Chapter 35: Mathematical Modelling Project: Room Renovation and Budget
Learn Grade 7 mathematics from very beginner explanations through worked examples, practice, reasoning, and review.
Chapter Overview
This chapter teaches Mathematical Modelling Project: Room Renovation and Budget with simple language, step-by-step reasoning, and original worked examples. Technical words are explained in plain language.
Key Technical Terms
- Unit Rate (a rate per one unit)
- Equation (a statement that two expressions are equal)
- Mathematical Model (a mathematical representation of a real situation)
- Surface Area (total area of the outside of a 3D object)
- 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.
35.1 Understanding the renovation problem
Understanding the renovation problem is an important Grade 7 idea in Mathematical Modelling Project: Room Renovation and Budget. 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
- Model wall area and paint coverage
- State assumptions.
- Calculate using the model.
- Check whether the result is realistic.
- Revise if needed.
Very beginner explanation: Understanding the renovation problem is an important Grade 7 idea in Mathematical Modelling Project: Room Renovation and Budget. 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
- Model floor area and tile area
- State assumptions.
- Calculate using the model.
- Check whether the result is realistic.
- Revise if needed.
Very beginner explanation: Understanding the renovation problem is an important Grade 7 idea in Mathematical Modelling Project: Room Renovation and Budget. 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
- Model attendance and cost per person
- State assumptions.
- Calculate using the model.
- Check whether the result is realistic.
- Revise if needed.
Very beginner explanation: Understanding the renovation problem is an important Grade 7 idea in Mathematical Modelling Project: Room Renovation and Budget. 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
- Model fixed and per-kilometre costs
- State assumptions.
- Calculate using the model.
- Check whether the result is realistic.
- Revise if needed.
Very beginner explanation: Understanding the renovation problem is an important Grade 7 idea in Mathematical Modelling Project: Room Renovation and Budget. 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
- Model starting savings plus monthly deposits
- State assumptions.
- Calculate using the model.
- Check whether the result is realistic.
- Revise if needed.
Very beginner explanation: Understanding the renovation problem is an important Grade 7 idea in Mathematical Modelling Project: Room Renovation and Budget. 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: Tile a floor
- Write a simple rule: area = length × width.
- State any assumptions.
- Use the rule on sample values.
- 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
- Write a simple rule: total = starting amount + monthly deposit × months.
- State any assumptions.
- Use the rule on sample values.
- 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
- Write a simple rule: generate a random whole number from 1 to 6.
- State any assumptions.
- Use the rule on sample values.
- 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
- Write a simple rule: new x = x + 3; new y = y - 2.
- State any assumptions.
- Use the rule on sample values.
- 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
- Write a simple rule: paint = wall area ÷ coverage per can.
- State any assumptions.
- Use the rule on sample values.
- 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 Understanding the renovation problem. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
35.2 Measuring room dimensions
Measuring room dimensions is an important Grade 7 idea in Mathematical Modelling Project: Room Renovation and Budget. 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: Measuring room dimensions: Identify a correct example.
- State the definition and show why the example fits.
- Show all important reasoning.
Very beginner explanation: Measuring room dimensions is an important Grade 7 idea in Mathematical Modelling Project: Room Renovation and Budget. 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: Measuring room dimensions: Identify a non-example.
- Explain which requirement is missing.
- Show all important reasoning.
Very beginner explanation: Measuring room dimensions is an important Grade 7 idea in Mathematical Modelling Project: Room Renovation and Budget. 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: Measuring room dimensions: Compare two cases.
- State one similarity and one difference.
- Show all important reasoning.
Very beginner explanation: Measuring room dimensions is an important Grade 7 idea in Mathematical Modelling Project: Room Renovation and Budget. 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: Measuring room dimensions: Apply the idea in a real situation.
- Translate the situation into mathematical language.
- Show all important reasoning.
Very beginner explanation: Measuring room dimensions is an important Grade 7 idea in Mathematical Modelling Project: Room Renovation and Budget. 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: Measuring room dimensions: Create and check your own example.
- Use the definition, then verify each condition.
- Show all important reasoning.
