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Chapter 21: Linear Patterns and Graphing

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

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

This chapter teaches Linear Patterns and Graphing in original, simple language with step-by-step reasoning, worked examples, practice exercises, common mistakes, and review questions.

Key Technical Terms

  • Linear pattern meaning (A pattern follows a rule that can be described, extended, and used to make predictions.)
  • Input and output (Input and output is a Grade 6 mathematics idea in Linear Patterns and Graphing. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.)
  • Tables of values (Tables of values is a Grade 6 mathematics idea in Linear Patterns and Graphing. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.)
  • Ordered pairs (Ordered pairs is a Grade 6 mathematics idea in Linear Patterns and Graphing. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.)
  • Rate of change (A rate compares two quantities measured in different units.)
  • Starting value (Starting value introduction is a Grade 6 mathematics idea in Linear Patterns and Graphing. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.)

How to Learn This Chapter

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

21.1 Linear pattern meaning

A pattern follows a rule that can be described, extended, and used to make predictions. In this Grade 6 lesson, focus on meaning first, then use a small example, show the steps, and check whether the result fits the question.

Beginner Note

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

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Continue the pattern 9, 12, 15, ... for two more terms.

  1. Find the constant change: +3.
  2. Add the same amount each time.

Very beginner explanation: A constant-rate numerical pattern changes by the same amount for each step.

Answer: 18, 21

Worked Example 2

Problem: Continue the pattern 6, 10, 14, ... for two more terms.

  1. Find the constant change: +4.
  2. Add the same amount each time.

Very beginner explanation: A constant-rate numerical pattern changes by the same amount for each step.

Answer: 18, 22

Worked Example 3

Problem: Continue the pattern 4, 8, 12, ... for two more terms.

  1. Find the constant change: +4.
  2. Add the same amount each time.

Very beginner explanation: A constant-rate numerical pattern changes by the same amount for each step.

Answer: 16, 20

Worked Example 4

Problem: Continue the pattern 0, 5, 10, ... for two more terms.

  1. Find the constant change: +5.
  2. Add the same amount each time.

Very beginner explanation: A constant-rate numerical pattern changes by the same amount for each step.

Answer: 15, 20

Worked Example 5

Problem: Continue the pattern 13, 21, 29, ... for two more terms.

  1. Find the constant change: +8.
  2. Add the same amount each time.

Very beginner explanation: A constant-rate numerical pattern changes by the same amount for each step.

Answer: 37, 45

Worked Example 6

Problem: Continue the pattern -4, 3, 10, ... for two more terms.

  1. Find the constant change: +7.
  2. Add the same amount each time.

Very beginner explanation: A constant-rate numerical pattern changes by the same amount for each step.

Answer: 17, 24

Worked Example 7

Problem: Continue the pattern 11, 17, 23, ... for two more terms.

  1. Find the constant change: +6.
  2. Add the same amount each time.

Very beginner explanation: A constant-rate numerical pattern changes by the same amount for each step.

Answer: 29, 35

Worked Example 8

Problem: Continue the pattern 9, 12, 15, ... for two more terms.

  1. Find the constant change: +3.
  2. Add the same amount each time.

Very beginner explanation: A constant-rate numerical pattern changes by the same amount for each step.

Answer: 18, 21

Worked Example 9

Problem: Continue the pattern -1, 2, 5, ... for two more terms.

  1. Find the constant change: +3.
  2. Add the same amount each time.

Very beginner explanation: A constant-rate numerical pattern changes by the same amount for each step.

Answer: 8, 11

Worked Example 10

Problem: Continue the pattern 3, 10, 17, ... for two more terms.

  1. Find the constant change: +7.
  2. Add the same amount each time.

Very beginner explanation: A constant-rate numerical pattern changes by the same amount for each step.

Answer: 24, 31

Practice Exercise

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

Common Mistake

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

21.2 Input and output

Input and output is a Grade 6 mathematics idea in Linear Patterns and Graphing. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable. In this Grade 6 lesson, focus on meaning first, then use a small example, show the steps, and check whether the result fits the question.

Beginner Note

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

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Use Input and output to analyze the values 22 and 6. What should be checked first?

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

Very beginner explanation: Input and output is a Grade 6 mathematics idea in Linear Patterns and Graphing. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

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

Worked Example 2

Problem: Use Input and output to analyze the values 23 and 4. What should be checked first?

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

Very beginner explanation: Input and output is a Grade 6 mathematics idea in Linear Patterns and Graphing. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

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

Worked Example 3

Problem: Use Input and output to analyze the values 9 and 7. What should be checked first?

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

Very beginner explanation: Input and output is a Grade 6 mathematics idea in Linear Patterns and Graphing. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

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

Worked Example 4

Problem: Use Input and output to analyze the values 22 and 6. What should be checked first?

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

Very beginner explanation: Input and output is a Grade 6 mathematics idea in Linear Patterns and Graphing. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

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

Worked Example 5

Problem: Use Input and output to analyze the values 10 and 10. What should be checked first?

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

Very beginner explanation: Input and output is a Grade 6 mathematics idea in Linear Patterns and Graphing. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

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

Worked Example 6

Problem: Use Input and output to analyze the values 4 and 13. What should be checked first?

