Chapter 31: Linear Relations and Rate of Change
Learn Grade 7 mathematics from very beginner explanations through worked examples, practice, reasoning, and review.
Chapter Overview
This chapter teaches Linear Relations and Rate of Change 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)
- Linear Relation (a relationship with a constant rate of change)
- Mathematical Model (a mathematical representation of a real situation)
- Circle Graph (a graph using sectors of a circle to show parts of a whole)
- Exchange Rate (the value of one currency expressed in another currency)
- Interest Rate (percentage used to calculate interest)
- 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.
31.1 Linear relation
A relation pairs input values with output values. This topic focuses on Linear relation .
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: For y=2x+(1), find y when x=3.
- Substitute x=3.
- y=2(3)+(1)=7.
Very beginner explanation: A relation pairs input values with output values. This topic focuses on Linear relation .
Answer: 7
Worked Example 2
Problem: For y=3x+(-2), find y when x=3.
- Substitute x=3.
- y=3(3)+(-2)=7.
Very beginner explanation: A relation pairs input values with output values. This topic focuses on Linear relation .
Answer: 7
Worked Example 3
Problem: For y=0.5x+(4), find y when x=3.
- Substitute x=3.
- y=0.5(3)+(4)=5.5.
Very beginner explanation: A relation pairs input values with output values. This topic focuses on Linear relation .
Answer: 5.5
Worked Example 4
Problem: For y=-1x+(5), find y when x=3.
- Substitute x=3.
- y=-1(3)+(5)=2.
Very beginner explanation: A relation pairs input values with output values. This topic focuses on Linear relation .
Answer: 2
Worked Example 5
Problem: For y=4x+(0), find y when x=3.
- Substitute x=3.
- y=4(3)+(0)=12.
Very beginner explanation: A relation pairs input values with output values. This topic focuses on Linear relation .
Answer: 12
Worked Example 6
Problem: For y = 2x + (1), find y when x = 3.
- Substitute x = 3.
- y = 2(3) + (1).
- Multiply first, then add.
Very beginner explanation: A relation pairs inputs with outputs. In a linear relation, the rate of change is constant.
Answer: 7
Worked Example 7
Problem: For y = -1x + (4), find y when x = 3.
- Substitute x = 3.
- y = -1(3) + (4).
- Multiply first, then add.
Very beginner explanation: A relation pairs inputs with outputs. In a linear relation, the rate of change is constant.
Answer: 1
Worked Example 8
Problem: For y = 0.5x + (-2), find y when x = 3.
- Substitute x = 3.
- y = 0.5(3) + (-2).
- Multiply first, then add.
Very beginner explanation: A relation pairs inputs with outputs. In a linear relation, the rate of change is constant.
Answer: -0.5
Worked Example 9
Problem: For y = 3x + (0), find y when x = 3.
- Substitute x = 3.
- y = 3(3) + (0).
- Multiply first, then add.
Very beginner explanation: A relation pairs inputs with outputs. In a linear relation, the rate of change is constant.
Answer: 9
Worked Example 10
Problem: For y = -2x + (5), find y when x = 3.
- Substitute x = 3.
- y = -2(3) + (5).
- Multiply first, then add.
Very beginner explanation: A relation pairs inputs with outputs. In a linear relation, the rate of change is constant.
Answer: -1
Practice Exercise
Create one new question about Linear relation. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
31.2 Constant rate of change
A rate compares two quantities measured in different units. This topic focuses on Constant rate of change .
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: 120 km in 2 h. Find the unit rate.
- Use rate = distance ÷ time.
- 120÷2=60.
Very beginner explanation: A rate compares two quantities measured in different units. This topic focuses on Constant rate of change .
Answer: 60 km/h
Worked Example 2
Problem: 180 km in 3 h. Find the unit rate.
- Use rate = distance ÷ time.
- 180÷3=60.
Very beginner explanation: A rate compares two quantities measured in different units. This topic focuses on Constant rate of change .
Answer: 60 km/h
Worked Example 3
Problem: 250 km in 5 h. Find the unit rate.
- Use rate = distance ÷ time.
- 250÷5=50.
Very beginner explanation: A rate compares two quantities measured in different units. This topic focuses on Constant rate of change .
Answer: 50 km/h
Worked Example 4
Problem: 72 km in 1.5 h. Find the unit rate.
- Use rate = distance ÷ time.
- 72÷1.5=48.
Very beginner explanation: A rate compares two quantities measured in different units. This topic focuses on Constant rate of change .
Answer: 48 km/h
Worked Example 5
Problem: 315 km in 4.5 h. Find the unit rate.
- Use rate = distance ÷ time.