Very beginner explanation: Measuring room dimensions is an important Grade 7 idea in Mathematical Modelling Project: Room Renovation and Budget. 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 Measuring room dimensions in one simple sentence.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Measuring room dimensions 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 Measuring room dimensions and explain why.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Measuring room dimensions 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 Measuring room dimensions using a table, diagram, number line, graph, or equation.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Measuring room dimensions 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 Measuring room dimensions problem, how can you check the answer?
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Measuring room dimensions 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 Measuring room dimensions could be useful.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Measuring room dimensions 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 Measuring room dimensions. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
35.3 Drawing a scale floor plan
Drawing a scale floor plan is an important Grade 7 idea in Mathematical Modelling Project: Room Renovation and Budget. 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: Scale factor 2; original side 6 cm. Find image side.
- Multiply 6 by 2.
Very beginner explanation: Drawing a scale floor plan is an important Grade 7 idea in Mathematical Modelling Project: Room Renovation and Budget. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: 12 cm
Worked Example 2
Problem: Scale factor 1.5; original side 8 cm. Find image side.
- Multiply 8 by 1.5.
Very beginner explanation: Drawing a scale floor plan is an important Grade 7 idea in Mathematical Modelling Project: Room Renovation and Budget. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: 12 cm
Worked Example 3
Problem: Scale factor 0.5; original side 14 cm. Find image side.
- Multiply 14 by 0.5.
Very beginner explanation: Drawing a scale floor plan is an important Grade 7 idea in Mathematical Modelling Project: Room Renovation and Budget. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: 7 cm
Worked Example 4
Problem: Scale factor 3; original side 5 cm. Find image side.
- Multiply 5 by 3.
Very beginner explanation: Drawing a scale floor plan is an important Grade 7 idea in Mathematical Modelling Project: Room Renovation and Budget. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: 15 cm
Worked Example 5
Problem: Scale factor 4; original side 2.5 cm. Find image side.
- Multiply 2.5 by 4.
Very beginner explanation: Drawing a scale floor plan is an important Grade 7 idea in Mathematical Modelling Project: Room Renovation and Budget. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: 10 cm
Worked Example 6
Problem: An original length is 8 cm and the scale factor is 0.5. Find the image length.
- Multiply the original length by the scale factor.
- 8 × 0.5 = 4.
Very beginner explanation: A scale factor multiplies every corresponding length by the same amount.
Answer: 4 cm
Worked Example 7
Problem: An original length is 12 cm and the scale factor is 1.5. Find the image length.
- Multiply the original length by the scale factor.
- 12 × 1.5 = 18.
Very beginner explanation: A scale factor multiplies every corresponding length by the same amount.
Answer: 18 cm
Worked Example 8
Problem: An original length is 19 cm and the scale factor is 2. Find the image length.
- Multiply the original length by the scale factor.
- 19 × 2 = 38.
Very beginner explanation: A scale factor multiplies every corresponding length by the same amount.
Answer: 38 cm
Worked Example 9
Problem: An original length is 22 cm and the scale factor is 2.5. Find the image length.
- Multiply the original length by the scale factor.
- 22 × 2.5 = 55.
Very beginner explanation: A scale factor multiplies every corresponding length by the same amount.
Answer: 55 cm
Worked Example 10
Problem: An original length is 25 cm and the scale factor is 3. Find the image length.
- Multiply the original length by the scale factor.
- 25 × 3 = 75.
Very beginner explanation: A scale factor multiplies every corresponding length by the same amount.
Answer: 75 cm
Practice Exercise
Create one new question about Drawing a scale floor plan. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
35.4 Calculating floor area
Calculating floor area is an important Grade 7 idea in Mathematical Modelling Project: Room Renovation and Budget. 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: Calculating floor area: Identify a correct example.
- State the definition and show why the example fits.
- Show all important reasoning.
Very beginner explanation: Calculating floor area is an important Grade 7 idea in Mathematical Modelling Project: Room Renovation and Budget. 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: Calculating floor area: Identify a non-example.
- Explain which requirement is missing.
- Show all important reasoning.
Very beginner explanation: Calculating floor area is an important Grade 7 idea in Mathematical Modelling Project: Room Renovation and Budget. 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: Calculating floor area: Compare two cases.
- State one similarity and one difference.
- Show all important reasoning.
Very beginner explanation: Calculating floor area is an important Grade 7 idea in Mathematical Modelling Project: Room Renovation and Budget. 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: Calculating floor area: Apply the idea in a real situation.
