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Chapter 31: Coding Curves and Trend Lines

Learn Grade 8 Math from beginner foundations through advanced Grade 8 problem solving with detailed explanations, at least five examples per topic, practice exercises, and review questions.

Grade 8Beginner Friendly5+ Examples Per TopicPractice30 Q&A
Estimated reading time0% read
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What this chapter covers

This chapter contains 13 topics. Technical terms are followed by plain-language meanings where they first appear. Each topic includes at least five worked examples. Coding is included only where it naturally supports the Grade 8 coding expectations.

31.1 Nonlinear data patterns

A pattern follows a rule that can be used to predict missing or future terms. In this section, the focus is Nonlinear data patterns.

For a beginner, focus on the meaning before memorizing a procedure. Start with a small example, name the quantities and units, apply one rule at a time, and connect the result back to the question.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: For y = 2x + (1), find y when x = 1.

  1. Substitute the x-value.
  2. y = 2(1) + (1).
  3. Multiply first, then add.

Very beginner explanation: In y = mx + b, m is the rate of change and b is the starting value or y-intercept.

Answer: 3

Worked Example 2

Problem: For y = -1x + (4), find y when x = 2.

  1. Substitute the x-value.
  2. y = -1(2) + (4).
  3. Multiply first, then add.

Very beginner explanation: In y = mx + b, m is the rate of change and b is the starting value or y-intercept.

Answer: 2

Worked Example 3

Problem: For y = 0.5x + (-2), find y when x = 3.

  1. Substitute the x-value.
  2. y = 0.5(3) + (-2).
  3. Multiply first, then add.

Very beginner explanation: In y = mx + b, m is the rate of change and b is the starting value or y-intercept.

Answer: -0.5

Worked Example 4

Problem: For y = 3x + (0), find y when x = 4.

  1. Substitute the x-value.
  2. y = 3(4) + (0).
  3. Multiply first, then add.

Very beginner explanation: In y = mx + b, m is the rate of change and b is the starting value or y-intercept.

Answer: 12

Worked Example 5

Problem: For y = -2x + (5), find y when x = 5.

  1. Substitute the x-value.
  2. y = -2(5) + (5).
  3. Multiply first, then add.

Very beginner explanation: In y = mx + b, m is the rate of change and b is the starting value or y-intercept.

Answer: -5

Worked Example 6

Problem: For y = 4x + (-3), find y when x = 1.

  1. Substitute the x-value.
  2. y = 4(1) + (-3).
  3. Multiply first, then add.

Very beginner explanation: In y = mx + b, m is the rate of change and b is the starting value or y-intercept.

Answer: 1

Worked Example 7

Problem: For y = 1.5x + (2), find y when x = 2.

  1. Substitute the x-value.
  2. y = 1.5(2) + (2).
  3. Multiply first, then add.

Very beginner explanation: In y = mx + b, m is the rate of change and b is the starting value or y-intercept.

Answer: 5

Worked Example 8

Problem: For y = -0.5x + (6), find y when x = 3.

  1. Substitute the x-value.
  2. y = -0.5(3) + (6).
  3. Multiply first, then add.

Very beginner explanation: In y = mx + b, m is the rate of change and b is the starting value or y-intercept.

Answer: 4.5

Worked Example 9

Problem: For y = 5x + (-1), find y when x = 4.

  1. Substitute the x-value.
  2. y = 5(4) + (-1).
  3. Multiply first, then add.

Very beginner explanation: In y = mx + b, m is the rate of change and b is the starting value or y-intercept.

Answer: 19

Worked Example 10

Problem: For y = 0x + (7), find y when x = 5.

  1. Substitute the x-value.
  2. y = 0(5) + (7).
  3. Multiply first, then add.

Very beginner explanation: In y = mx + b, m is the rate of change and b is the starting value or y-intercept.

Answer: 7

Practice exercise

Create one new Grade 8 problem involving Nonlinear data patterns. Show the important steps, include units when needed, and explain how you checked the answer.

31.2 Curve versus line

Curve versus line is an important Grade 8 concept in Coding Curves and Trend Lines. Start by identifying the quantities, relationships, units, or rules involved. Work with a small example first, show the important steps, and check whether the result is reasonable.

For a beginner, focus on the meaning before memorizing a procedure. Start with a small example, name the quantities and units, apply one rule at a time, and connect the result back to the question.

Coding example

// Curve versus line
const points = [[1,2],[2,4],[3,5],[4,8],[5,9]];
const model = x => 1.8 * x + 0.2;
const predictions = points.map(([x,y]) => ({x, observed:y, predicted:model(x)}));
console.log(predictions);

Code explanation

  1. The example stores a small set of coordinate pairs so each result can be checked manually.
  2. model() represents a simple line used to estimate y-values from x-values.
  3. map() applies the model to every point and keeps both observed and predicted values.
  4. Change the slope or intercept and compare the predictions to see whether the line follows the data more closely.

Expected result: A list of observed and predicted values is printed.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Predict taxi cost: use the model cost = 4 + 2.5d with input 8.

  1. Identify the input variable.
  2. Substitute 8 into the model.
  3. Calculate the predicted output.
  4. Ask whether the result makes sense in the real situation.

Very beginner explanation: A mathematical model is a simplified rule for representing a real situation. Predictions should always be checked for reasonableness.

Answer: 24

Worked Example 2

Problem: Predict fundraiser revenue: use the model revenue = 120 + 15s with input 10.

  1. Identify the input variable.
  2. Substitute 10 into the model.
  3. Calculate the predicted output.
  4. Ask whether the result makes sense in the real situation.

Very beginner explanation: A mathematical model is a simplified rule for representing a real situation. Predictions should always be checked for reasonableness.

Answer: 270

Worked Example 3

Problem: Predict distance: use the model distance = 60t with input 2.5.

  1. Identify the input variable.
  2. Substitute 2.5 into the model.
  3. Calculate the predicted output.
  4. Ask whether the result makes sense in the real situation.

Very beginner explanation: A mathematical model is a simplified rule for representing a real situation. Predictions should always be checked for reasonableness.

Answer: 150

Worked Example 4

Problem: Predict savings: use the model savings = 200 + 50m with input 6.

  1. Identify the input variable.
  2. Substitute 6 into the model.
  3. Calculate the predicted output.
  4. Ask whether the result makes sense in the real situation.

Very beginner explanation: A mathematical model is a simplified rule for representing a real situation. Predictions should always be checked for reasonableness.

Answer: 500

Worked Example 5

Problem: Predict temperature trend: use the model temp = 18 - 1.5h with input 4.

  1. Identify the input variable.
  2. Substitute 4 into the model.
  3. Calculate the predicted output.
  4. Ask whether the result makes sense in the real situation.

Very beginner explanation: A mathematical model is a simplified rule for representing a real situation. Predictions should always be checked for reasonableness.

Answer: 12

Worked Example 6

Problem: Predict plant height: use the model height = 5 + 2w with input 7.

  1. Identify the input variable.
  2. Substitute 7 into the model.
  3. Calculate the predicted output.
  4. Ask whether the result makes sense in the real situation.

Very beginner explanation: A mathematical model is a simplified rule for representing a real situation. Predictions should always be checked for reasonableness.

Answer: 19

Worked Example 7

Problem: Predict rental cost: use the model cost = 30 + 8d with input 5.

  1. Identify the input variable.
  2. Substitute 5 into the model.
  3. Calculate the predicted output.
  4. Ask whether the result makes sense in the real situation.

Very beginner explanation: A mathematical model is a simplified rule for representing a real situation. Predictions should always be checked for reasonableness.

Answer: 70

Worked Example 8

Problem: Predict page count: use the model pages = 12r with input 9.

  1. Identify the input variable.
  2. Substitute 9 into the model.
  3. Calculate the predicted output.
  4. Ask whether the result makes sense in the real situation.

