Chapter 6: Order of Operations
Learn Grade 7 mathematics from very beginner explanations through worked examples, practice, reasoning, and review.
Chapter Overview
This chapter teaches Order of Operations with simple language, step-by-step reasoning, and original worked examples. Technical words are explained in plain language.
Key Technical Terms
- Integer (a positive whole number, negative whole number, or zero)
- Ratio (a comparison of two quantities)
- Estimate (a close approximation used to check whether an answer is reasonable)
- Solution (a value or result that satisfies the problem)
- Representation (a way to show mathematics using symbols, diagrams, tables, graphs, or words)
- Reasonableness (whether an answer makes sense in the context of the problem)
How to Learn This Chapter
Read the explanation first, follow the examples one step at a time, cover the answer and retry the problem yourself, then use the practice and review questions to check understanding.
6.1 Grouping symbols
Grouping symbols is an important Grade 7 idea in Order of Operations. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Beginner Note
Identify what is given, what must be found, choose the correct rule or representation, work one step at a time, and check whether the result makes sense.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: List the factors of 18.
- Find all whole numbers that divide exactly.
Very beginner explanation: Grouping symbols is an important Grade 7 idea in Order of Operations. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: 1, 2, 3, 6, 9, 18
Worked Example 2
Problem: List the factors of 24.
- Find all whole numbers that divide exactly.
Very beginner explanation: Grouping symbols is an important Grade 7 idea in Order of Operations. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: 1, 2, 3, 4, 6, 8, 12, 24
Worked Example 3
Problem: List the factors of 36.
- Find all whole numbers that divide exactly.
Very beginner explanation: Grouping symbols is an important Grade 7 idea in Order of Operations. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: 1, 2, 3, 4, 6, 9, 12, 18, 36
Worked Example 4
Problem: List the factors of 45.
- Find all whole numbers that divide exactly.
Very beginner explanation: Grouping symbols is an important Grade 7 idea in Order of Operations. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: 1, 3, 5, 9, 15, 45
Worked Example 5
Problem: List the factors of 60.
- Find all whole numbers that divide exactly.
Very beginner explanation: Grouping symbols is an important Grade 7 idea in Order of Operations. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60
Worked Example 6
Problem: Explain Grouping symbols in one simple sentence.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Grouping symbols becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: State the meaning, then give one small example.
Worked Example 7
Problem: Give one example that does not satisfy Grouping symbols and explain why.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Grouping symbols becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: Identify the rule or condition that fails.
Worked Example 8
Problem: Show Grouping symbols using a table, diagram, number line, graph, or equation.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Grouping symbols becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: Choose the representation that makes the relationship easiest to see.
Worked Example 9
Problem: After solving a Grouping symbols problem, how can you check the answer?
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Grouping symbols becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: Use estimation, an inverse operation, substitution, or a second representation.
Worked Example 10
Problem: Give one real-life situation where Grouping symbols could be useful.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Grouping symbols becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: Connect the idea to money, measurement, data, design, or technology.
Practice Exercise
Create one new question about Grouping symbols. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
6.2 Parentheses and brackets
Parentheses and brackets is an important Grade 7 idea in Order of Operations. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Beginner Note
Identify what is given, what must be found, choose the correct rule or representation, work one step at a time, and check whether the result makes sense.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: Evaluate 3 + 4 × 5.
- Do grouping first.
- Then exponents.
- Then multiplication/division left to right.
- Then addition/subtraction left to right.
Very beginner explanation: Parentheses and brackets is an important Grade 7 idea in Order of Operations. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: 23
Worked Example 2
Problem: Evaluate (8 + 4) ÷ 3.
- Do grouping first.
- Then exponents.
- Then multiplication/division left to right.
- Then addition/subtraction left to right.
Very beginner explanation: Parentheses and brackets is an important Grade 7 idea in Order of Operations. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: 4
Worked Example 3
Problem: Evaluate 2³ + 6.
- Do grouping first.
- Then exponents.
- Then multiplication/division left to right.
- Then addition/subtraction left to right.
Very beginner explanation: Parentheses and brackets is an important Grade 7 idea in Order of Operations. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: 14
Worked Example 4
Problem: Evaluate 18 ÷ 3 × 2.
- Do grouping first.
- Then exponents.
- Then multiplication/division left to right.
- Then addition/subtraction left to right.
Very beginner explanation: Parentheses and brackets is an important Grade 7 idea in Order of Operations. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: 12
Worked Example 5
Problem: Evaluate 20 - 6 + 4.
- Do grouping first.
- Then exponents.
- Then multiplication/division left to right.
- Then addition/subtraction left to right.
Very beginner explanation: Parentheses and brackets is an important Grade 7 idea in Order of Operations. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: 18
Worked Example 6
Problem: Explain Parentheses and brackets in one simple sentence.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Parentheses and brackets becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: State the meaning, then give one small example.
Worked Example 7
Problem: Give one example that does not satisfy Parentheses and brackets and explain why.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Parentheses and brackets becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: Identify the rule or condition that fails.
Worked Example 8
Problem: Show Parentheses and brackets using a table, diagram, number line, graph, or equation.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Parentheses and brackets becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: Choose the representation that makes the relationship easiest to see.
Worked Example 9
Problem: After solving a Parentheses and brackets problem, how can you check the answer?