- Translate the situation into mathematical language.
- Show all important reasoning.
Very beginner explanation: Calculating floor area is an important Grade 7 idea in Mathematical Modelling Project: Room Renovation and Budget. 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: Calculating floor area: Create and check your own example.
- Use the definition, then verify each condition.
- Show all important reasoning.
Very beginner explanation: Calculating floor area is an important Grade 7 idea in Mathematical Modelling Project: Room Renovation and Budget. 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: Find the area of a rectangle 6 cm by 3 cm.
- Use A = length × width.
- A = 6 × 3.
Very beginner explanation: Area measures the 2D space inside a figure and uses square units.
Answer: 18 cm²
Worked Example 7
Problem: Find the area of a rectangle 7 cm by 4 cm.
- Use A = length × width.
- A = 7 × 4.
Very beginner explanation: Area measures the 2D space inside a figure and uses square units.
Answer: 28 cm²
Worked Example 8
Problem: Find the area of a rectangle 8 cm by 5 cm.
- Use A = length × width.
- A = 8 × 5.
Very beginner explanation: Area measures the 2D space inside a figure and uses square units.
Answer: 40 cm²
Worked Example 9
Problem: Find the area of a rectangle 9 cm by 6 cm.
- Use A = length × width.
- A = 9 × 6.
Very beginner explanation: Area measures the 2D space inside a figure and uses square units.
Answer: 54 cm²
Worked Example 10
Problem: Find the area of a rectangle 10 cm by 7 cm.
- Use A = length × width.
- A = 10 × 7.
Very beginner explanation: Area measures the 2D space inside a figure and uses square units.
Answer: 70 cm²
Practice Exercise
Create one new question about Calculating floor area. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
35.5 Calculating wall area
Calculating wall area is an important Grade 7 idea in Mathematical Modelling Project: Room Renovation and Budget. 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: Calculating wall area: Identify a correct example.
- State the definition and show why the example fits.
- Show all important reasoning.
Very beginner explanation: Calculating wall area is an important Grade 7 idea in Mathematical Modelling Project: Room Renovation and Budget. 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: Calculating wall area: Identify a non-example.
- Explain which requirement is missing.
- Show all important reasoning.
Very beginner explanation: Calculating wall area is an important Grade 7 idea in Mathematical Modelling Project: Room Renovation and Budget. 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: Calculating wall area: Compare two cases.
- State one similarity and one difference.
- Show all important reasoning.
Very beginner explanation: Calculating wall area is an important Grade 7 idea in Mathematical Modelling Project: Room Renovation and Budget. 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: Calculating wall area: Apply the idea in a real situation.
- Translate the situation into mathematical language.
- Show all important reasoning.
Very beginner explanation: Calculating wall area is an important Grade 7 idea in Mathematical Modelling Project: Room Renovation and Budget. 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: Calculating wall area: Create and check your own example.
- Use the definition, then verify each condition.
- Show all important reasoning.
Very beginner explanation: Calculating wall area is an important Grade 7 idea in Mathematical Modelling Project: Room Renovation and Budget. 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: Find the area of a rectangle 6 cm by 3 cm.
- Use A = length × width.
- A = 6 × 3.
Very beginner explanation: Area measures the 2D space inside a figure and uses square units.
Answer: 18 cm²
Worked Example 7
Problem: Find the area of a rectangle 7 cm by 4 cm.
- Use A = length × width.
- A = 7 × 4.
Very beginner explanation: Area measures the 2D space inside a figure and uses square units.
Answer: 28 cm²
Worked Example 8
Problem: Find the area of a rectangle 8 cm by 5 cm.
- Use A = length × width.
- A = 8 × 5.
Very beginner explanation: Area measures the 2D space inside a figure and uses square units.
Answer: 40 cm²
Worked Example 9
Problem: Find the area of a rectangle 9 cm by 6 cm.
- Use A = length × width.
- A = 9 × 6.
Very beginner explanation: Area measures the 2D space inside a figure and uses square units.
Answer: 54 cm²
Worked Example 10
Problem: Find the area of a rectangle 10 cm by 7 cm.
- Use A = length × width.
- A = 10 × 7.
Very beginner explanation: Area measures the 2D space inside a figure and uses square units.
