Chapter 2: Multiplication, Division, and Mental Math Fluency
Learn Grade 7 mathematics from very beginner explanations through worked examples, practice, reasoning, and review.
Chapter Overview
This chapter teaches Multiplication, Division, and Mental Math Fluency with simple language, step-by-step reasoning, and original worked examples. Technical words are explained in plain language.
Key Technical Terms
- Rational Number (a number that can be written as a fraction of integers with a nonzero denominator)
- Prime Number (a whole number greater than 1 with exactly two positive factors)
- Ratio (a comparison of two quantities)
- Unit Rate (a rate per one unit)
- Exchange Rate (the value of one currency expressed in another currency)
- Interest Rate (percentage used to calculate interest)
- 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.
2.1 Multiplication facts 0×0 to 12×12
Multiplication facts are basic products that should become automatic so larger calculations are easier. This topic focuses on Multiplication facts 0×0 to 12×12 .
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: Calculate 12 × 8.
- Use a known fact, distributive strategy, or standard multiplication.
- Check using division: 96 ÷ 8 = 12.
Very beginner explanation: Multiplication facts are basic products that should become automatic so larger calculations are easier. This topic focuses on Multiplication facts 0×0 to 12×12 .
Answer: 96
Worked Example 2
Problem: Calculate 9 × 7.
- Use a known fact, distributive strategy, or standard multiplication.
- Check using division: 63 ÷ 7 = 9.
Very beginner explanation: Multiplication facts are basic products that should become automatic so larger calculations are easier. This topic focuses on Multiplication facts 0×0 to 12×12 .
Answer: 63
Worked Example 3
Problem: Calculate 144 × 12.
- Use a known fact, distributive strategy, or standard multiplication.
- Check using division: 1728 ÷ 12 = 144.
Very beginner explanation: Multiplication facts are basic products that should become automatic so larger calculations are easier. This topic focuses on Multiplication facts 0×0 to 12×12 .
Answer: 1728
Worked Example 4
Problem: Calculate 325 × 6.
- Use a known fact, distributive strategy, or standard multiplication.
- Check using division: 1950 ÷ 6 = 325.
Very beginner explanation: Multiplication facts are basic products that should become automatic so larger calculations are easier. This topic focuses on Multiplication facts 0×0 to 12×12 .
Answer: 1950
Worked Example 5
Problem: Calculate 728 × 8.
- Use a known fact, distributive strategy, or standard multiplication.
- Check using division: 5824 ÷ 8 = 728.
Very beginner explanation: Multiplication facts are basic products that should become automatic so larger calculations are easier. This topic focuses on Multiplication facts 0×0 to 12×12 .
Answer: 5824
Worked Example 6
Problem: Calculate 84 × 7.
- Break one factor into easier parts if helpful.
- Multiply carefully.
- Check with division.
Very beginner explanation: Multiplication combines equal groups and can be checked with the inverse operation, division.
Answer: 588
Worked Example 7
Problem: Calculate 125 × 8.
- Break one factor into easier parts if helpful.
- Multiply carefully.
- Check with division.
Very beginner explanation: Multiplication combines equal groups and can be checked with the inverse operation, division.
Answer: 1000
Worked Example 8
Problem: Calculate 936 × 9.
- Break one factor into easier parts if helpful.
- Multiply carefully.
- Check with division.
Very beginner explanation: Multiplication combines equal groups and can be checked with the inverse operation, division.
Answer: 8424
Worked Example 9
Problem: Calculate 48 × 12.
- Break one factor into easier parts if helpful.
- Multiply carefully.
- Check with division.
Very beginner explanation: Multiplication combines equal groups and can be checked with the inverse operation, division.
Answer: 576
Worked Example 10
Problem: Calculate 315 × 5.
- Break one factor into easier parts if helpful.
- Multiply carefully.
- Check with division.
Very beginner explanation: Multiplication combines equal groups and can be checked with the inverse operation, division.
Answer: 1575
Practice Exercise
Create one new question about Multiplication facts 0×0 to 12×12. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
2.2 Related division facts
Division facts are related to multiplication facts and use the inverse relationship between multiplication and division. This topic focuses on Related division facts .
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: Calculate 12 ÷ 8.
- 8 fits into 12 1 whole times.
- Remainder = 4.
- Check by multiplication and addition.
Very beginner explanation: Division facts are related to multiplication facts and use the inverse relationship between multiplication and division. This topic focuses on Related division facts .