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

Very beginner explanation: Input and output is a Grade 6 mathematics idea in Linear Patterns and Graphing. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

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

Worked Example 7

Problem: Use Input and output to analyze the values 28 and 20. What should be checked first?

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

Very beginner explanation: Input and output is a Grade 6 mathematics idea in Linear Patterns and Graphing. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

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

Worked Example 8

Problem: Use Input and output to analyze the values 14 and 10. What should be checked first?

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

Very beginner explanation: Input and output is a Grade 6 mathematics idea in Linear Patterns and Graphing. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

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

Worked Example 9

Problem: Use Input and output to analyze the values 18 and 3. What should be checked first?

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

Very beginner explanation: Input and output is a Grade 6 mathematics idea in Linear Patterns and Graphing. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

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

Worked Example 10

Problem: Use Input and output to analyze the values 24 and 17. What should be checked first?

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

Very beginner explanation: Input and output is a Grade 6 mathematics idea in Linear Patterns and Graphing. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

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

Practice Exercise

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

Common Mistake

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

21.3 Tables of values

Tables of values is a Grade 6 mathematics idea in Linear Patterns and Graphing. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable. In this Grade 6 lesson, focus on meaning first, then use a small example, show the steps, and check whether the result fits the question.

Beginner Note

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

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Continue the pattern -5, -3, -1, ... for two more terms.

  1. Find the constant change: +2.
  2. Add the same amount each time.

Very beginner explanation: A constant-rate numerical pattern changes by the same amount for each step.

Answer: 1, 3

Worked Example 2

Problem: Continue the pattern -3, 4, 11, ... for two more terms.

  1. Find the constant change: +7.
  2. Add the same amount each time.

Very beginner explanation: A constant-rate numerical pattern changes by the same amount for each step.

Answer: 18, 25

Worked Example 3

Problem: Continue the pattern 10, 15, 20, ... for two more terms.

  1. Find the constant change: +5.
  2. Add the same amount each time.

Very beginner explanation: A constant-rate numerical pattern changes by the same amount for each step.

Answer: 25, 30

Worked Example 4

Problem: Continue the pattern 4, 6, 8, ... for two more terms.

  1. Find the constant change: +2.
  2. Add the same amount each time.

Very beginner explanation: A constant-rate numerical pattern changes by the same amount for each step.

Answer: 10, 12

Worked Example 5

Problem: Continue the pattern -1, 5, 11, ... for two more terms.

  1. Find the constant change: +6.
  2. Add the same amount each time.

Very beginner explanation: A constant-rate numerical pattern changes by the same amount for each step.

Answer: 17, 23

Worked Example 6

Problem: Continue the pattern 2, 5, 8, ... for two more terms.

  1. Find the constant change: +3.
  2. Add the same amount each time.

Very beginner explanation: A constant-rate numerical pattern changes by the same amount for each step.

Answer: 11, 14

Worked Example 7

Problem: Continue the pattern 1, 5, 9, ... for two more terms.

  1. Find the constant change: +4.
  2. Add the same amount each time.

Very beginner explanation: A constant-rate numerical pattern changes by the same amount for each step.

Answer: 13, 17

Worked Example 8

Problem: Continue the pattern 5, 13, 21, ... for two more terms.

  1. Find the constant change: +8.
  2. Add the same amount each time.

Very beginner explanation: A constant-rate numerical pattern changes by the same amount for each step.

Answer: 29, 37

Worked Example 9

Problem: Continue the pattern 0, 8, 16, ... for two more terms.

  1. Find the constant change: +8.
  2. Add the same amount each time.

Very beginner explanation: A constant-rate numerical pattern changes by the same amount for each step.

Answer: 24, 32

Worked Example 10

Problem: Continue the pattern 12, 19, 26, ... for two more terms.

  1. Find the constant change: +7.
  2. Add the same amount each time.

Very beginner explanation: A constant-rate numerical pattern changes by the same amount for each step.

Answer: 33, 40

Practice Exercise

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

Common Mistake

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

21.4 Ordered pairs

Ordered pairs is a Grade 6 mathematics idea in Linear Patterns and Graphing. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable. In this Grade 6 lesson, focus on meaning first, then use a small example, show the steps, and check whether the result fits the question.

Beginner Note

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

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Does the point (2, 3) satisfy x + y ≤ 6?

  1. Substitute the x- and y-values into the inequality.
  2. Evaluate both sides.
  3. Decide whether the statement is true.

Very beginner explanation: Testing an ordered pair tells whether that point belongs to the solution set.

Answer: Yes

Worked Example 2

Problem: Does the point (4, 4) satisfy x + y ≤ 6?

  1. Substitute the x- and y-values into the inequality.
  2. Evaluate both sides.
  3. Decide whether the statement is true.

Very beginner explanation: Testing an ordered pair tells whether that point belongs to the solution set.

Answer: No

Worked Example 3

Problem: Does the point (1, 5) satisfy x + y < 8?

  1. Substitute the x- and y-values into the inequality.
  2. Evaluate both sides.
  3. Decide whether the statement is true.

