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Chapter 26: Order of Operations with Variables

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 Order of Operations with Variables in original, simple language with step-by-step reasoning, worked examples, practice exercises, common mistakes, and review questions.

Key Technical Terms

  • Variables and grouping symbols (A variable is a symbol used to represent a number that may change or may be unknown.)
  • Multiplication before addition in algebra (Multiplication before addition in algebra is a Grade 6 mathematics idea in Order of Operations with Variables. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.)
  • Substitution before calculation (Substitution before calculation is a Grade 6 mathematics idea in Order of Operations with Variables. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.)
  • Order of operations after substitution (A ratio compares two quantities by division.)
  • Expressions with decimals (A decimal uses place value to show parts of one, such as tenths, hundredths, and thousandths.)
  • Expressions with brackets (An algebraic expression combines numbers, variables, and operations without an equals sign.)

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.

26.1 Variables and grouping symbols

A variable is a symbol used to represent a number that may change or may be unknown. 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: Evaluate 2 + 4 × 9.

  1. Do multiplication before addition.
  2. 4 × 9 = 36.
  3. Then add 2.

Very beginner explanation: Order of operations makes sure everyone interprets the same expression consistently.

Answer: 38

Worked Example 2

Problem: Evaluate 2 + 9 × 8.

  1. Do multiplication before addition.
  2. 9 × 8 = 72.
  3. Then add 2.

Very beginner explanation: Order of operations makes sure everyone interprets the same expression consistently.

Answer: 74

Worked Example 3

Problem: Evaluate 4 + 3 × 6.

  1. Do multiplication before addition.
  2. 3 × 6 = 18.
  3. Then add 4.

Very beginner explanation: Order of operations makes sure everyone interprets the same expression consistently.

Answer: 22

Worked Example 4

Problem: Evaluate 3 + 7 × 5.

  1. Do multiplication before addition.
  2. 7 × 5 = 35.
  3. Then add 3.

Very beginner explanation: Order of operations makes sure everyone interprets the same expression consistently.

Answer: 38

Worked Example 5

Problem: Evaluate 6 + 3 × 6.

  1. Do multiplication before addition.
  2. 3 × 6 = 18.
  3. Then add 6.

Very beginner explanation: Order of operations makes sure everyone interprets the same expression consistently.

Answer: 24

Worked Example 6

Problem: Evaluate 5 + 9 × 7.

  1. Do multiplication before addition.
  2. 9 × 7 = 63.
  3. Then add 5.

Very beginner explanation: Order of operations makes sure everyone interprets the same expression consistently.

Answer: 68

Worked Example 7

Problem: Evaluate 2 + 5 × 7.

  1. Do multiplication before addition.
  2. 5 × 7 = 35.
  3. Then add 2.

Very beginner explanation: Order of operations makes sure everyone interprets the same expression consistently.

Answer: 37

Worked Example 8

Problem: Evaluate 6 + 7 × 3.

  1. Do multiplication before addition.
  2. 7 × 3 = 21.
  3. Then add 6.

Very beginner explanation: Order of operations makes sure everyone interprets the same expression consistently.

Answer: 27

Worked Example 9

Problem: Evaluate 11 + 9 × 3.

  1. Do multiplication before addition.
  2. 9 × 3 = 27.
  3. Then add 11.

Very beginner explanation: Order of operations makes sure everyone interprets the same expression consistently.

Answer: 38

Worked Example 10

Problem: Evaluate 7 + 7 × 6.

  1. Do multiplication before addition.
  2. 7 × 6 = 42.
  3. Then add 7.

Very beginner explanation: Order of operations makes sure everyone interprets the same expression consistently.

Answer: 49

Practice Exercise

Create one new Grade 6 problem about Variables and grouping symbols. 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.

26.2 Multiplication before addition in algebra

Multiplication before addition in algebra is a Grade 6 mathematics idea in Order of Operations with Variables. 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: Find 1% of 80.

  1. Think of 1% as 1/100.
  2. Multiply the quantity by the percent as a decimal or fraction.

Very beginner explanation: Percent means per hundred, so benchmark percents can often be found mentally.

Answer: 0.8

Worked Example 2

Problem: Find 5% of 120.