- 315÷4.5=70.
Very beginner explanation: A rate compares two quantities measured in different units. This topic focuses on Constant rate of change .
Answer: 70 km/h
Worked Example 6
Problem: 120 km are travelled in 2 hours. Find the unit rate.
- Use rate = distance ÷ time.
- 120 ÷ 2 = 60.
Very beginner explanation: A unit rate compares a quantity with exactly one unit of another quantity.
Answer: 60 km/h
Worked Example 7
Problem: 180 km are travelled in 3 hours. Find the unit rate.
- Use rate = distance ÷ time.
- 180 ÷ 3 = 60.
Very beginner explanation: A unit rate compares a quantity with exactly one unit of another quantity.
Answer: 60 km/h
Worked Example 8
Problem: 240 km are travelled in 4 hours. Find the unit rate.
- Use rate = distance ÷ time.
- 240 ÷ 4 = 60.
Very beginner explanation: A unit rate compares a quantity with exactly one unit of another quantity.
Answer: 60 km/h
Worked Example 9
Problem: 300 km are travelled in 5 hours. Find the unit rate.
- Use rate = distance ÷ time.
- 300 ÷ 5 = 60.
Very beginner explanation: A unit rate compares a quantity with exactly one unit of another quantity.
Answer: 60 km/h
Worked Example 10
Problem: 360 km are travelled in 6 hours. Find the unit rate.
- Use rate = distance ÷ time.
- 360 ÷ 6 = 60.
Very beginner explanation: A unit rate compares a quantity with exactly one unit of another quantity.
Answer: 60 km/h
Practice Exercise
Create one new question about Constant rate of change. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
31.3 Tables of values
Tables of values is an important Grade 7 idea in Linear Relations and Rate of Change. 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: Tables of values: Identify a correct example.
- State the definition and show why the example fits.
- Show all important reasoning.
Very beginner explanation: Tables of values is an important Grade 7 idea in Linear Relations and Rate of Change. 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: Tables of values: Identify a non-example.
- Explain which requirement is missing.
- Show all important reasoning.
Very beginner explanation: Tables of values is an important Grade 7 idea in Linear Relations and Rate of Change. 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: Tables of values: Compare two cases.
- State one similarity and one difference.
- Show all important reasoning.
Very beginner explanation: Tables of values is an important Grade 7 idea in Linear Relations and Rate of Change. 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: Tables of values: Apply the idea in a real situation.
- Translate the situation into mathematical language.
- Show all important reasoning.
Very beginner explanation: Tables of values is an important Grade 7 idea in Linear Relations and Rate of Change. 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: Tables of values: Create and check your own example.
- Use the definition, then verify each condition.
- Show all important reasoning.
Very beginner explanation: Tables of values is an important Grade 7 idea in Linear Relations and Rate of Change. 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: Continue the pattern 2, 5, 8, ... for two more terms.
- The common difference is 3.
- Add the same difference each time.
Very beginner explanation: A constant-difference pattern changes by the same amount from one term to the next.
Answer: 11, 14
Worked Example 7
Problem: Continue the pattern 5, 9, 13, ... for two more terms.
- The common difference is 4.
- Add the same difference each time.
Very beginner explanation: A constant-difference pattern changes by the same amount from one term to the next.
Answer: 17, 21
Worked Example 8
Problem: Continue the pattern 10, 8, 6, ... for two more terms.
- The common difference is -2.
- Add the same difference each time.
Very beginner explanation: A constant-difference pattern changes by the same amount from one term to the next.
Answer: 4, 2
Worked Example 9
Problem: Continue the pattern 1.5, 2, 2.5, ... for two more terms.
- The common difference is 0.5.
- Add the same difference each time.
Very beginner explanation: A constant-difference pattern changes by the same amount from one term to the next.
Answer: 3, 3.5
Worked Example 10
Problem: Continue the pattern 20, 17.5, 15, ... for two more terms.
- The common difference is -2.5.
- Add the same difference each time.
Very beginner explanation: A constant-difference pattern changes by the same amount from one term to the next.
Answer: 12.5, 10
Practice Exercise
Create one new question about Tables of values. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
31.4 Graphing ordered pairs
Graphing ordered pairs is an important Grade 7 idea in Linear Relations and Rate of Change. 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: Locate (2,3).
- Move 2 units right.
- Move 3 units up.
Very beginner explanation: Graphing ordered pairs is an important Grade 7 idea in Linear Relations and Rate of Change. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: Quadrant I
Worked Example 2
Problem: Locate (-4,5).
- Move 4 units left.
- Move 5 units up.