Answer: 1 R 4
Worked Example 2
Problem: Calculate 9 ÷ 7.
- 7 fits into 9 1 whole times.
- Remainder = 2.
- Check by multiplication and addition.
Very beginner explanation: Division facts are related to multiplication facts and use the inverse relationship between multiplication and division. This topic focuses on Related division facts .
Answer: 1 R 2
Worked Example 3
Problem: Calculate 144 ÷ 12.
- 12 fits into 144 12 whole times.
- Remainder = 0.
- Check by multiplication and addition.
Very beginner explanation: Division facts are related to multiplication facts and use the inverse relationship between multiplication and division. This topic focuses on Related division facts .
Answer: 12
Worked Example 4
Problem: Calculate 325 ÷ 6.
- 6 fits into 325 54 whole times.
- Remainder = 1.
- Check by multiplication and addition.
Very beginner explanation: Division facts are related to multiplication facts and use the inverse relationship between multiplication and division. This topic focuses on Related division facts .
Answer: 54 R 1
Worked Example 5
Problem: Calculate 728 ÷ 8.
- 8 fits into 728 91 whole times.
- Remainder = 0.
- Check by multiplication and addition.
Very beginner explanation: Division facts are related to multiplication facts and use the inverse relationship between multiplication and division. This topic focuses on Related division facts .
Answer: 91
Worked Example 6
Problem: Calculate 84 ÷ 7.
- Ask how many groups of 7 fit in 84.
- 7 × 12 = 84.
- Remainder = 0.
Very beginner explanation: Division can be checked by multiplying the quotient by the divisor and adding any remainder.
Answer: 12
Worked Example 7
Problem: Calculate 125 ÷ 8.
- Ask how many groups of 8 fit in 125.
- 8 × 15 = 120.
- Remainder = 5.
Very beginner explanation: Division can be checked by multiplying the quotient by the divisor and adding any remainder.
Answer: 15 R 5
Worked Example 8
Problem: Calculate 936 ÷ 9.
- Ask how many groups of 9 fit in 936.
- 9 × 104 = 936.
- Remainder = 0.
Very beginner explanation: Division can be checked by multiplying the quotient by the divisor and adding any remainder.
Answer: 104
Worked Example 9
Problem: Calculate 48 ÷ 12.
- Ask how many groups of 12 fit in 48.
- 12 × 4 = 48.
- Remainder = 0.
Very beginner explanation: Division can be checked by multiplying the quotient by the divisor and adding any remainder.
Answer: 4
Worked Example 10
Problem: Calculate 315 ÷ 5.
- Ask how many groups of 5 fit in 315.
- 5 × 63 = 315.
- Remainder = 0.
Very beginner explanation: Division can be checked by multiplying the quotient by the divisor and adding any remainder.
Answer: 63
Practice Exercise
Create one new question about Related division facts. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
2.3 Mental multiplication strategies
A tip is an extra amount, often calculated as a percent of a bill. This topic focuses on Mental multiplication strategies .
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: Calculate 12 × 8.
- Use a known fact, distributive strategy, or standard multiplication.
- Check using division: 96 ÷ 8 = 12.
Very beginner explanation: A tip is an extra amount, often calculated as a percent of a bill. This topic focuses on Mental multiplication strategies .
Answer: 96
Worked Example 2
Problem: Calculate 9 × 7.
- Use a known fact, distributive strategy, or standard multiplication.
- Check using division: 63 ÷ 7 = 9.
Very beginner explanation: A tip is an extra amount, often calculated as a percent of a bill. This topic focuses on Mental multiplication strategies .
Answer: 63
Worked Example 3
Problem: Calculate 144 × 12.
- Use a known fact, distributive strategy, or standard multiplication.
- Check using division: 1728 ÷ 12 = 144.
Very beginner explanation: A tip is an extra amount, often calculated as a percent of a bill. This topic focuses on Mental multiplication strategies .
Answer: 1728
Worked Example 4
Problem: Calculate 325 × 6.
- Use a known fact, distributive strategy, or standard multiplication.
- Check using division: 1950 ÷ 6 = 325.
Very beginner explanation: A tip is an extra amount, often calculated as a percent of a bill. This topic focuses on Mental multiplication strategies .
Answer: 1950
Worked Example 5
Problem: Calculate 728 × 8.
- Use a known fact, distributive strategy, or standard multiplication.
- Check using division: 5824 ÷ 8 = 728.
