. Use the rule on sample values.
  4. Check whether the result is realistic.

Very beginner explanation: Coding and modelling organize a real situation into clear steps, variables, rules, and checks.

Answer: Estimate tile quantity and include a small waste allowance.

Worked Example 7

Problem: Plan monthly savings

  1. Write a simple rule: total = starting amount + monthly deposit × months.
  2. State any assumptions.
  3. Use the rule on sample values.
  4. Check whether the result is realistic.

Very beginner explanation: Coding and modelling organize a real situation into clear steps, variables, rules, and checks.

Answer: Predict when the savings goal is reached.

Worked Example 8

Problem: Simulate a die

  1. Write a simple rule: generate a random whole number from 1 to 6.
  2. State any assumptions.
  3. Use the rule on sample values.
  4. Check whether the result is realistic.

Very beginner explanation: Coding and modelling organize a real situation into clear steps, variables, rules, and checks.

Answer: Repeat many times and compare frequencies.

Worked Example 9

Problem: Translate a point

  1. Write a simple rule: new x = x + 3; new y = y - 2.
  2. State any assumptions.
  3. Use the rule on sample values.
  4. Check whether the result is realistic.

Very beginner explanation: Coding and modelling organize a real situation into clear steps, variables, rules, and checks.

Answer: Apply the same rule to every point.

Worked Example 10

Problem: Estimate paint needed

  1. Write a simple rule: paint = wall area ÷ coverage per can.
  2. State any assumptions.
  3. Use the rule on sample values.
  4. Check whether the result is realistic.

Very beginner explanation: Coding and modelling organize a real situation into clear steps, variables, rules, and checks.

Answer: Round up because a partial can may not be enough.

Practice Exercise

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

34.5 Defining variables

A variable is a letter or symbol representing a number that can change or may be unknown. This topic focuses on Defining variables .

Beginner Note

Identify what is given, what must be found, choose the correct rule or representation, work one step at a time, and check whether the result makes sense.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Simplify 3x+2x.

  1. Identify like terms or use distribution.
  2. Combine carefully.

Very beginner explanation: A variable is a letter or symbol representing a number that can change or may be unknown. This topic focuses on Defining variables .

Answer: 5x

Worked Example 2

Problem: Simplify 4(a+3).

  1. Identify like terms or use distribution.
  2. Combine carefully.

Very beginner explanation: A variable is a letter or symbol representing a number that can change or may be unknown. This topic focuses on Defining variables .

Answer: 4a+12

Worked Example 3

Problem: Simplify 7y-2y+5.

  1. Identify like terms or use distribution.
  2. Combine carefully.

Very beginner explanation: A variable is a letter or symbol representing a number that can change or may be unknown. This topic focuses on Defining variables .

Answer: 5y+5

Worked Example 4

Problem: Simplify 2(3x-4).

  1. Identify like terms or use distribution.
  2. Combine carefully.

Very beginner explanation: A variable is a letter or symbol representing a number that can change or may be unknown. This topic focuses on Defining variables .

Answer: 6x-8

Worked Example 5

Problem: Simplify 5m+3-2m.

  1. Identify like terms or use distribution.
  2. Combine carefully.

Very beginner explanation: A variable is a letter or symbol representing a number that can change or may be unknown. This topic focuses on Defining variables .

Answer: 3m+3

Worked Example 6

Problem: Simplify 3x + 4x.

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

Very beginner explanation: Like terms have the same variable part. The distributive property multiplies every term inside parentheses.

Answer: 7x

Worked Example 7

Problem: Simplify 8y - 3y + 2.

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

Very beginner explanation: Like terms have the same variable part. The distributive property multiplies every term inside parentheses.

Answer: 5y + 2

Worked Example 8

Problem: Simplify 4(a + 2).

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

Very beginner explanation: Like terms have the same variable part. The distributive property multiplies every term inside parentheses.

Answer: 4a + 8

Worked Example 9

Problem: Simplify 2(3x - 5) + x.

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

Very beginner explanation: Like terms have the same variable part. The distributive property multiplies every term inside parentheses.

Answer: 7x - 10

Worked Example 10

Problem: Simplify 6m + 7 - 2m - 3.

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

Very beginner explanation: Like terms have the same variable part. The distributive property multiplies every term inside parentheses.

Answer: 4m + 4

Practice Exercise

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

34.6 Creating a mathematical model

The mode is the value that occurs most often. This topic focuses on Creating a mathematical model .

Beginner Note

Identify what is given, what must be found, choose the correct rule or representation, work one step at a time, and check whether the result makes sense.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Paint a room

  1. Model wall area and paint coverage
  2. State assumptions.
  3. Calculate using the model.
  4. Check whether the result is realistic.
  5. Revise if needed.

Very beginner explanation: The mode is the value that occurs most often. This topic focuses on Creating a mathematical model .

Answer: Estimate litres and cost

Worked Example 2

Problem: Tile a floor

  1. Model floor area and tile area
  2. State assumptions.
  3. Calculate using the model.
  4. Check whether the result is realistic.
  5. Revise if needed.

