Chapter 13: Properties and Relationships of Operations
Learn Grade 4 mathematics with very-beginner explanations, original worked examples, practice, reasoning, and review.
Chapter Overview
This chapter teaches Properties and Relationships of Operations in original, simple language with step-by-step reasoning, worked examples, practice exercises, common mistakes, and review questions.
13.1 Addition and subtraction are related
Addition combines quantities. Align place values, regroup when needed, and estimate first to predict a reasonable answer. Work one step at a time, use a model or number sentence when useful, and check whether the answer is reasonable.
Beginner Explanation
First identify what is given and what must be found. Then choose a strategy, show the steps clearly, and check the final answer.
10 Worked Examples with Answers
Worked Example 1
Problem: Calculate 2,891 + 1,932.
- Line up place values.
- Add from right to left.
- Regroup when needed.
Very beginner explanation: Place-value alignment keeps like places together.
Answer: 4,823
Worked Example 2
Problem: Calculate 2,709 + 1,890.
- Line up place values.
- Add from right to left.
- Regroup when needed.
Very beginner explanation: Place-value alignment keeps like places together.
Answer: 4,599
Worked Example 3
Problem: Calculate 2,214 + 322.
- Line up place values.
- Add from right to left.
- Regroup when needed.
Very beginner explanation: Place-value alignment keeps like places together.
Answer: 2,536
Worked Example 4
Problem: Calculate 974 + 2,892.
- Line up place values.
- Add from right to left.
- Regroup when needed.
Very beginner explanation: Place-value alignment keeps like places together.
Answer: 3,866
Worked Example 5
Problem: Calculate 3,915 + 1,563.
- Line up place values.
- Add from right to left.
- Regroup when needed.
Very beginner explanation: Place-value alignment keeps like places together.
Answer: 5,478
Worked Example 6
Problem: Calculate 3,798 + 1,090.
- Line up place values.
- Add from right to left.
- Regroup when needed.
Very beginner explanation: Place-value alignment keeps like places together.
Answer: 4,888
Worked Example 7
Problem: Calculate 2,003 + 2,100.
- Line up place values.
- Add from right to left.
- Regroup when needed.
Very beginner explanation: Place-value alignment keeps like places together.
Answer: 4,103
Worked Example 8
Problem: Calculate 2,884 + 1,847.
- Line up place values.
- Add from right to left.
- Regroup when needed.
Very beginner explanation: Place-value alignment keeps like places together.
Answer: 4,731
Worked Example 9
Problem: Calculate 786 + 671.
- Line up place values.
- Add from right to left.
- Regroup when needed.
Very beginner explanation: Place-value alignment keeps like places together.
Answer: 1,457
Worked Example 10
Problem: Calculate 3,824 + 2,064.
- Line up place values.
- Add from right to left.
- Regroup when needed.
Very beginner explanation: Place-value alignment keeps like places together.
Answer: 5,888
Practice Exercise
Create and solve one new example of Addition and subtraction are related. Show the steps and explain how you checked it.
Common Mistake
Do not rush into a calculation before understanding the question. Check place values, labels, units, and whether your answer fits the situation.
13.2 Multiplication and division are related
Multiplication combines equal groups. Arrays, place value, known facts, and breaking apart numbers can all help. Work one step at a time, use a model or number sentence when useful, and check whether the answer is reasonable.
Beginner Explanation
First identify what is given and what must be found. Then choose a strategy, show the steps clearly, and check the final answer.
10 Worked Examples with Answers
Worked Example 1
Problem: Calculate 3 × 6.
- Break 3 apart if helpful.
- Multiply each part by the other factor.
- Combine the partial products.
Very beginner explanation: Multiplication finds a total made from equal groups.
Answer: 18
Worked Example 2
Problem: Calculate 6 × 4.
- Break 6 apart if helpful.
- Multiply each part by the other factor.
- Combine the partial products.
Very beginner explanation: Multiplication finds a total made from equal groups.
Answer: 24
Worked Example 3
Problem: Calculate 5 × 3.
- Break 5 apart if helpful.
- Multiply each part by the other factor.
- Combine the partial products.
Very beginner explanation: Multiplication finds a total made from equal groups.
Answer: 15
Worked Example 4
Problem: Calculate 10 × 5.
- Break 10 apart if helpful.