Very beginner explanation: Testing an ordered pair tells whether that point belongs to the solution set.

Answer: Yes

Worked Example 4

Problem: Does the point (3, 6) satisfy y ≥ 5?

  1. Substitute the x- and y-values into the inequality.
  2. Evaluate both sides.
  3. Decide whether the statement is true.

Very beginner explanation: Testing an ordered pair tells whether that point belongs to the solution set.

Answer: Yes

Worked Example 5

Problem: Does the point (7, 2) satisfy x < 5?

  1. Substitute the x- and y-values into the inequality.
  2. Evaluate both sides.
  3. Decide whether the statement is true.

Very beginner explanation: Testing an ordered pair tells whether that point belongs to the solution set.

Answer: No

Worked Example 6

Problem: Does the point (0, 0) satisfy x+y≥0?

  1. Substitute the x- and y-values into the inequality.
  2. Evaluate both sides.
  3. Decide whether the statement is true.

Very beginner explanation: Testing an ordered pair tells whether that point belongs to the solution set.

Answer: Yes

Worked Example 7

Problem: Does the point (-1, 4) satisfy y>2?

  1. Substitute the x- and y-values into the inequality.
  2. Evaluate both sides.
  3. Decide whether the statement is true.

Very beginner explanation: Testing an ordered pair tells whether that point belongs to the solution set.

Answer: Yes

Worked Example 8

Problem: Does the point (5, 1) satisfy x≤5?

  1. Substitute the x- and y-values into the inequality.
  2. Evaluate both sides.
  3. Decide whether the statement is true.

Very beginner explanation: Testing an ordered pair tells whether that point belongs to the solution set.

Answer: Yes

Worked Example 9

Problem: Does the point (2, 2) satisfy x+y>5?

  1. Substitute the x- and y-values into the inequality.
  2. Evaluate both sides.
  3. Decide whether the statement is true.

Very beginner explanation: Testing an ordered pair tells whether that point belongs to the solution set.

Answer: No

Worked Example 10

Problem: Does the point (6, 3) satisfy x-y≥2?

  1. Substitute the x- and y-values into the inequality.
  2. Evaluate both sides.
  3. Decide whether the statement is true.

Very beginner explanation: Testing an ordered pair tells whether that point belongs to the solution set.

Answer: Yes

Practice Exercise

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

Common Mistake

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

21.5 Plotting pattern points

A pattern follows a rule that can be described, extended, and used to make predictions. In this Grade 6 lesson, focus on meaning first, then use a small example, show the steps, and check whether the result fits the question.

Beginner Note

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

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Continue the pattern 8, 16, 24, ... for two more terms.

  1. Find the constant change: +8.
  2. Add the same amount each time.

Very beginner explanation: A constant-rate numerical pattern changes by the same amount for each step.

Answer: 32, 40

Worked Example 2

Problem: Continue the pattern 7, 13, 19, ... for two more terms.

  1. Find the constant change: +6.
  2. Add the same amount each time.

Very beginner explanation: A constant-rate numerical pattern changes by the same amount for each step.

Answer: 25, 31

Worked Example 3

Problem: Continue the pattern -4, 5, 14, ... for two more terms.

  1. Find the constant change: +9.
  2. Add the same amount each time.

Very beginner explanation: A constant-rate numerical pattern changes by the same amount for each step.

Answer: 23, 32

Worked Example 4

Problem: Continue the pattern 5, 11, 17, ... for two more terms.

  1. Find the constant change: +6.
  2. Add the same amount each time.

Very beginner explanation: A constant-rate numerical pattern changes by the same amount for each step.

Answer: 23, 29

Worked Example 5

Problem: Continue the pattern -1, 4, 9, ... for two more terms.

  1. Find the constant change: +5.
  2. Add the same amount each time.

Very beginner explanation: A constant-rate numerical pattern changes by the same amount for each step.

Answer: 14, 19

Worked Example 6

Problem: Continue the pattern 15, 20, 25, ... for two more terms.

  1. Find the constant change: +5.
  2. Add the same amount each time.

Very beginner explanation: A constant-rate numerical pattern changes by the same amount for each step.

Answer: 30, 35

Worked Example 7

Problem: Continue the pattern 9, 15, 21, ... for two more terms.

  1. Find the constant change: +6.
  2. Add the same amount each time.

Very beginner explanation: A constant-rate numerical pattern changes by the same amount for each step.

Answer: 27, 33

Worked Example 8

Problem: Continue the pattern -3, 4, 11, ... for two more terms.

  1. Find the constant change: +7.
  2. Add the same amount each time.

Very beginner explanation: A constant-rate numerical pattern changes by the same amount for each step.

Answer: 18, 25

Worked Example 9

Problem: Continue the pattern 5, 12, 19, ... for two more terms.

  1. Find the constant change: +7.
  2. Add the same amount each time.

Very beginner explanation: A constant-rate numerical pattern changes by the same amount for each step.

Answer: 26, 33

Worked Example 10

Problem: Continue the pattern -3, 3, 9, ... for two more terms.