Very beginner explanation: Graphing ordered pairs is an important Grade 7 idea in Linear Relations and Rate of Change. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: Quadrant II
Worked Example 3
Problem: Locate (-3,-2).
- Move 3 units left.
- Move 2 units down.
Very beginner explanation: Graphing ordered pairs is an important Grade 7 idea in Linear Relations and Rate of Change. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: Quadrant III
Worked Example 4
Problem: Locate (6,-1).
- Move 6 units right.
- Move 1 units down.
Very beginner explanation: Graphing ordered pairs is an important Grade 7 idea in Linear Relations and Rate of Change. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: Quadrant IV
Worked Example 5
Problem: Locate (0,4).
- Move 0 units right.
- Move 4 units up.
Very beginner explanation: Graphing ordered pairs is an important Grade 7 idea in Linear Relations and Rate of Change. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: on an axis
Worked Example 6
Problem: Explain Graphing ordered pairs 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: Graphing ordered pairs 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 Graphing ordered pairs 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: Graphing ordered pairs 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 Graphing ordered pairs 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: Graphing ordered pairs 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 Graphing ordered pairs 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: Graphing ordered pairs 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 Graphing ordered pairs 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: Graphing ordered pairs 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 Graphing ordered pairs. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
31.5 Straight-line graphs
Straight-line graphs is an important Grade 7 idea in Linear Relations and Rate of Change. 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: Evaluate this graph issue: Axis starts at 95 instead of 0.
- can exaggerate a small difference
- Inspect the axis minimum
Very beginner explanation: Straight-line graphs is an important Grade 7 idea in Linear Relations and Rate of Change. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: The display may be misleading; revise it for fair comparison.
Worked Example 2
Problem: Evaluate this graph issue: Intervals change from 1 to 10.
- creates unequal visual spacing
- Use equal intervals
Very beginner explanation: Straight-line graphs is an important Grade 7 idea in Linear Relations and Rate of Change. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: The display may be misleading; revise it for fair comparison.
Worked Example 3
Problem: Evaluate this graph issue: 3D pictures are scaled in height and width.
- can exaggerate area
- Use simple bars
Very beginner explanation: Straight-line graphs is an important Grade 7 idea in Linear Relations and Rate of Change. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: The display may be misleading; revise it for fair comparison.
Worked Example 4
Problem: Evaluate this graph issue: Only two favourable years are shown.
- cherry-picks data
- Look for the full time range
Very beginner explanation: Straight-line graphs is an important Grade 7 idea in Linear Relations and Rate of Change. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: The display may be misleading; revise it for fair comparison.
Worked Example 5
Problem: Evaluate this graph issue: Graph has no source.
- weakens trust
- Check source and context
Very beginner explanation: Straight-line graphs is an important Grade 7 idea in Linear Relations and Rate of Change. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: The display may be misleading; revise it for fair comparison.
Worked Example 6
Problem: Classify the data [4, 6, 8].
- These values are numerical.
- Decide whether they represent counts or measurements.
Very beginner explanation: Quantitative data are numerical values used for calculations or comparisons.
Answer: Quantitative data
Worked Example 7
Problem: Classify the data [5, 9, 10, 12].
- These values are numerical.
- Decide whether they represent counts or measurements.
Very beginner explanation: Quantitative data are numerical values used for calculations or comparisons.
Answer: Quantitative data
Worked Example 8
Problem: Classify the data [3, 7, 7, 11].
- These values are numerical.
- Decide whether they represent counts or measurements.
Very beginner explanation: Quantitative data are numerical values used for calculations or comparisons.
Answer: Quantitative data
Worked Example 9
Problem: Classify the data [20, 25, 30].
- These values are numerical.
- Decide whether they represent counts or measurements.
Very beginner explanation: Quantitative data are numerical values used for calculations or comparisons.
Answer: Quantitative data
Worked Example 10
Problem: Classify the data [6, 8, 9, 12, 15].
- These values are numerical.
- Decide whether they represent counts or measurements.
Very beginner explanation: Quantitative data are numerical values used for calculations or comparisons.
Answer: Quantitative data
Practice Exercise
Create one new question about Straight-line graphs. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
31.6 Increasing relations
A relation pairs input values with output values. This topic focuses on Increasing relations .
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: For y=2x+(1), find y when x=3.
- Substitute x=3.
- y=2(3)+(1)=7.
Very beginner explanation: A relation pairs input values with output values. This topic focuses on Increasing relations .
Answer: 7
Worked Example 2
Problem: For y=3x+(-2), find y when x=3.
- Substitute x=3.
- y=3(3)+(-2)=7.
Very beginner explanation: A relation pairs input values with output values. This topic focuses on Increasing relations .