  1. Think of 5% as 5/100.
  2. Multiply the quantity by the percent as a decimal or fraction.

Very beginner explanation: Percent means per hundred, so benchmark percents can often be found mentally.

Answer: 6

Worked Example 3

Problem: Find 10% of 64.

  1. Think of 10% as 10/100.
  2. Multiply the quantity by the percent as a decimal or fraction.

Very beginner explanation: Percent means per hundred, so benchmark percents can often be found mentally.

Answer: 6.4

Worked Example 4

Problem: Find 20% of 250.

  1. Think of 20% as 20/100.
  2. Multiply the quantity by the percent as a decimal or fraction.

Very beginner explanation: Percent means per hundred, so benchmark percents can often be found mentally.

Answer: 50

Worked Example 5

Problem: Find 25% of 96.

  1. Think of 25% as 25/100.
  2. Multiply the quantity by the percent as a decimal or fraction.

Very beginner explanation: Percent means per hundred, so benchmark percents can often be found mentally.

Answer: 24

Worked Example 6

Problem: Find 50% of 40.

  1. Think of 50% as 50/100.
  2. Multiply the quantity by the percent as a decimal or fraction.

Very beginner explanation: Percent means per hundred, so benchmark percents can often be found mentally.

Answer: 20

Worked Example 7

Problem: Find 75% of 200.

  1. Think of 75% as 75/100.
  2. Multiply the quantity by the percent as a decimal or fraction.

Very beginner explanation: Percent means per hundred, so benchmark percents can often be found mentally.

Answer: 150

Worked Example 8

Problem: Find 15% of 60.

  1. Think of 15% as 15/100.
  2. Multiply the quantity by the percent as a decimal or fraction.

Very beginner explanation: Percent means per hundred, so benchmark percents can often be found mentally.

Answer: 9

Worked Example 9

Problem: Find 30% of 150.

  1. Think of 30% as 30/100.
  2. Multiply the quantity by the percent as a decimal or fraction.

Very beginner explanation: Percent means per hundred, so benchmark percents can often be found mentally.

Answer: 45

Worked Example 10

Problem: Find 40% of 90.

  1. Think of 40% as 40/100.
  2. Multiply the quantity by the percent as a decimal or fraction.

Very beginner explanation: Percent means per hundred, so benchmark percents can often be found mentally.

Answer: 36

Practice Exercise

Create one new Grade 6 problem about Multiplication before addition in algebra. 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.

26.3 Substitution before calculation

Substitution before calculation is a Grade 6 mathematics idea in Order of Operations with Variables. 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: Evaluate 3x + 4 when x = 5.

  1. Replace x with 5.
  2. Follow order of operations.
  3. Calculate carefully.

Very beginner explanation: Substitution replaces a variable with a known value before calculating.

Answer: 19

Worked Example 2

Problem: Evaluate 2n + 7 when n = 6.

  1. Replace n with 6.
  2. Follow order of operations.
  3. Calculate carefully.

Very beginner explanation: Substitution replaces a variable with a known value before calculating.

Answer: 19

Worked Example 3

Problem: Evaluate 5a - 3 when a = 4.

  1. Replace a with 4.
  2. Follow order of operations.
  3. Calculate carefully.

Very beginner explanation: Substitution replaces a variable with a known value before calculating.

Answer: 17

Worked Example 4

Problem: Evaluate 4p + 1 when p = 8.

  1. Replace p with 8.
  2. Follow order of operations.
  3. Calculate carefully.

Very beginner explanation: Substitution replaces a variable with a known value before calculating.

Answer: 33

Worked Example 5

Problem: Evaluate 6m - 5 when m = 3.

  1. Replace m with 3.
  2. Follow order of operations.
  3. Calculate carefully.

Very beginner explanation: Substitution replaces a variable with a known value before calculating.

Answer: 13

Worked Example 6

Problem: Evaluate 2.5x + 1 when x = 4.

  1. Replace x with 4.
  2. Follow order of operations.
  3. Calculate carefully.

Very beginner explanation: Substitution replaces a variable with a known value before calculating.

Answer: 11

Worked Example 7

Problem: Evaluate 7q + 2 when q = 2.

