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Chapter 34: Optimization with Code

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 Optimization with Code in original, simple language with step-by-step reasoning, worked examples, practice exercises, common mistakes, and review questions.

Key Technical Terms

  • Meaning of optimization (Optimization means finding the best possible result while following given limits or conditions.)
  • Finding a maximum (Finding a maximum is a Grade 6 mathematics idea in Optimization with Code. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.)
  • Finding a minimum (Finding a minimum is a Grade 6 mathematics idea in Optimization with Code. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.)
  • Perimeter constraints (Perimeter is the total distance around a two-dimensional shape.)
  • Trying possible values systematically (Trying possible values systematically is a Grade 6 mathematics idea in Optimization with Code. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.)
  • Using a table to optimize (Using a table to optimize is a Grade 6 mathematics idea in Optimization with Code. 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.

34.1 Meaning of optimization

Optimization means finding the best possible result while following given limits or conditions. 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: A rectangle has perimeter 24 units. Test length 7 and width 5. What area does this candidate give?

  1. Check perimeter: 2(7 + 5) = 24.
  2. Calculate area: 7 × 5.

Very beginner explanation: Optimization compares valid choices and looks for the best result, such as the greatest area.

Answer: 35 square units

Worked Example 2

Problem: A rectangle has perimeter 26 units. Test length 7 and width 6. What area does this candidate give?

  1. Check perimeter: 2(7 + 6) = 26.
  2. Calculate area: 7 × 6.

Very beginner explanation: Optimization compares valid choices and looks for the best result, such as the greatest area.

Answer: 42 square units

Worked Example 3

Problem: A rectangle has perimeter 28 units. Test length 6 and width 8. What area does this candidate give?

  1. Check perimeter: 2(6 + 8) = 28.
  2. Calculate area: 6 × 8.

Very beginner explanation: Optimization compares valid choices and looks for the best result, such as the greatest area.

Answer: 48 square units

Worked Example 4

Problem: A rectangle has perimeter 30 units. Test length 9 and width 6. What area does this candidate give?

  1. Check perimeter: 2(9 + 6) = 30.
  2. Calculate area: 9 × 6.

Very beginner explanation: Optimization compares valid choices and looks for the best result, such as the greatest area.

Answer: 54 square units

Worked Example 5

Problem: A rectangle has perimeter 32 units. Test length 8 and width 8. What area does this candidate give?

  1. Check perimeter: 2(8 + 8) = 32.
  2. Calculate area: 8 × 8.

Very beginner explanation: Optimization compares valid choices and looks for the best result, such as the greatest area.

Answer: 64 square units

Worked Example 6

Problem: A rectangle has perimeter 34 units. Test length 8 and width 9. What area does this candidate give?

  1. Check perimeter: 2(8 + 9) = 34.
  2. Calculate area: 8 × 9.

Very beginner explanation: Optimization compares valid choices and looks for the best result, such as the greatest area.

Answer: 72 square units

Worked Example 7

Problem: A rectangle has perimeter 36 units. Test length 10 and width 8. What area does this candidate give?

  1. Check perimeter: 2(10 + 8) = 36.
  2. Calculate area: 10 × 8.

Very beginner explanation: Optimization compares valid choices and looks for the best result, such as the greatest area.

Answer: 80 square units

Worked Example 8

Problem: A rectangle has perimeter 38 units. Test length 10 and width 9. What area does this candidate give?

  1. Check perimeter: 2(10 + 9) = 38.
  2. Calculate area: 10 × 9.

Very beginner explanation: Optimization compares valid choices and looks for the best result, such as the greatest area.

Answer: 90 square units

Worked Example 9

Problem: A rectangle has perimeter 40 units. Test length 9 and width 11. What area does this candidate give?

  1. Check perimeter: 2(9 + 11) = 40.
  2. Calculate area: 9 × 11.

Very beginner explanation: Optimization compares valid choices and looks for the best result, such as the greatest area.

Answer: 99 square units

Worked Example 10

Problem: A rectangle has perimeter 42 units. Test length 12 and width 9. What area does this candidate give?

  1. Check perimeter: 2(12 + 9) = 42.
  2. Calculate area: 12 × 9.

Very beginner explanation: Optimization compares valid choices and looks for the best result, such as the greatest area.

Answer: 108 square units

Practice Exercise

Create one new Grade 6 problem about Meaning of optimization. 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.