Answer: 7
Worked Example 3
Problem: For y=0.5x+(4), find y when x=3.
- Substitute x=3.
- y=0.5(3)+(4)=5.5.
Very beginner explanation: A relation pairs input values with output values. This topic focuses on Increasing relations .
Answer: 5.5
Worked Example 4
Problem: For y=-1x+(5), find y when x=3.
- Substitute x=3.
- y=-1(3)+(5)=2.
Very beginner explanation: A relation pairs input values with output values. This topic focuses on Increasing relations .
Answer: 2
Worked Example 5
Problem: For y=4x+(0), find y when x=3.
- Substitute x=3.
- y=4(3)+(0)=12.
Very beginner explanation: A relation pairs input values with output values. This topic focuses on Increasing relations .
Answer: 12
Worked Example 6
Problem: For y = 2x + (1), find y when x = 3.
- Substitute x = 3.
- y = 2(3) + (1).
- Multiply first, then add.
Very beginner explanation: A relation pairs inputs with outputs. In a linear relation, the rate of change is constant.
Answer: 7
Worked Example 7
Problem: For y = -1x + (4), find y when x = 3.
- Substitute x = 3.
- y = -1(3) + (4).
- Multiply first, then add.
Very beginner explanation: A relation pairs inputs with outputs. In a linear relation, the rate of change is constant.
Answer: 1
Worked Example 8
Problem: For y = 0.5x + (-2), find y when x = 3.
- Substitute x = 3.
- y = 0.5(3) + (-2).
- Multiply first, then add.
Very beginner explanation: A relation pairs inputs with outputs. In a linear relation, the rate of change is constant.
Answer: -0.5
Worked Example 9
Problem: For y = 3x + (0), find y when x = 3.
- Substitute x = 3.
- y = 3(3) + (0).
- Multiply first, then add.
Very beginner explanation: A relation pairs inputs with outputs. In a linear relation, the rate of change is constant.
Answer: 9
Worked Example 10
Problem: For y = -2x + (5), find y when x = 3.
- Substitute x = 3.
- y = -2(3) + (5).
- Multiply first, then add.
Very beginner explanation: A relation pairs inputs with outputs. In a linear relation, the rate of change is constant.
Answer: -1
Practice Exercise
Create one new question about Increasing relations. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
31.7 Decreasing relations
A relation pairs input values with output values. This topic focuses on Decreasing relations .
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: For y=2x+(1), find y when x=3.
- Substitute x=3.
- y=2(3)+(1)=7.
Very beginner explanation: A relation pairs input values with output values. This topic focuses on Decreasing relations .
Answer: 7
Worked Example 2
Problem: For y=3x+(-2), find y when x=3.
- Substitute x=3.
- y=3(3)+(-2)=7.
Very beginner explanation: A relation pairs input values with output values. This topic focuses on Decreasing relations .
Answer: 7
Worked Example 3
Problem: For y=0.5x+(4), find y when x=3.
- Substitute x=3.
- y=0.5(3)+(4)=5.5.
Very beginner explanation: A relation pairs input values with output values. This topic focuses on Decreasing relations .
Answer: 5.5
Worked Example 4
Problem: For y=-1x+(5), find y when x=3.
- Substitute x=3.
- y=-1(3)+(5)=2.
Very beginner explanation: A relation pairs input values with output values. This topic focuses on Decreasing relations .
Answer: 2
Worked Example 5
Problem: For y=4x+(0), find y when x=3.
- Substitute x=3.
- y=4(3)+(0)=12.
Very beginner explanation: A relation pairs input values with output values. This topic focuses on Decreasing relations .
Answer: 12
Worked Example 6
Problem: For y = 2x + (1), find y when x = 3.
- Substitute x = 3.
- y = 2(3) + (1).
- Multiply first, then add.
Very beginner explanation: A relation pairs inputs with outputs. In a linear relation, the rate of change is constant.
Answer: 7
Worked Example 7
Problem: For y = -1x + (4), find y when x = 3.
- Substitute x = 3.
- y = -1(3) + (4).
- Multiply first, then add.
Very beginner explanation: A relation pairs inputs with outputs. In a linear relation, the rate of change is constant.
Answer: 1
Worked Example 8
Problem: For y = 0.5x + (-2), find y when x = 3.
- Substitute x = 3.
- y = 0.5(3) + (-2).
- Multiply first, then add.
Very beginner explanation: A relation pairs inputs with outputs. In a linear relation, the rate of change is constant.
Answer: -0.5
Worked Example 9
Problem: For y = 3x + (0), find y when x = 3.
- Substitute x = 3.