Very beginner explanation: A mathematical model is a simplified rule for representing a real situation. Predictions should always be checked for reasonableness.

Answer: 108

Worked Example 9

Problem: Predict points: use the model points = 6g + 3 with input 4.

  1. Identify the input variable.
  2. Substitute 4 into the model.
  3. Calculate the predicted output.
  4. Ask whether the result makes sense in the real situation.

Very beginner explanation: A mathematical model is a simplified rule for representing a real situation. Predictions should always be checked for reasonableness.

Answer: 27

Worked Example 10

Problem: Predict water use: use the model litres = 20 + 4m with input 5.

  1. Identify the input variable.
  2. Substitute 5 into the model.
  3. Calculate the predicted output.
  4. Ask whether the result makes sense in the real situation.

Very beginner explanation: A mathematical model is a simplified rule for representing a real situation. Predictions should always be checked for reasonableness.

Answer: 40

Practice exercise

Create one new Grade 8 problem involving Curve versus line. Show the important steps, include units when needed, and explain how you checked the answer.

31.3 Generating quadratic values

Generating quadratic values is an important Grade 8 concept in Coding Curves and Trend Lines. Start by identifying the quantities, relationships, units, or rules involved. Work with a small example first, show the important steps, and check whether the result is reasonable.

For a beginner, focus on the meaning before memorizing a procedure. Start with a small example, name the quantities and units, apply one rule at a time, and connect the result back to the question.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Generating quadratic values: Explain Generating quadratic values in one simple sentence.

  1. Look at the words in “Generating quadratic values”.
  2. State what the idea is used for.
  3. Give one small mathematical example.

Very beginner explanation: A beginner should first understand the meaning before memorizing a rule.

Answer: Generating quadratic values is a Grade 8 math idea used to describe or solve a specific mathematical relationship.

Worked Example 2

Problem: Generating quadratic values: A student says, “I can use Generating quadratic values without checking units or labels.” Is that a good method?

  1. Read the statement.
  2. Ask whether units, signs, labels, or conditions matter.
  3. Decide whether the method is safe.

Very beginner explanation: Many math mistakes come from using a correct calculation on the wrong quantity.

Answer: No. Important units, labels, signs, and conditions must be checked.

Worked Example 3

Problem: Generating quadratic values: What is the first step when solving a problem about Generating quadratic values?

  1. Read the whole question.
  2. Underline the known information.
  3. Identify what must be found.

Very beginner explanation: This prevents you from calculating before you understand the problem.

Answer: Identify the given information and the unknown before choosing a rule.

Worked Example 4

Problem: Generating quadratic values: After solving a Generating quadratic values problem, what should you do before accepting the answer?

  1. Estimate the expected size or direction.
  2. Check units and signs.
  3. Use an inverse method if possible.

Very beginner explanation: Checking is part of solving, not an optional extra.

Answer: Check whether the answer is reasonable and consistent with the problem.

Worked Example 5

Problem: Generating quadratic values: Give one way Generating quadratic values could appear outside a textbook.

  1. Think about money, measurements, data, maps, geometry, or technology.
  2. Connect the topic to one of those situations.

Very beginner explanation: Connecting math to real situations makes the rule easier to remember.

Answer: Generating quadratic values can be used in a real-life situation where quantities must be compared, measured, predicted, or calculated.

Worked Example 6

Problem: Generating quadratic values: Which representation could help explain Generating quadratic values: a table, graph, diagram, equation, or number line?

  1. Choose the representation that makes the relationship easiest to see.
  2. Label it clearly.

Very beginner explanation: Different representations show different features of the same mathematics.

Answer: Use the representation that best matches the problem; more than one may be valid.

Worked Example 7

Problem: Generating quadratic values: A student gets an answer for Generating quadratic values but cannot explain the steps. What should be improved?

  1. Rewrite the solution one step at a time.
  2. Name the rule used at each important step.

Very beginner explanation: A correct final number without reasoning may hide an error.

Answer: The reasoning should be shown so the solution can be checked.

Worked Example 8

Problem: Generating quadratic values: Why can estimation help before a detailed Generating quadratic values calculation?

  1. Round or use benchmark values.
  2. Predict the approximate answer.
  3. Compare the exact result with the estimate.

Very beginner explanation: If the exact answer is far from the estimate, recheck the work.

Answer: Estimation gives a target range for the final answer.

Worked Example 9

Problem: Generating quadratic values: How can you test whether your rule for Generating quadratic values works?

  1. Choose a very small easy example.
  2. Apply the rule.
  3. Check the result another way.

Very beginner explanation: Small examples expose mistakes quickly.

Answer: Test the rule on a simple case whose answer can be verified.

Worked Example 10

Problem: Generating quadratic values: How would you teach Generating quadratic values to someone seeing it for the first time?

  1. Define the idea.
  2. Show one easy example.
  3. Explain every step.
  4. Then let the learner try a similar problem.

Very beginner explanation: This sequence reduces memorization without understanding.

Answer: Teach meaning first, then a small worked example, then guided practice.

Practice exercise

Create one new Grade 8 problem involving Generating quadratic values. Show the important steps, include units when needed, and explain how you checked the answer.

31.4 Testing a simple curve

Testing a simple curve is an important Grade 8 concept in Coding Curves and Trend Lines. Start by identifying the quantities, relationships, units, or rules involved. Work with a small example first, show the important steps, and check whether the result is reasonable.

For a beginner, focus on the meaning before memorizing a procedure. Start with a small example, name the quantities and units, apply one rule at a time, and connect the result back to the question.

Coding example

// Testing a simple curve
const points = [[1,2],[2,4],[3,5],[4,8],[5,9]];
const model = x => 1.8 * x + 0.2;
const predictions = points.map(([x,y]) => ({x, observed:y, predicted:model(x)}));
console.log(predictions);

Code explanation

  1. The example stores a small set of coordinate pairs so each result can be checked manually.
  2. model() represents a simple line used to estimate y-values from x-values.
  3. map() applies the model to every point and keeps both observed and predicted values.
  4. Change the slope or intercept and compare the predictions to see whether the line follows the data more closely.

Expected result: A list of observed and predicted values is printed.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Predict taxi cost: use the model cost = 4 + 2.5d with input 8.

  1. Identify the input variable.
  2. Substitute 8 into the model.
  3. Calculate the predicted output.
  4. Ask whether the result makes sense in the real situation.

Very beginner explanation: A mathematical model is a simplified rule for representing a real situation. Predictions should always be checked for reasonableness.

Answer: 24

Worked Example 2

Problem: Predict fundraiser revenue: use the model revenue = 120 + 15s with input 10.

  1. Identify the input variable.
  2. Substitute 10 into the model.
  3. Calculate the predicted output.
  4. Ask whether the result makes sense in the real situation.

Very beginner explanation: A mathematical model is a simplified rule for representing a real situation. Predictions should always be checked for reasonableness.

Answer: 270

Worked Example 3

Problem: Predict distance: use the model distance = 60t with input 2.5.

  1. Identify the input variable.
  2. Substitute 2.5 into the model.
  3. Calculate the predicted output.
  4. Ask whether the result makes sense in the real situation.

Very beginner explanation: A mathematical model is a simplified rule for representing a real situation. Predictions should always be checked for reasonableness.

Answer: 150

Worked Example 4

Problem: Predict savings: use the model savings = 200 + 50m with input 6.

  1. Identify the input variable.
  2. Substitute 6 into the model.
  3. Calculate the predicted output.
  4. Ask whether the result makes sense in the real situation.

Very beginner explanation: A mathematical model is a simplified rule for representing a real situation. Predictions should always be checked for reasonableness.

Answer: 500

Worked Example 5

Problem: Predict temperature trend: use the model temp = 18 - 1.5h with input 4.