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Parentheses and brackets becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: Use estimation, an inverse operation, substitution, or a second representation.
Worked Example 10
Problem: Give one real-life situation where Parentheses and brackets could be useful.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Parentheses and brackets becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: Connect the idea to money, measurement, data, design, or technology.
Practice Exercise
Create one new question about Parentheses and brackets. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
6.3 Exponents in expressions
An exponent (a small raised number showing repeated multiplication) tells how many times the base is used as a factor. This topic focuses on Exponents in expressions .
Beginner Note
Identify what is given, what must be found, choose the correct rule or representation, work one step at a time, and check whether the result makes sense.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: Evaluate 2^3.
- Multiply 2 by itself 3 times.
Very beginner explanation: An exponent (a small raised number showing repeated multiplication) tells how many times the base is used as a factor. This topic focuses on Exponents in expressions .
Answer: 8
Worked Example 2
Problem: Evaluate 3^2.
- Multiply 3 by itself 2 times.
Very beginner explanation: An exponent (a small raised number showing repeated multiplication) tells how many times the base is used as a factor. This topic focuses on Exponents in expressions .
Answer: 9
Worked Example 3
Problem: Evaluate 4^2.
- Multiply 4 by itself 2 times.
Very beginner explanation: An exponent (a small raised number showing repeated multiplication) tells how many times the base is used as a factor. This topic focuses on Exponents in expressions .
Answer: 16
Worked Example 4
Problem: Evaluate 5^3.
- Multiply 5 by itself 3 times.
Very beginner explanation: An exponent (a small raised number showing repeated multiplication) tells how many times the base is used as a factor. This topic focuses on Exponents in expressions .
Answer: 125
Worked Example 5
Problem: Evaluate 10^2.
- Multiply 10 by itself 2 times.
Very beginner explanation: An exponent (a small raised number showing repeated multiplication) tells how many times the base is used as a factor. This topic focuses on Exponents in expressions .
Answer: 100
Worked Example 6
Problem: Simplify 3x + 4x.
- Identify like terms or distribute first if parentheses are present.
- Combine coefficients carefully.
Very beginner explanation: Like terms have the same variable part. The distributive property multiplies every term inside parentheses.
Answer: 7x
Worked Example 7
Problem: Simplify 8y - 3y + 2.
- Identify like terms or distribute first if parentheses are present.
- Combine coefficients carefully.
Very beginner explanation: Like terms have the same variable part. The distributive property multiplies every term inside parentheses.
Answer: 5y + 2
Worked Example 8
Problem: Simplify 4(a + 2).
- Identify like terms or distribute first if parentheses are present.
- Combine coefficients carefully.
Very beginner explanation: Like terms have the same variable part. The distributive property multiplies every term inside parentheses.
Answer: 4a + 8
Worked Example 9
Problem: Simplify 2(3x - 5) + x.
- Identify like terms or distribute first if parentheses are present.
- Combine coefficients carefully.
Very beginner explanation: Like terms have the same variable part. The distributive property multiplies every term inside parentheses.
Answer: 7x - 10
Worked Example 10
Problem: Simplify 6m + 7 - 2m - 3.
- Identify like terms or distribute first if parentheses are present.
- Combine coefficients carefully.
Very beginner explanation: Like terms have the same variable part. The distributive property multiplies every term inside parentheses.
Answer: 4m + 4
Practice Exercise
Create one new question about Exponents in expressions. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
6.4 Multiplication and division
A tip is an extra amount, often calculated as a percent of a bill. This topic focuses on Multiplication and division .
Beginner Note
Identify what is given, what must be found, choose the correct rule or representation, work one step at a time, and check whether the result makes sense.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: Find 10% of $80.
- Convert 10% to 0.1.
- Multiply by 80.
Very beginner explanation: A tip is an extra amount, often calculated as a percent of a bill. This topic focuses on Multiplication and division .
Answer: $8.00
Worked Example 2
Problem: Find 25% of $60.
- Convert 25% to 0.25.
- Multiply by 60.
Very beginner explanation: A tip is an extra amount, often calculated as a percent of a bill. This topic focuses on Multiplication and division .
Answer: $15.00
Worked Example 3
Problem: Find 15% of $120.
- Convert 15% to 0.15.
- Multiply by 120.
Very beginner explanation: A tip is an extra amount, often calculated as a percent of a bill. This topic focuses on Multiplication and division .
Answer: $18.00
Worked Example 4
Problem: Find 5% of $250.
- Convert 5% to 0.05.
- Multiply by 250.
Very beginner explanation: A tip is an extra amount, often calculated as a percent of a bill. This topic focuses on Multiplication and division .
Answer: $12.50
Worked Example 5
Problem: Find 13% of $75.
- Convert 13% to 0.13.
- Multiply by 75.
Very beginner explanation: A tip is an extra amount, often calculated as a percent of a bill. This topic focuses on Multiplication and division .
Answer: $9.75
Worked Example 6
Problem: Find 10% of $90.
- Convert 10% to 0.1.
- Multiply by 90.
Very beginner explanation: Percent of an amount is found by multiplying the amount by the percent written as a decimal.
Answer: $9.00
Worked Example 7
Problem: Find 25% of $64.
- Convert 25% to 0.25.
- Multiply by 64.