- Multiply each part by the other factor.
- Combine the partial products.
Very beginner explanation: Multiplication finds a total made from equal groups.
Answer: 50
Worked Example 5
Problem: Calculate 10 × 8.
- Break 10 apart if helpful.
- Multiply each part by the other factor.
- Combine the partial products.
Very beginner explanation: Multiplication finds a total made from equal groups.
Answer: 80
Worked Example 6
Problem: Calculate 8 × 2.
- Break 8 apart if helpful.
- Multiply each part by the other factor.
- Combine the partial products.
Very beginner explanation: Multiplication finds a total made from equal groups.
Answer: 16
Worked Example 7
Problem: Calculate 12 × 4.
- Break 12 apart if helpful.
- Multiply each part by the other factor.
- Combine the partial products.
Very beginner explanation: Multiplication finds a total made from equal groups.
Answer: 48
Worked Example 8
Problem: Calculate 7 × 6.
- Break 7 apart if helpful.
- Multiply each part by the other factor.
- Combine the partial products.
Very beginner explanation: Multiplication finds a total made from equal groups.
Answer: 42
Worked Example 9
Problem: Calculate 11 × 9.
- Break 11 apart if helpful.
- Multiply each part by the other factor.
- Combine the partial products.
Very beginner explanation: Multiplication finds a total made from equal groups.
Answer: 99
Worked Example 10
Problem: Calculate 7 × 4.
- Break 7 apart if helpful.
- Multiply each part by the other factor.
- Combine the partial products.
Very beginner explanation: Multiplication finds a total made from equal groups.
Answer: 28
Practice Exercise
Create and solve one new example of Multiplication and division are related. Show the steps and explain how you checked it.
Common Mistake
Do not rush into a calculation before understanding the question. Check place values, labels, units, and whether your answer fits the situation.
13.3 Commutative property of addition
Addition combines quantities. Align place values, regroup when needed, and estimate first to predict a reasonable answer. Work one step at a time, use a model or number sentence when useful, and check whether the answer is reasonable.
Beginner Explanation
First identify what is given and what must be found. Then choose a strategy, show the steps clearly, and check the final answer.
10 Worked Examples with Answers
Worked Example 1
Problem: Calculate 828 + 1,418.
- Line up place values.
- Add from right to left.
- Regroup when needed.
Very beginner explanation: Place-value alignment keeps like places together.
Answer: 2,246
Worked Example 2
Problem: Calculate 3,426 + 1,794.
- Line up place values.
- Add from right to left.
- Regroup when needed.
Very beginner explanation: Place-value alignment keeps like places together.
Answer: 5,220
Worked Example 3
Problem: Calculate 3,472 + 409.
- Line up place values.
- Add from right to left.
- Regroup when needed.
Very beginner explanation: Place-value alignment keeps like places together.
Answer: 3,881
Worked Example 4
Problem: Calculate 982 + 2,789.
- Line up place values.
- Add from right to left.
- Regroup when needed.
Very beginner explanation: Place-value alignment keeps like places together.
Answer: 3,771
Worked Example 5
Problem: Calculate 3,143 + 1,671.
- Line up place values.
- Add from right to left.
- Regroup when needed.
Very beginner explanation: Place-value alignment keeps like places together.
Answer: 4,814
Worked Example 6
Problem: Calculate 930 + 2,382.
- Line up place values.
- Add from right to left.
- Regroup when needed.
Very beginner explanation: Place-value alignment keeps like places together.
Answer: 3,312
Worked Example 7
Problem: Calculate 2,304 + 1,224.
- Line up place values.
- Add from right to left.
- Regroup when needed.
Very beginner explanation: Place-value alignment keeps like places together.
Answer: 3,528
Worked Example 8
Problem: Calculate 421 + 2,174.
- Line up place values.
- Add from right to left.
- Regroup when needed.
Very beginner explanation: Place-value alignment keeps like places together.
Answer: 2,595
Worked Example 9
Problem: Calculate 942 + 1,954.
- Line up place values.
- Add from right to left.
- Regroup when needed.
Very beginner explanation: Place-value alignment keeps like places together.
Answer: 2,896
Worked Example 10
Problem: Calculate 327 + 1,400.
- Line up place values.
- Add from right to left.
- Regroup when needed.