Very beginner explanation: A tip is an extra amount, often calculated as a percent of a bill. This topic focuses on Mental multiplication strategies .
Answer: 5824
Worked Example 6
Problem: Calculate 84 × 7.
- Break one factor into easier parts if helpful.
- Multiply carefully.
- Check with division.
Very beginner explanation: Multiplication combines equal groups and can be checked with the inverse operation, division.
Answer: 588
Worked Example 7
Problem: Calculate 125 × 8.
- Break one factor into easier parts if helpful.
- Multiply carefully.
- Check with division.
Very beginner explanation: Multiplication combines equal groups and can be checked with the inverse operation, division.
Answer: 1000
Worked Example 8
Problem: Calculate 936 × 9.
- Break one factor into easier parts if helpful.
- Multiply carefully.
- Check with division.
Very beginner explanation: Multiplication combines equal groups and can be checked with the inverse operation, division.
Answer: 8424
Worked Example 9
Problem: Calculate 48 × 12.
- Break one factor into easier parts if helpful.
- Multiply carefully.
- Check with division.
Very beginner explanation: Multiplication combines equal groups and can be checked with the inverse operation, division.
Answer: 576
Worked Example 10
Problem: Calculate 315 × 5.
- Break one factor into easier parts if helpful.
- Multiply carefully.
- Check with division.
Very beginner explanation: Multiplication combines equal groups and can be checked with the inverse operation, division.
Answer: 1575
Practice Exercise
Create one new question about Mental multiplication strategies. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
2.4 Mental division strategies
A rate compares two quantities measured in different units. This topic focuses on Mental division strategies .
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: Calculate 12 ÷ 8.
- 8 fits into 12 1 whole times.
- Remainder = 4.
- Check by multiplication and addition.
Very beginner explanation: A rate compares two quantities measured in different units. This topic focuses on Mental division strategies .
Answer: 1 R 4
Worked Example 2
Problem: Calculate 9 ÷ 7.
- 7 fits into 9 1 whole times.
- Remainder = 2.
- Check by multiplication and addition.
Very beginner explanation: A rate compares two quantities measured in different units. This topic focuses on Mental division strategies .
Answer: 1 R 2
Worked Example 3
Problem: Calculate 144 ÷ 12.
- 12 fits into 144 12 whole times.
- Remainder = 0.
- Check by multiplication and addition.
Very beginner explanation: A rate compares two quantities measured in different units. This topic focuses on Mental division strategies .
Answer: 12
Worked Example 4
Problem: Calculate 325 ÷ 6.
- 6 fits into 325 54 whole times.
- Remainder = 1.
- Check by multiplication and addition.
Very beginner explanation: A rate compares two quantities measured in different units. This topic focuses on Mental division strategies .
Answer: 54 R 1
Worked Example 5
Problem: Calculate 728 ÷ 8.
- 8 fits into 728 91 whole times.
- Remainder = 0.
- Check by multiplication and addition.
Very beginner explanation: A rate compares two quantities measured in different units. This topic focuses on Mental division strategies .
Answer: 91
Worked Example 6
Problem: Calculate 84 ÷ 7.
- Ask how many groups of 7 fit in 84.
- 7 × 12 = 84.
- Remainder = 0.
Very beginner explanation: Division can be checked by multiplying the quotient by the divisor and adding any remainder.
Answer: 12
Worked Example 7
Problem: Calculate 125 ÷ 8.
- Ask how many groups of 8 fit in 125.
- 8 × 15 = 120.
- Remainder = 5.
Very beginner explanation: Division can be checked by multiplying the quotient by the divisor and adding any remainder.
Answer: 15 R 5
Worked Example 8
Problem: Calculate 936 ÷ 9.
- Ask how many groups of 9 fit in 936.
- 9 × 104 = 936.
- Remainder = 0.
Very beginner explanation: Division can be checked by multiplying the quotient by the divisor and adding any remainder.
Answer: 104
Worked Example 9
Problem: Calculate 48 ÷ 12.
- Ask how many groups of 12 fit in 48.
- 12 × 4 = 48.
- Remainder = 0.
Very beginner explanation: Division can be checked by multiplying the quotient by the divisor and adding any remainder.
Answer: 4
Worked Example 10
Problem: Calculate 315 ÷ 5.
- Ask how many groups of 5 fit in 315.
- 5 × 63 = 315.
- Remainder = 0.