Very beginner explanation: The mode is the value that occurs most often. This topic focuses on Creating a mathematical model .

Answer: Estimate tile count plus waste

Worked Example 3

Problem: Plan a school event

  1. Model attendance and cost per person
  2. State assumptions.
  3. Calculate using the model.
  4. Check whether the result is realistic.
  5. Revise if needed.

Very beginner explanation: The mode is the value that occurs most often. This topic focuses on Creating a mathematical model .

Answer: Estimate total cost

Worked Example 4

Problem: Compare transportation

  1. Model fixed and per-kilometre costs
  2. State assumptions.
  3. Calculate using the model.
  4. Check whether the result is realistic.
  5. Revise if needed.

Very beginner explanation: The mode is the value that occurs most often. This topic focuses on Creating a mathematical model .

Answer: Find a break-even distance

Worked Example 5

Problem: Plan monthly savings

  1. Model starting savings plus monthly deposits
  2. State assumptions.
  3. Calculate using the model.
  4. Check whether the result is realistic.
  5. Revise if needed.

Very beginner explanation: The mode is the value that occurs most often. This topic focuses on Creating a mathematical model .

Answer: Predict when a goal is reached

Worked Example 6

Problem: Tile a floor

  1. Write a simple rule: area = length × width.
  2. State any assumptions.
  3. Use the rule on sample values.
  4. Check whether the result is realistic.

Very beginner explanation: Coding and modelling organize a real situation into clear steps, variables, rules, and checks.

Answer: Estimate tile quantity and include a small waste allowance.

Worked Example 7

Problem: Plan monthly savings

  1. Write a simple rule: total = starting amount + monthly deposit × months.
  2. State any assumptions.
  3. Use the rule on sample values.
  4. Check whether the result is realistic.

Very beginner explanation: Coding and modelling organize a real situation into clear steps, variables, rules, and checks.

Answer: Predict when the savings goal is reached.

Worked Example 8

Problem: Simulate a die

  1. Write a simple rule: generate a random whole number from 1 to 6.
  2. State any assumptions.
  3. Use the rule on sample values.
  4. Check whether the result is realistic.

Very beginner explanation: Coding and modelling organize a real situation into clear steps, variables, rules, and checks.

Answer: Repeat many times and compare frequencies.

Worked Example 9

Problem: Translate a point

  1. Write a simple rule: new x = x + 3; new y = y - 2.
  2. State any assumptions.
  3. Use the rule on sample values.
  4. Check whether the result is realistic.

Very beginner explanation: Coding and modelling organize a real situation into clear steps, variables, rules, and checks.

Answer: Apply the same rule to every point.

Worked Example 10

Problem: Estimate paint needed

  1. Write a simple rule: paint = wall area ÷ coverage per can.
  2. State any assumptions.
  3. Use the rule on sample values.
  4. Check whether the result is realistic.

Very beginner explanation: Coding and modelling organize a real situation into clear steps, variables, rules, and checks.

Answer: Round up because a partial can may not be enough.

Practice Exercise

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

34.7 Using tables in models

The mode is the value that occurs most often. This topic focuses on Using tables in models .

Beginner Note

Identify what is given, what must be found, choose the correct rule or representation, work one step at a time, and check whether the result makes sense.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Paint a room

  1. Model wall area and paint coverage
  2. State assumptions.
  3. Calculate using the model.
  4. Check whether the result is realistic.
  5. Revise if needed.

Very beginner explanation: The mode is the value that occurs most often. This topic focuses on Using tables in models .

Answer: Estimate litres and cost

Worked Example 2

Problem: Tile a floor

  1. Model floor area and tile area
  2. State assumptions.
  3. Calculate using the model.
  4. Check whether the result is realistic.
  5. Revise if needed.

Very beginner explanation: The mode is the value that occurs most often. This topic focuses on Using tables in models .

Answer: Estimate tile count plus waste

Worked Example 3

Problem: Plan a school event

  1. Model attendance and cost per person
  2. State assumptions.
  3. Calculate using the model.
  4. Check whether the result is realistic.
  5. Revise if needed.

Very beginner explanation: The mode is the value that occurs most often. This topic focuses on Using tables in models .

Answer: Estimate total cost

Worked Example 4

Problem: Compare transportation

  1. Model fixed and per-kilometre costs
  2. State assumptions.
  3. Calculate using the model.
  4. Check whether the result is realistic.
  5. Revise if needed.

Very beginner explanation: The mode is the value that occurs most often. This topic focuses on Using tables in models .

Answer: Find a break-even distance

Worked Example 5

Problem: Plan monthly savings

  1. Model starting savings plus monthly deposits
  2. State assumptions.
  3. Calculate using the model.
  4. Check whether the result is realistic.
  5. Revise if needed.

Very beginner explanation: The mode is the value that occurs most often. This topic focuses on Using tables in models .