- y = 3(3) + (0).
- Multiply first, then add.
Very beginner explanation: A relation pairs inputs with outputs. In a linear relation, the rate of change is constant.
Answer: 9
Worked Example 10
Problem: For y = -2x + (5), find y when x = 3.
- Substitute x = 3.
- y = -2(3) + (5).
- Multiply first, then add.
Very beginner explanation: A relation pairs inputs with outputs. In a linear relation, the rate of change is constant.
Answer: -1
Practice Exercise
Create one new question about Decreasing relations. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
31.8 Horizontal relations
A relation pairs input values with output values. This topic focuses on Horizontal relations .
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: For y=2x+(1), find y when x=3.
- Substitute x=3.
- y=2(3)+(1)=7.
Very beginner explanation: A relation pairs input values with output values. This topic focuses on Horizontal relations .
Answer: 7
Worked Example 2
Problem: For y=3x+(-2), find y when x=3.
- Substitute x=3.
- y=3(3)+(-2)=7.
Very beginner explanation: A relation pairs input values with output values. This topic focuses on Horizontal relations .
Answer: 7
Worked Example 3
Problem: For y=0.5x+(4), find y when x=3.
- Substitute x=3.
- y=0.5(3)+(4)=5.5.
Very beginner explanation: A relation pairs input values with output values. This topic focuses on Horizontal relations .
Answer: 5.5
Worked Example 4
Problem: For y=-1x+(5), find y when x=3.
- Substitute x=3.
- y=-1(3)+(5)=2.
Very beginner explanation: A relation pairs input values with output values. This topic focuses on Horizontal relations .
Answer: 2
Worked Example 5
Problem: For y=4x+(0), find y when x=3.
- Substitute x=3.
- y=4(3)+(0)=12.
Very beginner explanation: A relation pairs input values with output values. This topic focuses on Horizontal relations .
Answer: 12
Worked Example 6
Problem: For y = 2x + (1), find y when x = 3.
- Substitute x = 3.
- y = 2(3) + (1).
- Multiply first, then add.
Very beginner explanation: A relation pairs inputs with outputs. In a linear relation, the rate of change is constant.
Answer: 7
Worked Example 7
Problem: For y = -1x + (4), find y when x = 3.
- Substitute x = 3.
- y = -1(3) + (4).
- Multiply first, then add.
Very beginner explanation: A relation pairs inputs with outputs. In a linear relation, the rate of change is constant.
Answer: 1
Worked Example 8
Problem: For y = 0.5x + (-2), find y when x = 3.
- Substitute x = 3.
- y = 0.5(3) + (-2).
- Multiply first, then add.
Very beginner explanation: A relation pairs inputs with outputs. In a linear relation, the rate of change is constant.
Answer: -0.5
Worked Example 9
Problem: For y = 3x + (0), find y when x = 3.
- Substitute x = 3.
- y = 3(3) + (0).
- Multiply first, then add.
Very beginner explanation: A relation pairs inputs with outputs. In a linear relation, the rate of change is constant.
Answer: 9
Worked Example 10
Problem: For y = -2x + (5), find y when x = 3.
- Substitute x = 3.
- y = -2(3) + (5).
- Multiply first, then add.
Very beginner explanation: A relation pairs inputs with outputs. In a linear relation, the rate of change is constant.
Answer: -1
Practice Exercise
Create one new question about Horizontal relations. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
31.9 Slope introduction
Slope is a measure of rate of change, commonly found as rise divided by run. This topic focuses on Slope introduction .
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: For y=2x+(1), find y when x=3.
- Substitute x=3.
- y=2(3)+(1)=7.
Very beginner explanation: Slope is a measure of rate of change, commonly found as rise divided by run. This topic focuses on Slope introduction .
Answer: 7
Worked Example 2
Problem: For y=3x+(-2), find y when x=3.
- Substitute x=3.
- y=3(3)+(-2)=7.
Very beginner explanation: Slope is a measure of rate of change, commonly found as rise divided by run. This topic focuses on Slope introduction .
Answer: 7
Worked Example 3
Problem: For y=0.5x+(4), find y when x=3.
- Substitute x=3.
- y=0.5(3)+(4)=5.5.
Very beginner explanation: Slope is a measure of rate of change, commonly found as rise divided by run. This topic focuses on Slope introduction .
Answer: 5.5
Worked Example 4
Problem: For y=-1x+(5), find y when x=3.
- Substitute x=3.
- y=-1(3)+(5)=2.
Very beginner explanation: Slope is a measure of rate of change, commonly found as rise divided by run. This topic focuses on Slope introduction .