  1. Identify the input variable.
  2. Substitute 4 into the model.
  3. Calculate the predicted output.
  4. Ask whether the result makes sense in the real situation.

Very beginner explanation: A mathematical model is a simplified rule for representing a real situation. Predictions should always be checked for reasonableness.

Answer: 12

Worked Example 6

Problem: Predict plant height: use the model height = 5 + 2w with input 7.

  1. Identify the input variable.
  2. Substitute 7 into the model.
  3. Calculate the predicted output.
  4. Ask whether the result makes sense in the real situation.

Very beginner explanation: A mathematical model is a simplified rule for representing a real situation. Predictions should always be checked for reasonableness.

Answer: 19

Worked Example 7

Problem: Predict rental cost: use the model cost = 30 + 8d with input 5.

  1. Identify the input variable.
  2. Substitute 5 into the model.
  3. Calculate the predicted output.
  4. Ask whether the result makes sense in the real situation.

Very beginner explanation: A mathematical model is a simplified rule for representing a real situation. Predictions should always be checked for reasonableness.

Answer: 70

Worked Example 8

Problem: Predict page count: use the model pages = 12r with input 9.

  1. Identify the input variable.
  2. Substitute 9 into the model.
  3. Calculate the predicted output.
  4. Ask whether the result makes sense in the real situation.

Very beginner explanation: A mathematical model is a simplified rule for representing a real situation. Predictions should always be checked for reasonableness.

Answer: 108

Worked Example 9

Problem: Predict points: use the model points = 6g + 3 with input 4.

  1. Identify the input variable.
  2. Substitute 4 into the model.
  3. Calculate the predicted output.
  4. Ask whether the result makes sense in the real situation.

Very beginner explanation: A mathematical model is a simplified rule for representing a real situation. Predictions should always be checked for reasonableness.

Answer: 27

Worked Example 10

Problem: Predict water use: use the model litres = 20 + 4m with input 5.

  1. Identify the input variable.
  2. Substitute 5 into the model.
  3. Calculate the predicted output.
  4. Ask whether the result makes sense in the real situation.

Very beginner explanation: A mathematical model is a simplified rule for representing a real situation. Predictions should always be checked for reasonableness.

Answer: 40

Practice exercise

Create one new Grade 8 problem involving Testing a simple curve. Show the important steps, include units when needed, and explain how you checked the answer.

31.5 Comparing line and curve fit

Comparing line and curve fit is an important Grade 8 concept in Coding Curves and Trend Lines. Start by identifying the quantities, relationships, units, or rules involved. Work with a small example first, show the important steps, and check whether the result is reasonable.

For a beginner, focus on the meaning before memorizing a procedure. Start with a small example, name the quantities and units, apply one rule at a time, and connect the result back to the question.

Coding example

// Comparing line and curve fit
const points = [[1,2],[2,4],[3,5],[4,8],[5,9]];
const model = x => 1.8 * x + 0.2;
const predictions = points.map(([x,y]) => ({x, observed:y, predicted:model(x)}));
console.log(predictions);

Code explanation

  1. The example stores a small set of coordinate pairs so each result can be checked manually.
  2. model() represents a simple line used to estimate y-values from x-values.
  3. map() applies the model to every point and keeps both observed and predicted values.
  4. Change the slope or intercept and compare the predictions to see whether the line follows the data more closely.

Expected result: A list of observed and predicted values is printed.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Predict taxi cost: use the model cost = 4 + 2.5d with input 8.

  1. Identify the input variable.
  2. Substitute 8 into the model.
  3. Calculate the predicted output.
  4. Ask whether the result makes sense in the real situation.

Very beginner explanation: A mathematical model is a simplified rule for representing a real situation. Predictions should always be checked for reasonableness.

Answer: 24

Worked Example 2

Problem: Predict fundraiser revenue: use the model revenue = 120 + 15s with input 10.

  1. Identify the input variable.
  2. Substitute 10 into the model.
  3. Calculate the predicted output.
  4. Ask whether the result makes sense in the real situation.

Very beginner explanation: A mathematical model is a simplified rule for representing a real situation. Predictions should always be checked for reasonableness.

Answer: 270

Worked Example 3

Problem: Predict distance: use the model distance = 60t with input 2.5.

  1. Identify the input variable.
  2. Substitute 2.5 into the model.
  3. Calculate the predicted output.
  4. Ask whether the result makes sense in the real situation.

Very beginner explanation: A mathematical model is a simplified rule for representing a real situation. Predictions should always be checked for reasonableness.

Answer: 150

Worked Example 4

Problem: Predict savings: use the model savings = 200 + 50m with input 6.

  1. Identify the input variable.
  2. Substitute 6 into the model.
  3. Calculate the predicted output.
  4. Ask whether the result makes sense in the real situation.

Very beginner explanation: A mathematical model is a simplified rule for representing a real situation. Predictions should always be checked for reasonableness.

Answer: 500

Worked Example 5

Problem: Predict temperature trend: use the model temp = 18 - 1.5h with input 4.

  1. Identify the input variable.
  2. Substitute 4 into the model.
  3. Calculate the predicted output.
  4. Ask whether the result makes sense in the real situation.

Very beginner explanation: A mathematical model is a simplified rule for representing a real situation. Predictions should always be checked for reasonableness.

Answer: 12

Worked Example 6

Problem: Predict plant height: use the model height = 5 + 2w with input 7.

  1. Identify the input variable.
  2. Substitute 7 into the model.
  3. Calculate the predicted output.
  4. Ask whether the result makes sense in the real situation.

Very beginner explanation: A mathematical model is a simplified rule for representing a real situation. Predictions should always be checked for reasonableness.

Answer: 19

Worked Example 7

Problem: Predict rental cost: use the model cost = 30 + 8d with input 5.

  1. Identify the input variable.
  2. Substitute 5 into the model.
  3. Calculate the predicted output.
  4. Ask whether the result makes sense in the real situation.

Very beginner explanation: A mathematical model is a simplified rule for representing a real situation. Predictions should always be checked for reasonableness.

Answer: 70

Worked Example 8

Problem: Predict page count: use the model pages = 12r with input 9.

  1. Identify the input variable.
  2. Substitute 9 into the model.
  3. Calculate the predicted output.
  4. Ask whether the result makes sense in the real situation.

Very beginner explanation: A mathematical model is a simplified rule for representing a real situation. Predictions should always be checked for reasonableness.

Answer: 108

Worked Example 9

Problem: Predict points: use the model points = 6g + 3 with input 4.

  1. Identify the input variable.
  2. Substitute 4 into the model.
  3. Calculate the predicted output.
  4. Ask whether the result makes sense in the real situation.

Very beginner explanation: A mathematical model is a simplified rule for representing a real situation. Predictions should always be checked for reasonableness.

Answer: 27

Worked Example 10

Problem: Predict water use: use the model litres = 20 + 4m with input 5.

  1. Identify the input variable.
  2. Substitute 5 into the model.
  3. Calculate the predicted output.
  4. Ask whether the result makes sense in the real situation.

Very beginner explanation: A mathematical model is a simplified rule for representing a real situation. Predictions should always be checked for reasonableness.

Answer: 40

Practice exercise

Create one new Grade 8 problem involving Comparing line and curve fit. Show the important steps, include units when needed, and explain how you checked the answer.

31.6 Prediction from a curve

Prediction from a curve is an important Grade 8 concept in Coding Curves and Trend Lines. Start by identifying the quantities, relationships, units, or rules involved. Work with a small example first, show the important steps, and check whether the result is reasonable.

For a beginner, focus on the meaning before memorizing a procedure. Start with a small example, name the quantities and units, apply one rule at a time, and connect the result back to the question.