34.2 Finding a maximum

Finding a maximum is a Grade 6 mathematics idea in Optimization with Code. 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 Finding a maximum to analyze the values 21 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: Finding a maximum is a Grade 6 mathematics idea in Optimization with Code. 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 Finding a maximum to analyze the values 14 and 11. 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: Finding a maximum is a Grade 6 mathematics idea in Optimization with Code. 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 Finding a maximum to analyze the values 26 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: Finding a maximum is a Grade 6 mathematics idea in Optimization with Code. 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 Finding a maximum to analyze the values 4 and 16. 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: Finding a maximum is a Grade 6 mathematics idea in Optimization with Code. 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 Finding a maximum to analyze the values 28 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: Finding a maximum is a Grade 6 mathematics idea in Optimization with Code. 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 Finding a maximum to analyze the values 20 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: Finding a maximum is a Grade 6 mathematics idea in Optimization with Code. 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 Finding a maximum to analyze the values 14 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: Finding a maximum is a Grade 6 mathematics idea in Optimization with Code. 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 Finding a maximum to analyze the values 11 and 2. 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: Finding a maximum is a Grade 6 mathematics idea in Optimization with Code. 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 Finding a maximum to analyze the values 20 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: Finding a maximum is a Grade 6 mathematics idea in Optimization with Code. 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 Finding a maximum to analyze the values 24 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: Finding a maximum is a Grade 6 mathematics idea in Optimization with Code. 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 Finding a maximum. 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.

34.3 Finding a minimum

Finding a minimum is a Grade 6 mathematics idea in Optimization with Code. 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 Finding a minimum to analyze the values 27 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: Finding a minimum is a Grade 6 mathematics idea in Optimization with Code. 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 Finding a minimum to analyze the values 29 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: Finding a minimum is a Grade 6 mathematics idea in Optimization with Code. 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 Finding a minimum to analyze the values 21 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: Finding a minimum is a Grade 6 mathematics idea in Optimization with Code. 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 Finding a minimum to analyze the values 14 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: Finding a minimum is a Grade 6 mathematics idea in Optimization with Code. 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 Finding a minimum to analyze the values 19 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: Finding a minimum is a Grade 6 mathematics idea in Optimization with Code. 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 Finding a minimum to analyze the values 18 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: Finding a minimum is a Grade 6 mathematics idea in Optimization with Code. 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 Finding a minimum to analyze the values 22 and 8. 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: Finding a minimum is a Grade 6 mathematics idea in Optimization with Code. 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 Finding a minimum to analyze the values 4 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: Finding a minimum is a Grade 6 mathematics idea in Optimization with Code. 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 Finding a minimum to analyze the values 30 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: Finding a minimum is a Grade 6 mathematics idea in Optimization with Code. 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 Finding a minimum to analyze the values 14 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: Finding a minimum is a Grade 6 mathematics idea in Optimization with Code. 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 Finding a minimum. 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.

34.4 Perimeter constraints

Perimeter is the total distance around a two-dimensional shape. 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: Find the perimeter of a rectangle 5 cm by 2 cm.

  1. Use P = 2(l + w).
  2. P = 2(5+2).

Very beginner explanation: Perimeter is a length around a boundary, so it uses linear units.

Answer: 14 cm

Worked Example 2

Problem: Find the perimeter of a rectangle 6 cm by 3 cm.

  1. Use P = 2(l + w).
  2. P = 2(6+3).

Very beginner explanation: Perimeter is a length around a boundary, so it uses linear units.

Answer: 18 cm

Worked Example 3

Problem: Find the perimeter of a rectangle 7 cm by 4 cm.

  1. Use P = 2(l + w).
  2. P = 2(7+4).

Very beginner explanation: Perimeter is a length around a boundary, so it uses linear units.

Answer: 22 cm

Worked Example 4

Problem: Find the perimeter of a rectangle 8 cm by 5 cm.

  1. Use P = 2(l + w).
  2. P = 2(8+5).

Very beginner explanation: Perimeter is a length around a boundary, so it uses linear units.

Answer: 26 cm

Worked Example 5

Problem: Find the perimeter of a rectangle 9 cm by 6 cm.

  1. Use P = 2(l + w).
  2. P = 2(9+6).

Very beginner explanation: Perimeter is a length around a boundary, so it uses linear units.