  1. Find the constant change: +6.
  2. Add the same amount each time.

Very beginner explanation: A constant-rate numerical pattern changes by the same amount for each step.

Answer: 15, 21

Practice Exercise

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

Common Mistake

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

21.6 Straight-line patterns

A pattern follows a rule that can be described, extended, and used to make predictions. In this Grade 6 lesson, focus on meaning first, then use a small example, show the steps, and check whether the result fits the question.

Beginner Note

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

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Continue the pattern 12, 14, 16, ... for two more terms.

  1. Find the constant change: +2.
  2. Add the same amount each time.

Very beginner explanation: A constant-rate numerical pattern changes by the same amount for each step.

Answer: 18, 20

Worked Example 2

Problem: Continue the pattern -2, 1, 4, ... for two more terms.

  1. Find the constant change: +3.
  2. Add the same amount each time.

Very beginner explanation: A constant-rate numerical pattern changes by the same amount for each step.

Answer: 7, 10

Worked Example 3

Problem: Continue the pattern -5, -1, 3, ... for two more terms.

  1. Find the constant change: +4.
  2. Add the same amount each time.

Very beginner explanation: A constant-rate numerical pattern changes by the same amount for each step.

Answer: 7, 11

Worked Example 4

Problem: Continue the pattern 3, 8, 13, ... for two more terms.

  1. Find the constant change: +5.
  2. Add the same amount each time.

Very beginner explanation: A constant-rate numerical pattern changes by the same amount for each step.

Answer: 18, 23

Worked Example 5

Problem: Continue the pattern 3, 11, 19, ... for two more terms.

  1. Find the constant change: +8.
  2. Add the same amount each time.

Very beginner explanation: A constant-rate numerical pattern changes by the same amount for each step.

Answer: 27, 35

Worked Example 6

Problem: Continue the pattern 11, 20, 29, ... for two more terms.

  1. Find the constant change: +9.
  2. Add the same amount each time.

Very beginner explanation: A constant-rate numerical pattern changes by the same amount for each step.

Answer: 38, 47

Worked Example 7

Problem: Continue the pattern 12, 21, 30, ... for two more terms.

  1. Find the constant change: +9.
  2. Add the same amount each time.

Very beginner explanation: A constant-rate numerical pattern changes by the same amount for each step.

Answer: 39, 48

Worked Example 8

Problem: Continue the pattern -5, 4, 13, ... for two more terms.

  1. Find the constant change: +9.
  2. Add the same amount each time.

Very beginner explanation: A constant-rate numerical pattern changes by the same amount for each step.

Answer: 22, 31

Worked Example 9

Problem: Continue the pattern -5, 1, 7, ... for two more terms.

  1. Find the constant change: +6.
  2. Add the same amount each time.

Very beginner explanation: A constant-rate numerical pattern changes by the same amount for each step.

Answer: 13, 19

Worked Example 10

Problem: Continue the pattern 7, 14, 21, ... for two more terms.

  1. Find the constant change: +7.
  2. Add the same amount each time.

Very beginner explanation: A constant-rate numerical pattern changes by the same amount for each step.

Answer: 28, 35

Practice Exercise

Create one new Grade 6 problem about Straight-line patterns. Solve it step by step, label any units, and explain how you checked your answer.

Common Mistake

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

21.7 Increasing linear patterns

A pattern follows a rule that can be described, extended, and used to make predictions. In this Grade 6 lesson, focus on meaning first, then use a small example, show the steps, and check whether the result fits the question.

Beginner Note

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

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Continue the pattern 6, 9, 12, ... for two more terms.

  1. Find the constant change: +3.
  2. Add the same amount each time.

Very beginner explanation: A constant-rate numerical pattern changes by the same amount for each step.

Answer: 15, 18

Worked Example 2

Problem: Continue the pattern 7, 14, 21, ... for two more terms.

  1. Find the constant change: +7.
  2. Add the same amount each time.

Very beginner explanation: A constant-rate numerical pattern changes by the same amount for each step.

Answer: 28, 35

Worked Example 3

Problem: Continue the pattern 8, 10, 12, ... for two more terms.

  1. Find the constant change: +2.
  2. Add the same amount each time.

Very beginner explanation: A constant-rate numerical pattern changes by the same amount for each step.

Answer: 14, 16

Worked Example 4

Problem: Continue the pattern 11, 16, 21, ... for two more terms.

  1. Find the constant change: +5.
  2. Add the same amount each time.

Very beginner explanation: A constant-rate numerical pattern changes by the same amount for each step.

Answer: 26, 31

Worked Example 5

Problem: Continue the pattern 2, 9, 16, ... for two more terms.

  1. Find the constant change: +7.
  2. Add the same amount each time.

Very beginner explanation: A constant-rate numerical pattern changes by the same amount for each step.

Answer: 23, 30

Worked Example 6

Problem: Continue the pattern 11, 17, 23, ... for two more terms.

  1. Find the constant change: +6.
  2. Add the same amount each time.

Very beginner explanation: A constant-rate numerical pattern changes by the same amount for each step.

Answer: 29, 35

Worked Example 7

Problem: Continue the pattern -1, 2, 5, ... for two more terms.