Coding example

// Prediction from a curve
const points = [[1,2],[2,4],[3,5],[4,8],[5,9]];
const model = x => 1.8 * x + 0.2;
const predictions = points.map(([x,y]) => ({x, observed:y, predicted:model(x)}));
console.log(predictions);

Code explanation

  1. The example stores a small set of coordinate pairs so each result can be checked manually.
  2. model() represents a simple line used to estimate y-values from x-values.
  3. map() applies the model to every point and keeps both observed and predicted values.
  4. Change the slope or intercept and compare the predictions to see whether the line follows the data more closely.

Expected result: A list of observed and predicted values is printed.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Predict taxi cost: use the model cost = 4 + 2.5d with input 8.

  1. Identify the input variable.
  2. Substitute 8 into the model.
  3. Calculate the predicted output.
  4. Ask whether the result makes sense in the real situation.

Very beginner explanation: A mathematical model is a simplified rule for representing a real situation. Predictions should always be checked for reasonableness.

Answer: 24

Worked Example 2

Problem: Predict fundraiser revenue: use the model revenue = 120 + 15s with input 10.

  1. Identify the input variable.
  2. Substitute 10 into the model.
  3. Calculate the predicted output.
  4. Ask whether the result makes sense in the real situation.

Very beginner explanation: A mathematical model is a simplified rule for representing a real situation. Predictions should always be checked for reasonableness.

Answer: 270

Worked Example 3

Problem: Predict distance: use the model distance = 60t with input 2.5.

  1. Identify the input variable.
  2. Substitute 2.5 into the model.
  3. Calculate the predicted output.
  4. Ask whether the result makes sense in the real situation.

Very beginner explanation: A mathematical model is a simplified rule for representing a real situation. Predictions should always be checked for reasonableness.

Answer: 150

Worked Example 4

Problem: Predict savings: use the model savings = 200 + 50m with input 6.

  1. Identify the input variable.
  2. Substitute 6 into the model.
  3. Calculate the predicted output.
  4. Ask whether the result makes sense in the real situation.

Very beginner explanation: A mathematical model is a simplified rule for representing a real situation. Predictions should always be checked for reasonableness.

Answer: 500

Worked Example 5

Problem: Predict temperature trend: use the model temp = 18 - 1.5h with input 4.

  1. Identify the input variable.
  2. Substitute 4 into the model.
  3. Calculate the predicted output.
  4. Ask whether the result makes sense in the real situation.

Very beginner explanation: A mathematical model is a simplified rule for representing a real situation. Predictions should always be checked for reasonableness.

Answer: 12

Worked Example 6

Problem: Predict plant height: use the model height = 5 + 2w with input 7.

  1. Identify the input variable.
  2. Substitute 7 into the model.
  3. Calculate the predicted output.
  4. Ask whether the result makes sense in the real situation.

Very beginner explanation: A mathematical model is a simplified rule for representing a real situation. Predictions should always be checked for reasonableness.

Answer: 19

Worked Example 7

Problem: Predict rental cost: use the model cost = 30 + 8d with input 5.

  1. Identify the input variable.
  2. Substitute 5 into the model.
  3. Calculate the predicted output.
  4. Ask whether the result makes sense in the real situation.

Very beginner explanation: A mathematical model is a simplified rule for representing a real situation. Predictions should always be checked for reasonableness.

Answer: 70

Worked Example 8

Problem: Predict page count: use the model pages = 12r with input 9.

  1. Identify the input variable.
  2. Substitute 9 into the model.
  3. Calculate the predicted output.
  4. Ask whether the result makes sense in the real situation.

Very beginner explanation: A mathematical model is a simplified rule for representing a real situation. Predictions should always be checked for reasonableness.

Answer: 108

Worked Example 9

Problem: Predict points: use the model points = 6g + 3 with input 4.

  1. Identify the input variable.
  2. Substitute 4 into the model.
  3. Calculate the predicted output.
  4. Ask whether the result makes sense in the real situation.

Very beginner explanation: A mathematical model is a simplified rule for representing a real situation. Predictions should always be checked for reasonableness.

Answer: 27

Worked Example 10

Problem: Predict water use: use the model litres = 20 + 4m with input 5.

  1. Identify the input variable.
  2. Substitute 5 into the model.
  3. Calculate the predicted output.
  4. Ask whether the result makes sense in the real situation.

Very beginner explanation: A mathematical model is a simplified rule for representing a real situation. Predictions should always be checked for reasonableness.

Answer: 40

Practice exercise

Create one new Grade 8 problem involving Prediction from a curve. Show the important steps, include units when needed, and explain how you checked the answer.

31.7 Loops for candidate models

A loop repeats a set of instructions in code. In this section, the focus is Loops for candidate models.

For a beginner, focus on the meaning before memorizing a procedure. Start with a small example, name the quantities and units, apply one rule at a time, and connect the result back to the question.

Coding example

// Loops for candidate models
const points = [[1,2],[2,4],[3,5],[4,8],[5,9]];
const model = x => 1.8 * x + 0.2;
const predictions = points.map(([x,y]) => ({x, observed:y, predicted:model(x)}));
console.log(predictions);

Code explanation

  1. The example stores a small set of coordinate pairs so each result can be checked manually.
  2. model() represents a simple line used to estimate y-values from x-values.
  3. map() applies the model to every point and keeps both observed and predicted values.
  4. Change the slope or intercept and compare the predictions to see whether the line follows the data more closely.

Expected result: A list of observed and predicted values is printed.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Predict taxi cost: use the model cost = 4 + 2.5d with input 8.

  1. Identify the input variable.
  2. Substitute 8 into the model.
  3. Calculate the predicted output.
  4. Ask whether the result makes sense in the real situation.

Very beginner explanation: A mathematical model is a simplified rule for representing a real situation. Predictions should always be checked for reasonableness.

Answer: 24

Worked Example 2

Problem: Predict fundraiser revenue: use the model revenue = 120 + 15s with input 10.

  1. Identify the input variable.
  2. Substitute 10 into the model.
  3. Calculate the predicted output.
  4. Ask whether the result makes sense in the real situation.

Very beginner explanation: A mathematical model is a simplified rule for representing a real situation. Predictions should always be checked for reasonableness.

Answer: 270

Worked Example 3

Problem: Predict distance: use the model distance = 60t with input 2.5.

  1. Identify the input variable.
  2. Substitute 2.5 into the model.
  3. Calculate the predicted output.
  4. Ask whether the result makes sense in the real situation.

Very beginner explanation: A mathematical model is a simplified rule for representing a real situation. Predictions should always be checked for reasonableness.

Answer: 150

Worked Example 4

Problem: Predict savings: use the model savings = 200 + 50m with input 6.

  1. Identify the input variable.
  2. Substitute 6 into the model.
  3. Calculate the predicted output.
  4. Ask whether the result makes sense in the real situation.

Very beginner explanation: A mathematical model is a simplified rule for representing a real situation. Predictions should always be checked for reasonableness.

Answer: 500

Worked Example 5

Problem: Predict temperature trend: use the model temp = 18 - 1.5h with input 4.

  1. Identify the input variable.
  2. Substitute 4 into the model.
  3. Calculate the predicted output.
  4. Ask whether the result makes sense in the real situation.

Very beginner explanation: A mathematical model is a simplified rule for representing a real situation. Predictions should always be checked for reasonableness.

Answer: 12

Worked Example 6

Problem: Predict plant height: use the model height = 5 + 2w with input 7.

  1. Identify the input variable.
  2. Substitute 7 into the model.
  3. Calculate the predicted output.
  4. Ask whether the result makes sense in the real situation.

Very beginner explanation: A mathematical model is a simplified rule for representing a real situation. Predictions should always be checked for reasonableness.

Answer: 19

Worked Example 7

Problem: Predict rental cost: use the model cost = 30 + 8d with input 5.