Very beginner explanation: Percent of an amount is found by multiplying the amount by the percent written as a decimal.
Answer: $16.00
Worked Example 8
Problem: Find 15% of $140.
- Convert 15% to 0.15.
- Multiply by 140.
Very beginner explanation: Percent of an amount is found by multiplying the amount by the percent written as a decimal.
Answer: $21.00
Worked Example 9
Problem: Find 5% of $260.
- Convert 5% to 0.05.
- Multiply by 260.
Very beginner explanation: Percent of an amount is found by multiplying the amount by the percent written as a decimal.
Answer: $13.00
Worked Example 10
Problem: Find 20% of $75.
- Convert 20% to 0.2.
- Multiply by 75.
Very beginner explanation: Percent of an amount is found by multiplying the amount by the percent written as a decimal.
Answer: $15.00
Practice Exercise
Create one new question about Multiplication and division. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
6.5 Addition and subtraction
Addition and subtraction is an important Grade 7 idea in Order of Operations. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Beginner Note
Identify what is given, what must be found, choose the correct rule or representation, work one step at a time, and check whether the result makes sense.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: Addition and subtraction: Identify a correct example.
- State the definition and show why the example fits.
- Show all important reasoning.
Very beginner explanation: Addition and subtraction is an important Grade 7 idea in Order of Operations. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: A correct response must satisfy the definition or rule and include a check.
Worked Example 2
Problem: Addition and subtraction: Identify a non-example.
- Explain which requirement is missing.
- Show all important reasoning.
Very beginner explanation: Addition and subtraction is an important Grade 7 idea in Order of Operations. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: A correct response must satisfy the definition or rule and include a check.
Worked Example 3
Problem: Addition and subtraction: Compare two cases.
- State one similarity and one difference.
- Show all important reasoning.
Very beginner explanation: Addition and subtraction is an important Grade 7 idea in Order of Operations. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: A correct response must satisfy the definition or rule and include a check.
Worked Example 4
Problem: Addition and subtraction: Apply the idea in a real situation.
- Translate the situation into mathematical language.
- Show all important reasoning.
Very beginner explanation: Addition and subtraction is an important Grade 7 idea in Order of Operations. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: A correct response must satisfy the definition or rule and include a check.
Worked Example 5
Problem: Addition and subtraction: Create and check your own example.
- Use the definition, then verify each condition.
- Show all important reasoning.
Very beginner explanation: Addition and subtraction is an important Grade 7 idea in Order of Operations. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: A correct response must satisfy the definition or rule and include a check.
Worked Example 6
Problem: Explain Addition and subtraction in one simple sentence.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Addition and subtraction becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: State the meaning, then give one small example.
Worked Example 7
Problem: Give one example that does not satisfy Addition and subtraction and explain why.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Addition and subtraction becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: Identify the rule or condition that fails.
Worked Example 8
Problem: Show Addition and subtraction using a table, diagram, number line, graph, or equation.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Addition and subtraction becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: Choose the representation that makes the relationship easiest to see.
Worked Example 9
Problem: After solving a Addition and subtraction problem, how can you check the answer?
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Addition and subtraction becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: Use estimation, an inverse operation, substitution, or a second representation.
Worked Example 10
Problem: Give one real-life situation where Addition and subtraction could be useful.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Addition and subtraction becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: Connect the idea to money, measurement, data, design, or technology.
Practice Exercise
Create one new question about Addition and subtraction. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
6.6 BEDMAS/PEMDAS
BEDMAS/PEMDAS is an important Grade 7 idea in Order of Operations. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Beginner Note
Identify what is given, what must be found, choose the correct rule or representation, work one step at a time, and check whether the result makes sense.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: Evaluate 3 + 4 × 5.
- Do grouping first.
- Then exponents.
- Then multiplication/division left to right.
- Then addition/subtraction left to right.
Very beginner explanation: BEDMAS/PEMDAS is an important Grade 7 idea in Order of Operations. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: 23
Worked Example 2
Problem: Evaluate (8 + 4) ÷ 3.
- Do grouping first.
- Then exponents.
- Then multiplication/division left to right.
- Then addition/subtraction left to right.
Very beginner explanation: BEDMAS/PEMDAS is an important Grade 7 idea in Order of Operations. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: 4
Worked Example 3
Problem: Evaluate 2³ + 6.
- Do grouping first.
- Then exponents.
- Then multiplication/division left to right.
- Then addition/subtraction left to right.
Very beginner explanation: BEDMAS/PEMDAS is an important Grade 7 idea in Order of Operations. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: 14
Worked Example 4
Problem: Evaluate 18 ÷ 3 × 2.
- Do grouping first.
- Then exponents.
- Then multiplication/division left to right.
- Then addition/subtraction left to right.
Very beginner explanation: BEDMAS/PEMDAS is an important Grade 7 idea in Order of Operations. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: 12
Worked Example 5
Problem: Evaluate 20 - 6 + 4.
- Do grouping first.
- Then exponents.
- Then multiplication/division left to right.
- Then addition/subtraction left to right.
Very beginner explanation: BEDMAS/PEMDAS is an important Grade 7 idea in Order of Operations. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: 18
Worked Example 6
Problem: Explain BEDMAS/PEMDAS in one simple sentence.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: BEDMAS/PEMDAS becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: State the meaning, then give one small example.