Answer: 70 cm²
Practice Exercise
Create one new question about Calculating wall area. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
35.6 Estimating material quantities
Estimating material quantities is an important Grade 7 idea in Mathematical Modelling Project: Room Renovation and Budget. 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
- Model wall area and paint coverage
- State assumptions.
- Calculate using the model.
- Check whether the result is realistic.
- Revise if needed.
Very beginner explanation: Estimating material quantities is an important Grade 7 idea in Mathematical Modelling Project: Room Renovation and Budget. 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
- Model floor area and tile area
- State assumptions.
- Calculate using the model.
- Check whether the result is realistic.
- Revise if needed.
Very beginner explanation: Estimating material quantities is an important Grade 7 idea in Mathematical Modelling Project: Room Renovation and Budget. 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
- Model attendance and cost per person
- State assumptions.
- Calculate using the model.
- Check whether the result is realistic.
- Revise if needed.
Very beginner explanation: Estimating material quantities is an important Grade 7 idea in Mathematical Modelling Project: Room Renovation and Budget. 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
- Model fixed and per-kilometre costs
- State assumptions.
- Calculate using the model.
- Check whether the result is realistic.
- Revise if needed.
Very beginner explanation: Estimating material quantities is an important Grade 7 idea in Mathematical Modelling Project: Room Renovation and Budget. 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
- Model starting savings plus monthly deposits
- State assumptions.
- Calculate using the model.
- Check whether the result is realistic.
- Revise if needed.
Very beginner explanation: Estimating material quantities is an important Grade 7 idea in Mathematical Modelling Project: Room Renovation and Budget. 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 Estimating material quantities in one simple sentence.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Estimating material quantities 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 Estimating material quantities and explain why.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Estimating material quantities 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 Estimating material quantities using a table, diagram, number line, graph, or equation.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Estimating material quantities 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 Estimating material quantities problem, how can you check the answer?
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Estimating material quantities 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 Estimating material quantities could be useful.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Estimating material quantities 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 Estimating material quantities. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
35.7 Comparing unit prices
Comparing unit prices is an important Grade 7 idea in Mathematical Modelling Project: Room Renovation and Budget. 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: Comparing unit prices: Identify a correct example.
- State the definition and show why the example fits.
- Show all important reasoning.
Very beginner explanation: Comparing unit prices is an important Grade 7 idea in Mathematical Modelling Project: Room Renovation and Budget. 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: Comparing unit prices: Identify a non-example.
- Explain which requirement is missing.
- Show all important reasoning.
Very beginner explanation: Comparing unit prices is an important Grade 7 idea in Mathematical Modelling Project: Room Renovation and Budget. 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: Comparing unit prices: Compare two cases.
- State one similarity and one difference.
- Show all important reasoning.
Very beginner explanation: Comparing unit prices is an important Grade 7 idea in Mathematical Modelling Project: Room Renovation and Budget. 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: Comparing unit prices: Apply the idea in a real situation.
- Translate the situation into mathematical language.
- Show all important reasoning.
Very beginner explanation: Comparing unit prices is an important Grade 7 idea in Mathematical Modelling Project: Room Renovation and Budget. 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: Comparing unit prices: Create and check your own example.
- Use the definition, then verify each condition.
- Show all important reasoning.
Very beginner explanation: Comparing unit prices is an important Grade 7 idea in Mathematical Modelling Project: Room Renovation and Budget. 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 Comparing unit prices in one simple sentence.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Comparing unit prices 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 Comparing unit prices and explain why.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Comparing unit prices 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 Comparing unit prices using a table, diagram, number line, graph, or equation.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Comparing unit prices 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 Comparing unit prices problem, how can you check the answer?
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Comparing unit prices 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 Comparing unit prices could be useful.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Comparing unit prices 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 Comparing unit prices. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
35.8 Applying discounts and tax
A discount is an amount taken off an original price, usually expressed as a percent. This topic focuses on Applying discounts and tax .
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: $80 item with 10% discount. Find sale price.
- Discount = 0.1×80 = 8.00.
- Sale price = 80-8.00.
Very beginner explanation: A discount is an amount taken off an original price, usually expressed as a percent. This topic focuses on Applying discounts and tax .
Answer: $72.00
Worked Example 2
Problem: $60 item with 25% discount. Find sale price.
- Discount = 0.25×60 = 15.00.
- Sale price = 60-15.00.