  1. Replace q with 2.
  2. Follow order of operations.
  3. Calculate carefully.

Very beginner explanation: Substitution replaces a variable with a known value before calculating.

Answer: 16

Worked Example 8

Problem: Evaluate 3r - 1 when r = 9.

  1. Replace r with 9.
  2. Follow order of operations.
  3. Calculate carefully.

Very beginner explanation: Substitution replaces a variable with a known value before calculating.

Answer: 26

Worked Example 9

Problem: Evaluate 4y + 6 when y = 5.

  1. Replace y with 5.
  2. Follow order of operations.
  3. Calculate carefully.

Very beginner explanation: Substitution replaces a variable with a known value before calculating.

Answer: 26

Worked Example 10

Problem: Evaluate 8k - 4 when k = 3.

  1. Replace k with 3.
  2. Follow order of operations.
  3. Calculate carefully.

Very beginner explanation: Substitution replaces a variable with a known value before calculating.

Answer: 20

Practice Exercise

Create one new Grade 6 problem about Substitution before calculation. 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.

26.4 Order of operations after substitution

A ratio compares two quantities by division. 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: Simplify the ratio 10:10.

  1. Find the greatest common factor, 10.
  2. Divide both parts by the same factor.

Very beginner explanation: A ratio is simplified the same way as a fraction.

Answer: 1:1

Worked Example 2

Problem: Simplify the ratio 6:4.

  1. Find the greatest common factor, 2.
  2. Divide both parts by the same factor.

Very beginner explanation: A ratio is simplified the same way as a fraction.

Answer: 3:2

Worked Example 3

Problem: Simplify the ratio 12:4.

  1. Find the greatest common factor, 4.
  2. Divide both parts by the same factor.

Very beginner explanation: A ratio is simplified the same way as a fraction.

Answer: 3:1

Worked Example 4

Problem: Simplify the ratio 6:12.

  1. Find the greatest common factor, 6.
  2. Divide both parts by the same factor.

Very beginner explanation: A ratio is simplified the same way as a fraction.

Answer: 1:2

Worked Example 5

Problem: Simplify the ratio 7:4.

  1. Find the greatest common factor, 1.
  2. Divide both parts by the same factor.

Very beginner explanation: A ratio is simplified the same way as a fraction.

Answer: 7:4

Worked Example 6

Problem: Simplify the ratio 7:7.

  1. Find the greatest common factor, 7.
  2. Divide both parts by the same factor.

Very beginner explanation: A ratio is simplified the same way as a fraction.

Answer: 1:1

Worked Example 7

Problem: Simplify the ratio 12:3.

  1. Find the greatest common factor, 3.
  2. Divide both parts by the same factor.

Very beginner explanation: A ratio is simplified the same way as a fraction.

Answer: 4:1

Worked Example 8

Problem: Simplify the ratio 7:4.

  1. Find the greatest common factor, 1.
  2. Divide both parts by the same factor.

Very beginner explanation: A ratio is simplified the same way as a fraction.

Answer: 7:4

Worked Example 9

Problem: Simplify the ratio 5:14.

  1. Find the greatest common factor, 1.
  2. Divide both parts by the same factor.

Very beginner explanation: A ratio is simplified the same way as a fraction.

Answer: 5:14

Worked Example 10

Problem: Simplify the ratio 9:14.

  1. Find the greatest common factor, 1.
  2. Divide both parts by the same factor.

Very beginner explanation: A ratio is simplified the same way as a fraction.

Answer: 9:14

Practice Exercise

Create one new Grade 6 problem about Order of operations after substitution. 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.

26.5 Expressions with two operations

A ratio compares two quantities by division. 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: Simplify the ratio 6:8.

  1. Find the greatest common factor, 2.
  2. Divide both parts by the same factor.

Very beginner explanation: A ratio is simplified the same way as a fraction.

Answer: 3:4

Worked Example 2

Problem: Simplify the ratio 10:5.

  1. Find the greatest common factor, 5.
  2. Divide both parts by the same factor.

Very beginner explanation: A ratio is simplified the same way as a fraction.

Answer: 2:1

Worked Example 3

Problem: Simplify the ratio 3:13.