Answer: 2
Worked Example 5
Problem: For y=4x+(0), find y when x=3.
- Substitute x=3.
- y=4(3)+(0)=12.
Very beginner explanation: Slope is a measure of rate of change, commonly found as rise divided by run. This topic focuses on Slope introduction .
Answer: 12
Worked Example 6
Problem: For y = 2x + (1), find y when x = 3.
- Substitute x = 3.
- y = 2(3) + (1).
- Multiply first, then add.
Very beginner explanation: A relation pairs inputs with outputs. In a linear relation, the rate of change is constant.
Answer: 7
Worked Example 7
Problem: For y = -1x + (4), find y when x = 3.
- Substitute x = 3.
- y = -1(3) + (4).
- Multiply first, then add.
Very beginner explanation: A relation pairs inputs with outputs. In a linear relation, the rate of change is constant.
Answer: 1
Worked Example 8
Problem: For y = 0.5x + (-2), find y when x = 3.
- Substitute x = 3.
- y = 0.5(3) + (-2).
- Multiply first, then add.
Very beginner explanation: A relation pairs inputs with outputs. In a linear relation, the rate of change is constant.
Answer: -0.5
Worked Example 9
Problem: For y = 3x + (0), find y when x = 3.
- Substitute x = 3.
- y = 3(3) + (0).
- Multiply first, then add.
Very beginner explanation: A relation pairs inputs with outputs. In a linear relation, the rate of change is constant.
Answer: 9
Worked Example 10
Problem: For y = -2x + (5), find y when x = 3.
- Substitute x = 3.
- y = -2(3) + (5).
- Multiply first, then add.
Very beginner explanation: A relation pairs inputs with outputs. In a linear relation, the rate of change is constant.
Answer: -1
Practice Exercise
Create one new question about Slope introduction. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
31.10 Rise and run
Rise and run is an important Grade 7 idea in Linear Relations and Rate of Change. 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: For y=2x+(1), find y when x=3.
- Substitute x=3.
- y=2(3)+(1)=7.
Very beginner explanation: Rise and run is an important Grade 7 idea in Linear Relations and Rate of Change. 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
Worked Example 2
Problem: For y=3x+(-2), find y when x=3.
- Substitute x=3.
- y=3(3)+(-2)=7.
Very beginner explanation: Rise and run is an important Grade 7 idea in Linear Relations and Rate of Change. 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
Worked Example 3
Problem: For y=0.5x+(4), find y when x=3.
- Substitute x=3.
- y=0.5(3)+(4)=5.5.
Very beginner explanation: Rise and run is an important Grade 7 idea in Linear Relations and Rate of Change. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: 5.5
Worked Example 4
Problem: For y=-1x+(5), find y when x=3.
- Substitute x=3.
- y=-1(3)+(5)=2.
Very beginner explanation: Rise and run is an important Grade 7 idea in Linear Relations and Rate of Change. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: 2
Worked Example 5
Problem: For y=4x+(0), find y when x=3.
- Substitute x=3.
- y=4(3)+(0)=12.
Very beginner explanation: Rise and run is an important Grade 7 idea in Linear Relations and Rate of Change. 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
Worked Example 6
Problem: Explain Rise and run 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: Rise and run 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 Rise and run 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: Rise and run 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 Rise and run 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: Rise and run 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 Rise and run 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: Rise and run 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 Rise and run 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: Rise and run 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 Rise and run. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
31.11 Comparing linear relations
A relation pairs input values with output values. This topic focuses on Comparing linear relations .
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: For y=2x+(1), find y when x=3.
- Substitute x=3.
- y=2(3)+(1)=7.
Very beginner explanation: A relation pairs input values with output values. This topic focuses on Comparing linear relations .
Answer: 7
Worked Example 2
Problem: For y=3x+(-2), find y when x=3.
- Substitute x=3.
- y=3(3)+(-2)=7.
Very beginner explanation: A relation pairs input values with output values. This topic focuses on Comparing linear relations .
Answer: 7
Worked Example 3
Problem: For y=0.5x+(4), find y when x=3.
- Substitute x=3.
- y=0.5(3)+(4)=5.5.
Very beginner explanation: A relation pairs input values with output values. This topic focuses on Comparing linear relations .
Answer: 5.5
Worked Example 4
Problem: For y=-1x+(5), find y when x=3.
- Substitute x=3.
- y=-1(3)+(5)=2.
Very beginner explanation: A relation pairs input values with output values. This topic focuses on Comparing linear relations .
Answer: 2
Worked Example 5
Problem: For y=4x+(0), find y when x=3.
- Substitute x=3.
- y=4(3)+(0)=12.