Answer: Predict when a goal is reached

Worked Example 6

Problem: Tile a floor

  1. Write a simple rule: area = length × width.
  2. State any assumptions.
  3. Use the rule on sample values.
  4. Check whether the result is realistic.

Very beginner explanation: Coding and modelling organize a real situation into clear steps, variables, rules, and checks.

Answer: Estimate tile quantity and include a small waste allowance.

Worked Example 7

Problem: Plan monthly savings

  1. Write a simple rule: total = starting amount + monthly deposit × months.
  2. State any assumptions.
  3. Use the rule on sample values.
  4. Check whether the result is realistic.

Very beginner explanation: Coding and modelling organize a real situation into clear steps, variables, rules, and checks.

Answer: Predict when the savings goal is reached.

Worked Example 8

Problem: Simulate a die

  1. Write a simple rule: generate a random whole number from 1 to 6.
  2. State any assumptions.
  3. Use the rule on sample values.
  4. Check whether the result is realistic.

Very beginner explanation: Coding and modelling organize a real situation into clear steps, variables, rules, and checks.

Answer: Repeat many times and compare frequencies.

Worked Example 9

Problem: Translate a point

  1. Write a simple rule: new x = x + 3; new y = y - 2.
  2. State any assumptions.
  3. Use the rule on sample values.
  4. Check whether the result is realistic.

Very beginner explanation: Coding and modelling organize a real situation into clear steps, variables, rules, and checks.

Answer: Apply the same rule to every point.

Worked Example 10

Problem: Estimate paint needed

  1. Write a simple rule: paint = wall area ÷ coverage per can.
  2. State any assumptions.
  3. Use the rule on sample values.
  4. Check whether the result is realistic.

Very beginner explanation: Coding and modelling organize a real situation into clear steps, variables, rules, and checks.

Answer: Round up because a partial can may not be enough.

Practice Exercise

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

34.8 Using equations in models

An equation says that two expressions are equal. This topic focuses on Using equations in models .

Beginner Note

Identify what is given, what must be found, choose the correct rule or representation, work one step at a time, and check whether the result makes sense.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Solve 2x+5=3x-1.

  1. Simplify each side if needed.
  2. Collect variable terms on one side.
  3. Collect constants on the other side.
  4. Divide to isolate the variable.
  5. Check by substitution.

Very beginner explanation: An equation says that two expressions are equal. This topic focuses on Using equations in models .

Answer: x=6

Worked Example 2

Problem: Solve 4x-7=13.

  1. Simplify each side if needed.
  2. Collect variable terms on one side.
  3. Collect constants on the other side.
  4. Divide to isolate the variable.
  5. Check by substitution.

Very beginner explanation: An equation says that two expressions are equal. This topic focuses on Using equations in models .

Answer: x=5

Worked Example 3

Problem: Solve 3x+2=2x+9.

  1. Simplify each side if needed.
  2. Collect variable terms on one side.
  3. Collect constants on the other side.
  4. Divide to isolate the variable.
  5. Check by substitution.

Very beginner explanation: An equation says that two expressions are equal. This topic focuses on Using equations in models .

Answer: x=7

Worked Example 4

Problem: Solve 5x+4=2x+19.

  1. Simplify each side if needed.
  2. Collect variable terms on one side.
  3. Collect constants on the other side.
  4. Divide to isolate the variable.
  5. Check by substitution.

Very beginner explanation: An equation says that two expressions are equal. This topic focuses on Using equations in models .

Answer: x=5

Worked Example 5

Problem: Solve 2(x+3)=18.

  1. Simplify each side if needed.
  2. Collect variable terms on one side.
  3. Collect constants on the other side.
  4. Divide to isolate the variable.
  5. Check by substitution.

Very beginner explanation: An equation says that two expressions are equal. This topic focuses on Using equations in models .

Answer: x=6

Worked Example 6

Problem: Solve 2x + 5 = 17.

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

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

Answer: x = 6

Worked Example 7

Problem: Solve 3x - 4 = 11.

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

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

Answer: x = 5

Worked Example 8

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

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

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

Answer: x = 5

Worked Example 9

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

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

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

Answer: x = 4

Worked Example 10

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

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

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

Answer: x = 5

Practice Exercise

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

34.9 Using graphs in models

The mode is the value that occurs most often. This topic focuses on Using graphs in models .

Beginner Note

Identify what is given, what must be found, choose the correct rule or representation, work one step at a time, and check whether the result makes sense.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Paint a room

  1. Model wall area and paint coverage
  2. State assumptions.
  3. Calculate using the model.
  4. Check whether the result is realistic.
  5. Revise if needed.

Very beginner explanation: The mode is the value that occurs most often. This topic focuses on Using graphs in models .

Answer: Estimate litres and cost

Worked Example 2

Problem: Tile a floor

  1. Model floor area and tile area
  2. State assumptions.
  3. Calculate using the model.
  4. Check whether the result is realistic.
  5. Revise if needed.

Very beginner explanation: The mode is the value that occurs most often. This topic focuses on Using graphs in models .