  1. Identify the input variable.
  2. Substitute 5 into the model.
  3. Calculate the predicted output.
  4. Ask whether the result makes sense in the real situation.

Very beginner explanation: A mathematical model is a simplified rule for representing a real situation. Predictions should always be checked for reasonableness.

Answer: 70

Worked Example 8

Problem: Predict page count: use the model pages = 12r with input 9.

  1. Identify the input variable.
  2. Substitute 9 into the model.
  3. Calculate the predicted output.
  4. Ask whether the result makes sense in the real situation.

Very beginner explanation: A mathematical model is a simplified rule for representing a real situation. Predictions should always be checked for reasonableness.

Answer: 108

Worked Example 9

Problem: Predict points: use the model points = 6g + 3 with input 4.

  1. Identify the input variable.
  2. Substitute 4 into the model.
  3. Calculate the predicted output.
  4. Ask whether the result makes sense in the real situation.

Very beginner explanation: A mathematical model is a simplified rule for representing a real situation. Predictions should always be checked for reasonableness.

Answer: 27

Worked Example 10

Problem: Predict water use: use the model litres = 20 + 4m with input 5.

  1. Identify the input variable.
  2. Substitute 5 into the model.
  3. Calculate the predicted output.
  4. Ask whether the result makes sense in the real situation.

Very beginner explanation: A mathematical model is a simplified rule for representing a real situation. Predictions should always be checked for reasonableness.

Answer: 40

Practice exercise

Create one new Grade 8 problem involving Loops for candidate models. Show the important steps, include units when needed, and explain how you checked the answer.

31.8 Simple error measure

Simple error measure is an important Grade 8 concept in Coding Curves and Trend Lines. Start by identifying the quantities, relationships, units, or rules involved. Work with a small example first, show the important steps, and check whether the result is reasonable.

For a beginner, focus on the meaning before memorizing a procedure. Start with a small example, name the quantities and units, apply one rule at a time, and connect the result back to the question.

Coding example

// Simple error measure
const points = [[1,2],[2,4],[3,5],[4,8],[5,9]];
const model = x => 1.8 * x + 0.2;
const predictions = points.map(([x,y]) => ({x, observed:y, predicted:model(x)}));
console.log(predictions);

Code explanation

  1. The example stores a small set of coordinate pairs so each result can be checked manually.
  2. model() represents a simple line used to estimate y-values from x-values.
  3. map() applies the model to every point and keeps both observed and predicted values.
  4. Change the slope or intercept and compare the predictions to see whether the line follows the data more closely.

Expected result: A list of observed and predicted values is printed.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Classify this data set: [4, 6, 8].

  1. The values are numerical.
  2. Ask whether they are counts or measurements.

Very beginner explanation: Quantitative data are numbers that represent counts or measurements.

Answer: Quantitative data

Worked Example 2

Problem: Classify this data set: [5, 9, 10, 12].

  1. The values are numerical.
  2. Ask whether they are counts or measurements.

Very beginner explanation: Quantitative data are numbers that represent counts or measurements.

Answer: Quantitative data

Worked Example 3

Problem: Classify this data set: [3, 7, 7, 11].

  1. The values are numerical.
  2. Ask whether they are counts or measurements.

Very beginner explanation: Quantitative data are numbers that represent counts or measurements.

Answer: Quantitative data

Worked Example 4

Problem: Classify this data set: [20, 25, 30].

  1. The values are numerical.
  2. Ask whether they are counts or measurements.

Very beginner explanation: Quantitative data are numbers that represent counts or measurements.

Answer: Quantitative data

Worked Example 5

Problem: Classify this data set: [6, 8, 9, 12, 15].

  1. The values are numerical.
  2. Ask whether they are counts or measurements.

Very beginner explanation: Quantitative data are numbers that represent counts or measurements.

Answer: Quantitative data

Worked Example 6

Problem: Classify this data set: [2, 4, 4, 5, 20].

  1. The values are numerical.
  2. Ask whether they are counts or measurements.

Very beginner explanation: Quantitative data are numbers that represent counts or measurements.

Answer: Quantitative data

Worked Example 7

Problem: Classify this data set: [10, 11, 12, 13, 14].

  1. The values are numerical.
  2. Ask whether they are counts or measurements.

Very beginner explanation: Quantitative data are numbers that represent counts or measurements.

Answer: Quantitative data

Worked Example 8

Problem: Classify this data set: [1, 3, 5, 7, 9].

  1. The values are numerical.
  2. Ask whether they are counts or measurements.

Very beginner explanation: Quantitative data are numbers that represent counts or measurements.

Answer: Quantitative data

Worked Example 9

Problem: Classify this data set: [8, 8, 8, 9, 10].

  1. The values are numerical.
  2. Ask whether they are counts or measurements.

Very beginner explanation: Quantitative data are numbers that represent counts or measurements.

Answer: Quantitative data

Worked Example 10

Problem: Classify this data set: [15, 18, 21, 24, 27].

  1. The values are numerical.
  2. Ask whether they are counts or measurements.

Very beginner explanation: Quantitative data are numbers that represent counts or measurements.

Answer: Quantitative data

Practice exercise

Create one new Grade 8 problem involving Simple error measure. Show the important steps, include units when needed, and explain how you checked the answer.

31.9 Choosing a better-fitting model

The mode is the value that occurs most often. In this section, the focus is Choosing a better-fitting model.

For a beginner, focus on the meaning before memorizing a procedure. Start with a small example, name the quantities and units, apply one rule at a time, and connect the result back to the question.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Predict taxi cost: use the model cost = 4 + 2.5d with input 8.

  1. Identify the input variable.
  2. Substitute 8 into the model.
  3. Calculate the predicted output.
  4. Ask whether the result makes sense in the real situation.

Very beginner explanation: A mathematical model is a simplified rule for representing a real situation. Predictions should always be checked for reasonableness.

Answer: 24

Worked Example 2

Problem: Predict fundraiser revenue: use the model revenue = 120 + 15s with input 10.

  1. Identify the input variable.
  2. Substitute 10 into the model.
  3. Calculate the predicted output.
  4. Ask whether the result makes sense in the real situation.

Very beginner explanation: A mathematical model is a simplified rule for representing a real situation. Predictions should always be checked for reasonableness.

Answer: 270

Worked Example 3

Problem: Predict distance: use the model distance = 60t with input 2.5.

  1. Identify the input variable.
  2. Substitute 2.5 into the model.
  3. Calculate the predicted output.
  4. Ask whether the result makes sense in the real situation.

Very beginner explanation: A mathematical model is a simplified rule for representing a real situation. Predictions should always be checked for reasonableness.

Answer: 150

Worked Example 4

Problem: Predict savings: use the model savings = 200 + 50m with input 6.

  1. Identify the input variable.
  2. Substitute 6 into the model.
  3. Calculate the predicted output.
  4. Ask whether the result makes sense in the real situation.

Very beginner explanation: A mathematical model is a simplified rule for representing a real situation. Predictions should always be checked for reasonableness.

Answer: 500

Worked Example 5

Problem: Predict temperature trend: use the model temp = 18 - 1.5h with input 4.

  1. Identify the input variable.
  2. Substitute 4 into the model.
  3. Calculate the predicted output.
  4. Ask whether the result makes sense in the real situation.

Very beginner explanation: A mathematical model is a simplified rule for representing a real situation. Predictions should always be checked for reasonableness.

Answer: 12

Worked Example 6

Problem: Predict plant height: use the model height = 5 + 2w with input 7.

  1. Identify the input variable.
  2. Substitute 7 into the model.
  3. Calculate the predicted output.
  4. Ask whether the result makes sense in the real situation.

Very beginner explanation: A mathematical model is a simplified rule for representing a real situation. Predictions should always be checked for reasonableness.