Worked Example 7
Problem: Give one example that does not satisfy BEDMAS/PEMDAS and explain why.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: BEDMAS/PEMDAS becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: Identify the rule or condition that fails.
Worked Example 8
Problem: Show BEDMAS/PEMDAS using a table, diagram, number line, graph, or equation.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: BEDMAS/PEMDAS becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: Choose the representation that makes the relationship easiest to see.
Worked Example 9
Problem: After solving a BEDMAS/PEMDAS problem, how can you check the answer?
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: BEDMAS/PEMDAS becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: Use estimation, an inverse operation, substitution, or a second representation.
Worked Example 10
Problem: Give one real-life situation where BEDMAS/PEMDAS could be useful.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: BEDMAS/PEMDAS becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: Connect the idea to money, measurement, data, design, or technology.
Practice Exercise
Create one new question about BEDMAS/PEMDAS. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
6.7 Left-to-right rules
Left-to-right rules is an important Grade 7 idea in Order of Operations. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Beginner Note
Identify what is given, what must be found, choose the correct rule or representation, work one step at a time, and check whether the result makes sense.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: Evaluate 3 + 4 × 5.
- Do grouping first.
- Then exponents.
- Then multiplication/division left to right.
- Then addition/subtraction left to right.
Very beginner explanation: Left-to-right rules is an important Grade 7 idea in Order of Operations. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: 23
Worked Example 2
Problem: Evaluate (8 + 4) ÷ 3.
- Do grouping first.
- Then exponents.
- Then multiplication/division left to right.
- Then addition/subtraction left to right.
Very beginner explanation: Left-to-right rules is an important Grade 7 idea in Order of Operations. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: 4
Worked Example 3
Problem: Evaluate 2³ + 6.
- Do grouping first.
- Then exponents.
- Then multiplication/division left to right.
- Then addition/subtraction left to right.
Very beginner explanation: Left-to-right rules is an important Grade 7 idea in Order of Operations. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: 14
Worked Example 4
Problem: Evaluate 18 ÷ 3 × 2.
- Do grouping first.
- Then exponents.
- Then multiplication/division left to right.
- Then addition/subtraction left to right.
Very beginner explanation: Left-to-right rules is an important Grade 7 idea in Order of Operations. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: 12
Worked Example 5
Problem: Evaluate 20 - 6 + 4.
- Do grouping first.
- Then exponents.
- Then multiplication/division left to right.
- Then addition/subtraction left to right.
Very beginner explanation: Left-to-right rules is an important Grade 7 idea in Order of Operations. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: 18
Worked Example 6
Problem: Explain Left-to-right rules in one simple sentence.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Left-to-right rules becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: State the meaning, then give one small example.
Worked Example 7
Problem: Give one example that does not satisfy Left-to-right rules and explain why.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Left-to-right rules becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: Identify the rule or condition that fails.
Worked Example 8
Problem: Show Left-to-right rules using a table, diagram, number line, graph, or equation.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Left-to-right rules becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: Choose the representation that makes the relationship easiest to see.
Worked Example 9
Problem: After solving a Left-to-right rules problem, how can you check the answer?
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Left-to-right rules becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: Use estimation, an inverse operation, substitution, or a second representation.
Worked Example 10
Problem: Give one real-life situation where Left-to-right rules could be useful.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Left-to-right rules becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: Connect the idea to money, measurement, data, design, or technology.
Practice Exercise
Create one new question about Left-to-right rules. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
6.8 Expressions with integers
An integer (a positive whole number, a negative whole number, or zero) is useful for temperatures, elevations, gains, and losses. This topic focuses on Expressions with integers .
Beginner Note
Identify what is given, what must be found, choose the correct rule or representation, work one step at a time, and check whether the result makes sense.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: Place -7 on a number line.
- Start at 0.
- Move 7 units left.
Very beginner explanation: An integer (a positive whole number, a negative whole number, or zero) is useful for temperatures, elevations, gains, and losses. This topic focuses on Expressions with integers .
Answer: -7 is negative.
Worked Example 2
Problem: Place 4 on a number line.
- Start at 0.
- Move 4 units right.
Very beginner explanation: An integer (a positive whole number, a negative whole number, or zero) is useful for temperatures, elevations, gains, and losses. This topic focuses on Expressions with integers .
Answer: 4 is positive.
Worked Example 3
Problem: Place -6 on a number line.
- Start at 0.
- Move 6 units left.
Very beginner explanation: An integer (a positive whole number, a negative whole number, or zero) is useful for temperatures, elevations, gains, and losses. This topic focuses on Expressions with integers .
Answer: -6 is negative.
Worked Example 4
Problem: Place 12 on a number line.
- Start at 0.
- Move 12 units right.
Very beginner explanation: An integer (a positive whole number, a negative whole number, or zero) is useful for temperatures, elevations, gains, and losses. This topic focuses on Expressions with integers .
Answer: 12 is positive.
Worked Example 5
Problem: Place -15 on a number line.
- Start at 0.
- Move 15 units left.
Very beginner explanation: An integer (a positive whole number, a negative whole number, or zero) is useful for temperatures, elevations, gains, and losses. This topic focuses on Expressions with integers .
Answer: -15 is negative.