Answer: Estimate tile count plus waste

Worked Example 3

Problem: Plan a school event

  1. Model attendance and cost per person
  2. State assumptions.
  3. Calculate using the model.
  4. Check whether the result is realistic.
  5. Revise if needed.

Very beginner explanation: The mode is the value that occurs most often. This topic focuses on Using graphs in models .

Answer: Estimate total cost

Worked Example 4

Problem: Compare transportation

  1. Model fixed and per-kilometre costs
  2. State assumptions.
  3. Calculate using the model.
  4. Check whether the result is realistic.
  5. Revise if needed.

Very beginner explanation: The mode is the value that occurs most often. This topic focuses on Using graphs in models .

Answer: Find a break-even distance

Worked Example 5

Problem: Plan monthly savings

  1. Model starting savings plus monthly deposits
  2. State assumptions.
  3. Calculate using the model.
  4. Check whether the result is realistic.
  5. Revise if needed.

Very beginner explanation: The mode is the value that occurs most often. This topic focuses on Using graphs in models .

Answer: Predict when a goal is reached

Worked Example 6

Problem: Tile a floor

  1. Write a simple rule: area = length × width.
  2. State any assumptions.
  3. Use the rule on sample values.
  4. Check whether the result is realistic.

Very beginner explanation: Coding and modelling organize a real situation into clear steps, variables, rules, and checks.

Answer: Estimate tile quantity and include a small waste allowance.

Worked Example 7

Problem: Plan monthly savings

  1. Write a simple rule: total = starting amount + monthly deposit × months.
  2. State any assumptions.
  3. Use the rule on sample values.
  4. Check whether the result is realistic.

Very beginner explanation: Coding and modelling organize a real situation into clear steps, variables, rules, and checks.

Answer: Predict when the savings goal is reached.

Worked Example 8

Problem: Simulate a die

  1. Write a simple rule: generate a random whole number from 1 to 6.
  2. State any assumptions.
  3. Use the rule on sample values.
  4. Check whether the result is realistic.

Very beginner explanation: Coding and modelling organize a real situation into clear steps, variables, rules, and checks.

Answer: Repeat many times and compare frequencies.

Worked Example 9

Problem: Translate a point

  1. Write a simple rule: new x = x + 3; new y = y - 2.
  2. State any assumptions.
  3. Use the rule on sample values.
  4. Check whether the result is realistic.

Very beginner explanation: Coding and modelling organize a real situation into clear steps, variables, rules, and checks.

Answer: Apply the same rule to every point.

Worked Example 10

Problem: Estimate paint needed

  1. Write a simple rule: paint = wall area ÷ coverage per can.
  2. State any assumptions.
  3. Use the rule on sample values.
  4. Check whether the result is realistic.

Very beginner explanation: Coding and modelling organize a real situation into clear steps, variables, rules, and checks.

Answer: Round up because a partial can may not be enough.

Practice Exercise

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

34.10 Testing a model

The mode is the value that occurs most often. This topic focuses on Testing a model .

Beginner Note

Identify what is given, what must be found, choose the correct rule or representation, work one step at a time, and check whether the result makes sense.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Paint a room

  1. Model wall area and paint coverage
  2. State assumptions.
  3. Calculate using the model.
  4. Check whether the result is realistic.
  5. Revise if needed.

Very beginner explanation: The mode is the value that occurs most often. This topic focuses on Testing a model .

Answer: Estimate litres and cost

Worked Example 2

Problem: Tile a floor

  1. Model floor area and tile area
  2. State assumptions.
  3. Calculate using the model.
  4. Check whether the result is realistic.
  5. Revise if needed.

Very beginner explanation: The mode is the value that occurs most often. This topic focuses on Testing a model .

Answer: Estimate tile count plus waste

Worked Example 3

Problem: Plan a school event

  1. Model attendance and cost per person
  2. State assumptions.
  3. Calculate using the model.
  4. Check whether the result is realistic.
  5. Revise if needed.

Very beginner explanation: The mode is the value that occurs most often. This topic focuses on Testing a model .

Answer: Estimate total cost

Worked Example 4

Problem: Compare transportation

  1. Model fixed and per-kilometre costs
  2. State assumptions.
  3. Calculate using the model.
  4. Check whether the result is realistic.
  5. Revise if needed.

Very beginner explanation: The mode is the value that occurs most often. This topic focuses on Testing a model .

Answer: Find a break-even distance

Worked Example 5

Problem: Plan monthly savings

  1. Model starting savings plus monthly deposits
  2. State assumptions.
  3. Calculate using the model.
  4. Check whether the result is realistic.
  5. Revise if needed.

Very beginner explanation: The mode is the value that occurs most often. This topic focuses on Testing a model .

Answer: Predict when a goal is reached

Worked Example 6

Problem: Tile a floor

  1. Write a simple rule: area = length × width.
  2. State any assumptions.
  3. Use the rule on sample values.
  4. Check whether the result is realistic.