Very beginner explanation: Place-value alignment keeps like places together.
Answer: 1,727
Practice Exercise
Create and solve one new example of Commutative property of addition. Show the steps and explain how you checked it.
Common Mistake
Do not rush into a calculation before understanding the question. Check place values, labels, units, and whether your answer fits the situation.
13.4 Commutative property of multiplication
Multiplication combines equal groups. Arrays, place value, known facts, and breaking apart numbers can all help. Work one step at a time, use a model or number sentence when useful, and check whether the answer is reasonable.
Beginner Explanation
First identify what is given and what must be found. Then choose a strategy, show the steps clearly, and check the final answer.
10 Worked Examples with Answers
Worked Example 1
Problem: Calculate 10 × 3.
- Break 10 apart if helpful.
- Multiply each part by the other factor.
- Combine the partial products.
Very beginner explanation: Multiplication finds a total made from equal groups.
Answer: 30
Worked Example 2
Problem: Calculate 5 × 9.
- Break 5 apart if helpful.
- Multiply each part by the other factor.
- Combine the partial products.
Very beginner explanation: Multiplication finds a total made from equal groups.
Answer: 45
Worked Example 3
Problem: Calculate 3 × 6.
- Break 3 apart if helpful.
- Multiply each part by the other factor.
- Combine the partial products.
Very beginner explanation: Multiplication finds a total made from equal groups.
Answer: 18
Worked Example 4
Problem: Calculate 7 × 7.
- Break 7 apart if helpful.
- Multiply each part by the other factor.
- Combine the partial products.
Very beginner explanation: Multiplication finds a total made from equal groups.
Answer: 49
Worked Example 5
Problem: Calculate 2 × 6.
- Break 2 apart if helpful.
- Multiply each part by the other factor.
- Combine the partial products.
Very beginner explanation: Multiplication finds a total made from equal groups.
Answer: 12
Worked Example 6
Problem: Calculate 8 × 8.
- Break 8 apart if helpful.
- Multiply each part by the other factor.
- Combine the partial products.
Very beginner explanation: Multiplication finds a total made from equal groups.
Answer: 64
Worked Example 7
Problem: Calculate 4 × 3.
- Break 4 apart if helpful.
- Multiply each part by the other factor.
- Combine the partial products.
Very beginner explanation: Multiplication finds a total made from equal groups.
Answer: 12
Worked Example 8
Problem: Calculate 12 × 6.
- Break 12 apart if helpful.
- Multiply each part by the other factor.
- Combine the partial products.
Very beginner explanation: Multiplication finds a total made from equal groups.
Answer: 72
Worked Example 9
Problem: Calculate 6 × 9.
- Break 6 apart if helpful.
- Multiply each part by the other factor.
- Combine the partial products.
Very beginner explanation: Multiplication finds a total made from equal groups.
Answer: 54
Worked Example 10
Problem: Calculate 8 × 3.
- Break 8 apart if helpful.
- Multiply each part by the other factor.
- Combine the partial products.
Very beginner explanation: Multiplication finds a total made from equal groups.
Answer: 24
Practice Exercise
Create and solve one new example of Commutative property of multiplication. Show the steps and explain how you checked it.
Common Mistake
Do not rush into a calculation before understanding the question. Check place values, labels, units, and whether your answer fits the situation.
13.5 Associative property of addition
Addition combines quantities. Align place values, regroup when needed, and estimate first to predict a reasonable answer. Work one step at a time, use a model or number sentence when useful, and check whether the answer is reasonable.
Beginner Explanation
First identify what is given and what must be found. Then choose a strategy, show the steps clearly, and check the final answer.
10 Worked Examples with Answers
Worked Example 1
Problem: Calculate 1,293 + 120.
- Line up place values.
- Add from right to left.
- Regroup when needed.
Very beginner explanation: Place-value alignment keeps like places together.
Answer: 1,413
Worked Example 2
Problem: Calculate 1,949 + 2,100.
- Line up place values.
- Add from right to left.
- Regroup when needed.
Very beginner explanation: Place-value alignment keeps like places together.
Answer: 4,049
Worked Example 3
Problem: Calculate 519 + 2,754.
- Line up place values.
- Add from right to left.
- Regroup when needed.
Very beginner explanation: Place-value alignment keeps like places together.