Answer: 30 cm

Worked Example 6

Problem: Find the perimeter of a rectangle 10 cm by 7 cm.

  1. Use P = 2(l + w).
  2. P = 2(10+7).

Very beginner explanation: Perimeter is a length around a boundary, so it uses linear units.

Answer: 34 cm

Worked Example 7

Problem: Find the perimeter of a rectangle 11 cm by 8 cm.

  1. Use P = 2(l + w).
  2. P = 2(11+8).

Very beginner explanation: Perimeter is a length around a boundary, so it uses linear units.

Answer: 38 cm

Worked Example 8

Problem: Find the perimeter of a rectangle 12 cm by 9 cm.

  1. Use P = 2(l + w).
  2. P = 2(12+9).

Very beginner explanation: Perimeter is a length around a boundary, so it uses linear units.

Answer: 42 cm

Worked Example 9

Problem: Find the perimeter of a rectangle 13 cm by 10 cm.

  1. Use P = 2(l + w).
  2. P = 2(13+10).

Very beginner explanation: Perimeter is a length around a boundary, so it uses linear units.

Answer: 46 cm

Worked Example 10

Problem: Find the perimeter of a rectangle 14 cm by 11 cm.

  1. Use P = 2(l + w).
  2. P = 2(14+11).

Very beginner explanation: Perimeter is a length around a boundary, so it uses linear units.

Answer: 50 cm

Practice Exercise

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

Common Mistake

A common mistake is using the wrong formula or forgetting square units for area.

34.5 Maximum area for a given perimeter

Perimeter is the total distance around a two-dimensional shape. 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: Find the perimeter of a rectangle 5 cm by 2 cm.

  1. Use P = 2(l + w).
  2. P = 2(5+2).

Very beginner explanation: Perimeter is a length around a boundary, so it uses linear units.

Answer: 14 cm

Worked Example 2

Problem: Find the perimeter of a rectangle 6 cm by 3 cm.

  1. Use P = 2(l + w).
  2. P = 2(6+3).

Very beginner explanation: Perimeter is a length around a boundary, so it uses linear units.

Answer: 18 cm

Worked Example 3

Problem: Find the perimeter of a rectangle 7 cm by 4 cm.

  1. Use P = 2(l + w).
  2. P = 2(7+4).

Very beginner explanation: Perimeter is a length around a boundary, so it uses linear units.

Answer: 22 cm

Worked Example 4

Problem: Find the perimeter of a rectangle 8 cm by 5 cm.

  1. Use P = 2(l + w).
  2. P = 2(8+5).

Very beginner explanation: Perimeter is a length around a boundary, so it uses linear units.

Answer: 26 cm

Worked Example 5

Problem: Find the perimeter of a rectangle 9 cm by 6 cm.

  1. Use P = 2(l + w).
  2. P = 2(9+6).

Very beginner explanation: Perimeter is a length around a boundary, so it uses linear units.

Answer: 30 cm

Worked Example 6

Problem: Find the perimeter of a rectangle 10 cm by 7 cm.

  1. Use P = 2(l + w).
  2. P = 2(10+7).

Very beginner explanation: Perimeter is a length around a boundary, so it uses linear units.

Answer: 34 cm

Worked Example 7

Problem: Find the perimeter of a rectangle 11 cm by 8 cm.

  1. Use P = 2(l + w).
  2. P = 2(11+8).

Very beginner explanation: Perimeter is a length around a boundary, so it uses linear units.

Answer: 38 cm

Worked Example 8

Problem: Find the perimeter of a rectangle 12 cm by 9 cm.

  1. Use P = 2(l + w).
  2. P = 2(12+9).

Very beginner explanation: Perimeter is a length around a boundary, so it uses linear units.

Answer: 42 cm

Worked Example 9

Problem: Find the perimeter of a rectangle 13 cm by 10 cm.

  1. Use P = 2(l + w).
  2. P = 2(13+10).

Very beginner explanation: Perimeter is a length around a boundary, so it uses linear units.

Answer: 46 cm

Worked Example 10

Problem: Find the perimeter of a rectangle 14 cm by 11 cm.

  1. Use P = 2(l + w).
  2. P = 2(14+11).

Very beginner explanation: Perimeter is a length around a boundary, so it uses linear units.