  1. Find the constant change: +3.
  2. Add the same amount each time.

Very beginner explanation: A constant-rate numerical pattern changes by the same amount for each step.

Answer: 8, 11

Worked Example 8

Problem: Continue the pattern -3, 0, 3, ... for two more terms.

  1. Find the constant change: +3.
  2. Add the same amount each time.

Very beginner explanation: A constant-rate numerical pattern changes by the same amount for each step.

Answer: 6, 9

Worked Example 9

Problem: Continue the pattern -5, 3, 11, ... for two more terms.

  1. Find the constant change: +8.
  2. Add the same amount each time.

Very beginner explanation: A constant-rate numerical pattern changes by the same amount for each step.

Answer: 19, 27

Worked Example 10

Problem: Continue the pattern -4, -1, 2, ... for two more terms.

  1. Find the constant change: +3.
  2. Add the same amount each time.

Very beginner explanation: A constant-rate numerical pattern changes by the same amount for each step.

Answer: 5, 8

Practice Exercise

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

Common Mistake

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

21.8 Decreasing linear patterns

A pattern follows a rule that can be described, extended, and used to make predictions. In this Grade 6 lesson, focus on meaning first, then use a small example, show the steps, and check whether the result fits the question.

Beginner Note

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

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Continue the pattern -3, 5, 13, ... for two more terms.

  1. Find the constant change: +8.
  2. Add the same amount each time.

Very beginner explanation: A constant-rate numerical pattern changes by the same amount for each step.

Answer: 21, 29

Worked Example 2

Problem: Continue the pattern -1, 4, 9, ... for two more terms.

  1. Find the constant change: +5.
  2. Add the same amount each time.

Very beginner explanation: A constant-rate numerical pattern changes by the same amount for each step.

Answer: 14, 19

Worked Example 3

Problem: Continue the pattern -3, 6, 15, ... for two more terms.

  1. Find the constant change: +9.
  2. Add the same amount each time.

Very beginner explanation: A constant-rate numerical pattern changes by the same amount for each step.

Answer: 24, 33

Worked Example 4

Problem: Continue the pattern 10, 12, 14, ... for two more terms.

  1. Find the constant change: +2.
  2. Add the same amount each time.

Very beginner explanation: A constant-rate numerical pattern changes by the same amount for each step.

Answer: 16, 18

Worked Example 5

Problem: Continue the pattern 5, 9, 13, ... for two more terms.

  1. Find the constant change: +4.
  2. Add the same amount each time.

Very beginner explanation: A constant-rate numerical pattern changes by the same amount for each step.

Answer: 17, 21

Worked Example 6

Problem: Continue the pattern 1, 6, 11, ... for two more terms.

  1. Find the constant change: +5.
  2. Add the same amount each time.

Very beginner explanation: A constant-rate numerical pattern changes by the same amount for each step.

Answer: 16, 21

Worked Example 7

Problem: Continue the pattern 1, 6, 11, ... for two more terms.

  1. Find the constant change: +5.
  2. Add the same amount each time.

Very beginner explanation: A constant-rate numerical pattern changes by the same amount for each step.

Answer: 16, 21

Worked Example 8

Problem: Continue the pattern 6, 14, 22, ... for two more terms.

  1. Find the constant change: +8.
  2. Add the same amount each time.

Very beginner explanation: A constant-rate numerical pattern changes by the same amount for each step.

Answer: 30, 38

Worked Example 9

Problem: Continue the pattern -5, 0, 5, ... for two more terms.

  1. Find the constant change: +5.
  2. Add the same amount each time.

Very beginner explanation: A constant-rate numerical pattern changes by the same amount for each step.

Answer: 10, 15

Worked Example 10

Problem: Continue the pattern -5, -1, 3, ... for two more terms.

  1. Find the constant change: +4.
  2. Add the same amount each time.

Very beginner explanation: A constant-rate numerical pattern changes by the same amount for each step.

Answer: 7, 11

Practice Exercise

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

Common Mistake

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

21.9 Rate of change introduction

A rate compares two quantities measured in different units. In this Grade 6 lesson, focus on meaning first, then use a small example, show the steps, and check whether the result fits the question.

Beginner Note

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

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: 80 units are used in 8 equal groups. Find the unit rate.

  1. Divide the first quantity by the second.
  2. 80 ÷ 8 = 10.

Very beginner explanation: A unit rate tells the amount for one unit of the second quantity.

Answer: 10 per group

Worked Example 2

Problem: 48 units are used in 8 equal groups. Find the unit rate.

  1. Divide the first quantity by the second.
  2. 48 ÷ 8 = 6.

Very beginner explanation: A unit rate tells the amount for one unit of the second quantity.

Answer: 6 per group

Worked Example 3

Problem: 112 units are used in 14 equal groups. Find the unit rate.

  1. Divide the first quantity by the second.
  2. 112 ÷ 14 = 8.

Very beginner explanation: A unit rate tells the amount for one unit of the second quantity.

Answer: 8 per group

Worked Example 4

Problem: 84 units are used in 14 equal groups. Find the unit rate.