Very beginner explanation: Division can be checked by multiplying the quotient by the divisor and adding any remainder.
Answer: 63
Practice Exercise
Create one new question about Mental division strategies. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
2.5 Multiplying multi-digit whole numbers
A tip is an extra amount, often calculated as a percent of a bill. This topic focuses on Multiplying multi-digit whole numbers .
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: Calculate 12 × 8.
- Use a known fact, distributive strategy, or standard multiplication.
- Check using division: 96 ÷ 8 = 12.
Very beginner explanation: A tip is an extra amount, often calculated as a percent of a bill. This topic focuses on Multiplying multi-digit whole numbers .
Answer: 96
Worked Example 2
Problem: Calculate 9 × 7.
- Use a known fact, distributive strategy, or standard multiplication.
- Check using division: 63 ÷ 7 = 9.
Very beginner explanation: A tip is an extra amount, often calculated as a percent of a bill. This topic focuses on Multiplying multi-digit whole numbers .
Answer: 63
Worked Example 3
Problem: Calculate 144 × 12.
- Use a known fact, distributive strategy, or standard multiplication.
- Check using division: 1728 ÷ 12 = 144.
Very beginner explanation: A tip is an extra amount, often calculated as a percent of a bill. This topic focuses on Multiplying multi-digit whole numbers .
Answer: 1728
Worked Example 4
Problem: Calculate 325 × 6.
- Use a known fact, distributive strategy, or standard multiplication.
- Check using division: 1950 ÷ 6 = 325.
Very beginner explanation: A tip is an extra amount, often calculated as a percent of a bill. This topic focuses on Multiplying multi-digit whole numbers .
Answer: 1950
Worked Example 5
Problem: Calculate 728 × 8.
- Use a known fact, distributive strategy, or standard multiplication.
- Check using division: 5824 ÷ 8 = 728.
Very beginner explanation: A tip is an extra amount, often calculated as a percent of a bill. This topic focuses on Multiplying multi-digit whole numbers .
Answer: 5824
Worked Example 6
Problem: What is the value of the first digit in 638,420,715?
- Find the position of the first digit.
- Multiply the digit by its place value.
Very beginner explanation: Place value tells the value of a digit because of where it is located.
Answer: 600,000,000
Worked Example 7
Problem: What is the value of the first digit in 92,305,004?
- Find the position of the first digit.
- Multiply the digit by its place value.
Very beginner explanation: Place value tells the value of a digit because of where it is located.
Answer: 90,000,000
Worked Example 8
Problem: What is the value of the first digit in 704,090,650?
- Find the position of the first digit.
- Multiply the digit by its place value.
Very beginner explanation: Place value tells the value of a digit because of where it is located.
Answer: 700,000,000
Worked Example 9
Problem: What is the value of the first digit in 18,765,432?
- Find the position of the first digit.
- Multiply the digit by its place value.
Very beginner explanation: Place value tells the value of a digit because of where it is located.
Answer: 10,000,000
Worked Example 10
Problem: What is the value of the first digit in 999,500,001?
- Find the position of the first digit.
- Multiply the digit by its place value.
Very beginner explanation: Place value tells the value of a digit because of where it is located.
Answer: 900,000,000
Practice Exercise
Create one new question about Multiplying multi-digit whole numbers. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
2.6 Dividing whole numbers
Dividing whole numbers is an important Grade 7 idea in Multiplication, Division, and Mental Math Fluency. 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: Calculate 12 × 8.
- Use a known fact, distributive strategy, or standard multiplication.
- Check using division: 96 ÷ 8 = 12.
Very beginner explanation: Dividing whole numbers is an important Grade 7 idea in Multiplication, Division, and Mental Math Fluency. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: 96
Worked Example 2
Problem: Calculate 9 × 7.
- Use a known fact, distributive strategy, or standard multiplication.
- Check using division: 63 ÷ 7 = 9.
Very beginner explanation: Dividing whole numbers is an important Grade 7 idea in Multiplication, Division, and Mental Math Fluency. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: 63
Worked Example 3
Problem: Calculate 144 × 12.
- Use a known fact, distributive strategy, or standard multiplication.
- Check using division: 1728 ÷ 12 = 144.
Very beginner explanation: Dividing whole numbers is an important Grade 7 idea in Multiplication, Division, and Mental Math Fluency. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: 1728
Worked Example 4
Problem: Calculate 325 × 6.
- Use a known fact, distributive strategy, or standard multiplication.