  1. Find the greatest common factor, 1.
  2. Divide both parts by the same factor.

Very beginner explanation: A ratio is simplified the same way as a fraction.

Answer: 3:13

Worked Example 4

Problem: Simplify the ratio 4:6.

  1. Find the greatest common factor, 2.
  2. Divide both parts by the same factor.

Very beginner explanation: A ratio is simplified the same way as a fraction.

Answer: 2:3

Worked Example 5

Problem: Simplify the ratio 8:8.

  1. Find the greatest common factor, 8.
  2. Divide both parts by the same factor.

Very beginner explanation: A ratio is simplified the same way as a fraction.

Answer: 1:1

Worked Example 6

Problem: Simplify the ratio 2:11.

  1. Find the greatest common factor, 1.
  2. Divide both parts by the same factor.

Very beginner explanation: A ratio is simplified the same way as a fraction.

Answer: 2:11

Worked Example 7

Problem: Simplify the ratio 3:5.

  1. Find the greatest common factor, 1.
  2. Divide both parts by the same factor.

Very beginner explanation: A ratio is simplified the same way as a fraction.

Answer: 3:5

Worked Example 8

Problem: Simplify the ratio 4:8.

  1. Find the greatest common factor, 4.
  2. Divide both parts by the same factor.

Very beginner explanation: A ratio is simplified the same way as a fraction.

Answer: 1:2

Worked Example 9

Problem: Simplify the ratio 5:4.

  1. Find the greatest common factor, 1.
  2. Divide both parts by the same factor.

Very beginner explanation: A ratio is simplified the same way as a fraction.

Answer: 5:4

Worked Example 10

Problem: Simplify the ratio 4:5.

  1. Find the greatest common factor, 1.
  2. Divide both parts by the same factor.

Very beginner explanation: A ratio is simplified the same way as a fraction.

Answer: 4:5

Practice Exercise

Create one new Grade 6 problem about Expressions with two operations. 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.

26.6 Expressions with three operations

A ratio compares two quantities by division. 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: Simplify the ratio 5:4.

  1. Find the greatest common factor, 1.
  2. Divide both parts by the same factor.

Very beginner explanation: A ratio is simplified the same way as a fraction.

Answer: 5:4

Worked Example 2

Problem: Simplify the ratio 10:7.

  1. Find the greatest common factor, 1.
  2. Divide both parts by the same factor.

Very beginner explanation: A ratio is simplified the same way as a fraction.

Answer: 10:7

Worked Example 3

Problem: Simplify the ratio 10:5.

  1. Find the greatest common factor, 5.
  2. Divide both parts by the same factor.

Very beginner explanation: A ratio is simplified the same way as a fraction.

Answer: 2:1

Worked Example 4

Problem: Simplify the ratio 8:12.

  1. Find the greatest common factor, 4.
  2. Divide both parts by the same factor.

Very beginner explanation: A ratio is simplified the same way as a fraction.

Answer: 2:3

Worked Example 5

Problem: Simplify the ratio 8:10.

  1. Find the greatest common factor, 2.
  2. Divide both parts by the same factor.

Very beginner explanation: A ratio is simplified the same way as a fraction.

Answer: 4:5

Worked Example 6

Problem: Simplify the ratio 7:11.

  1. Find the greatest common factor, 1.
  2. Divide both parts by the same factor.

Very beginner explanation: A ratio is simplified the same way as a fraction.

Answer: 7:11

Worked Example 7

Problem: Simplify the ratio 9:14.

  1. Find the greatest common factor, 1.
  2. Divide both parts by the same factor.

Very beginner explanation: A ratio is simplified the same way as a fraction.

Answer: 9:14

Worked Example 8

Problem: Simplify the ratio 4:8.

  1. Find the greatest common factor, 4.
  2. Divide both parts by the same factor.

Very beginner explanation: A ratio is simplified the same way as a fraction.

Answer: 1:2

Worked Example 9

Problem: Simplify the ratio 2:5.

  1. Find the greatest common factor, 1.
  2. Divide both parts by the same factor.

Very beginner explanation: A ratio is simplified the same way as a fraction.

Answer: 2:5

Worked Example 10

Problem: Simplify the ratio 8:15.