Answer: 50 cm

Practice Exercise

Create one new Grade 6 problem about Maximum area for a given perimeter. Solve it step by step, label any units, and explain how you checked your answer.

Common Mistake

A common mistake is using the wrong formula or forgetting square units for area.

34.6 Trying possible values systematically

Trying possible values systematically is a Grade 6 mathematics idea in Optimization with Code. 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 Trying possible values systematically to analyze the values 8 and 18. 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: Trying possible values systematically is a Grade 6 mathematics idea in Optimization with Code. 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 Trying possible values systematically to analyze the values 16 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: Trying possible values systematically is a Grade 6 mathematics idea in Optimization with Code. 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 Trying possible values systematically to analyze the values 15 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: Trying possible values systematically is a Grade 6 mathematics idea in Optimization with Code. 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 Trying possible values systematically 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: Trying possible values systematically is a Grade 6 mathematics idea in Optimization with Code. 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 Trying possible values systematically to analyze the values 9 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: Trying possible values systematically is a Grade 6 mathematics idea in Optimization with Code. 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 Trying possible values systematically to analyze the values 4 and 2. 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: Trying possible values systematically is a Grade 6 mathematics idea in Optimization with Code. 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 Trying possible values systematically to analyze the values 10 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: Trying possible values systematically is a Grade 6 mathematics idea in Optimization with Code. 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 Trying possible values systematically 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: Trying possible values systematically is a Grade 6 mathematics idea in Optimization with Code. 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 Trying possible values systematically to analyze the values 18 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: Trying possible values systematically is a Grade 6 mathematics idea in Optimization with Code. 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 Trying possible values systematically to analyze the values 22 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: Trying possible values systematically is a Grade 6 mathematics idea in Optimization with Code. 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 Trying possible values systematically. 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.

34.7 Using a table to optimize

Using a table to optimize is a Grade 6 mathematics idea in Optimization with Code. 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 Using a table to optimize to analyze the values 5 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: Using a table to optimize is a Grade 6 mathematics idea in Optimization with Code. 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 Using a table to optimize to analyze the values 11 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: Using a table to optimize is a Grade 6 mathematics idea in Optimization with Code. 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 Using a table to optimize 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: Using a table to optimize is a Grade 6 mathematics idea in Optimization with Code. 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 Using a table to optimize to analyze the values 22 and 8. 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: Using a table to optimize is a Grade 6 mathematics idea in Optimization with Code. 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 Using a table to optimize to analyze the values 9 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: Using a table to optimize is a Grade 6 mathematics idea in Optimization with Code. 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 Using a table to optimize to analyze the values 19 and 11. 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: Using a table to optimize is a Grade 6 mathematics idea in Optimization with Code. 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 Using a table to optimize 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: Using a table to optimize is a Grade 6 mathematics idea in Optimization with Code. 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 Using a table to optimize to analyze the values 11 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: Using a table to optimize is a Grade 6 mathematics idea in Optimization with Code. 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 Using a table to optimize to analyze the values 9 and 8. 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: Using a table to optimize is a Grade 6 mathematics idea in Optimization with Code. 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 Using a table to optimize to analyze the values 16 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: Using a table to optimize is a Grade 6 mathematics idea in Optimization with Code. 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 Using a table to optimize. 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.

34.8 Using a loop to test possibilities

Using a loop to test possibilities is a Grade 6 mathematics idea in Optimization with Code. 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: A loop adds 4 exactly 6 times. What total is produced?

  1. Recognize repeated addition.
  2. Multiply the repeated amount by the number of repeats.

Very beginner explanation: Loops replace repeated instructions with one reusable rule.

Answer: 24

Worked Example 2

Problem: A loop adds 3 exactly 3 times. What total is produced?

  1. Recognize repeated addition.
  2. Multiply the repeated amount by the number of repeats.

Very beginner explanation: Loops replace repeated instructions with one reusable rule.

Answer: 9

Worked Example 3

Problem: A loop adds 2 exactly 6 times. What total is produced?

  1. Recognize repeated addition.
  2. Multiply the repeated amount by the number of repeats.

Very beginner explanation: Loops replace repeated instructions with one reusable rule.

Answer: 12

Worked Example 4

Problem: A loop adds 6 exactly 8 times. What total is produced?

  1. Recognize repeated addition.
  2. Multiply the repeated amount by the number of repeats.

Very beginner explanation: Loops replace repeated instructions with one reusable rule.