Very beginner explanation: A relation pairs input values with output values. This topic focuses on Comparing linear relations .
Answer: 12
Worked Example 6
Problem: For y = 2x + (1), find y when x = 3.
- Substitute x = 3.
- y = 2(3) + (1).
- Multiply first, then add.
Very beginner explanation: A relation pairs inputs with outputs. In a linear relation, the rate of change is constant.
Answer: 7
Worked Example 7
Problem: For y = -1x + (4), find y when x = 3.
- Substitute x = 3.
- y = -1(3) + (4).
- Multiply first, then add.
Very beginner explanation: A relation pairs inputs with outputs. In a linear relation, the rate of change is constant.
Answer: 1
Worked Example 8
Problem: For y = 0.5x + (-2), find y when x = 3.
- Substitute x = 3.
- y = 0.5(3) + (-2).
- Multiply first, then add.
Very beginner explanation: A relation pairs inputs with outputs. In a linear relation, the rate of change is constant.
Answer: -0.5
Worked Example 9
Problem: For y = 3x + (0), find y when x = 3.
- Substitute x = 3.
- y = 3(3) + (0).
- Multiply first, then add.
Very beginner explanation: A relation pairs inputs with outputs. In a linear relation, the rate of change is constant.
Answer: 9
Worked Example 10
Problem: For y = -2x + (5), find y when x = 3.
- Substitute x = 3.
- y = -2(3) + (5).
- Multiply first, then add.
Very beginner explanation: A relation pairs inputs with outputs. In a linear relation, the rate of change is constant.
Answer: -1
Practice Exercise
Create one new question about Comparing linear relations. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
31.12 Interpolation introduction
Interpolation introduction is an important Grade 7 idea in Linear Relations and Rate of Change. 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: For y=2x+(1), find y when x=3.
- Substitute x=3.
- y=2(3)+(1)=7.
Very beginner explanation: Interpolation introduction is an important Grade 7 idea in Linear Relations and Rate of Change. 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
Worked Example 2
Problem: For y=3x+(-2), find y when x=3.
- Substitute x=3.
- y=3(3)+(-2)=7.
Very beginner explanation: Interpolation introduction is an important Grade 7 idea in Linear Relations and Rate of Change. 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
Worked Example 3
Problem: For y=0.5x+(4), find y when x=3.
- Substitute x=3.
- y=0.5(3)+(4)=5.5.
Very beginner explanation: Interpolation introduction is an important Grade 7 idea in Linear Relations and Rate of Change. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: 5.5
Worked Example 4
Problem: For y=-1x+(5), find y when x=3.
- Substitute x=3.
- y=-1(3)+(5)=2.
Very beginner explanation: Interpolation introduction is an important Grade 7 idea in Linear Relations and Rate of Change. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: 2
Worked Example 5
Problem: For y=4x+(0), find y when x=3.
- Substitute x=3.
- y=4(3)+(0)=12.
Very beginner explanation: Interpolation introduction is an important Grade 7 idea in Linear Relations and Rate of Change. 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
Worked Example 6
Problem: Explain Interpolation introduction 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: Interpolation introduction 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 Interpolation introduction 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: Interpolation introduction 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 Interpolation introduction 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: Interpolation introduction 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 Interpolation introduction 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: Interpolation introduction 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 Interpolation introduction 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: Interpolation introduction 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 Interpolation introduction. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
31.13 Real-life linear models
The mode is the value that occurs most often. This topic focuses on Real-life linear 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: For y=2x+(1), find y when x=3.
- Substitute x=3.
- y=2(3)+(1)=7.
Very beginner explanation: The mode is the value that occurs most often. This topic focuses on Real-life linear models .
Answer: 7
Worked Example 2
Problem: For y=3x+(-2), find y when x=3.
- Substitute x=3.
- y=3(3)+(-2)=7.
Very beginner explanation: The mode is the value that occurs most often. This topic focuses on Real-life linear models .
Answer: 7
Worked Example 3
Problem: For y=0.5x+(4), find y when x=3.
- Substitute x=3.
- y=0.5(3)+(4)=5.5.
Very beginner explanation: The mode is the value that occurs most often. This topic focuses on Real-life linear models .
Answer: 5.5
Worked Example 4
Problem: For y=-1x+(5), find y when x=3.
- Substitute x=3.
- y=-1(3)+(5)=2.
Very beginner explanation: The mode is the value that occurs most often. This topic focuses on Real-life linear models .
Answer: 2
Worked Example 5
Problem: For y=4x+(0), find y when x=3.
- Substitute x=3.
- y=4(3)+(0)=12.