Very beginner explanation: A discount is an amount taken off an original price, usually expressed as a percent. This topic focuses on Applying discounts and tax .
Answer: $45.00
Worked Example 3
Problem: $120 item with 15% discount. Find sale price.
- Discount = 0.15×120 = 18.00.
- Sale price = 120-18.00.
Very beginner explanation: A discount is an amount taken off an original price, usually expressed as a percent. This topic focuses on Applying discounts and tax .
Answer: $102.00
Worked Example 4
Problem: $250 item with 5% discount. Find sale price.
- Discount = 0.05×250 = 12.50.
- Sale price = 250-12.50.
Very beginner explanation: A discount is an amount taken off an original price, usually expressed as a percent. This topic focuses on Applying discounts and tax .
Answer: $237.50
Worked Example 5
Problem: $75 item with 13% discount. Find sale price.
- Discount = 0.13×75 = 9.75.
- Sale price = 75-9.75.
Very beginner explanation: A discount is an amount taken off an original price, usually expressed as a percent. This topic focuses on Applying discounts and tax .
Answer: $65.25
Worked Example 6
Problem: An item costs $90. It is discounted 10%. Find the sale price.
- Discount = 0.1 × 90 = 9.00.
- Sale price = 90 - 9.00.
Very beginner explanation: A discount removes a percentage of the original price.
Answer: $81.00
Worked Example 7
Problem: An item costs $64. It is discounted 25%. Find the sale price.
- Discount = 0.25 × 64 = 16.00.
- Sale price = 64 - 16.00.
Very beginner explanation: A discount removes a percentage of the original price.
Answer: $48.00
Worked Example 8
Problem: An item costs $140. It is discounted 15%. Find the sale price.
- Discount = 0.15 × 140 = 21.00.
- Sale price = 140 - 21.00.
Very beginner explanation: A discount removes a percentage of the original price.
Answer: $119.00
Worked Example 9
Problem: An item costs $260. It is discounted 5%. Find the sale price.
- Discount = 0.05 × 260 = 13.00.
- Sale price = 260 - 13.00.
Very beginner explanation: A discount removes a percentage of the original price.
Answer: $247.00
Worked Example 10
Problem: An item costs $75. It is discounted 20%. Find the sale price.
- Discount = 0.2 × 75 = 15.00.
- Sale price = 75 - 15.00.
Very beginner explanation: A discount removes a percentage of the original price.
Answer: $60.00
Practice Exercise
Create one new question about Applying discounts and tax. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
35.9 Creating a budget
A budget is a plan that compares income, spending, and saving. This topic focuses on Creating a budget .
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
- Model wall area and paint coverage
- State assumptions.
- Calculate using the model.
- Check whether the result is realistic.
- Revise if needed.
Very beginner explanation: A budget is a plan that compares income, spending, and saving. This topic focuses on Creating a budget .
Answer: Estimate litres and cost
Worked Example 2
Problem: Tile a floor
- Model floor area and tile area
- State assumptions.
- Calculate using the model.
- Check whether the result is realistic.
- Revise if needed.
Very beginner explanation: A budget is a plan that compares income, spending, and saving. This topic focuses on Creating a budget .
Answer: Estimate tile count plus waste
Worked Example 3
Problem: Plan a school event
- Model attendance and cost per person
- State assumptions.
- Calculate using the model.
- Check whether the result is realistic.
- Revise if needed.
Very beginner explanation: A budget is a plan that compares income, spending, and saving. This topic focuses on Creating a budget .
Answer: Estimate total cost
Worked Example 4
Problem: Compare transportation
- Model fixed and per-kilometre costs
- State assumptions.
- Calculate using the model.
- Check whether the result is realistic.
- Revise if needed.
Very beginner explanation: A budget is a plan that compares income, spending, and saving. This topic focuses on Creating a budget .
Answer: Find a break-even distance
Worked Example 5
Problem: Plan monthly savings
- Model starting savings plus monthly deposits
- State assumptions.
- Calculate using the model.
- Check whether the result is realistic.
- Revise if needed.
Very beginner explanation: A budget is a plan that compares income, spending, and saving. This topic focuses on Creating a budget .
Answer: Predict when a goal is reached
Worked Example 6
Problem: Tile a floor
- Write a simple rule: area = length × width.
- State any assumptions.