Answer: 19

Worked Example 7

Problem: Predict rental cost: use the model cost = 30 + 8d with input 5.

  1. Identify the input variable.
  2. Substitute 5 into the model.
  3. Calculate the predicted output.
  4. Ask whether the result makes sense in the real situation.

Very beginner explanation: A mathematical model is a simplified rule for representing a real situation. Predictions should always be checked for reasonableness.

Answer: 70

Worked Example 8

Problem: Predict page count: use the model pages = 12r with input 9.

  1. Identify the input variable.
  2. Substitute 9 into the model.
  3. Calculate the predicted output.
  4. Ask whether the result makes sense in the real situation.

Very beginner explanation: A mathematical model is a simplified rule for representing a real situation. Predictions should always be checked for reasonableness.

Answer: 108

Worked Example 9

Problem: Predict points: use the model points = 6g + 3 with input 4.

  1. Identify the input variable.
  2. Substitute 4 into the model.
  3. Calculate the predicted output.
  4. Ask whether the result makes sense in the real situation.

Very beginner explanation: A mathematical model is a simplified rule for representing a real situation. Predictions should always be checked for reasonableness.

Answer: 27

Worked Example 10

Problem: Predict water use: use the model litres = 20 + 4m with input 5.

  1. Identify the input variable.
  2. Substitute 5 into the model.
  3. Calculate the predicted output.
  4. Ask whether the result makes sense in the real situation.

Very beginner explanation: A mathematical model is a simplified rule for representing a real situation. Predictions should always be checked for reasonableness.

Answer: 40

Practice exercise

Create one new Grade 8 problem involving Choosing a better-fitting model. Show the important steps, include units when needed, and explain how you checked the answer.

31.10 Avoiding overfitting introduction

Avoiding overfitting introduction is an important Grade 8 concept in Coding Curves and Trend Lines. Start by identifying the quantities, relationships, units, or rules involved. Work with a small example first, show the important steps, and check whether the result is reasonable.

For a beginner, focus on the meaning before memorizing a procedure. Start with a small example, name the quantities and units, apply one rule at a time, and connect the result back to the question.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Predict taxi cost: use the model cost = 4 + 2.5d with input 8.

  1. Identify the input variable.
  2. Substitute 8 into the model.
  3. Calculate the predicted output.
  4. Ask whether the result makes sense in the real situation.

Very beginner explanation: A mathematical model is a simplified rule for representing a real situation. Predictions should always be checked for reasonableness.

Answer: 24

Worked Example 2

Problem: Predict fundraiser revenue: use the model revenue = 120 + 15s with input 10.

  1. Identify the input variable.
  2. Substitute 10 into the model.
  3. Calculate the predicted output.
  4. Ask whether the result makes sense in the real situation.

Very beginner explanation: A mathematical model is a simplified rule for representing a real situation. Predictions should always be checked for reasonableness.

Answer: 270

Worked Example 3

Problem: Predict distance: use the model distance = 60t with input 2.5.

  1. Identify the input variable.
  2. Substitute 2.5 into the model.
  3. Calculate the predicted output.
  4. Ask whether the result makes sense in the real situation.

Very beginner explanation: A mathematical model is a simplified rule for representing a real situation. Predictions should always be checked for reasonableness.

Answer: 150

Worked Example 4

Problem: Predict savings: use the model savings = 200 + 50m with input 6.

  1. Identify the input variable.
  2. Substitute 6 into the model.
  3. Calculate the predicted output.
  4. Ask whether the result makes sense in the real situation.

Very beginner explanation: A mathematical model is a simplified rule for representing a real situation. Predictions should always be checked for reasonableness.

Answer: 500

Worked Example 5

Problem: Predict temperature trend: use the model temp = 18 - 1.5h with input 4.

  1. Identify the input variable.
  2. Substitute 4 into the model.
  3. Calculate the predicted output.
  4. Ask whether the result makes sense in the real situation.

Very beginner explanation: A mathematical model is a simplified rule for representing a real situation. Predictions should always be checked for reasonableness.

Answer: 12

Worked Example 6

Problem: Predict plant height: use the model height = 5 + 2w with input 7.

  1. Identify the input variable.
  2. Substitute 7 into the model.
  3. Calculate the predicted output.
  4. Ask whether the result makes sense in the real situation.

Very beginner explanation: A mathematical model is a simplified rule for representing a real situation. Predictions should always be checked for reasonableness.

Answer: 19

Worked Example 7

Problem: Predict rental cost: use the model cost = 30 + 8d with input 5.

  1. Identify the input variable.
  2. Substitute 5 into the model.
  3. Calculate the predicted output.
  4. Ask whether the result makes sense in the real situation.

Very beginner explanation: A mathematical model is a simplified rule for representing a real situation. Predictions should always be checked for reasonableness.

Answer: 70

Worked Example 8

Problem: Predict page count: use the model pages = 12r with input 9.

  1. Identify the input variable.
  2. Substitute 9 into the model.
  3. Calculate the predicted output.
  4. Ask whether the result makes sense in the real situation.

Very beginner explanation: A mathematical model is a simplified rule for representing a real situation. Predictions should always be checked for reasonableness.

Answer: 108

Worked Example 9

Problem: Predict points: use the model points = 6g + 3 with input 4.

  1. Identify the input variable.
  2. Substitute 4 into the model.
  3. Calculate the predicted output.
  4. Ask whether the result makes sense in the real situation.

Very beginner explanation: A mathematical model is a simplified rule for representing a real situation. Predictions should always be checked for reasonableness.

Answer: 27

Worked Example 10

Problem: Predict water use: use the model litres = 20 + 4m with input 5.

  1. Identify the input variable.
  2. Substitute 5 into the model.
  3. Calculate the predicted output.
  4. Ask whether the result makes sense in the real situation.

Very beginner explanation: A mathematical model is a simplified rule for representing a real situation. Predictions should always be checked for reasonableness.

Answer: 40

Practice exercise

Create one new Grade 8 problem involving Avoiding overfitting introduction. Show the important steps, include units when needed, and explain how you checked the answer.

31.11 Visualizing trends

Visualizing trends is an important Grade 8 concept in Coding Curves and Trend Lines. Start by identifying the quantities, relationships, units, or rules involved. Work with a small example first, show the important steps, and check whether the result is reasonable.

For a beginner, focus on the meaning before memorizing a procedure. Start with a small example, name the quantities and units, apply one rule at a time, and connect the result back to the question.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Predict taxi cost: use the model cost = 4 + 2.5d with input 8.

  1. Identify the input variable.
  2. Substitute 8 into the model.
  3. Calculate the predicted output.
  4. Ask whether the result makes sense in the real situation.

Very beginner explanation: A mathematical model is a simplified rule for representing a real situation. Predictions should always be checked for reasonableness.

Answer: 24

Worked Example 2

Problem: Predict fundraiser revenue: use the model revenue = 120 + 15s with input 10.

  1. Identify the input variable.
  2. Substitute 10 into the model.
  3. Calculate the predicted output.
  4. Ask whether the result makes sense in the real situation.

Very beginner explanation: A mathematical model is a simplified rule for representing a real situation. Predictions should always be checked for reasonableness.

Answer: 270

Worked Example 3

Problem: Predict distance: use the model distance = 60t with input 2.5.

  1. Identify the input variable.
  2. Substitute 2.5 into the model.
  3. Calculate the predicted output.
  4. Ask whether the result makes sense in the real situation.

Very beginner explanation: A mathematical model is a simplified rule for representing a real situation. Predictions should always be checked for reasonableness.

Answer: 150

Worked Example 4

Problem: Predict savings: use the model savings = 200 + 50m with input 6.

  1. Identify the input variable.
  2. Substitute 6 into the model.
  3. Calculate the predicted output.
  4. Ask whether the result makes sense in the real situation.