Worked Example 6
Problem: Calculate -8 + (5).
- Use a number line or sign rules.
- If signs differ, subtract absolute values and keep the sign of the larger absolute value.
Very beginner explanation: Integer addition can be understood as movement left and right on a number line.
Answer: -3
Worked Example 7
Problem: Calculate 7 + (-12).
- Use a number line or sign rules.
- If signs differ, subtract absolute values and keep the sign of the larger absolute value.
Very beginner explanation: Integer addition can be understood as movement left and right on a number line.
Answer: -5
Worked Example 8
Problem: Calculate -4 + (-9).
- Use a number line or sign rules.
- If signs differ, subtract absolute values and keep the sign of the larger absolute value.
Very beginner explanation: Integer addition can be understood as movement left and right on a number line.
Answer: -13
Worked Example 9
Problem: Calculate 15 + (-6).
- Use a number line or sign rules.
- If signs differ, subtract absolute values and keep the sign of the larger absolute value.
Very beginner explanation: Integer addition can be understood as movement left and right on a number line.
Answer: 9
Worked Example 10
Problem: Calculate -20 + (13).
- Use a number line or sign rules.
- If signs differ, subtract absolute values and keep the sign of the larger absolute value.
Very beginner explanation: Integer addition can be understood as movement left and right on a number line.
Answer: -7
Practice Exercise
Create one new question about Expressions with integers. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
6.9 Expressions with fractions
A fraction (a numerator over a denominator) can represent part of a whole, a quotient, or a ratio. This topic focuses on Expressions with fractions .
Beginner Note
Identify what is given, what must be found, choose the correct rule or representation, work one step at a time, and check whether the result makes sense.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: Write 1/2 as a decimal.
- Divide 1 by 2.
Very beginner explanation: A fraction (a numerator over a denominator) can represent part of a whole, a quotient, or a ratio. This topic focuses on Expressions with fractions .
Answer: 0.5
Worked Example 2
Problem: Write 2/3 as a decimal.
- Divide 2 by 3.
Very beginner explanation: A fraction (a numerator over a denominator) can represent part of a whole, a quotient, or a ratio. This topic focuses on Expressions with fractions .
Answer: 0.6667
Worked Example 3
Problem: Write 3/4 as a decimal.
- Divide 3 by 4.
Very beginner explanation: A fraction (a numerator over a denominator) can represent part of a whole, a quotient, or a ratio. This topic focuses on Expressions with fractions .
Answer: 0.75
Worked Example 4
Problem: Write 5/6 as a decimal.
- Divide 5 by 6.
Very beginner explanation: A fraction (a numerator over a denominator) can represent part of a whole, a quotient, or a ratio. This topic focuses on Expressions with fractions .
Answer: 0.8333
Worked Example 5
Problem: Write 7/8 as a decimal.
- Divide 7 by 8.
Very beginner explanation: A fraction (a numerator over a denominator) can represent part of a whole, a quotient, or a ratio. This topic focuses on Expressions with fractions .
Answer: 0.875
Worked Example 6
Problem: Write 1/3 as a decimal.
- A fraction bar means division.
- Divide 1 by 3.
Very beginner explanation: Fractions and decimals can represent the same rational number.
Answer: 0.3333
Worked Example 7
Problem: Write 2/5 as a decimal.
- A fraction bar means division.
- Divide 2 by 5.
Very beginner explanation: Fractions and decimals can represent the same rational number.
Answer: 0.4
Worked Example 8
Problem: Write 3/8 as a decimal.
- A fraction bar means division.
- Divide 3 by 8.
Very beginner explanation: Fractions and decimals can represent the same rational number.
Answer: 0.375
Worked Example 9
Problem: Write 5/12 as a decimal.
- A fraction bar means division.
- Divide 5 by 12.
Very beginner explanation: Fractions and decimals can represent the same rational number.
Answer: 0.4167
Worked Example 10
Problem: Write 7/9 as a decimal.
- A fraction bar means division.
- Divide 7 by 9.
Very beginner explanation: Fractions and decimals can represent the same rational number.
Answer: 0.7778
Practice Exercise
Create one new question about Expressions with fractions. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
6.10 Expressions with decimals
A decimal uses place value to represent parts of one, such as tenths, hundredths, and thousandths. This topic focuses on Expressions with decimals .
Beginner Note
Identify what is given, what must be found, choose the correct rule or representation, work one step at a time, and check whether the result makes sense.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: Identify the tenths digit in 3.47.
- The tenths digit is the first digit to the right of the decimal point.
Very beginner explanation: A decimal uses place value to represent parts of one, such as tenths, hundredths, and thousandths. This topic focuses on Expressions with decimals .
Answer: 4
Worked Example 2
Problem: Identify the tenths digit in 8.205.
- The tenths digit is the first digit to the right of the decimal point.
Very beginner explanation: A decimal uses place value to represent parts of one, such as tenths, hundredths, and thousandths. This topic focuses on Expressions with decimals .
Answer: 2
Worked Example 3
Problem: Identify the tenths digit in 0.96.
- The tenths digit is the first digit to the right of the decimal point.
Very beginner explanation: A decimal uses place value to represent parts of one, such as tenths, hundredths, and thousandths. This topic focuses on Expressions with decimals .
Answer: 9
Worked Example 4
Problem: Identify the tenths digit in 12.375.