  1. Find the greatest common factor, 1.
  2. Divide both parts by the same factor.

Very beginner explanation: A ratio is simplified the same way as a fraction.

Answer: 8:15

Practice Exercise

Create one new Grade 6 problem about Expressions with three operations. 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.

26.7 Expressions with decimals

A decimal uses place value to show parts of one, such as tenths, hundredths, and thousandths. 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: What digit is in the tenths place of 3.854?

  1. The first digit to the right of the decimal point is the tenths digit.

Very beginner explanation: Decimal places move right from tenths to hundredths to thousandths.

Answer: 8

Worked Example 2

Problem: What digit is in the tenths place of 48.512?

  1. The first digit to the right of the decimal point is the tenths digit.

Very beginner explanation: Decimal places move right from tenths to hundredths to thousandths.

Answer: 5

Worked Example 3

Problem: What digit is in the tenths place of 95.686?

  1. The first digit to the right of the decimal point is the tenths digit.

Very beginner explanation: Decimal places move right from tenths to hundredths to thousandths.

Answer: 6

Worked Example 4

Problem: What digit is in the tenths place of 29.793?

  1. The first digit to the right of the decimal point is the tenths digit.

Very beginner explanation: Decimal places move right from tenths to hundredths to thousandths.

Answer: 7

Worked Example 5

Problem: What digit is in the tenths place of 24.874?

  1. The first digit to the right of the decimal point is the tenths digit.

Very beginner explanation: Decimal places move right from tenths to hundredths to thousandths.

Answer: 8

Worked Example 6

Problem: What digit is in the tenths place of 97.821?

  1. The first digit to the right of the decimal point is the tenths digit.

Very beginner explanation: Decimal places move right from tenths to hundredths to thousandths.

Answer: 8

Worked Example 7

Problem: What digit is in the tenths place of 85.74?

  1. The first digit to the right of the decimal point is the tenths digit.

Very beginner explanation: Decimal places move right from tenths to hundredths to thousandths.

Answer: 7

Worked Example 8

Problem: What digit is in the tenths place of 57.118?

  1. The first digit to the right of the decimal point is the tenths digit.

Very beginner explanation: Decimal places move right from tenths to hundredths to thousandths.

Answer: 1

Worked Example 9

Problem: What digit is in the tenths place of 24.654?

  1. The first digit to the right of the decimal point is the tenths digit.

Very beginner explanation: Decimal places move right from tenths to hundredths to thousandths.

Answer: 6

Worked Example 10

Problem: What digit is in the tenths place of 90.623?

  1. The first digit to the right of the decimal point is the tenths digit.

Very beginner explanation: Decimal places move right from tenths to hundredths to thousandths.

Answer: 6

Practice Exercise

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

Common Mistake

A common mistake is lining up the last digits instead of lining up decimal points and place values.

26.8 Expressions with brackets

An algebraic expression combines numbers, variables, and operations without an equals sign. 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: Evaluate 3x + 4 when x = 5.

  1. Replace x with 5.
  2. Follow order of operations.
  3. Calculate carefully.

Very beginner explanation: Substitution replaces a variable with a known value before calculating.

Answer: 19

Worked Example 2

Problem: Evaluate 2n + 7 when n = 6.

  1. Replace n with 6.
  2. Follow order of operations.
  3. Calculate carefully.

Very beginner explanation: Substitution replaces a variable with a known value before calculating.

Answer: 19

Worked Example 3

Problem: Evaluate 5a - 3 when a = 4.

  1. Replace a with 4.
  2. Follow order of operations.
  3. Calculate carefully.

Very beginner explanation: Substitution replaces a variable with a known value before calculating.

Answer: 17

Worked Example 4

Problem: Evaluate 4p + 1 when p = 8.

  1. Replace p with 8.
  2. Follow order of operations.
  3. Calculate carefully.

Very beginner explanation: Substitution replaces a variable with a known value before calculating.

Answer: 33

Worked Example 5

Problem: Evaluate 6m - 5 when m = 3.

  1. Replace m with 3.
  2. Follow order of operations.
  3. Calculate carefully.

Very beginner explanation: Substitution replaces a variable with a known value before calculating.