Answer: 3,273
Worked Example 4
Problem: Calculate 2,972 + 2,975.
- Line up place values.
- Add from right to left.
- Regroup when needed.
Very beginner explanation: Place-value alignment keeps like places together.
Answer: 5,947
Worked Example 5
Problem: Calculate 1,810 + 1,944.
- Line up place values.
- Add from right to left.
- Regroup when needed.
Very beginner explanation: Place-value alignment keeps like places together.
Answer: 3,754
Worked Example 6
Problem: Calculate 834 + 1,949.
- Line up place values.
- Add from right to left.
- Regroup when needed.
Very beginner explanation: Place-value alignment keeps like places together.
Answer: 2,783
Worked Example 7
Problem: Calculate 2,696 + 1,279.
- Line up place values.
- Add from right to left.
- Regroup when needed.
Very beginner explanation: Place-value alignment keeps like places together.
Answer: 3,975
Worked Example 8
Problem: Calculate 114 + 2,104.
- Line up place values.
- Add from right to left.
- Regroup when needed.
Very beginner explanation: Place-value alignment keeps like places together.
Answer: 2,218
Worked Example 9
Problem: Calculate 580 + 1,951.
- Line up place values.
- Add from right to left.
- Regroup when needed.
Very beginner explanation: Place-value alignment keeps like places together.
Answer: 2,531
Worked Example 10
Problem: Calculate 3,985 + 1,958.
- Line up place values.
- Add from right to left.
- Regroup when needed.
Very beginner explanation: Place-value alignment keeps like places together.
Answer: 5,943
Practice Exercise
Create and solve one new example of Associative property of addition. Show the steps and explain how you checked it.
Common Mistake
Do not rush into a calculation before understanding the question. Check place values, labels, units, and whether your answer fits the situation.
13.6 Associative property of multiplication
Multiplication combines equal groups. Arrays, place value, known facts, and breaking apart numbers can all help. Work one step at a time, use a model or number sentence when useful, and check whether the answer is reasonable.
Beginner Explanation
First identify what is given and what must be found. Then choose a strategy, show the steps clearly, and check the final answer.
10 Worked Examples with Answers
Worked Example 1
Problem: Calculate 3 × 7.
- Break 3 apart if helpful.
- Multiply each part by the other factor.
- Combine the partial products.
Very beginner explanation: Multiplication finds a total made from equal groups.
Answer: 21
Worked Example 2
Problem: Calculate 3 × 5.
- Break 3 apart if helpful.
- Multiply each part by the other factor.
- Combine the partial products.
Very beginner explanation: Multiplication finds a total made from equal groups.
Answer: 15
Worked Example 3
Problem: Calculate 7 × 8.
- Break 7 apart if helpful.
- Multiply each part by the other factor.
- Combine the partial products.
Very beginner explanation: Multiplication finds a total made from equal groups.
Answer: 56
Worked Example 4
Problem: Calculate 4 × 6.
- Break 4 apart if helpful.
- Multiply each part by the other factor.
- Combine the partial products.
Very beginner explanation: Multiplication finds a total made from equal groups.
Answer: 24
Worked Example 5
Problem: Calculate 2 × 5.
- Break 2 apart if helpful.
- Multiply each part by the other factor.
- Combine the partial products.
Very beginner explanation: Multiplication finds a total made from equal groups.
Answer: 10
Worked Example 6
Problem: Calculate 7 × 8.
- Break 7 apart if helpful.
- Multiply each part by the other factor.
- Combine the partial products.
Very beginner explanation: Multiplication finds a total made from equal groups.
Answer: 56
Worked Example 7
Problem: Calculate 10 × 9.
- Break 10 apart if helpful.
- Multiply each part by the other factor.
- Combine the partial products.
Very beginner explanation: Multiplication finds a total made from equal groups.
Answer: 90
Worked Example 8
Problem: Calculate 6 × 4.
- Break 6 apart if helpful.
- Multiply each part by the other factor.
- Combine the partial products.
Very beginner explanation: Multiplication finds a total made from equal groups.
Answer: 24
Worked Example 9
Problem: Calculate 3 × 8.
- Break 3 apart if helpful.
- Multiply each part by the other factor.
- Combine the partial products.
Very beginner explanation: Multiplication finds a total made from equal groups.