- Check using division: 1950 ÷ 6 = 325.
Very beginner explanation: Dividing whole numbers is an important Grade 7 idea in Multiplication, Division, and Mental Math Fluency. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: 1950
Worked Example 5
Problem: Calculate 728 × 8.
- Use a known fact, distributive strategy, or standard multiplication.
- Check using division: 5824 ÷ 8 = 728.
Very beginner explanation: Dividing whole numbers is an important Grade 7 idea in Multiplication, Division, and Mental Math Fluency. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: 5824
Worked Example 6
Problem: What is the value of the first digit in 638,420,715?
- Find the position of the first digit.
- Multiply the digit by its place value.
Very beginner explanation: Place value tells the value of a digit because of where it is located.
Answer: 600,000,000
Worked Example 7
Problem: What is the value of the first digit in 92,305,004?
- Find the position of the first digit.
- Multiply the digit by its place value.
Very beginner explanation: Place value tells the value of a digit because of where it is located.
Answer: 90,000,000
Worked Example 8
Problem: What is the value of the first digit in 704,090,650?
- Find the position of the first digit.
- Multiply the digit by its place value.
Very beginner explanation: Place value tells the value of a digit because of where it is located.
Answer: 700,000,000
Worked Example 9
Problem: What is the value of the first digit in 18,765,432?
- Find the position of the first digit.
- Multiply the digit by its place value.
Very beginner explanation: Place value tells the value of a digit because of where it is located.
Answer: 10,000,000
Worked Example 10
Problem: What is the value of the first digit in 999,500,001?
- Find the position of the first digit.
- Multiply the digit by its place value.
Very beginner explanation: Place value tells the value of a digit because of where it is located.
Answer: 900,000,000
Practice Exercise
Create one new question about Dividing whole numbers. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
2.7 Remainders
Remainders is an important Grade 7 idea in Multiplication, Division, and Mental Math Fluency. 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: Calculate 12 ÷ 8.
- 8 fits into 12 1 whole times.
- Remainder = 4.
- Check by multiplication and addition.
Very beginner explanation: Remainders is an important Grade 7 idea in Multiplication, Division, and Mental Math Fluency. 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 R 4
Worked Example 2
Problem: Calculate 9 ÷ 7.
- 7 fits into 9 1 whole times.
- Remainder = 2.
- Check by multiplication and addition.
Very beginner explanation: Remainders is an important Grade 7 idea in Multiplication, Division, and Mental Math Fluency. 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 R 2
Worked Example 3
Problem: Calculate 144 ÷ 12.
- 12 fits into 144 12 whole times.
- Remainder = 0.
- Check by multiplication and addition.
Very beginner explanation: Remainders is an important Grade 7 idea in Multiplication, Division, and Mental Math Fluency. 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 4
Problem: Calculate 325 ÷ 6.
- 6 fits into 325 54 whole times.
- Remainder = 1.
- Check by multiplication and addition.
Very beginner explanation: Remainders is an important Grade 7 idea in Multiplication, Division, and Mental Math Fluency. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: 54 R 1
Worked Example 5
Problem: Calculate 728 ÷ 8.
- 8 fits into 728 91 whole times.
- Remainder = 0.
- Check by multiplication and addition.
Very beginner explanation: Remainders is an important Grade 7 idea in Multiplication, Division, and Mental Math Fluency. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: 91
Worked Example 6
Problem: Calculate 84 ÷ 7.
- Ask how many groups of 7 fit in 84.
- 7 × 12 = 84.
- Remainder = 0.
Very beginner explanation: Division can be checked by multiplying the quotient by the divisor and adding any remainder.
Answer: 12
Worked Example 7
Problem: Calculate 125 ÷ 8.
- Ask how many groups of 8 fit in 125.
- 8 × 15 = 120.
- Remainder = 5.
Very beginner explanation: Division can be checked by multiplying the quotient by the divisor and adding any remainder.
Answer: 15 R 5
Worked Example 8
Problem: Calculate 936 ÷ 9.
- Ask how many groups of 9 fit in 936.
- 9 × 104 = 936.
- Remainder = 0.
Very beginner explanation: Division can be checked by multiplying the quotient by the divisor and adding any remainder.
Answer: 104
Worked Example 9
Problem: Calculate 48 ÷ 12.
- Ask how many groups of 12 fit in 48.
- 12 × 4 = 48.
- Remainder = 0.