Answer: 13

Worked Example 6

Problem: Evaluate 2.5x + 1 when x = 4.

  1. Replace x with 4.
  2. Follow order of operations.
  3. Calculate carefully.

Very beginner explanation: Substitution replaces a variable with a known value before calculating.

Answer: 11

Worked Example 7

Problem: Evaluate 7q + 2 when q = 2.

  1. Replace q with 2.
  2. Follow order of operations.
  3. Calculate carefully.

Very beginner explanation: Substitution replaces a variable with a known value before calculating.

Answer: 16

Worked Example 8

Problem: Evaluate 3r - 1 when r = 9.

  1. Replace r with 9.
  2. Follow order of operations.
  3. Calculate carefully.

Very beginner explanation: Substitution replaces a variable with a known value before calculating.

Answer: 26

Worked Example 9

Problem: Evaluate 4y + 6 when y = 5.

  1. Replace y with 5.
  2. Follow order of operations.
  3. Calculate carefully.

Very beginner explanation: Substitution replaces a variable with a known value before calculating.

Answer: 26

Worked Example 10

Problem: Evaluate 8k - 4 when k = 3.

  1. Replace k with 3.
  2. Follow order of operations.
  3. Calculate carefully.

Very beginner explanation: Substitution replaces a variable with a known value before calculating.

Answer: 20

Practice Exercise

Create one new Grade 6 problem about Expressions with brackets. 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.

26.9 Checking with a second method

Checking with a second method is a Grade 6 mathematics idea in Order of Operations with Variables. 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 with a second method 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: Checking with a second method is a Grade 6 mathematics idea in Order of Operations with Variables. 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 with a second method to analyze the values 19 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: Checking with a second method is a Grade 6 mathematics idea in Order of Operations with Variables. 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 with a second method 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: Checking with a second method is a Grade 6 mathematics idea in Order of Operations with Variables. 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 with a second method 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: Checking with a second method is a Grade 6 mathematics idea in Order of Operations with Variables. 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 with a second method to analyze the values 6 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: Checking with a second method is a Grade 6 mathematics idea in Order of Operations with Variables. 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 with a second method to analyze the values 19 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 with a second method is a Grade 6 mathematics idea in Order of Operations with Variables. 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 with a second method 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: Checking with a second method is a Grade 6 mathematics idea in Order of Operations with Variables. 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 with a second method to analyze the values 10 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: Checking with a second method is a Grade 6 mathematics idea in Order of Operations with Variables. 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 with a second method to analyze the values 8 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 with a second method is a Grade 6 mathematics idea in Order of Operations with Variables. 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 with a second method to analyze the values 14 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: Checking with a second method is a Grade 6 mathematics idea in Order of Operations with Variables. 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 with a second method. 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.

26.10 Writing step-by-step algebra work

Writing step-by-step algebra work is a Grade 6 mathematics idea in Order of Operations with Variables. 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: Evaluate 3x + 4 when x = 5.

  1. Replace x with 5.
  2. Follow order of operations.
  3. Calculate carefully.

Very beginner explanation: Substitution replaces a variable with a known value before calculating.

Answer: 19

Worked Example 2

Problem: Evaluate 2n + 7 when n = 6.

  1. Replace n with 6.
  2. Follow order of operations.
  3. Calculate carefully.

Very beginner explanation: Substitution replaces a variable with a known value before calculating.

Answer: 19

Worked Example 3

Problem: Evaluate 5a - 3 when a = 4.

  1. Replace a with 4.
  2. Follow order of operations.
  3. Calculate carefully.

Very beginner explanation: Substitution replaces a variable with a known value before calculating.

Answer: 17

Worked Example 4

Problem: Evaluate 4p + 1 when p = 8.

  1. Replace p with 8.
  2. Follow order of operations.
  3. Calculate carefully.

Very beginner explanation: Substitution replaces a variable with a known value before calculating.

Answer: 33

Worked Example 5

Problem: Evaluate 6m - 5 when m = 3.

  1. Replace m with 3.
  2. Follow order of operations.
  3. Calculate carefully.

Very beginner explanation: Substitution replaces a variable with a known value before calculating.

Answer: 13

Worked Example 6

Problem: Evaluate 2.5x + 1 when x = 4.