Very beginner explanation: Coding and modelling organize a real situation into clear steps, variables, rules, and checks.

Answer: Estimate tile quantity and include a small waste allowance.

Worked Example 7

Problem: Plan monthly savings

  1. Write a simple rule: total = starting amount + monthly deposit × months.
  2. State any assumptions.
  3. Use the rule on sample values.
  4. Check whether the result is realistic.

Very beginner explanation: Coding and modelling organize a real situation into clear steps, variables, rules, and checks.

Answer: Predict when the savings goal is reached.

Worked Example 8

Problem: Simulate a die

  1. Write a simple rule: generate a random whole number from 1 to 6.
  2. State any assumptions.
  3. Use the rule on sample values.
  4. Check whether the result is realistic.

Very beginner explanation: Coding and modelling organize a real situation into clear steps, variables, rules, and checks.

Answer: Repeat many times and compare frequencies.

Worked Example 9

Problem: Translate a point

  1. Write a simple rule: new x = x + 3; new y = y - 2.
  2. State any assumptions.
  3. Use the rule on sample values.
  4. Check whether the result is realistic.

Very beginner explanation: Coding and modelling organize a real situation into clear steps, variables, rules, and checks.

Answer: Apply the same rule to every point.

Worked Example 10

Problem: Estimate paint needed

  1. Write a simple rule: paint = wall area ÷ coverage per can.
  2. State any assumptions.
  3. Use the rule on sample values.
  4. Check whether the result is realistic.

Very beginner explanation: Coding and modelling organize a real situation into clear steps, variables, rules, and checks.

Answer: Round up because a partial can may not be enough.

Practice Exercise

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

34.11 Revising a model

The mode is the value that occurs most often. This topic focuses on Revising a model .

Beginner Note

Identify what is given, what must be found, choose the correct rule or representation, work one step at a time, and check whether the result makes sense.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Paint a room

  1. Model wall area and paint coverage
  2. State assumptions.
  3. Calculate using the model.
  4. Check whether the result is realistic.
  5. Revise if needed.

Very beginner explanation: The mode is the value that occurs most often. This topic focuses on Revising a model .

Answer: Estimate litres and cost

Worked Example 2

Problem: Tile a floor

  1. Model floor area and tile area
  2. State assumptions.
  3. Calculate using the model.
  4. Check whether the result is realistic.
  5. Revise if needed.

Very beginner explanation: The mode is the value that occurs most often. This topic focuses on Revising a model .

Answer: Estimate tile count plus waste

Worked Example 3

Problem: Plan a school event

  1. Model attendance and cost per person
  2. State assumptions.
  3. Calculate using the model.
  4. Check whether the result is realistic.
  5. Revise if needed.

Very beginner explanation: The mode is the value that occurs most often. This topic focuses on Revising a model .

Answer: Estimate total cost

Worked Example 4

Problem: Compare transportation

  1. Model fixed and per-kilometre costs
  2. State assumptions.
  3. Calculate using the model.
  4. Check whether the result is realistic.
  5. Revise if needed.

Very beginner explanation: The mode is the value that occurs most often. This topic focuses on Revising a model .

Answer: Find a break-even distance

Worked Example 5

Problem: Plan monthly savings

  1. Model starting savings plus monthly deposits
  2. State assumptions.
  3. Calculate using the model.
  4. Check whether the result is realistic.
  5. Revise if needed.

Very beginner explanation: The mode is the value that occurs most often. This topic focuses on Revising a model .

Answer: Predict when a goal is reached

Worked Example 6

Problem: Tile a floor

  1. Write a simple rule: area = length × width.
  2. State any assumptions.
  3. Use the rule on sample values.
  4. Check whether the result is realistic.

Very beginner explanation: Coding and modelling organize a real situation into clear steps, variables, rules, and checks.

Answer: Estimate tile quantity and include a small waste allowance.

Worked Example 7

Problem: Plan monthly savings

  1. Write a simple rule: total = starting amount + monthly deposit × months.
  2. State any assumptions.
  3. Use the rule on sample values.
  4. Check whether the result is realistic.

Very beginner explanation: Coding and modelling organize a real situation into clear steps, variables, rules, and checks.

Answer: Predict when the savings goal is reached.

Worked Example 8

Problem: Simulate a die

  1. Write a simple rule: generate a random whole number from 1 to 6.
  2. State any assumptions.
  3. Use the rule on sample values.
  4. Check whether the result is realistic.

Very beginner explanation: Coding and modelling organize a real situation into clear steps, variables, rules, and checks.

Answer: Repeat many times and compare frequencies.

Worked Example 9

Problem: Translate a point

  1. Write a simple rule: new x = x + 3; new y = y - 2.
  2. State any assumptions.
  3. Use the rule on sample values.
  4. Check whether the result is realistic.

Very beginner explanation: Coding and modelling organize a real situation into clear steps, variables, rules, and checks.

Answer: Apply the same rule to every point.