  1. Divide the first quantity by the second.
  2. 84 ÷ 14 = 6.

Very beginner explanation: A unit rate tells the amount for one unit of the second quantity.

Answer: 6 per group

Worked Example 5

Problem: 135 units are used in 15 equal groups. Find the unit rate.

  1. Divide the first quantity by the second.
  2. 135 ÷ 15 = 9.

Very beginner explanation: A unit rate tells the amount for one unit of the second quantity.

Answer: 9 per group

Worked Example 6

Problem: 60 units are used in 15 equal groups. Find the unit rate.

  1. Divide the first quantity by the second.
  2. 60 ÷ 15 = 4.

Very beginner explanation: A unit rate tells the amount for one unit of the second quantity.

Answer: 4 per group

Worked Example 7

Problem: 28 units are used in 4 equal groups. Find the unit rate.

  1. Divide the first quantity by the second.
  2. 28 ÷ 4 = 7.

Very beginner explanation: A unit rate tells the amount for one unit of the second quantity.

Answer: 7 per group

Worked Example 8

Problem: 40 units are used in 4 equal groups. Find the unit rate.

  1. Divide the first quantity by the second.
  2. 40 ÷ 4 = 10.

Very beginner explanation: A unit rate tells the amount for one unit of the second quantity.

Answer: 10 per group

Worked Example 9

Problem: 30 units are used in 3 equal groups. Find the unit rate.

  1. Divide the first quantity by the second.
  2. 30 ÷ 3 = 10.

Very beginner explanation: A unit rate tells the amount for one unit of the second quantity.

Answer: 10 per group

Worked Example 10

Problem: 18 units are used in 3 equal groups. Find the unit rate.

  1. Divide the first quantity by the second.
  2. 18 ÷ 3 = 6.

Very beginner explanation: A unit rate tells the amount for one unit of the second quantity.

Answer: 6 per group

Practice Exercise

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

Common Mistake

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

21.10 Starting value introduction

Starting value introduction is a Grade 6 mathematics idea in Linear Patterns and Graphing. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable. In this Grade 6 lesson, focus on meaning first, then use a small example, show the steps, and check whether the result fits the question.

Beginner Note

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

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Use Starting value introduction to analyze the values 5 and 7. What should be checked first?

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

Very beginner explanation: Starting value introduction is a Grade 6 mathematics idea in Linear Patterns and Graphing. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

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

Worked Example 2

Problem: Use Starting value introduction to analyze the values 16 and 19. What should be checked first?

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

Very beginner explanation: Starting value introduction is a Grade 6 mathematics idea in Linear Patterns and Graphing. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

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

Worked Example 3

Problem: Use Starting value introduction to analyze the values 25 and 10. What should be checked first?

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

Very beginner explanation: Starting value introduction is a Grade 6 mathematics idea in Linear Patterns and Graphing. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

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

Worked Example 4

Problem: Use Starting value introduction to analyze the values 6 and 10. What should be checked first?

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

Very beginner explanation: Starting value introduction is a Grade 6 mathematics idea in Linear Patterns and Graphing. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

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

Worked Example 5

Problem: Use Starting value introduction to analyze the values 6 and 15. What should be checked first?

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

Very beginner explanation: Starting value introduction is a Grade 6 mathematics idea in Linear Patterns and Graphing. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

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

Worked Example 6

Problem: Use Starting value introduction to analyze the values 3 and 19. What should be checked first?

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

Very beginner explanation: Starting value introduction is a Grade 6 mathematics idea in Linear Patterns and Graphing. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

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

Worked Example 7

Problem: Use Starting value introduction to analyze the values 29 and 12. What should be checked first?

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

Very beginner explanation: Starting value introduction is a Grade 6 mathematics idea in Linear Patterns and Graphing. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

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

Worked Example 8

Problem: Use Starting value introduction to analyze the values 27 and 12. What should be checked first?

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

Very beginner explanation: Starting value introduction is a Grade 6 mathematics idea in Linear Patterns and Graphing. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

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

Worked Example 9

Problem: Use Starting value introduction to analyze the values 12 and 19. What should be checked first?

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

Very beginner explanation: Starting value introduction is a Grade 6 mathematics idea in Linear Patterns and Graphing. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

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

Worked Example 10

Problem: Use Starting value introduction to analyze the values 10 and 17. What should be checked first?

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

Very beginner explanation: Starting value introduction is a Grade 6 mathematics idea in Linear Patterns and Graphing. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

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

Practice Exercise

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

Common Mistake

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

21.11 Reading values from graphs

Reading values from graphs is a Grade 6 mathematics idea in Linear Patterns and Graphing. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable. In this Grade 6 lesson, focus on meaning first, then use a small example, show the steps, and check whether the result fits the question.

Beginner Note

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

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Use Reading values from graphs to analyze the values 18 and 19. What should be checked first?

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

Very beginner explanation: Reading values from graphs is a Grade 6 mathematics idea in Linear Patterns and Graphing. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

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

Worked Example 2

Problem: Use Reading values from graphs to analyze the values 10 and 5. What should be checked first?