  1. Replace x with 4.
  2. Follow order of operations.
  3. Calculate carefully.

Very beginner explanation: Substitution replaces a variable with a known value before calculating.

Answer: 11

Worked Example 7

Problem: Evaluate 7q + 2 when q = 2.

  1. Replace q with 2.
  2. Follow order of operations.
  3. Calculate carefully.

Very beginner explanation: Substitution replaces a variable with a known value before calculating.

Answer: 16

Worked Example 8

Problem: Evaluate 3r - 1 when r = 9.

  1. Replace r with 9.
  2. Follow order of operations.
  3. Calculate carefully.

Very beginner explanation: Substitution replaces a variable with a known value before calculating.

Answer: 26

Worked Example 9

Problem: Evaluate 4y + 6 when y = 5.

  1. Replace y with 5.
  2. Follow order of operations.
  3. Calculate carefully.

Very beginner explanation: Substitution replaces a variable with a known value before calculating.

Answer: 26

Worked Example 10

Problem: Evaluate 8k - 4 when k = 3.

  1. Replace k with 3.
  2. Follow order of operations.
  3. Calculate carefully.

Very beginner explanation: Substitution replaces a variable with a known value before calculating.

Answer: 20

Practice Exercise

Create one new Grade 6 problem about Writing step-by-step algebra work. 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.

26.11 Common algebra order mistakes

Common algebra order mistakes is a Grade 6 mathematics idea in Order of Operations with Variables. 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: Evaluate 3x + 4 when x = 5.

  1. Replace x with 5.
  2. Follow order of operations.
  3. Calculate carefully.

Very beginner explanation: Substitution replaces a variable with a known value before calculating.

Answer: 19

Worked Example 2

Problem: Evaluate 2n + 7 when n = 6.

  1. Replace n with 6.
  2. Follow order of operations.
  3. Calculate carefully.

Very beginner explanation: Substitution replaces a variable with a known value before calculating.

Answer: 19

Worked Example 3

Problem: Evaluate 5a - 3 when a = 4.

  1. Replace a with 4.
  2. Follow order of operations.
  3. Calculate carefully.

Very beginner explanation: Substitution replaces a variable with a known value before calculating.

Answer: 17

Worked Example 4

Problem: Evaluate 4p + 1 when p = 8.

  1. Replace p with 8.
  2. Follow order of operations.
  3. Calculate carefully.

Very beginner explanation: Substitution replaces a variable with a known value before calculating.

Answer: 33

Worked Example 5

Problem: Evaluate 6m - 5 when m = 3.

  1. Replace m with 3.
  2. Follow order of operations.
  3. Calculate carefully.

Very beginner explanation: Substitution replaces a variable with a known value before calculating.

Answer: 13

Worked Example 6

Problem: Evaluate 2.5x + 1 when x = 4.

  1. Replace x with 4.
  2. Follow order of operations.
  3. Calculate carefully.

Very beginner explanation: Substitution replaces a variable with a known value before calculating.

Answer: 11

Worked Example 7

Problem: Evaluate 7q + 2 when q = 2.

  1. Replace q with 2.
  2. Follow order of operations.
  3. Calculate carefully.

Very beginner explanation: Substitution replaces a variable with a known value before calculating.

Answer: 16

Worked Example 8

Problem: Evaluate 3r - 1 when r = 9.

  1. Replace r with 9.
  2. Follow order of operations.
  3. Calculate carefully.

Very beginner explanation: Substitution replaces a variable with a known value before calculating.

Answer: 26

Worked Example 9

Problem: Evaluate 4y + 6 when y = 5.

  1. Replace y with 5.
  2. Follow order of operations.
  3. Calculate carefully.

Very beginner explanation: Substitution replaces a variable with a known value before calculating.

Answer: 26

Worked Example 10

Problem: Evaluate 8k - 4 when k = 3.

  1. Replace k with 3.
  2. Follow order of operations.
  3. Calculate carefully.

Very beginner explanation: Substitution replaces a variable with a known value before calculating.

Answer: 20

Practice Exercise

Create one new Grade 6 problem about Common algebra order mistakes. 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.