Worked Example 10

Problem: Estimate paint needed

  1. Write a simple rule: paint = wall area ÷ coverage per can.
  2. State any assumptions.
  3. Use the rule on sample values.
  4. Check whether the result is realistic.

Very beginner explanation: Coding and modelling organize a real situation into clear steps, variables, rules, and checks.

Answer: Round up because a partial can may not be enough.

Practice Exercise

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

34.12 Interpreting results

Interpreting results is an important Grade 7 idea in Mathematical Modelling Process. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Beginner Note

Identify what is given, what must be found, choose the correct rule or representation, work one step at a time, and check whether the result makes sense.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Interpreting results: Identify a correct example.

  1. State the definition and show why the example fits.
  2. Show all important reasoning.

Very beginner explanation: Interpreting results is an important Grade 7 idea in Mathematical Modelling Process. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Answer: A correct response must satisfy the definition or rule and include a check.

Worked Example 2

Problem: Interpreting results: Identify a non-example.

  1. Explain which requirement is missing.
  2. Show all important reasoning.

Very beginner explanation: Interpreting results is an important Grade 7 idea in Mathematical Modelling Process. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Answer: A correct response must satisfy the definition or rule and include a check.

Worked Example 3

Problem: Interpreting results: Compare two cases.

  1. State one similarity and one difference.
  2. Show all important reasoning.

Very beginner explanation: Interpreting results is an important Grade 7 idea in Mathematical Modelling Process. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Answer: A correct response must satisfy the definition or rule and include a check.

Worked Example 4

Problem: Interpreting results: Apply the idea in a real situation.

  1. Translate the situation into mathematical language.
  2. Show all important reasoning.

Very beginner explanation: Interpreting results is an important Grade 7 idea in Mathematical Modelling Process. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Answer: A correct response must satisfy the definition or rule and include a check.

Worked Example 5

Problem: Interpreting results: Create and check your own example.

  1. Use the definition, then verify each condition.
  2. Show all important reasoning.

Very beginner explanation: Interpreting results is an important Grade 7 idea in Mathematical Modelling Process. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Answer: A correct response must satisfy the definition or rule and include a check.

Worked Example 6

Problem: Explain Interpreting results in one simple sentence.

  1. Read the definition of the topic.
  2. Use the definition to build the response.
  3. Check that the response matches every important condition.

Very beginner explanation: Interpreting results 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 Interpreting results and explain why.

  1. Read the definition of the topic.
  2. Use the definition to build the response.
  3. Check that the response matches every important condition.

Very beginner explanation: Interpreting results 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 Interpreting results using a table, diagram, number line, graph, or equation.

  1. Read the definition of the topic.
  2. Use the definition to build the response.
  3. Check that the response matches every important condition.

Very beginner explanation: Interpreting results 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 Interpreting results problem, how can you check the answer?

  1. Read the definition of the topic.
  2. Use the definition to build the response.
  3. Check that the response matches every important condition.

Very beginner explanation: Interpreting results 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 Interpreting results could be useful.

  1. Read the definition of the topic.
  2. Use the definition to build the response.
  3. Check that the response matches every important condition.

Very beginner explanation: Interpreting results 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 Interpreting results. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.

34.13 Communicating conclusions

Mathematical communication means clearly explaining ideas using words, numbers, diagrams, symbols, tables, or graphs. This topic focuses on Communicating conclusions .

Beginner Note

Identify what is given, what must be found, choose the correct rule or representation, work one step at a time, and check whether the result makes sense.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Number problem

  1. Estimate first, calculate, then check
  2. Use an inverse operation
  3. Explain why the strategy fits the problem.

Very beginner explanation: Mathematical communication means clearly explaining ideas using words, numbers, diagrams, symbols, tables, or graphs. This topic focuses on Communicating conclusions .

Answer: A complete solution includes reasoning, calculations, and a check.

Worked Example 2

Problem: Geometry problem

  1. Draw and label a diagram
  2. Check units
  3. Explain why the strategy fits the problem.

Very beginner explanation: Mathematical communication means clearly explaining ideas using words, numbers, diagrams, symbols, tables, or graphs. This topic focuses on Communicating conclusions .

Answer: A complete solution includes reasoning, calculations, and a check.

Worked Example 3

Problem: Data problem

  1. Choose a suitable graph or measure
  2. Explain what the result means
  3. Explain why the strategy fits the problem.

Very beginner explanation: Mathematical communication means clearly explaining ideas using words, numbers, diagrams, symbols, tables, or graphs. This topic focuses on Communicating conclusions .

Answer: A complete solution includes reasoning, calculations, and a check.

Worked Example 4

Problem: Financial problem

  1. Write amounts and rates clearly
  2. Compare options
  3. Explain why the strategy fits the problem.

Very beginner explanation: Mathematical communication means clearly explaining ideas using words, numbers, diagrams, symbols, tables, or graphs. This topic focuses on Communicating conclusions .

Answer: A complete solution includes reasoning, calculations, and a check.