- Use the rule on sample values.
- 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
- Write a simple rule: total = starting amount + monthly deposit × months.
- State any assumptions.
- Use the rule on sample values.
- 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
- Write a simple rule: generate a random whole number from 1 to 6.
- State any assumptions.
- Use the rule on sample values.
- 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
- Write a simple rule: new x = x + 3; new y = y - 2.
- State any assumptions.
- Use the rule on sample values.
- 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
- Write a simple rule: paint = wall area ÷ coverage per can.
- State any assumptions.
- Use the rule on sample values.
- 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 budget. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
35.10 Writing equations for costs
An equation says that two expressions are equal. This topic focuses on Writing equations for costs .
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.
- Simplify each side if needed.
- Collect variable terms on one side.
- Collect constants on the other side.
- Divide to isolate the variable.
- Check by substitution.
Very beginner explanation: An equation says that two expressions are equal. This topic focuses on Writing equations for costs .
Answer: x=6
Worked Example 2
Problem: Solve 4x-7=13.
- Simplify each side if needed.
- Collect variable terms on one side.
- Collect constants on the other side.
- Divide to isolate the variable.
- Check by substitution.
Very beginner explanation: An equation says that two expressions are equal. This topic focuses on Writing equations for costs .
Answer: x=5
Worked Example 3
Problem: Solve 3x+2=2x+9.
- Simplify each side if needed.
- Collect variable terms on one side.
- Collect constants on the other side.
- Divide to isolate the variable.
- Check by substitution.
Very beginner explanation: An equation says that two expressions are equal. This topic focuses on Writing equations for costs .
Answer: x=7
Worked Example 4
Problem: Solve 5x+4=2x+19.
- Simplify each side if needed.
- Collect variable terms on one side.
- Collect constants on the other side.
- Divide to isolate the variable.
- Check by substitution.
Very beginner explanation: An equation says that two expressions are equal. This topic focuses on Writing equations for costs .
Answer: x=5
Worked Example 5
Problem: Solve 2(x+3)=18.
- Simplify each side if needed.
- Collect variable terms on one side.
- Collect constants on the other side.
- Divide to isolate the variable.
- Check by substitution.
Very beginner explanation: An equation says that two expressions are equal. This topic focuses on Writing equations for costs .
Answer: x=6
Worked Example 6
Problem: Solve 2x + 5 = 17.
- Simplify each side if needed.
- Use inverse operations to collect variable terms and constants.
- Keep both sides balanced.
- 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.
- Simplify each side if needed.
- Use inverse operations to collect variable terms and constants.
- Keep both sides balanced.
- 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.
- Simplify each side if needed.
- Use inverse operations to collect variable terms and constants.
- Keep both sides balanced.
- 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.
- Simplify each side if needed.
- Use inverse operations to collect variable terms and constants.
- Keep both sides balanced.
- 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.
- Simplify each side if needed.
- Use inverse operations to collect variable terms and constants.
- Keep both sides balanced.
- 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 Writing equations for costs. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
35.11 Testing different options
Testing different options is an important Grade 7 idea in Mathematical Modelling Project: Room Renovation and Budget. 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: Testing different options: Identify a correct example.
- State the definition and show why the example fits.
- Show all important reasoning.
Very beginner explanation: Testing different options is an important Grade 7 idea in Mathematical Modelling Project: Room Renovation and Budget. 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: Testing different options: Identify a non-example.
- Explain which requirement is missing.
- Show all important reasoning.
Very beginner explanation: Testing different options is an important Grade 7 idea in Mathematical Modelling Project: Room Renovation and Budget. 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: Testing different options: Compare two cases.
- State one similarity and one difference.
- Show all important reasoning.
Very beginner explanation: Testing different options is an important Grade 7 idea in Mathematical Modelling Project: Room Renovation and Budget. 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: Testing different options: Apply the idea in a real situation.
- Translate the situation into mathematical language.
- Show all important reasoning.
Very beginner explanation: Testing different options is an important Grade 7 idea in Mathematical Modelling Project: Room Renovation and Budget. 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: Testing different options: Create and check your own example.
- Use the definition, then verify each condition.
- Show all important reasoning.