- The tenths digit is the first digit to the right of the decimal point.
Very beginner explanation: A decimal uses place value to represent parts of one, such as tenths, hundredths, and thousandths. This topic focuses on Expressions with decimals .
Answer: 3
Worked Example 5
Problem: Identify the tenths digit in 5.004.
- The tenths digit is the first digit to the right of the decimal point.
Very beginner explanation: A decimal uses place value to represent parts of one, such as tenths, hundredths, and thousandths. This topic focuses on Expressions with decimals .
Answer: 0
Worked Example 6
Problem: Round 3.75 to the nearest tenth.
- Find the tenths digit.
- Look at the hundredths digit.
- Round up if the hundredths digit is 5 or more.
Very beginner explanation: Rounding uses the next digit to decide whether the target digit stays or increases.
Answer: 3.8
Worked Example 7
Problem: Round 8.4 to the nearest tenth.
- Find the tenths digit.
- Look at the hundredths digit.
- Round up if the hundredths digit is 5 or more.
Very beginner explanation: Rounding uses the next digit to decide whether the target digit stays or increases.
Answer: 8.4
Worked Example 8
Problem: Round 12.05 to the nearest tenth.
- Find the tenths digit.
- Look at the hundredths digit.
- Round up if the hundredths digit is 5 or more.
Very beginner explanation: Rounding uses the next digit to decide whether the target digit stays or increases.
Answer: 12.1
Worked Example 9
Problem: Round 0.96 to the nearest tenth.
- Find the tenths digit.
- Look at the hundredths digit.
- Round up if the hundredths digit is 5 or more.
Very beginner explanation: Rounding uses the next digit to decide whether the target digit stays or increases.
Answer: 1
Worked Example 10
Problem: Round 5.125 to the nearest tenth.
- Find the tenths digit.
- Look at the hundredths digit.
- Round up if the hundredths digit is 5 or more.
Very beginner explanation: Rounding uses the next digit to decide whether the target digit stays or increases.
Answer: 5.1
Practice Exercise
Create one new question about Expressions with decimals. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
6.11 Multi-step word calculations
Multi-step word calculations is an important Grade 7 idea in Order of Operations. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Beginner Note
Identify what is given, what must be found, choose the correct rule or representation, work one step at a time, and check whether the result makes sense.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: Evaluate 3 + 4 × 5.
- Do grouping first.
- Then exponents.
- Then multiplication/division left to right.
- Then addition/subtraction left to right.
Very beginner explanation: Multi-step word calculations is an important Grade 7 idea in Order of Operations. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: 23
Worked Example 2
Problem: Evaluate (8 + 4) ÷ 3.
- Do grouping first.
- Then exponents.
- Then multiplication/division left to right.
- Then addition/subtraction left to right.
Very beginner explanation: Multi-step word calculations is an important Grade 7 idea in Order of Operations. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: 4
Worked Example 3
Problem: Evaluate 2³ + 6.
- Do grouping first.
- Then exponents.
- Then multiplication/division left to right.
- Then addition/subtraction left to right.
Very beginner explanation: Multi-step word calculations is an important Grade 7 idea in Order of Operations. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: 14
Worked Example 4
Problem: Evaluate 18 ÷ 3 × 2.
- Do grouping first.
- Then exponents.
- Then multiplication/division left to right.
- Then addition/subtraction left to right.
Very beginner explanation: Multi-step word calculations is an important Grade 7 idea in Order of Operations. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: 12
Worked Example 5
Problem: Evaluate 20 - 6 + 4.
- Do grouping first.
- Then exponents.
- Then multiplication/division left to right.
- Then addition/subtraction left to right.
Very beginner explanation: Multi-step word calculations is an important Grade 7 idea in Order of Operations. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: 18
Worked Example 6
Problem: Explain Multi-step word calculations in one simple sentence.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Multi-step word calculations becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: State the meaning, then give one small example.
Worked Example 7
Problem: Give one example that does not satisfy Multi-step word calculations and explain why.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Multi-step word calculations becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: Identify the rule or condition that fails.
Worked Example 8
Problem: Show Multi-step word calculations using a table, diagram, number line, graph, or equation.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Multi-step word calculations becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: Choose the representation that makes the relationship easiest to see.
Worked Example 9
Problem: After solving a Multi-step word calculations problem, how can you check the answer?
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Multi-step word calculations becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: Use estimation, an inverse operation, substitution, or a second representation.
Worked Example 10
Problem: Give one real-life situation where Multi-step word calculations could be useful.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Multi-step word calculations becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: Connect the idea to money, measurement, data, design, or technology.
Practice Exercise
Create one new question about Multi-step word calculations. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
6.12 Estimating before calculating
Estimating before calculating is an important Grade 7 idea in Order of Operations. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Beginner Note
Identify what is given, what must be found, choose the correct rule or representation, work one step at a time, and check whether the result makes sense.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: Estimating before calculating: Identify a correct example.
- State the definition and show why the example fits.
- Show all important reasoning.
Very beginner explanation: Estimating before calculating is an important Grade 7 idea in Order of Operations. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: A correct response must satisfy the definition or rule and include a check.
Worked Example 2
Problem: Estimating before calculating: Identify a non-example.
- Explain which requirement is missing.
- Show all important reasoning.