Answer: 48

Worked Example 5

Problem: A loop adds 2 exactly 8 times. What total is produced?

  1. Recognize repeated addition.
  2. Multiply the repeated amount by the number of repeats.

Very beginner explanation: Loops replace repeated instructions with one reusable rule.

Answer: 16

Worked Example 6

Problem: A loop adds 4 exactly 8 times. What total is produced?

  1. Recognize repeated addition.
  2. Multiply the repeated amount by the number of repeats.

Very beginner explanation: Loops replace repeated instructions with one reusable rule.

Answer: 32

Worked Example 7

Problem: A loop adds 5 exactly 5 times. What total is produced?

  1. Recognize repeated addition.
  2. Multiply the repeated amount by the number of repeats.

Very beginner explanation: Loops replace repeated instructions with one reusable rule.

Answer: 25

Worked Example 8

Problem: A loop adds 5 exactly 6 times. What total is produced?

  1. Recognize repeated addition.
  2. Multiply the repeated amount by the number of repeats.

Very beginner explanation: Loops replace repeated instructions with one reusable rule.

Answer: 30

Worked Example 9

Problem: A loop adds 5 exactly 7 times. What total is produced?

  1. Recognize repeated addition.
  2. Multiply the repeated amount by the number of repeats.

Very beginner explanation: Loops replace repeated instructions with one reusable rule.

Answer: 35

Worked Example 10

Problem: A loop adds 6 exactly 3 times. What total is produced?

  1. Recognize repeated addition.
  2. Multiply the repeated amount by the number of repeats.

Very beginner explanation: Loops replace repeated instructions with one reusable rule.

Answer: 18

Practice Exercise

Create one new Grade 6 problem about Using a loop to test possibilities. 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.

34.9 Comparing candidate solutions

Comparing candidate solutions is a Grade 6 mathematics idea in Optimization with Code. 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 Comparing candidate solutions to analyze the values 10 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: Comparing candidate solutions is a Grade 6 mathematics idea in Optimization with Code. 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 Comparing candidate solutions to analyze the values 21 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: Comparing candidate solutions is a Grade 6 mathematics idea in Optimization with Code. 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 Comparing candidate solutions to analyze the values 25 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: Comparing candidate solutions is a Grade 6 mathematics idea in Optimization with Code. 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 Comparing candidate solutions to analyze the values 9 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: Comparing candidate solutions is a Grade 6 mathematics idea in Optimization with Code. 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 Comparing candidate solutions to analyze the values 9 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: Comparing candidate solutions is a Grade 6 mathematics idea in Optimization with Code. 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 Comparing candidate solutions to analyze the values 4 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: Comparing candidate solutions is a Grade 6 mathematics idea in Optimization with Code. 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 Comparing candidate solutions to analyze the values 16 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: Comparing candidate solutions is a Grade 6 mathematics idea in Optimization with Code. 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 Comparing candidate solutions to analyze the values 26 and 8. 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: Comparing candidate solutions is a Grade 6 mathematics idea in Optimization with Code. 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 Comparing candidate solutions to analyze the values 27 and 18. 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: Comparing candidate solutions is a Grade 6 mathematics idea in Optimization with Code. 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 Comparing candidate solutions to analyze the values 22 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: Comparing candidate solutions is a Grade 6 mathematics idea in Optimization with Code. 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 Comparing candidate solutions. 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.

34.10 Checking constraints

Checking constraints is a Grade 6 mathematics idea in Optimization with Code. 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 Checking constraints to analyze the values 27 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: Checking constraints is a Grade 6 mathematics idea in Optimization with Code. 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 Checking constraints to analyze the values 30 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: Checking constraints is a Grade 6 mathematics idea in Optimization with Code. 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 Checking constraints to analyze the values 27 and 16. 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: Checking constraints is a Grade 6 mathematics idea in Optimization with Code. 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 Checking constraints 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: Checking constraints is a Grade 6 mathematics idea in Optimization with Code. 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 Checking constraints to analyze the values 15 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: Checking constraints is a Grade 6 mathematics idea in Optimization with Code. 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 Checking constraints to analyze the values 4 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: Checking constraints is a Grade 6 mathematics idea in Optimization with Code. 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 Checking constraints to analyze the values 7 and 11. 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: Checking constraints is a Grade 6 mathematics idea in Optimization with Code. 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 Checking constraints to analyze the values 4 and 16. 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: Checking constraints is a Grade 6 mathematics idea in Optimization with Code. 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 Checking constraints to analyze the values 12 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: Checking constraints is a Grade 6 mathematics idea in Optimization with Code. 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 Checking constraints to analyze the values 4 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: Checking constraints is a Grade 6 mathematics idea in Optimization with Code. 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 Checking constraints. 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.