Very beginner explanation: Testing different options is an important Grade 7 idea in Mathematical Modelling Project: Room Renovation and Budget. 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 Testing different options in one simple sentence.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Testing different options 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 Testing different options and explain why.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Testing different options 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 Testing different options using a table, diagram, number line, graph, or equation.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Testing different options 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 Testing different options problem, how can you check the answer?
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Testing different options 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 Testing different options could be useful.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Testing different options 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 Testing different options. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
35.12 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
- Model wall area and paint coverage
- State assumptions.
- Calculate using the model.
- Check whether the result is realistic.
- 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
- Model floor area and tile area
- State assumptions.
- Calculate using the model.
- Check whether the result is realistic.
- 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
- Model attendance and cost per person
- State assumptions.
- Calculate using the model.
- Check whether the result is realistic.
- 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
- Model fixed and per-kilometre costs
- State assumptions.
- Calculate using the model.
- Check whether the result is realistic.
- 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
- Model starting savings plus monthly deposits
- State assumptions.
- Calculate using the model.
- Check whether the result is realistic.
- 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
- Write a simple rule: area = length × width.
- State any assumptions.
- Use the rule on sample values.
- 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
- Write a simple rule: total = starting amount + monthly deposit × months.
- State any assumptions.
- Use the rule on sample values.
- 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
- Write a simple rule: generate a random whole number from 1 to 6.
- State any assumptions.
- Use the rule on sample values.
- 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
- Write a simple rule: new x = x + 3; new y = y - 2.
- State any assumptions.
- Use the rule on sample values.
- 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
- Write a simple rule: paint = wall area ÷ coverage per can.
- State any assumptions.
- Use the rule on sample values.
- 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.
35.13 Revising the plan
Revising the plan is an important Grade 7 idea in Mathematical Modelling Project: Room Renovation and Budget. 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: Revising the plan: Identify a correct example.
- State the definition and show why the example fits.
- Show all important reasoning.
Very beginner explanation: Revising the plan is an important Grade 7 idea in Mathematical Modelling Project: Room Renovation and Budget. 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: Revising the plan: Identify a non-example.
- Explain which requirement is missing.
- Show all important reasoning.
Very beginner explanation: Revising the plan is an important Grade 7 idea in Mathematical Modelling Project: Room Renovation and Budget. 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: Revising the plan: Compare two cases.
- State one similarity and one difference.
- Show all important reasoning.
Very beginner explanation: Revising the plan is an important Grade 7 idea in Mathematical Modelling Project: Room Renovation and Budget. 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: Revising the plan: Apply the idea in a real situation.
- Translate the situation into mathematical language.
- Show all important reasoning.
Very beginner explanation: Revising the plan is an important Grade 7 idea in Mathematical Modelling Project: Room Renovation and Budget. 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: Revising the plan: Create and check your own example.
- Use the definition, then verify each condition.
- Show all important reasoning.
Very beginner explanation: Revising the plan is an important Grade 7 idea in Mathematical Modelling Project: Room Renovation and Budget. 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 Revising the plan in one simple sentence.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Revising the plan 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 Revising the plan and explain why.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Revising the plan 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 Revising the plan using a table, diagram, number line, graph, or equation.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Revising the plan 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 Revising the plan problem, how can you check the answer?
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Revising the plan 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 Revising the plan could be useful.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Revising the plan 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 Revising the plan. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
35.14 Presenting a recommendation
Presenting a recommendation is an important Grade 7 idea in Mathematical Modelling Project: Room Renovation and Budget. 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
- Model wall area and paint coverage
- State assumptions.
- Calculate using the model.
- Check whether the result is realistic.
- Revise if needed.
Very beginner explanation: Presenting a recommendation is an important Grade 7 idea in Mathematical Modelling Project: Room Renovation and Budget. 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
- Model floor area and tile area
- State assumptions.
- Calculate using the model.
- Check whether the result is realistic.
- Revise if needed.
Very beginner explanation: Presenting a recommendation is an important Grade 7 idea in Mathematical Modelling Project: Room Renovation and Budget. 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
- Model attendance and cost per person
- State assumptions.
- Calculate using the model.
- Check whether the result is realistic.
- Revise if needed.
Very beginner explanation: Presenting a recommendation is an important Grade 7 idea in Mathematical Modelling Project: Room Renovation and Budget. 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
- Model fixed and per-kilometre costs
- State assumptions.
- Calculate using the model.
- Check whether the result is realistic.