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

Very beginner explanation: Reading values from graphs is a Grade 6 mathematics idea in Linear Patterns and Graphing. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

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

Worked Example 3

Problem: Use Reading values from graphs to analyze the values 24 and 10. What should be checked first?

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

Very beginner explanation: Reading values from graphs is a Grade 6 mathematics idea in Linear Patterns and Graphing. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

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

Worked Example 4

Problem: Use Reading values from graphs to analyze the values 7 and 20. What should be checked first?

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

Very beginner explanation: Reading values from graphs is a Grade 6 mathematics idea in Linear Patterns and Graphing. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

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

Worked Example 5

Problem: Use Reading values from graphs to analyze the values 2 and 14. What should be checked first?

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

Very beginner explanation: Reading values from graphs is a Grade 6 mathematics idea in Linear Patterns and Graphing. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

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

Worked Example 6

Problem: Use Reading values from graphs to analyze the values 22 and 19. What should be checked first?

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

Very beginner explanation: Reading values from graphs is a Grade 6 mathematics idea in Linear Patterns and Graphing. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

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

Worked Example 7

Problem: Use Reading values from graphs to analyze the values 9 and 13. What should be checked first?

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

Very beginner explanation: Reading values from graphs is a Grade 6 mathematics idea in Linear Patterns and Graphing. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

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

Worked Example 8

Problem: Use Reading values from graphs to analyze the values 30 and 3. What should be checked first?

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

Very beginner explanation: Reading values from graphs is a Grade 6 mathematics idea in Linear Patterns and Graphing. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

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

Worked Example 9

Problem: Use Reading values from graphs to analyze the values 13 and 6. What should be checked first?

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

Very beginner explanation: Reading values from graphs is a Grade 6 mathematics idea in Linear Patterns and Graphing. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

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

Worked Example 10

Problem: Use Reading values from graphs to analyze the values 30 and 6. What should be checked first?

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

Very beginner explanation: Reading values from graphs is a Grade 6 mathematics idea in Linear Patterns and Graphing. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

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

Practice Exercise

Create one new Grade 6 problem about Reading values from graphs. Solve it step by step, label any units, and explain how you checked your answer.

Common Mistake

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

21.12 Creating graphs from tables

Creating graphs from tables is a Grade 6 mathematics idea in Linear Patterns and Graphing. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable. In this Grade 6 lesson, focus on meaning first, then use a small example, show the steps, and check whether the result fits the question.

Beginner Note

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

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Use Creating graphs from tables to analyze the values 25 and 7. What should be checked first?

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

Very beginner explanation: Creating graphs from tables is a Grade 6 mathematics idea in Linear Patterns and Graphing. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

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

Worked Example 2

Problem: Use Creating graphs from tables to analyze the values 3 and 4. What should be checked first?

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

Very beginner explanation: Creating graphs from tables is a Grade 6 mathematics idea in Linear Patterns and Graphing. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

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

Worked Example 3

Problem: Use Creating graphs from tables to analyze the values 30 and 9. What should be checked first?

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

Very beginner explanation: Creating graphs from tables is a Grade 6 mathematics idea in Linear Patterns and Graphing. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

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

Worked Example 4

Problem: Use Creating graphs from tables to analyze the values 27 and 12. What should be checked first?

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

Very beginner explanation: Creating graphs from tables is a Grade 6 mathematics idea in Linear Patterns and Graphing. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

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

Worked Example 5

Problem: Use Creating graphs from tables to analyze the values 17 and 15. What should be checked first?

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

Very beginner explanation: Creating graphs from tables is a Grade 6 mathematics idea in Linear Patterns and Graphing. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

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

Worked Example 6

Problem: Use Creating graphs from tables to analyze the values 2 and 3. What should be checked first?

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

Very beginner explanation: Creating graphs from tables is a Grade 6 mathematics idea in Linear Patterns and Graphing. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

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

Worked Example 7

Problem: Use Creating graphs from tables to analyze the values 27 and 6. What should be checked first?

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

Very beginner explanation: Creating graphs from tables is a Grade 6 mathematics idea in Linear Patterns and Graphing. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

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

Worked Example 8

Problem: Use Creating graphs from tables to analyze the values 24 and 3. What should be checked first?

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

Very beginner explanation: Creating graphs from tables is a Grade 6 mathematics idea in Linear Patterns and Graphing. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

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

Worked Example 9

Problem: Use Creating graphs from tables to analyze the values 30 and 3. What should be checked first?

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

Very beginner explanation: Creating graphs from tables is a Grade 6 mathematics idea in Linear Patterns and Graphing. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

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

Worked Example 10

Problem: Use Creating graphs from tables to analyze the values 16 and 13. What should be checked first?

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

Very beginner explanation: Creating graphs from tables is a Grade 6 mathematics idea in Linear Patterns and Graphing. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

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

Practice Exercise

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

Common Mistake

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

21.13 Interpreting linear graphs

Interpreting linear graphs is a Grade 6 mathematics idea in Linear Patterns and Graphing. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable. In this Grade 6 lesson, focus on meaning first, then use a small example, show the steps, and check whether the result fits the question.

Beginner Note

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

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Continue the pattern 4, 13, 22, ... for two more terms.