26.12 Real-life variable expressions

A variable is a symbol used to represent a number that may change or may be unknown. 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: Evaluate 3x + 4 when x = 5.

  1. Replace x with 5.
  2. Follow order of operations.
  3. Calculate carefully.

Very beginner explanation: Substitution replaces a variable with a known value before calculating.

Answer: 19

Worked Example 2

Problem: Evaluate 2n + 7 when n = 6.

  1. Replace n with 6.
  2. Follow order of operations.
  3. Calculate carefully.

Very beginner explanation: Substitution replaces a variable with a known value before calculating.

Answer: 19

Worked Example 3

Problem: Evaluate 5a - 3 when a = 4.

  1. Replace a with 4.
  2. Follow order of operations.
  3. Calculate carefully.

Very beginner explanation: Substitution replaces a variable with a known value before calculating.

Answer: 17

Worked Example 4

Problem: Evaluate 4p + 1 when p = 8.

  1. Replace p with 8.
  2. Follow order of operations.
  3. Calculate carefully.

Very beginner explanation: Substitution replaces a variable with a known value before calculating.

Answer: 33

Worked Example 5

Problem: Evaluate 6m - 5 when m = 3.

  1. Replace m with 3.
  2. Follow order of operations.
  3. Calculate carefully.

Very beginner explanation: Substitution replaces a variable with a known value before calculating.

Answer: 13

Worked Example 6

Problem: Evaluate 2.5x + 1 when x = 4.

  1. Replace x with 4.
  2. Follow order of operations.
  3. Calculate carefully.

Very beginner explanation: Substitution replaces a variable with a known value before calculating.

Answer: 11

Worked Example 7

Problem: Evaluate 7q + 2 when q = 2.

  1. Replace q with 2.
  2. Follow order of operations.
  3. Calculate carefully.

Very beginner explanation: Substitution replaces a variable with a known value before calculating.

Answer: 16

Worked Example 8

Problem: Evaluate 3r - 1 when r = 9.

  1. Replace r with 9.
  2. Follow order of operations.
  3. Calculate carefully.

Very beginner explanation: Substitution replaces a variable with a known value before calculating.

Answer: 26

Worked Example 9

Problem: Evaluate 4y + 6 when y = 5.

  1. Replace y with 5.
  2. Follow order of operations.
  3. Calculate carefully.

Very beginner explanation: Substitution replaces a variable with a known value before calculating.

Answer: 26

Worked Example 10

Problem: Evaluate 8k - 4 when k = 3.

  1. Replace k with 3.
  2. Follow order of operations.
  3. Calculate carefully.

Very beginner explanation: Substitution replaces a variable with a known value before calculating.

Answer: 20

Practice Exercise

Create one new Grade 6 problem about Real-life variable expressions. 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 Variables and grouping symbols?

Answer: A variable is a symbol used to represent a number that may change or may be unknown.

Q2. What is important to understand about Multiplication before addition in algebra?

Answer: Multiplication before addition in algebra is a Grade 6 mathematics idea in Order of Operations with Variables. 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 Substitution before calculation?

Answer: Substitution before calculation is a Grade 6 mathematics idea in Order of Operations with Variables. 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 Order of operations after substitution?

Answer: A ratio compares two quantities by division.

Q5. What is important to understand about Expressions with two operations?

Answer: A ratio compares two quantities by division.

Q6. What is important to understand about Expressions with three operations?

Answer: A ratio compares two quantities by division.

Q7. What is important to understand about Expressions with decimals?

Answer: A decimal uses place value to show parts of one, such as tenths, hundredths, and thousandths.

Q8. What is important to understand about Expressions with brackets?

Answer: An algebraic expression combines numbers, variables, and operations without an equals sign.

Q9. What is important to understand about Checking with a second method?

Answer: Checking with a second method is a Grade 6 mathematics idea in Order of Operations with Variables. 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 Writing step-by-step algebra work?

Answer: Writing step-by-step algebra work is a Grade 6 mathematics idea in Order of Operations with Variables. 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 Common algebra order mistakes?

Answer: Common algebra order mistakes is a Grade 6 mathematics idea in Order of Operations with Variables. 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 variable expressions?

Answer: A variable is a symbol used to represent a number that may change or may be unknown.

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.