Worked Example 5

Problem: Algebra problem

  1. Define the variable and write an equation
  2. Substitute to check
  3. Explain why the strategy fits the problem.

Very beginner explanation: Mathematical communication means clearly explaining ideas using words, numbers, diagrams, symbols, tables, or graphs. This topic focuses on Communicating conclusions .

Answer: A complete solution includes reasoning, calculations, and a check.

Worked Example 6

Problem: Explain Communicating conclusions in one simple sentence.

  1. Read the definition of the topic.
  2. Use the definition to build the response.
  3. Check that the response matches every important condition.

Very beginner explanation: Communicating conclusions 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 Communicating conclusions and explain why.

  1. Read the definition of the topic.
  2. Use the definition to build the response.
  3. Check that the response matches every important condition.

Very beginner explanation: Communicating conclusions 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 Communicating conclusions using a table, diagram, number line, graph, or equation.

  1. Read the definition of the topic.
  2. Use the definition to build the response.
  3. Check that the response matches every important condition.

Very beginner explanation: Communicating conclusions 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 Communicating conclusions problem, how can you check the answer?

  1. Read the definition of the topic.
  2. Use the definition to build the response.
  3. Check that the response matches every important condition.

Very beginner explanation: Communicating conclusions 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 Communicating conclusions could be useful.

  1. Read the definition of the topic.
  2. Use the definition to build the response.
  3. Check that the response matches every important condition.

Very beginner explanation: Communicating conclusions 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 Communicating conclusions. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.

34.14 Limits of a model

The mode is the value that occurs most often. This topic focuses on Limits of a model .

Beginner Note

Identify what is given, what must be found, choose the correct rule or representation, work one step at a time, and check whether the result makes sense.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Paint a room

  1. Model wall area and paint coverage
  2. State assumptions.
  3. Calculate using the model.
  4. Check whether the result is realistic.
  5. Revise if needed.

Very beginner explanation: The mode is the value that occurs most often. This topic focuses on Limits of a model .

Answer: Estimate litres and cost

Worked Example 2

Problem: Tile a floor

  1. Model floor area and tile area
  2. State assumptions.
  3. Calculate using the model.
  4. Check whether the result is realistic.
  5. Revise if needed.

Very beginner explanation: The mode is the value that occurs most often. This topic focuses on Limits of a model .

Answer: Estimate tile count plus waste

Worked Example 3

Problem: Plan a school event

  1. Model attendance and cost per person
  2. State assumptions.
  3. Calculate using the model.
  4. Check whether the result is realistic.
  5. Revise if needed.

Very beginner explanation: The mode is the value that occurs most often. This topic focuses on Limits of a model .

Answer: Estimate total cost

Worked Example 4

Problem: Compare transportation

  1. Model fixed and per-kilometre costs
  2. State assumptions.
  3. Calculate using the model.
  4. Check whether the result is realistic.
  5. Revise if needed.

Very beginner explanation: The mode is the value that occurs most often. This topic focuses on Limits of a model .

Answer: Find a break-even distance

Worked Example 5

Problem: Plan monthly savings

  1. Model starting savings plus monthly deposits
  2. State assumptions.
  3. Calculate using the model.
  4. Check whether the result is realistic.
  5. Revise if needed.

Very beginner explanation: The mode is the value that occurs most often. This topic focuses on Limits of a model .

Answer: Predict when a goal is reached

Worked Example 6

Problem: Tile a floor

  1. Write a simple rule: area = length × width.
  2. State any assumptions.
  3. Use the rule on sample values.
  4. Check whether the result is realistic.

Very beginner explanation: Coding and modelling organize a real situation into clear steps, variables, rules, and checks.

Answer: Estimate tile quantity and include a small waste allowance.

Worked Example 7

Problem: Plan monthly savings

  1. Write a simple rule: total = starting amount + monthly deposit × months.
  2. State any assumptions.
  3. Use the rule on sample values.
  4. Check whether the result is realistic.

Very beginner explanation: Coding and modelling organize a real situation into clear steps, variables, rules, and checks.

Answer: Predict when the savings goal is reached.

Worked Example 8

Problem: Simulate a die

  1. Write a simple rule: generate a random whole number from 1 to 6.
  2. State any assumptions.
  3. Use the rule on sample values.
  4. Check whether the result is realistic.

Very beginner explanation: Coding and modelling organize a real situation into clear steps, variables, rules, and checks.

Answer: Repeat many times and compare frequencies.

Worked Example 9

Problem: Translate a point

  1. Write a simple rule: new x = x + 3; new y = y - 2.
  2. State any assumptions.
  3. Use the rule on sample values.
  4. Check whether the result is realistic.

Very beginner explanation: Coding and modelling organize a real situation into clear steps, variables, rules, and checks.

Answer: Apply the same rule to every point.

Worked Example 10

Problem: Estimate paint needed

  1. Write a simple rule: paint = wall area ÷ coverage per can.
  2. State any assumptions.
  3. Use the rule on sample values.
  4. Check whether the result is realistic.