Answer: 24
Worked Example 10
Problem: Calculate 3 × 8.
- Break 3 apart if helpful.
- Multiply each part by the other factor.
- Combine the partial products.
Very beginner explanation: Multiplication finds a total made from equal groups.
Answer: 24
Practice Exercise
Create and solve one new example of Associative property of multiplication. Show the steps and explain how you checked it.
Common Mistake
Do not rush into a calculation before understanding the question. Check place values, labels, units, and whether your answer fits the situation.
13.7 Distributive thinking
Distributive thinking is an important Grade 4 mathematics idea in Properties and Relationships of Operations. Understand the meaning first, then use a clear representation or strategy and check whether the answer makes sense. Work one step at a time, use a model or number sentence when useful, and check whether the answer is reasonable.
Beginner Explanation
First identify what is given and what must be found. Then choose a strategy, show the steps clearly, and check the final answer.
10 Worked Examples with Answers
Worked Example 1
Problem: Create a simple Grade 4 example about Distributive thinking using the numbers 7 and 18.
- Identify what the topic means.
- Use the numbers in a way that follows the rule.
- Check that the result or representation makes sense.
Very beginner explanation: Meaning comes first; the numbers should illustrate the concept clearly.
Answer: A correct example that demonstrates Distributive thinking.
Worked Example 2
Problem: Create a simple Grade 4 example about Distributive thinking using the numbers 2 and 7.
- Identify what the topic means.
- Use the numbers in a way that follows the rule.
- Check that the result or representation makes sense.
Very beginner explanation: Meaning comes first; the numbers should illustrate the concept clearly.
Answer: A correct example that demonstrates Distributive thinking.
Worked Example 3
Problem: Create a simple Grade 4 example about Distributive thinking using the numbers 20 and 6.
- Identify what the topic means.
- Use the numbers in a way that follows the rule.
- Check that the result or representation makes sense.
Very beginner explanation: Meaning comes first; the numbers should illustrate the concept clearly.
Answer: A correct example that demonstrates Distributive thinking.
Worked Example 4
Problem: Create a simple Grade 4 example about Distributive thinking using the numbers 7 and 7.
- Identify what the topic means.
- Use the numbers in a way that follows the rule.
- Check that the result or representation makes sense.
Very beginner explanation: Meaning comes first; the numbers should illustrate the concept clearly.
Answer: A correct example that demonstrates Distributive thinking.
Worked Example 5
Problem: Create a simple Grade 4 example about Distributive thinking using the numbers 10 and 15.
- Identify what the topic means.
- Use the numbers in a way that follows the rule.
- Check that the result or representation makes sense.
Very beginner explanation: Meaning comes first; the numbers should illustrate the concept clearly.
Answer: A correct example that demonstrates Distributive thinking.
Worked Example 6
Problem: Create a simple Grade 4 example about Distributive thinking using the numbers 10 and 16.
- Identify what the topic means.
- Use the numbers in a way that follows the rule.
- Check that the result or representation makes sense.
Very beginner explanation: Meaning comes first; the numbers should illustrate the concept clearly.
Answer: A correct example that demonstrates Distributive thinking.
Worked Example 7
Problem: Create a simple Grade 4 example about Distributive thinking using the numbers 12 and 15.
- Identify what the topic means.
- Use the numbers in a way that follows the rule.
- Check that the result or representation makes sense.
Very beginner explanation: Meaning comes first; the numbers should illustrate the concept clearly.
Answer: A correct example that demonstrates Distributive thinking.
Worked Example 8
Problem: Create a simple Grade 4 example about Distributive thinking using the numbers 19 and 15.
- Identify what the topic means.
- Use the numbers in a way that follows the rule.
- Check that the result or representation makes sense.
Very beginner explanation: Meaning comes first; the numbers should illustrate the concept clearly.
Answer: A correct example that demonstrates Distributive thinking.
Worked Example 9
Problem: Create a simple Grade 4 example about Distributive thinking using the numbers 20 and 18.
- Identify what the topic means.
- Use the numbers in a way that follows the rule.
- Check that the result or representation makes sense.
Very beginner explanation: Meaning comes first; the numbers should illustrate the concept clearly.
Answer: A correct example that demonstrates Distributive thinking.