Very beginner explanation: Division can be checked by multiplying the quotient by the divisor and adding any remainder.
Answer: 4
Worked Example 10
Problem: Calculate 315 ÷ 5.
- Ask how many groups of 5 fit in 315.
- 5 × 63 = 315.
- Remainder = 0.
Very beginner explanation: Division can be checked by multiplying the quotient by the divisor and adding any remainder.
Answer: 63
Practice Exercise
Create one new question about Remainders. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
2.8 Estimating products
Estimating products is an important Grade 7 idea in Multiplication, Division, and Mental Math Fluency. 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: Calculate 12 × 8.
- Use a known fact, distributive strategy, or standard multiplication.
- Check using division: 96 ÷ 8 = 12.
Very beginner explanation: Estimating products is an important Grade 7 idea in Multiplication, Division, and Mental Math Fluency. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: 96
Worked Example 2
Problem: Calculate 9 × 7.
- Use a known fact, distributive strategy, or standard multiplication.
- Check using division: 63 ÷ 7 = 9.
Very beginner explanation: Estimating products is an important Grade 7 idea in Multiplication, Division, and Mental Math Fluency. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: 63
Worked Example 3
Problem: Calculate 144 × 12.
- Use a known fact, distributive strategy, or standard multiplication.
- Check using division: 1728 ÷ 12 = 144.
Very beginner explanation: Estimating products is an important Grade 7 idea in Multiplication, Division, and Mental Math Fluency. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: 1728
Worked Example 4
Problem: Calculate 325 × 6.
- Use a known fact, distributive strategy, or standard multiplication.
- Check using division: 1950 ÷ 6 = 325.
Very beginner explanation: Estimating products is an important Grade 7 idea in Multiplication, Division, and Mental Math Fluency. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: 1950
Worked Example 5
Problem: Calculate 728 × 8.
- Use a known fact, distributive strategy, or standard multiplication.
- Check using division: 5824 ÷ 8 = 728.
Very beginner explanation: Estimating products is an important Grade 7 idea in Multiplication, Division, and Mental Math Fluency. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: 5824
Worked Example 6
Problem: Calculate 84 × 7.
- Break one factor into easier parts if helpful.
- Multiply carefully.
- Check with division.
Very beginner explanation: Multiplication combines equal groups and can be checked with the inverse operation, division.
Answer: 588
Worked Example 7
Problem: Calculate 125 × 8.
- Break one factor into easier parts if helpful.
- Multiply carefully.
- Check with division.
Very beginner explanation: Multiplication combines equal groups and can be checked with the inverse operation, division.
Answer: 1000
Worked Example 8
Problem: Calculate 936 × 9.
- Break one factor into easier parts if helpful.
- Multiply carefully.
- Check with division.
Very beginner explanation: Multiplication combines equal groups and can be checked with the inverse operation, division.
Answer: 8424
Worked Example 9
Problem: Calculate 48 × 12.
- Break one factor into easier parts if helpful.
- Multiply carefully.
- Check with division.
Very beginner explanation: Multiplication combines equal groups and can be checked with the inverse operation, division.
Answer: 576
Worked Example 10
Problem: Calculate 315 × 5.
- Break one factor into easier parts if helpful.
- Multiply carefully.
- Check with division.
Very beginner explanation: Multiplication combines equal groups and can be checked with the inverse operation, division.
Answer: 1575
Practice Exercise
Create one new question about Estimating products. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
2.9 Estimating quotients
Estimating quotients is an important Grade 7 idea in Multiplication, Division, and Mental Math Fluency. 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: Calculate 12 ÷ 8.
- 8 fits into 12 1 whole times.
- Remainder = 4.
- Check by multiplication and addition.
Very beginner explanation: Estimating quotients is an important Grade 7 idea in Multiplication, Division, and Mental Math Fluency. 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 R 4
Worked Example 2
Problem: Calculate 9 ÷ 7.
- 7 fits into 9 1 whole times.
- Remainder = 2.
- Check by multiplication and addition.
Very beginner explanation: Estimating quotients is an important Grade 7 idea in Multiplication, Division, and Mental Math Fluency. 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 R 2
Worked Example 3
Problem: Calculate 144 ÷ 12.
- 12 fits into 144 12 whole times.
- Remainder = 0.
- Check by multiplication and addition.
Very beginner explanation: Estimating quotients is an important Grade 7 idea in Multiplication, Division, and Mental Math Fluency. 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 4
Problem: Calculate 325 ÷ 6.