34.11 Explaining why a solution is best

Explaining why a solution is best is a Grade 6 mathematics idea in Optimization with Code. 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 Explaining why a solution is best to analyze the values 11 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: Explaining why a solution is best is a Grade 6 mathematics idea in Optimization with Code. 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 Explaining why a solution is best to analyze the values 12 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: Explaining why a solution is best is a Grade 6 mathematics idea in Optimization with Code. 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 Explaining why a solution is best to analyze the values 25 and 18. 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: Explaining why a solution is best is a Grade 6 mathematics idea in Optimization with Code. 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 Explaining why a solution is best to analyze the values 3 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: Explaining why a solution is best is a Grade 6 mathematics idea in Optimization with Code. 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 Explaining why a solution is best to analyze the values 26 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: Explaining why a solution is best is a Grade 6 mathematics idea in Optimization with Code. 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 Explaining why a solution is best to analyze the values 11 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: Explaining why a solution is best is a Grade 6 mathematics idea in Optimization with Code. 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 Explaining why a solution is best to analyze the values 16 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: Explaining why a solution is best is a Grade 6 mathematics idea in Optimization with Code. 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 Explaining why a solution is best to analyze the values 11 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: Explaining why a solution is best is a Grade 6 mathematics idea in Optimization with Code. 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 Explaining why a solution is best to analyze the values 13 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: Explaining why a solution is best is a Grade 6 mathematics idea in Optimization with Code. 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 Explaining why a solution is best 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: Explaining why a solution is best is a Grade 6 mathematics idea in Optimization with Code. 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 Explaining why a solution is best. 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.

34.12 Real-life optimization problems

Optimization means finding the best possible result while following given limits or conditions. 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: A rectangle has perimeter 24 units. Test length 7 and width 5. What area does this candidate give?

  1. Check perimeter: 2(7 + 5) = 24.
  2. Calculate area: 7 × 5.

Very beginner explanation: Optimization compares valid choices and looks for the best result, such as the greatest area.

Answer: 35 square units

Worked Example 2

Problem: A rectangle has perimeter 26 units. Test length 7 and width 6. What area does this candidate give?

  1. Check perimeter: 2(7 + 6) = 26.
  2. Calculate area: 7 × 6.

Very beginner explanation: Optimization compares valid choices and looks for the best result, such as the greatest area.

Answer: 42 square units

Worked Example 3

Problem: A rectangle has perimeter 28 units. Test length 6 and width 8. What area does this candidate give?

  1. Check perimeter: 2(6 + 8) = 28.
  2. Calculate area: 6 × 8.

Very beginner explanation: Optimization compares valid choices and looks for the best result, such as the greatest area.

Answer: 48 square units

Worked Example 4

Problem: A rectangle has perimeter 30 units. Test length 9 and width 6. What area does this candidate give?

  1. Check perimeter: 2(9 + 6) = 30.
  2. Calculate area: 9 × 6.

Very beginner explanation: Optimization compares valid choices and looks for the best result, such as the greatest area.

Answer: 54 square units

Worked Example 5

Problem: A rectangle has perimeter 32 units. Test length 8 and width 8. What area does this candidate give?

  1. Check perimeter: 2(8 + 8) = 32.
  2. Calculate area: 8 × 8.

Very beginner explanation: Optimization compares valid choices and looks for the best result, such as the greatest area.

Answer: 64 square units

Worked Example 6

Problem: A rectangle has perimeter 34 units. Test length 8 and width 9. What area does this candidate give?

  1. Check perimeter: 2(8 + 9) = 34.
  2. Calculate area: 8 × 9.

Very beginner explanation: Optimization compares valid choices and looks for the best result, such as the greatest area.

Answer: 72 square units

Worked Example 7

Problem: A rectangle has perimeter 36 units. Test length 10 and width 8. What area does this candidate give?