Worked Example 10
Problem: Create a simple Grade 4 example about Distributive thinking using the numbers 18 and 6.
- Identify what the topic means.
- Use the numbers in a way that follows the rule.
- Check that the result or representation makes sense.
Very beginner explanation: Meaning comes first; the numbers should illustrate the concept clearly.
Answer: A correct example that demonstrates Distributive thinking.
Practice Exercise
Create and solve one new example of Distributive thinking. Show the steps and explain how you checked it.
Common Mistake
Do not rush into a calculation before understanding the question. Check place values, labels, units, and whether your answer fits the situation.
13.8 Identity in addition
Addition combines quantities. Align place values, regroup when needed, and estimate first to predict a reasonable answer. Work one step at a time, use a model or number sentence when useful, and check whether the answer is reasonable.
Beginner Explanation
First identify what is given and what must be found. Then choose a strategy, show the steps clearly, and check the final answer.
10 Worked Examples with Answers
Worked Example 1
Problem: Calculate 921 + 2,860.
- Line up place values.
- Add from right to left.
- Regroup when needed.
Very beginner explanation: Place-value alignment keeps like places together.
Answer: 3,781
Worked Example 2
Problem: Calculate 3,480 + 2,569.
- Line up place values.
- Add from right to left.
- Regroup when needed.
Very beginner explanation: Place-value alignment keeps like places together.
Answer: 6,049
Worked Example 3
Problem: Calculate 2,781 + 675.
- Line up place values.
- Add from right to left.
- Regroup when needed.
Very beginner explanation: Place-value alignment keeps like places together.
Answer: 3,456
Worked Example 4
Problem: Calculate 1,198 + 2,053.
- Line up place values.
- Add from right to left.
- Regroup when needed.
Very beginner explanation: Place-value alignment keeps like places together.
Answer: 3,251
Worked Example 5
Problem: Calculate 3,886 + 1,282.
- Line up place values.
- Add from right to left.
- Regroup when needed.
Very beginner explanation: Place-value alignment keeps like places together.
Answer: 5,168
Worked Example 6
Problem: Calculate 2,036 + 557.
- Line up place values.
- Add from right to left.
- Regroup when needed.
Very beginner explanation: Place-value alignment keeps like places together.
Answer: 2,593
Worked Example 7
Problem: Calculate 3,720 + 622.
- Line up place values.
- Add from right to left.
- Regroup when needed.
Very beginner explanation: Place-value alignment keeps like places together.
Answer: 4,342
Worked Example 8
Problem: Calculate 1,432 + 1,179.
- Line up place values.
- Add from right to left.
- Regroup when needed.
Very beginner explanation: Place-value alignment keeps like places together.
Answer: 2,611
Worked Example 9
Problem: Calculate 1,015 + 552.
- Line up place values.
- Add from right to left.
- Regroup when needed.
Very beginner explanation: Place-value alignment keeps like places together.
Answer: 1,567
Worked Example 10
Problem: Calculate 1,469 + 2,482.
- Line up place values.
- Add from right to left.
- Regroup when needed.
Very beginner explanation: Place-value alignment keeps like places together.
Answer: 3,951
Practice Exercise
Create and solve one new example of Identity in addition. Show the steps and explain how you checked it.
Common Mistake
Do not rush into a calculation before understanding the question. Check place values, labels, units, and whether your answer fits the situation.
13.9 Identity in multiplication
Multiplication combines equal groups. Arrays, place value, known facts, and breaking apart numbers can all help. Work one step at a time, use a model or number sentence when useful, and check whether the answer is reasonable.
Beginner Explanation
First identify what is given and what must be found. Then choose a strategy, show the steps clearly, and check the final answer.
10 Worked Examples with Answers
Worked Example 1
Problem: Calculate 4 × 3.
- Break 4 apart if helpful.
- Multiply each part by the other factor.
- Combine the partial products.
Very beginner explanation: Multiplication finds a total made from equal groups.
Answer: 12
Worked Example 2
Problem: Calculate 5 × 9.
- Break 5 apart if helpful.
- Multiply each part by the other factor.
- Combine the partial products.
Very beginner explanation: Multiplication finds a total made from equal groups.
Answer: 45
Worked Example 3
Problem: Calculate 12 × 4.
- Break 12 apart if helpful.