- 6 fits into 325 54 whole times.
- Remainder = 1.
- Check by multiplication and addition.
Very beginner explanation: Estimating quotients is an important Grade 7 idea in Multiplication, Division, and Mental Math Fluency. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: 54 R 1
Worked Example 5
Problem: Calculate 728 ÷ 8.
- 8 fits into 728 91 whole times.
- Remainder = 0.
- Check by multiplication and addition.
Very beginner explanation: Estimating quotients is an important Grade 7 idea in Multiplication, Division, and Mental Math Fluency. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: 91
Worked Example 6
Problem: Calculate 84 ÷ 7.
- Ask how many groups of 7 fit in 84.
- 7 × 12 = 84.
- Remainder = 0.
Very beginner explanation: Division can be checked by multiplying the quotient by the divisor and adding any remainder.
Answer: 12
Worked Example 7
Problem: Calculate 125 ÷ 8.
- Ask how many groups of 8 fit in 125.
- 8 × 15 = 120.
- Remainder = 5.
Very beginner explanation: Division can be checked by multiplying the quotient by the divisor and adding any remainder.
Answer: 15 R 5
Worked Example 8
Problem: Calculate 936 ÷ 9.
- Ask how many groups of 9 fit in 936.
- 9 × 104 = 936.
- Remainder = 0.
Very beginner explanation: Division can be checked by multiplying the quotient by the divisor and adding any remainder.
Answer: 104
Worked Example 9
Problem: Calculate 48 ÷ 12.
- Ask how many groups of 12 fit in 48.
- 12 × 4 = 48.
- Remainder = 0.
Very beginner explanation: Division can be checked by multiplying the quotient by the divisor and adding any remainder.
Answer: 4
Worked Example 10
Problem: Calculate 315 ÷ 5.
- Ask how many groups of 5 fit in 315.
- 5 × 63 = 315.
- Remainder = 0.
Very beginner explanation: Division can be checked by multiplying the quotient by the divisor and adding any remainder.
Answer: 63
Practice Exercise
Create one new question about Estimating quotients. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
2.10 Checking with inverse operations
A ratio compares two quantities in a fixed order. This topic focuses on Checking with inverse operations .
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: Calculate 12 × 8.
- Use a known fact, distributive strategy, or standard multiplication.
- Check using division: 96 ÷ 8 = 12.
Very beginner explanation: A ratio compares two quantities in a fixed order. This topic focuses on Checking with inverse operations .
Answer: 96
Worked Example 2
Problem: Calculate 9 × 7.
- Use a known fact, distributive strategy, or standard multiplication.
- Check using division: 63 ÷ 7 = 9.
Very beginner explanation: A ratio compares two quantities in a fixed order. This topic focuses on Checking with inverse operations .
Answer: 63
Worked Example 3
Problem: Calculate 144 × 12.
- Use a known fact, distributive strategy, or standard multiplication.
- Check using division: 1728 ÷ 12 = 144.
Very beginner explanation: A ratio compares two quantities in a fixed order. This topic focuses on Checking with inverse operations .
Answer: 1728
Worked Example 4
Problem: Calculate 325 × 6.
- Use a known fact, distributive strategy, or standard multiplication.
- Check using division: 1950 ÷ 6 = 325.
Very beginner explanation: A ratio compares two quantities in a fixed order. This topic focuses on Checking with inverse operations .
Answer: 1950
Worked Example 5
Problem: Calculate 728 × 8.
- Use a known fact, distributive strategy, or standard multiplication.
- Check using division: 5824 ÷ 8 = 728.
Very beginner explanation: A ratio compares two quantities in a fixed order. This topic focuses on Checking with inverse operations .
Answer: 5824
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 Checking with inverse operations. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
2.11 Real-life multiplication and division
A tip is an extra amount, often calculated as a percent of a bill. This topic focuses on Real-life 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 Real-life 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 Real-life 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 Real-life 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 Real-life 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 Real-life 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 Real-life multiplication and division. 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 Multiplication facts 0×0 to 12×12 . Show your reasoning and check your answer.
- Create and solve a new question about Related division facts . Show your reasoning and check your answer.
- Create and solve a new question about Mental multiplication strategies . Show your reasoning and check your answer.
- Create and solve a new question about Mental division strategies . Show your reasoning and check your answer.
- Create and solve a new question about Multiplying multi-digit whole numbers . Show your reasoning and check your answer.