Very beginner explanation: A mathematical model is a simplified rule for representing a real situation. Predictions should always be checked for reasonableness.

Answer: 500

Worked Example 5

Problem: Predict temperature trend: use the model temp = 18 - 1.5h with input 4.

  1. Identify the input variable.
  2. Substitute 4 into the model.
  3. Calculate the predicted output.
  4. Ask whether the result makes sense in the real situation.

Very beginner explanation: A mathematical model is a simplified rule for representing a real situation. Predictions should always be checked for reasonableness.

Answer: 12

Worked Example 6

Problem: Predict plant height: use the model height = 5 + 2w with input 7.

  1. Identify the input variable.
  2. Substitute 7 into the model.
  3. Calculate the predicted output.
  4. Ask whether the result makes sense in the real situation.

Very beginner explanation: A mathematical model is a simplified rule for representing a real situation. Predictions should always be checked for reasonableness.

Answer: 19

Worked Example 7

Problem: Predict rental cost: use the model cost = 30 + 8d with input 5.

  1. Identify the input variable.
  2. Substitute 5 into the model.
  3. Calculate the predicted output.
  4. Ask whether the result makes sense in the real situation.

Very beginner explanation: A mathematical model is a simplified rule for representing a real situation. Predictions should always be checked for reasonableness.

Answer: 70

Worked Example 8

Problem: Predict page count: use the model pages = 12r with input 9.

  1. Identify the input variable.
  2. Substitute 9 into the model.
  3. Calculate the predicted output.
  4. Ask whether the result makes sense in the real situation.

Very beginner explanation: A mathematical model is a simplified rule for representing a real situation. Predictions should always be checked for reasonableness.

Answer: 108

Worked Example 9

Problem: Predict points: use the model points = 6g + 3 with input 4.

  1. Identify the input variable.
  2. Substitute 4 into the model.
  3. Calculate the predicted output.
  4. Ask whether the result makes sense in the real situation.

Very beginner explanation: A mathematical model is a simplified rule for representing a real situation. Predictions should always be checked for reasonableness.

Answer: 27

Worked Example 10

Problem: Predict water use: use the model litres = 20 + 4m with input 5.

  1. Identify the input variable.
  2. Substitute 5 into the model.
  3. Calculate the predicted output.
  4. Ask whether the result makes sense in the real situation.

Very beginner explanation: A mathematical model is a simplified rule for representing a real situation. Predictions should always be checked for reasonableness.

Answer: 40

Practice exercise

Create one new Grade 8 problem involving Visualizing trends. Show the important steps, include units when needed, and explain how you checked the answer.

31.12 Debugging trend code

Debugging means finding and fixing errors in code or logic. In this section, the focus is Debugging trend code.

For a beginner, focus on the meaning before memorizing a procedure. Start with a small example, name the quantities and units, apply one rule at a time, and connect the result back to the question.

Coding example

// Debugging trend code
const points = [[1,2],[2,4],[3,5],[4,8],[5,9]];
const model = x => 1.8 * x + 0.2;
const predictions = points.map(([x,y]) => ({x, observed:y, predicted:model(x)}));
console.log(predictions);

Code explanation

  1. The example stores a small set of coordinate pairs so each result can be checked manually.
  2. model() represents a simple line used to estimate y-values from x-values.
  3. map() applies the model to every point and keeps both observed and predicted values.
  4. Change the slope or intercept and compare the predictions to see whether the line follows the data more closely.

Expected result: A list of observed and predicted values is printed.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Predict taxi cost: use the model cost = 4 + 2.5d with input 8.

  1. Identify the input variable.
  2. Substitute 8 into the model.
  3. Calculate the predicted output.
  4. Ask whether the result makes sense in the real situation.

Very beginner explanation: A mathematical model is a simplified rule for representing a real situation. Predictions should always be checked for reasonableness.

Answer: 24

Worked Example 2

Problem: Predict fundraiser revenue: use the model revenue = 120 + 15s with input 10.

  1. Identify the input variable.
  2. Substitute 10 into the model.
  3. Calculate the predicted output.
  4. Ask whether the result makes sense in the real situation.

Very beginner explanation: A mathematical model is a simplified rule for representing a real situation. Predictions should always be checked for reasonableness.

Answer: 270

Worked Example 3

Problem: Predict distance: use the model distance = 60t with input 2.5.

  1. Identify the input variable.
  2. Substitute 2.5 into the model.
  3. Calculate the predicted output.
  4. Ask whether the result makes sense in the real situation.

Very beginner explanation: A mathematical model is a simplified rule for representing a real situation. Predictions should always be checked for reasonableness.

Answer: 150

Worked Example 4

Problem: Predict savings: use the model savings = 200 + 50m with input 6.

  1. Identify the input variable.
  2. Substitute 6 into the model.
  3. Calculate the predicted output.
  4. Ask whether the result makes sense in the real situation.

Very beginner explanation: A mathematical model is a simplified rule for representing a real situation. Predictions should always be checked for reasonableness.

Answer: 500

Worked Example 5

Problem: Predict temperature trend: use the model temp = 18 - 1.5h with input 4.

  1. Identify the input variable.
  2. Substitute 4 into the model.
  3. Calculate the predicted output.
  4. Ask whether the result makes sense in the real situation.

Very beginner explanation: A mathematical model is a simplified rule for representing a real situation. Predictions should always be checked for reasonableness.

Answer: 12

Worked Example 6

Problem: Predict plant height: use the model height = 5 + 2w with input 7.

  1. Identify the input variable.
  2. Substitute 7 into the model.
  3. Calculate the predicted output.
  4. Ask whether the result makes sense in the real situation.

Very beginner explanation: A mathematical model is a simplified rule for representing a real situation. Predictions should always be checked for reasonableness.

Answer: 19

Worked Example 7

Problem: Predict rental cost: use the model cost = 30 + 8d with input 5.

  1. Identify the input variable.
  2. Substitute 5 into the model.
  3. Calculate the predicted output.
  4. Ask whether the result makes sense in the real situation.

Very beginner explanation: A mathematical model is a simplified rule for representing a real situation. Predictions should always be checked for reasonableness.

Answer: 70

Worked Example 8

Problem: Predict page count: use the model pages = 12r with input 9.

  1. Identify the input variable.
  2. Substitute 9 into the model.
  3. Calculate the predicted output.
  4. Ask whether the result makes sense in the real situation.

Very beginner explanation: A mathematical model is a simplified rule for representing a real situation. Predictions should always be checked for reasonableness.

Answer: 108

Worked Example 9

Problem: Predict points: use the model points = 6g + 3 with input 4.

  1. Identify the input variable.
  2. Substitute 4 into the model.
  3. Calculate the predicted output.
  4. Ask whether the result makes sense in the real situation.

Very beginner explanation: A mathematical model is a simplified rule for representing a real situation. Predictions should always be checked for reasonableness.

Answer: 27

Worked Example 10

Problem: Predict water use: use the model litres = 20 + 4m with input 5.

  1. Identify the input variable.
  2. Substitute 5 into the model.
  3. Calculate the predicted output.
  4. Ask whether the result makes sense in the real situation.

Very beginner explanation: A mathematical model is a simplified rule for representing a real situation. Predictions should always be checked for reasonableness.

Answer: 40

Practice exercise

Create one new Grade 8 problem involving Debugging trend code. Show the important steps, include units when needed, and explain how you checked the answer.

31.13 Explaining model limitations

The mode is the value that occurs most often. In this section, the focus is Explaining model limitations.

For a beginner, focus on the meaning before memorizing a procedure. Start with a small example, name the quantities and units, apply one rule at a time, and connect the result back to the question.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Predict taxi cost: use the model cost = 4 + 2.5d with input 8.