  1. Check perimeter: 2(10 + 8) = 36.
  2. Calculate area: 10 × 8.

Very beginner explanation: Optimization compares valid choices and looks for the best result, such as the greatest area.

Answer: 80 square units

Worked Example 8

Problem: A rectangle has perimeter 38 units. Test length 10 and width 9. What area does this candidate give?

  1. Check perimeter: 2(10 + 9) = 38.
  2. Calculate area: 10 × 9.

Very beginner explanation: Optimization compares valid choices and looks for the best result, such as the greatest area.

Answer: 90 square units

Worked Example 9

Problem: A rectangle has perimeter 40 units. Test length 9 and width 11. What area does this candidate give?

  1. Check perimeter: 2(9 + 11) = 40.
  2. Calculate area: 9 × 11.

Very beginner explanation: Optimization compares valid choices and looks for the best result, such as the greatest area.

Answer: 99 square units

Worked Example 10

Problem: A rectangle has perimeter 42 units. Test length 12 and width 9. What area does this candidate give?

  1. Check perimeter: 2(12 + 9) = 42.
  2. Calculate area: 12 × 9.

Very beginner explanation: Optimization compares valid choices and looks for the best result, such as the greatest area.

Answer: 108 square units

Practice Exercise

Create one new Grade 6 problem about Real-life optimization problems. 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.

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30 Review Questions and Answers

Q1. What is important to understand about Meaning of optimization?

Answer: Optimization means finding the best possible result while following given limits or conditions.

Q2. What is important to understand about Finding a maximum?

Answer: Finding a maximum is a Grade 6 mathematics idea in Optimization with Code. 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 Finding a minimum?

Answer: Finding a minimum is a Grade 6 mathematics idea in Optimization with Code. 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 Perimeter constraints?

Answer: Perimeter is the total distance around a two-dimensional shape.

Q5. What is important to understand about Maximum area for a given perimeter?

Answer: Perimeter is the total distance around a two-dimensional shape.

Q6. What is important to understand about Trying possible values systematically?

Answer: Trying possible values systematically is a Grade 6 mathematics idea in Optimization with Code. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

Q7. What is important to understand about Using a table to optimize?

Answer: Using a table to optimize is a Grade 6 mathematics idea in Optimization with Code. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

Q8. What is important to understand about Using a loop to test possibilities?

Answer: Using a loop to test possibilities is a Grade 6 mathematics idea in Optimization with Code. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

Q9. What is important to understand about Comparing candidate solutions?

Answer: Comparing candidate solutions is a Grade 6 mathematics idea in Optimization with Code. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

Q10. What is important to understand about Checking constraints?

Answer: Checking constraints is a Grade 6 mathematics idea in Optimization with Code. 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 Explaining why a solution is best?

Answer: Explaining why a solution is best is a Grade 6 mathematics idea in Optimization with Code. 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 Real-life optimization problems?

Answer: Optimization means finding the best possible result while following given limits or conditions.

Q13. Why should you show your steps?

Answer: Showing steps makes reasoning easier to verify and helps locate mistakes.

Q14. Why is estimation useful?

Answer: Estimation helps you judge whether a final answer is reasonable before accepting it.

Q15. Why do units matter?

Answer: Units tell what a value measures and help prevent incompatible quantities from being mixed.

Q16. When should you round?

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

Q17. How can you check an answer?

Answer: Use an inverse operation, substitution, estimation, a graph, a table, or a second method.

Q18. What should you identify first in a word problem?

Answer: Identify what is known, what is unknown, and the relationship connecting them.

Q19. What makes a final answer complete?

Answer: Give the value, correct units when needed, and a brief interpretation in context.

Q20. What should you do if an answer seems unreasonable?

Answer: Re-read the question and check copied values, operations, signs, units, rules, and calculations.

Q21. Why are diagrams useful?

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

Q22. Why are tables useful?

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

Q23. Why are graphs useful?

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

Q24. Why should you explain your reasoning?

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

Q25. Why can more than one strategy be correct?

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

Q26. What should you do before using a formula or rule?

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

Q27. How does practice help in mathematics?

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

Q28. 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.

Q29. Why should you compare an exact answer with an estimate?

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

Q30. What does representing mathematics mean?

Answer: It means showing an idea with numbers, symbols, words, diagrams, tables, graphs, or models.