  1. Find the constant change: +9.
  2. Add the same amount each time.

Very beginner explanation: A constant-rate numerical pattern changes by the same amount for each step.

Answer: 31, 40

Worked Example 2

Problem: Continue the pattern -3, 6, 15, ... for two more terms.

  1. Find the constant change: +9.
  2. Add the same amount each time.

Very beginner explanation: A constant-rate numerical pattern changes by the same amount for each step.

Answer: 24, 33

Worked Example 3

Problem: Continue the pattern 1, 4, 7, ... for two more terms.

  1. Find the constant change: +3.
  2. Add the same amount each time.

Very beginner explanation: A constant-rate numerical pattern changes by the same amount for each step.

Answer: 10, 13

Worked Example 4

Problem: Continue the pattern 12, 14, 16, ... for two more terms.

  1. Find the constant change: +2.
  2. Add the same amount each time.

Very beginner explanation: A constant-rate numerical pattern changes by the same amount for each step.

Answer: 18, 20

Worked Example 5

Problem: Continue the pattern -4, -1, 2, ... for two more terms.

  1. Find the constant change: +3.
  2. Add the same amount each time.

Very beginner explanation: A constant-rate numerical pattern changes by the same amount for each step.

Answer: 5, 8

Worked Example 6

Problem: Continue the pattern 6, 13, 20, ... for two more terms.

  1. Find the constant change: +7.
  2. Add the same amount each time.

Very beginner explanation: A constant-rate numerical pattern changes by the same amount for each step.

Answer: 27, 34

Worked Example 7

Problem: Continue the pattern 7, 12, 17, ... for two more terms.

  1. Find the constant change: +5.
  2. Add the same amount each time.

Very beginner explanation: A constant-rate numerical pattern changes by the same amount for each step.

Answer: 22, 27

Worked Example 8

Problem: Continue the pattern -2, 6, 14, ... for two more terms.

  1. Find the constant change: +8.
  2. Add the same amount each time.

Very beginner explanation: A constant-rate numerical pattern changes by the same amount for each step.

Answer: 22, 30

Worked Example 9

Problem: Continue the pattern 6, 13, 20, ... for two more terms.

  1. Find the constant change: +7.
  2. Add the same amount each time.

Very beginner explanation: A constant-rate numerical pattern changes by the same amount for each step.

Answer: 27, 34

Worked Example 10

Problem: Continue the pattern 15, 24, 33, ... for two more terms.

  1. Find the constant change: +9.
  2. Add the same amount each time.

Very beginner explanation: A constant-rate numerical pattern changes by the same amount for each step.

Answer: 42, 51

Practice Exercise

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

Common Mistake

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

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30 Review Questions and Answers

Q1. What is important to understand about Linear pattern meaning?

Answer: A pattern follows a rule that can be described, extended, and used to make predictions.

Q2. What is important to understand about Input and output?

Answer: Input and output is a Grade 6 mathematics idea in Linear Patterns and Graphing. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

Q3. What is important to understand about Tables of values?

Answer: Tables of values is a Grade 6 mathematics idea in Linear Patterns and Graphing. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

Q4. What is important to understand about Ordered pairs?

Answer: Ordered pairs is a Grade 6 mathematics idea in Linear Patterns and Graphing. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

Q5. What is important to understand about Plotting pattern points?

Answer: A pattern follows a rule that can be described, extended, and used to make predictions.

Q6. What is important to understand about Straight-line patterns?

Answer: A pattern follows a rule that can be described, extended, and used to make predictions.

Q7. What is important to understand about Increasing linear patterns?

Answer: A pattern follows a rule that can be described, extended, and used to make predictions.

Q8. What is important to understand about Decreasing linear patterns?

Answer: A pattern follows a rule that can be described, extended, and used to make predictions.

Q9. What is important to understand about Rate of change introduction?

Answer: A rate compares two quantities measured in different units.

Q10. What is important to understand about Starting value introduction?

Answer: Starting value introduction is a Grade 6 mathematics idea in Linear Patterns and Graphing. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

Q11. What is important to understand about Reading values from graphs?

Answer: Reading values from graphs is a Grade 6 mathematics idea in Linear Patterns and Graphing. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

Q12. What is important to understand about Creating graphs from tables?

Answer: Creating graphs from tables is a Grade 6 mathematics idea in Linear Patterns and Graphing. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

Q13. What is important to understand about Interpreting linear graphs?

Answer: Interpreting linear graphs is a Grade 6 mathematics idea in Linear Patterns and Graphing. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

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 help 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, a table, 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 and check copied values, operations, signs, units, rules, and calculations.

Q22. Why are diagrams useful?

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

Q23. Why are tables useful?

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

Q24. Why are graphs useful?

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

Q25. Why should you explain your reasoning?

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

Q26. Why can more than one strategy be correct?

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

Q27. What should you do before using a formula or rule?

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

Q28. How does practice help in mathematics?

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

Q29. What is a good way to learn from a mistake?

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

Q30. Why should you compare an exact answer with an estimate?

Answer: The estimate gives a quick reasonableness check for the exact calculation.