- Create and solve a new question about Dividing whole numbers . Show your reasoning and check your answer.
- Create and solve a new question about Remainders . Show your reasoning and check your answer.
- Create and solve a new question about Estimating products . Show your reasoning and check your answer.
- Create and solve a new question about Estimating quotients . Show your reasoning and check your answer.
- Create and solve a new question about Checking with inverse operations . Show your reasoning and check your answer.
- Create and solve a new question about Real-life multiplication and division . 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 Multiplication facts 0×0 to 12×12?
Answer: Multiplication facts are basic products that should become automatic so larger calculations are easier. This topic focuses on Multiplication facts 0×0 to 12×12.
Q2. What is important to remember about Related division facts?
Answer: Division facts are related to multiplication facts and use the inverse relationship between multiplication and division. This topic focuses on Related division facts.
Q3. What is important to remember about Mental multiplication strategies?
Answer: A tip is an extra amount, often calculated as a percent of a bill. This topic focuses on Mental multiplication strategies.
Q4. What is important to remember about Mental division strategies?
Answer: A rate compares two quantities measured in different units. This topic focuses on Mental division strategies.
Q5. What is important to remember about Multiplying multi-digit whole numbers?
Answer: A tip is an extra amount, often calculated as a percent of a bill. This topic focuses on Multiplying multi-digit whole numbers.
Q6. What is important to remember about Dividing whole numbers?
Answer: Dividing whole numbers is an important Grade 7 idea in Multiplication, Division, and Mental Math Fluency. 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 Remainders?
Answer: Remainders is an important Grade 7 idea in Multiplication, Division, and Mental Math Fluency. 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 Estimating products?
Answer: Estimating products is an important Grade 7 idea in Multiplication, Division, and Mental Math Fluency. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Q9. What is important to remember about Estimating quotients?
Answer: Estimating quotients is an important Grade 7 idea in Multiplication, Division, and Mental Math Fluency. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Q10. What is important to remember about Checking with inverse operations?
Answer: A ratio compares two quantities in a fixed order. This topic focuses on Checking with inverse operations.
Q11. What is important to remember about Real-life multiplication and division?
Answer: A tip is an extra amount, often calculated as a percent of a bill. This topic focuses on Real-life multiplication and division.
Q12. Why should you show your steps?
Answer: Showing steps makes reasoning easier to verify and helps locate mistakes.
Q13. Why is estimation useful?
Answer: Estimation helps you judge whether a final answer is reasonable before accepting it.
Q14. Why do units matter?
Answer: Units tell what a value measures and prevent incompatible quantities from being mixed.
Q15. When should you round?
Answer: Usually round near the end of a calculation unless the question specifically asks for earlier rounding.
Q16. How can you check an answer?
Answer: Use an inverse operation, substitution, estimation, a graph, or a second method.
Q17. What should you identify first in a word problem?
Answer: Identify what is known, what is unknown, and the relationship connecting them.
Q18. What makes a final answer complete?
Answer: Give the value, correct units when needed, and a brief interpretation in context.
Q19. What should you do if an answer seems unreasonable?
Answer: Re-read the question, check copied values, signs, units, formulas, and calculations.
Q20. Why are diagrams useful?
Answer: A labelled diagram can reveal relationships between lengths, angles, areas, and other quantities.
Q21. Why are tables useful?
Answer: Tables organize values and help reveal patterns, relationships, rates, and missing information.
Q22. Why should you explain your reasoning?
Answer: Reasoning shows why a method works, not just what answer you obtained.
Q23. How do mistakes help learning?
Answer: Analyzing a mistake identifies the misunderstood rule or step and helps prevent the same error later.
Q24. How should you study this chapter?
Answer: Review definitions, redo worked examples without looking, practise mixed questions, and explain solutions in your own words.
Q25. 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.
Q26. Why compare more than one strategy?
Answer: Different strategies can make a problem easier and provide a way to verify the result.
Q27. What is a mathematical model?
Answer: It is a simplified mathematical representation of a real situation used to analyze, explain, or predict.
Q28. Why should assumptions be stated?
Answer: Assumptions show what conditions the solution depends on and help readers judge whether the model is reasonable.
Q29. Why should sources be checked in data problems?
Answer: Reliable sources and fair collection methods make conclusions more trustworthy.
Q30. What is mathematical communication?
Answer: It is presenting mathematical ideas clearly using words, symbols, diagrams, tables, graphs, and justified steps.