Very beginner explanation: The mode is the value that occurs most often. This topic focuses on Real-life linear models .
Answer: 12
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 Real-life linear models. 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 Linear relation . Show your reasoning and check your answer.
- Create and solve a new question about Constant rate of change . Show your reasoning and check your answer.
- Create and solve a new question about Tables of values . Show your reasoning and check your answer.
- Create and solve a new question about Graphing ordered pairs . Show your reasoning and check your answer.
- Create and solve a new question about Straight-line graphs . Show your reasoning and check your answer.
- Create and solve a new question about Increasing relations . Show your reasoning and check your answer.
- Create and solve a new question about Decreasing relations . Show your reasoning and check your answer.
- Create and solve a new question about Horizontal relations . Show your reasoning and check your answer.
- Create and solve a new question about Slope introduction . Show your reasoning and check your answer.
- Create and solve a new question about Rise and run . Show your reasoning and check your answer.
- Create and solve a new question about Comparing linear relations . Show your reasoning and check your answer.
- Create and solve a new question about Interpolation introduction . 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 Linear relation?
Answer: A relation pairs input values with output values. This topic focuses on Linear relation.
Q2. What is important to remember about Constant rate of change?
Answer: A rate compares two quantities measured in different units. This topic focuses on Constant rate of change.
Q3. What is important to remember about Tables of values?
Answer: Tables of values is an important Grade 7 idea in Linear Relations and Rate of Change. 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 Graphing ordered pairs?
Answer: Graphing ordered pairs is an important Grade 7 idea in Linear Relations and Rate of Change. 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 Straight-line graphs?
Answer: Straight-line graphs is an important Grade 7 idea in Linear Relations and Rate of Change. 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 Increasing relations?
Answer: A relation pairs input values with output values. This topic focuses on Increasing relations.
Q7. What is important to remember about Decreasing relations?
Answer: A relation pairs input values with output values. This topic focuses on Decreasing relations.
Q8. What is important to remember about Horizontal relations?
Answer: A relation pairs input values with output values. This topic focuses on Horizontal relations.
Q9. What is important to remember about Slope introduction?
Answer: Slope is a measure of rate of change, commonly found as rise divided by run. This topic focuses on Slope introduction.
Q10. What is important to remember about Rise and run?
Answer: Rise and run is an important Grade 7 idea in Linear Relations and Rate of Change. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Q11. What is important to remember about Comparing linear relations?
Answer: A relation pairs input values with output values. This topic focuses on Comparing linear relations.
Q12. What is important to remember about Interpolation introduction?
Answer: Interpolation introduction is an important Grade 7 idea in Linear Relations and Rate of Change. 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 Real-life linear models?
Answer: The mode is the value that occurs most often. This topic focuses on Real-life linear models.
Q14. Why should you show your steps?
Answer: Showing steps makes reasoning easier to verify and helps locate mistakes.
Q15. Why is estimation useful?
Answer: Estimation helps you judge whether a final answer is reasonable before accepting it.
Q16. Why do units matter?
Answer: Units tell what a value measures and prevent incompatible quantities from being mixed.
Q17. When should you round?
Answer: Usually round near the end of a calculation unless the question specifically asks for earlier rounding.
Q18. How can you check an answer?
Answer: Use an inverse operation, substitution, estimation, a graph, or a second method.
Q19. What should you identify first in a word problem?
Answer: Identify what is known, what is unknown, and the relationship connecting them.
Q20. What makes a final answer complete?
Answer: Give the value, correct units when needed, and a brief interpretation in context.
Q21. What should you do if an answer seems unreasonable?
Answer: Re-read the question, check copied values, signs, units, formulas, and calculations.
Q22. Why are diagrams useful?
Answer: A labelled diagram can reveal relationships between lengths, angles, areas, and other quantities.
Q23. Why are tables useful?
Answer: Tables organize values and help reveal patterns, relationships, rates, and missing information.
Q24. Why should you explain your reasoning?
Answer: Reasoning shows why a method works, not just what answer you obtained.
Q25. How do mistakes help learning?
Answer: Analyzing a mistake identifies the misunderstood rule or step and helps prevent the same error later.
Q26. How should you study this chapter?
Answer: Review definitions, redo worked examples without looking, practise mixed questions, and explain solutions in your own words.
Q27. 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.
Q28. Why compare more than one strategy?
Answer: Different strategies can make a problem easier and provide a way to verify the result.
Q29. What is a mathematical model?
Answer: It is a simplified mathematical representation of a real situation used to analyze, explain, or predict.
Q30. Why should assumptions be stated?
Answer: Assumptions show what conditions the solution depends on and help readers judge whether the model is reasonable.