Very beginner explanation: Estimating before calculating is an important Grade 7 idea in Order of Operations. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: A correct response must satisfy the definition or rule and include a check.
Worked Example 3
Problem: Estimating before calculating: Compare two cases.
- State one similarity and one difference.
- Show all important reasoning.
Very beginner explanation: Estimating before calculating is an important Grade 7 idea in Order of Operations. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: A correct response must satisfy the definition or rule and include a check.
Worked Example 4
Problem: Estimating before calculating: Apply the idea in a real situation.
- Translate the situation into mathematical language.
- Show all important reasoning.
Very beginner explanation: Estimating before calculating is an important Grade 7 idea in Order of Operations. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: A correct response must satisfy the definition or rule and include a check.
Worked Example 5
Problem: Estimating before calculating: Create and check your own example.
- Use the definition, then verify each condition.
- Show all important reasoning.
Very beginner explanation: Estimating before calculating is an important Grade 7 idea in Order of Operations. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: A correct response must satisfy the definition or rule and include a check.
Worked Example 6
Problem: Explain Estimating before calculating in one simple sentence.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Estimating before calculating becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: State the meaning, then give one small example.
Worked Example 7
Problem: Give one example that does not satisfy Estimating before calculating and explain why.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Estimating before calculating becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: Identify the rule or condition that fails.
Worked Example 8
Problem: Show Estimating before calculating using a table, diagram, number line, graph, or equation.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Estimating before calculating becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: Choose the representation that makes the relationship easiest to see.
Worked Example 9
Problem: After solving a Estimating before calculating problem, how can you check the answer?
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Estimating before calculating becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: Use estimation, an inverse operation, substitution, or a second representation.
Worked Example 10
Problem: Give one real-life situation where Estimating before calculating could be useful.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Estimating before calculating becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: Connect the idea to money, measurement, data, design, or technology.
Practice Exercise
Create one new question about Estimating before calculating. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
6.13 Common order-of-operations errors
A ratio compares two quantities in a fixed order. This topic focuses on Common order-of-operations errors .
Beginner Note
Identify what is given, what must be found, choose the correct rule or representation, work one step at a time, and check whether the result makes sense.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: Simplify 4:6.
- GCF(4,6)=2.
- Divide both terms by 2.
Very beginner explanation: A ratio compares two quantities in a fixed order. This topic focuses on Common order-of-operations errors .
Answer: 2:3
Worked Example 2
Problem: Simplify 8:12.
- GCF(8,12)=4.
- Divide both terms by 4.
Very beginner explanation: A ratio compares two quantities in a fixed order. This topic focuses on Common order-of-operations errors .
Answer: 2:3
Worked Example 3
Problem: Simplify 15:25.
- GCF(15,25)=5.
- Divide both terms by 5.
Very beginner explanation: A ratio compares two quantities in a fixed order. This topic focuses on Common order-of-operations errors .
Answer: 3:5
Worked Example 4
Problem: Simplify 21:28.
- GCF(21,28)=7.
- Divide both terms by 7.
Very beginner explanation: A ratio compares two quantities in a fixed order. This topic focuses on Common order-of-operations errors .
Answer: 3:4
Worked Example 5
Problem: Simplify 18:30.
- GCF(18,30)=6.
- Divide both terms by 6.
Very beginner explanation: A ratio compares two quantities in a fixed order. This topic focuses on Common order-of-operations errors .
Answer: 3:5
Worked Example 6
Problem: Simplify the ratio 4:6.
- Find the greatest common factor, 2.
- Divide both terms by 2.
Very beginner explanation: Simplifying a ratio is like simplifying a fraction: divide both quantities by the same common factor.
Answer: 2:3
Worked Example 7
Problem: Simplify the ratio 8:12.
- Find the greatest common factor, 4.
- Divide both terms by 4.
Very beginner explanation: Simplifying a ratio is like simplifying a fraction: divide both quantities by the same common factor.
Answer: 2:3
Worked Example 8
Problem: Simplify the ratio 15:25.
- Find the greatest common factor, 5.
- Divide both terms by 5.
Very beginner explanation: Simplifying a ratio is like simplifying a fraction: divide both quantities by the same common factor.
Answer: 3:5
Worked Example 9
Problem: Simplify the ratio 18:30.
- Find the greatest common factor, 6.
- Divide both terms by 6.
Very beginner explanation: Simplifying a ratio is like simplifying a fraction: divide both quantities by the same common factor.
Answer: 3:5
Worked Example 10
Problem: Simplify the ratio 21:28.
- Find the greatest common factor, 7.
- Divide both terms by 7.
Very beginner explanation: Simplifying a ratio is like simplifying a fraction: divide both quantities by the same common factor.
Answer: 3:4
Practice Exercise
Create one new question about Common order-of-operations errors. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
Chapter Notes, Practice, and Common Mistakes
Chapter Notes
- Read the complete question before choosing an operation, formula, graph, or model.
- Write technical words together with their meaning until you are comfortable using them.
- Show all important steps so another student can follow your reasoning.
- Keep units, labels, signs, axes, variables, and mathematical symbols clear.
- Estimate before or after calculating when an estimate can help check reasonableness.
- For real-life problems, explain what the final number means in the situation.