  1. Identify the input variable.
  2. Substitute 8 into the model.
  3. Calculate the predicted output.
  4. Ask whether the result makes sense in the real situation.

Very beginner explanation: A mathematical model is a simplified rule for representing a real situation. Predictions should always be checked for reasonableness.

Answer: 24

Worked Example 2

Problem: Predict fundraiser revenue: use the model revenue = 120 + 15s with input 10.

  1. Identify the input variable.
  2. Substitute 10 into the model.
  3. Calculate the predicted output.
  4. Ask whether the result makes sense in the real situation.

Very beginner explanation: A mathematical model is a simplified rule for representing a real situation. Predictions should always be checked for reasonableness.

Answer: 270

Worked Example 3

Problem: Predict distance: use the model distance = 60t with input 2.5.

  1. Identify the input variable.
  2. Substitute 2.5 into the model.
  3. Calculate the predicted output.
  4. Ask whether the result makes sense in the real situation.

Very beginner explanation: A mathematical model is a simplified rule for representing a real situation. Predictions should always be checked for reasonableness.

Answer: 150

Worked Example 4

Problem: Predict savings: use the model savings = 200 + 50m with input 6.

  1. Identify the input variable.
  2. Substitute 6 into the model.
  3. Calculate the predicted output.
  4. Ask whether the result makes sense in the real situation.

Very beginner explanation: A mathematical model is a simplified rule for representing a real situation. Predictions should always be checked for reasonableness.

Answer: 500

Worked Example 5

Problem: Predict temperature trend: use the model temp = 18 - 1.5h with input 4.

  1. Identify the input variable.
  2. Substitute 4 into the model.
  3. Calculate the predicted output.
  4. Ask whether the result makes sense in the real situation.

Very beginner explanation: A mathematical model is a simplified rule for representing a real situation. Predictions should always be checked for reasonableness.

Answer: 12

Worked Example 6

Problem: Predict plant height: use the model height = 5 + 2w with input 7.

  1. Identify the input variable.
  2. Substitute 7 into the model.
  3. Calculate the predicted output.
  4. Ask whether the result makes sense in the real situation.

Very beginner explanation: A mathematical model is a simplified rule for representing a real situation. Predictions should always be checked for reasonableness.

Answer: 19

Worked Example 7

Problem: Predict rental cost: use the model cost = 30 + 8d with input 5.

  1. Identify the input variable.
  2. Substitute 5 into the model.
  3. Calculate the predicted output.
  4. Ask whether the result makes sense in the real situation.

Very beginner explanation: A mathematical model is a simplified rule for representing a real situation. Predictions should always be checked for reasonableness.

Answer: 70

Worked Example 8

Problem: Predict page count: use the model pages = 12r with input 9.

  1. Identify the input variable.
  2. Substitute 9 into the model.
  3. Calculate the predicted output.
  4. Ask whether the result makes sense in the real situation.

Very beginner explanation: A mathematical model is a simplified rule for representing a real situation. Predictions should always be checked for reasonableness.

Answer: 108

Worked Example 9

Problem: Predict points: use the model points = 6g + 3 with input 4.

  1. Identify the input variable.
  2. Substitute 4 into the model.
  3. Calculate the predicted output.
  4. Ask whether the result makes sense in the real situation.

Very beginner explanation: A mathematical model is a simplified rule for representing a real situation. Predictions should always be checked for reasonableness.

Answer: 27

Worked Example 10

Problem: Predict water use: use the model litres = 20 + 4m with input 5.

  1. Identify the input variable.
  2. Substitute 5 into the model.
  3. Calculate the predicted output.
  4. Ask whether the result makes sense in the real situation.

Very beginner explanation: A mathematical model is a simplified rule for representing a real situation. Predictions should always be checked for reasonableness.

Answer: 40

Practice exercise

Create one new Grade 8 problem involving Explaining model limitations. Show the important steps, include units when needed, and explain how you checked the answer.

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Chapter 31 Review Questions and Answers

Q1. What is important to remember about Nonlinear data patterns?

Answer: A pattern follows a rule that can be used to predict missing or future terms. In this section, the focus is Nonlinear data patterns.

Q2. What is important to remember about Curve versus line?

Answer: Curve versus line is an important Grade 8 concept in Coding Curves and Trend Lines. Start by identifying the quantities, relationships, units, or rules involved. Work with a small example first, show the important steps, and check whether the result is reasonable.

Q3. What is important to remember about Generating quadratic values?

Answer: Generating quadratic values is an important Grade 8 concept in Coding Curves and Trend Lines. Start by identifying the quantities, relationships, units, or rules involved. Work with a small example first, show the important steps, and check whether the result is reasonable.

Q4. What is important to remember about Testing a simple curve?

Answer: Testing a simple curve is an important Grade 8 concept in Coding Curves and Trend Lines. Start by identifying the quantities, relationships, units, or rules involved. Work with a small example first, show the important steps, and check whether the result is reasonable.

Q5. What is important to remember about Comparing line and curve fit?

Answer: Comparing line and curve fit is an important Grade 8 concept in Coding Curves and Trend Lines. Start by identifying the quantities, relationships, units, or rules involved. Work with a small example first, show the important steps, and check whether the result is reasonable.

Q6. What is important to remember about Prediction from a curve?

Answer: Prediction from a curve is an important Grade 8 concept in Coding Curves and Trend Lines. Start by identifying the quantities, relationships, units, or rules involved. Work with a small example first, show the important steps, and check whether the result is reasonable.

Q7. What is important to remember about Loops for candidate models?

Answer: A loop repeats a set of instructions in code. In this section, the focus is Loops for candidate models.

Q8. What is important to remember about Simple error measure?

Answer: Simple error measure is an important Grade 8 concept in Coding Curves and Trend Lines. Start by identifying the quantities, relationships, units, or rules involved. Work with a small example first, show the important steps, and check whether the result is reasonable.

Q9. What is important to remember about Choosing a better-fitting model?

Answer: The mode is the value that occurs most often. In this section, the focus is Choosing a better-fitting model.

Q10. What is important to remember about Avoiding overfitting introduction?

Answer: Avoiding overfitting introduction is an important Grade 8 concept in Coding Curves and Trend Lines. Start by identifying the quantities, relationships, units, or rules involved. Work with a small example first, show the important steps, and check whether the result is reasonable.

Q11. What is important to remember about Visualizing trends?

Answer: Visualizing trends is an important Grade 8 concept in Coding Curves and Trend Lines. Start by identifying the quantities, relationships, units, or rules involved. Work with a small example first, show the important steps, and check whether the result is reasonable.

Q12. What is important to remember about Debugging trend code?

Answer: Debugging means finding and fixing errors in code or logic. In this section, the focus is Debugging trend code.

Q13. What is important to remember about Explaining model limitations?

Answer: The mode is the value that occurs most often. In this section, the focus is Explaining model limitations.

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 calculated answer is reasonable.

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 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 mathematical 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 and check copied values, signs, units, formulas, and calculations.

Q22. Why are diagrams useful?

Answer: A labelled diagram can reveal relationships among lengths, angles, areas, and coordinates.

Q23. Why are tables useful?

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

Q24. Why should you explain your reasoning?

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

Q25. How do mistakes help learning?

Answer: Analyzing a mistake identifies the misunderstood rule or step and helps prevent repetition.

Q26. When is a calculator useful?

Answer: A calculator helps with lengthy arithmetic after the mathematical setup is understood and estimated.

Q27. Why compare more than one strategy?

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

Q28. What is mathematical communication?

Answer: It is presenting ideas clearly with words, symbols, diagrams, tables, graphs, and justified steps.

Q29. What is mathematical modelling?

Answer: It is representing a real situation with mathematics, testing the model, and revising it if needed.

Q30. Why should assumptions be stated?

Answer: Assumptions show the conditions the model depends on and help readers judge whether it is reasonable.