Very beginner explanation: Coding and modelling organize a real situation into clear steps, variables, rules, and checks.

Answer: Round up because a partial can may not be enough.

Practice Exercise

Create one new question about Limits of a model. 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 complete question before choosing an operation, formula, graph, or model.
  • Write technical words together with their meaning until you are comfortable using them.
  • Show all important steps so another student can follow your reasoning.
  • Keep units, labels, signs, axes, variables, and mathematical symbols clear.
  • Estimate before or after calculating when an estimate can help check reasonableness.
  • For real-life problems, explain what the final number means in the situation.

Extra Practice

  1. Create and solve a new question about What is mathematical modelling? . Show your reasoning and check your answer.
  2. Create and solve a new question about Identifying a real-life problem . Show your reasoning and check your answer.
  3. Create and solve a new question about Choosing important information . Show your reasoning and check your answer.
  4. Create and solve a new question about Making assumptions . Show your reasoning and check your answer.
  5. Create and solve a new question about Defining variables . Show your reasoning and check your answer.
  6. Create and solve a new question about Creating a mathematical model . Show your reasoning and check your answer.
  7. Create and solve a new question about Using tables in models . Show your reasoning and check your answer.
  8. Create and solve a new question about Using equations in models . Show your reasoning and check your answer.
  9. Create and solve a new question about Using graphs in models . Show your reasoning and check your answer.
  10. Create and solve a new question about Testing a model . Show your reasoning and check your answer.
  11. Create and solve a new question about Revising a model . Show your reasoning and check your answer.
  12. Create and solve a new question about Interpreting results . Show your reasoning and check your answer.

Common Mistakes

  • Combining unlike terms.
  • Changing only one side of an equation.
  • Forgetting to distribute to every term.
  • Using a pattern rule that works only for the first few terms.
  • Writing code without tracing variable values.
  • Using a mathematical model without stating assumptions.
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30 Review Questions and Answers

Q1. What is important to remember about What is mathematical modelling??

Answer: Mathematical modelling is the process of representing a real situation with mathematics, testing the model, and improving it when needed. This topic focuses on What is mathematical modelling?.

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

Answer: Identifying a real-life problem is an important Grade 7 idea in Mathematical Modelling Process. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

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

Answer: Choosing important information is an important Grade 7 idea in Mathematical Modelling Process. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Q4. What is important to remember about Making assumptions?

Answer: An assumption is a reasonable statement accepted for the purpose of building a model. This topic focuses on Making assumptions.

Q5. What is important to remember about Defining variables?

Answer: A variable is a letter or symbol representing a number that can change or may be unknown. This topic focuses on Defining variables.

Q6. What is important to remember about Creating a mathematical model?

Answer: The mode is the value that occurs most often. This topic focuses on Creating a mathematical model.

Q7. What is important to remember about Using tables in models?

Answer: The mode is the value that occurs most often. This topic focuses on Using tables in models.

Q8. What is important to remember about Using equations in models?

Answer: An equation says that two expressions are equal. This topic focuses on Using equations in models.

Q9. What is important to remember about Using graphs in models?

Answer: The mode is the value that occurs most often. This topic focuses on Using graphs in models.

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

Answer: The mode is the value that occurs most often. This topic focuses on Testing a model.

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

Answer: The mode is the value that occurs most often. This topic focuses on Revising a model.

Q12. What is important to remember about Interpreting results?

Answer: Interpreting results is an important Grade 7 idea in Mathematical Modelling Process. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Q13. What is important to remember about Communicating conclusions?

Answer: Mathematical communication means clearly explaining ideas using words, numbers, diagrams, symbols, tables, or graphs. This topic focuses on Communicating conclusions.

Q14. What is important to remember about Limits of a model?

Answer: The mode is the value that occurs most often. This topic focuses on Limits of a model.

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 prevent incompatible quantities from being mixed.

Q18. When should you round?

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

Q19. How can you check an answer?

Answer: Use an inverse operation, substitution, estimation, a graph, 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, check copied values, signs, units, formulas, and calculations.

Q23. Why are diagrams useful?

Answer: A labelled diagram can reveal relationships between lengths, angles, areas, and other quantities.

Q24. Why are tables useful?

Answer: Tables organize values and help reveal patterns, relationships, rates, and missing information.

Q25. Why should you explain your reasoning?

Answer: Reasoning shows why a method works, not just what answer you obtained.

Q26. How do mistakes help learning?

Answer: Analyzing a mistake identifies the misunderstood rule or step and helps prevent the same error later.

Q27. How should you study this chapter?

Answer: Review definitions, redo worked examples without looking, practise mixed questions, and explain solutions in your own words.

Q28. When is a calculator useful?

Answer: A calculator is useful for lengthy arithmetic after you understand the mathematical setup and can estimate the expected size of the answer.

Q29. Why compare more than one strategy?

Answer: Different strategies can make a problem easier and provide a way to verify the result.

Q30. What is a mathematical model?

Answer: It is a simplified mathematical representation of a real situation used to analyze, explain, or predict.