Extra Practice
- Create and solve a new question about Grouping symbols . Show your reasoning and check your answer.
- Create and solve a new question about Parentheses and brackets . Show your reasoning and check your answer.
- Create and solve a new question about Exponents in expressions . Show your reasoning and check your answer.
- Create and solve a new question about Multiplication and division . Show your reasoning and check your answer.
- Create and solve a new question about Addition and subtraction . Show your reasoning and check your answer.
- Create and solve a new question about BEDMAS/PEMDAS . Show your reasoning and check your answer.
- Create and solve a new question about Left-to-right rules . Show your reasoning and check your answer.
- Create and solve a new question about Expressions with integers . Show your reasoning and check your answer.
- Create and solve a new question about Expressions with fractions . Show your reasoning and check your answer.
- Create and solve a new question about Expressions with decimals . Show your reasoning and check your answer.
- Create and solve a new question about Multi-step word calculations . Show your reasoning and check your answer.
- Create and solve a new question about Estimating before calculating . Show your reasoning and check your answer.
Common Mistakes
- Ignoring place value or signs.
- Using an operation before checking what the question asks.
- Skipping estimation and accepting an unreasonable result.
- Forgetting the correct order of operations.
30 Review Questions and Answers
Q1. What is important to remember about Grouping symbols?
Answer: Grouping symbols is an important Grade 7 idea in Order of Operations. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Q2. What is important to remember about Parentheses and brackets?
Answer: Parentheses and brackets is an important Grade 7 idea in Order of Operations. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Q3. What is important to remember about Exponents in expressions?
Answer: An exponent (a small raised number showing repeated multiplication) tells how many times the base is used as a factor. This topic focuses on Exponents in expressions.
Q4. What is important to remember about Multiplication and division?
Answer: A tip is an extra amount, often calculated as a percent of a bill. This topic focuses on Multiplication and division.
Q5. What is important to remember about Addition and subtraction?
Answer: Addition and subtraction is an important Grade 7 idea in Order of Operations. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Q6. What is important to remember about BEDMAS/PEMDAS?
Answer: BEDMAS/PEMDAS is an important Grade 7 idea in Order of Operations. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Q7. What is important to remember about Left-to-right rules?
Answer: Left-to-right rules is an important Grade 7 idea in Order of Operations. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Q8. What is important to remember about Expressions with integers?
Answer: An integer (a positive whole number, a negative whole number, or zero) is useful for temperatures, elevations, gains, and losses. This topic focuses on Expressions with integers.
Q9. What is important to remember about Expressions with fractions?
Answer: A fraction (a numerator over a denominator) can represent part of a whole, a quotient, or a ratio. This topic focuses on Expressions with fractions.
Q10. What is important to remember about Expressions with decimals?
Answer: A decimal uses place value to represent parts of one, such as tenths, hundredths, and thousandths. This topic focuses on Expressions with decimals.
Q11. What is important to remember about Multi-step word calculations?
Answer: Multi-step word calculations is an important Grade 7 idea in Order of Operations. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Q12. What is important to remember about Estimating before calculating?
Answer: Estimating before calculating is an important Grade 7 idea in Order of Operations. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Q13. What is important to remember about Common order-of-operations errors?
Answer: A ratio compares two quantities in a fixed order. This topic focuses on Common order-of-operations errors.
Q14. Why should you show your steps?
Answer: Showing steps makes reasoning easier to verify and helps locate mistakes.
Q15. Why is estimation useful?
Answer: Estimation helps you judge whether a final answer is reasonable before accepting it.
Q16. Why do units matter?
Answer: Units tell what a value measures and prevent incompatible quantities from being mixed.
Q17. When should you round?
Answer: Usually round near the end of a calculation unless the question specifically asks for earlier rounding.
Q18. How can you check an answer?
Answer: Use an inverse operation, substitution, estimation, a graph, or a second method.
Q19. What should you identify first in a word problem?
Answer: Identify what is known, what is unknown, and the relationship connecting them.
Q20. What makes a final answer complete?
Answer: Give the value, correct units when needed, and a brief interpretation in context.
Q21. What should you do if an answer seems unreasonable?
Answer: Re-read the question, check copied values, signs, units, formulas, and calculations.
Q22. Why are diagrams useful?
Answer: A labelled diagram can reveal relationships between lengths, angles, areas, and other quantities.
Q23. Why are tables useful?
Answer: Tables organize values and help reveal patterns, relationships, rates, and missing information.
Q24. Why should you explain your reasoning?
Answer: Reasoning shows why a method works, not just what answer you obtained.
Q25. How do mistakes help learning?
Answer: Analyzing a mistake identifies the misunderstood rule or step and helps prevent the same error later.
Q26. How should you study this chapter?
Answer: Review definitions, redo worked examples without looking, practise mixed questions, and explain solutions in your own words.
Q27. When is a calculator useful?
Answer: A calculator is useful for lengthy arithmetic after you understand the mathematical setup and can estimate the expected size of the answer.
Q28. Why compare more than one strategy?
Answer: Different strategies can make a problem easier and provide a way to verify the result.
Q29. What is a mathematical model?
Answer: It is a simplified mathematical representation of a real situation used to analyze, explain, or predict.
Q30. Why should assumptions be stated?
Answer: Assumptions show what conditions the solution depends on and help readers judge whether the model is reasonable.