- Revise if needed.
Very beginner explanation: Presenting a recommendation is an important Grade 7 idea in Mathematical Modelling Project: Room Renovation and Budget. 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
- Model starting savings plus monthly deposits
- State assumptions.
- Calculate using the model.
- Check whether the result is realistic.
- Revise if needed.
Very beginner explanation: Presenting a recommendation is an important Grade 7 idea in Mathematical Modelling Project: Room Renovation and Budget. 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: Tile a floor
- Write a simple rule: area = length × width.
- State any assumptions.
- Use the rule on sample values.
- 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
- Write a simple rule: total = starting amount + monthly deposit × months.
- State any assumptions.
- Use the rule on sample values.
- 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
- Write a simple rule: generate a random whole number from 1 to 6.
- State any assumptions.
- Use the rule on sample values.
- 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
- Write a simple rule: new x = x + 3; new y = y - 2.
- State any assumptions.
- Use the rule on sample values.
- 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
- Write a simple rule: paint = wall area ÷ coverage per can.
- State any assumptions.
- Use the rule on sample values.
- 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 Presenting a recommendation. 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
- Create and solve a new question about Understanding the renovation problem . Show your reasoning and check your answer.
- Create and solve a new question about Measuring room dimensions . Show your reasoning and check your answer.
- Create and solve a new question about Drawing a scale floor plan . Show your reasoning and check your answer.
- Create and solve a new question about Calculating floor area . Show your reasoning and check your answer.
- Create and solve a new question about Calculating wall area . Show your reasoning and check your answer.
- Create and solve a new question about Estimating material quantities . Show your reasoning and check your answer.
- Create and solve a new question about Comparing unit prices . Show your reasoning and check your answer.
- Create and solve a new question about Applying discounts and tax . Show your reasoning and check your answer.
- Create and solve a new question about Creating a budget . Show your reasoning and check your answer.
- Create and solve a new question about Writing equations for costs . Show your reasoning and check your answer.
- Create and solve a new question about Testing different options . Show your reasoning and check your answer.
- Create and solve a new question about Making assumptions . 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.
30 Review Questions and Answers
Q1. What is important to remember about Understanding the renovation problem?
Answer: Understanding the renovation problem is an important Grade 7 idea in Mathematical Modelling Project: Room Renovation and Budget. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Q2. What is important to remember about Measuring room dimensions?
Answer: Measuring room dimensions is an important Grade 7 idea in Mathematical Modelling Project: Room Renovation and Budget. 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 Drawing a scale floor plan?
Answer: Drawing a scale floor plan is an important Grade 7 idea in Mathematical Modelling Project: Room Renovation and Budget. 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 Calculating floor area?
Answer: Calculating floor area is an important Grade 7 idea in Mathematical Modelling Project: Room Renovation and Budget. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Q5. What is important to remember about Calculating wall area?
Answer: Calculating wall area is an important Grade 7 idea in Mathematical Modelling Project: Room Renovation and Budget. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Q6. What is important to remember about Estimating material quantities?
Answer: Estimating material quantities is an important Grade 7 idea in Mathematical Modelling Project: Room Renovation and Budget. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Q7. What is important to remember about Comparing unit prices?
Answer: Comparing unit prices is an important Grade 7 idea in Mathematical Modelling Project: Room Renovation and Budget. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Q8. What is important to remember about Applying discounts and tax?
Answer: A discount is an amount taken off an original price, usually expressed as a percent. This topic focuses on Applying discounts and tax.
Q9. What is important to remember about Creating a budget?
Answer: A budget is a plan that compares income, spending, and saving. This topic focuses on Creating a budget.
Q10. What is important to remember about Writing equations for costs?
Answer: An equation says that two expressions are equal. This topic focuses on Writing equations for costs.
Q11. What is important to remember about Testing different options?
Answer: Testing different options is an important Grade 7 idea in Mathematical Modelling Project: Room Renovation and Budget. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Q12. 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.
Q13. What is important to remember about Revising the plan?
Answer: Revising the plan is an important Grade 7 idea in Mathematical Modelling Project: Room Renovation and Budget. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Q14. What is important to remember about Presenting a recommendation?
Answer: Presenting a recommendation is an important Grade 7 idea in Mathematical Modelling Project: Room Renovation and Budget. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether 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 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.