- Multiply each part by the other factor.
- Combine the partial products.
Very beginner explanation: Multiplication finds a total made from equal groups.
Answer: 48
Worked Example 4
Problem: Calculate 9 × 7.
- Break 9 apart if helpful.
- Multiply each part by the other factor.
- Combine the partial products.
Very beginner explanation: Multiplication finds a total made from equal groups.
Answer: 63
Worked Example 5
Problem: Calculate 7 × 8.
- Break 7 apart if helpful.
- Multiply each part by the other factor.
- Combine the partial products.
Very beginner explanation: Multiplication finds a total made from equal groups.
Answer: 56
Worked Example 6
Problem: Calculate 9 × 8.
- Break 9 apart if helpful.
- Multiply each part by the other factor.
- Combine the partial products.
Very beginner explanation: Multiplication finds a total made from equal groups.
Answer: 72
Worked Example 7
Problem: Calculate 11 × 9.
- Break 11 apart if helpful.
- Multiply each part by the other factor.
- Combine the partial products.
Very beginner explanation: Multiplication finds a total made from equal groups.
Answer: 99
Worked Example 8
Problem: Calculate 7 × 9.
- Break 7 apart if helpful.
- Multiply each part by the other factor.
- Combine the partial products.
Very beginner explanation: Multiplication finds a total made from equal groups.
Answer: 63
Worked Example 9
Problem: Calculate 2 × 8.
- Break 2 apart if helpful.
- Multiply each part by the other factor.
- Combine the partial products.
Very beginner explanation: Multiplication finds a total made from equal groups.
Answer: 16
Worked Example 10
Problem: Calculate 9 × 5.
- Break 9 apart if helpful.
- Multiply each part by the other factor.
- Combine the partial products.
Very beginner explanation: Multiplication finds a total made from equal groups.
Answer: 45
Practice Exercise
Create and solve one new example of Identity in multiplication. Show the steps and explain how you checked it.
Common Mistake
Do not rush into a calculation before understanding the question. Check place values, labels, units, and whether your answer fits the situation.
13.10 Inverse operations
Inverse operations is an important Grade 4 mathematics idea in Properties and Relationships of Operations. Understand the meaning first, then use a clear representation or strategy and check whether the answer makes sense. Work one step at a time, use a model or number sentence when useful, and check whether the answer is reasonable.
Beginner Explanation
First identify what is given and what must be found. Then choose a strategy, show the steps clearly, and check the final answer.
10 Worked Examples with Answers
Worked Example 1
Problem: Create a simple Grade 4 example about Inverse operations using the numbers 14 and 17.
- Identify what the topic means.
- Use the numbers in a way that follows the rule.
- Check that the result or representation makes sense.
Very beginner explanation: Meaning comes first; the numbers should illustrate the concept clearly.
Answer: A correct example that demonstrates Inverse operations.
Worked Example 2
Problem: Create a simple Grade 4 example about Inverse operations using the numbers 5 and 4.
- Identify what the topic means.
- Use the numbers in a way that follows the rule.
- Check that the result or representation makes sense.
Very beginner explanation: Meaning comes first; the numbers should illustrate the concept clearly.
Answer: A correct example that demonstrates Inverse operations.
Worked Example 3
Problem: Create a simple Grade 4 example about Inverse operations using the numbers 17 and 15.
- Identify what the topic means.
- Use the numbers in a way that follows the rule.
- Check that the result or representation makes sense.
Very beginner explanation: Meaning comes first; the numbers should illustrate the concept clearly.
Answer: A correct example that demonstrates Inverse operations.
Worked Example 4
Problem: Create a simple Grade 4 example about Inverse operations using the numbers 5 and 12.
- Identify what the topic means.
- Use the numbers in a way that follows the rule.
- Check that the result or representation makes sense.
Very beginner explanation: Meaning comes first; the numbers should illustrate the concept clearly.
Answer: A correct example that demonstrates Inverse operations.
Worked Example 5
Problem: Create a simple Grade 4 example about Inverse operations using the numbers 14 and 8.
- Identify what the topic means.
- Use the numbers in a way that follows the rule.
- Check that the result or representation makes sense.
Very beginner explanation: Meaning comes first; the numbers should illustrate the concept clearly.
Answer: A correct example that demonstrates Inverse operations.
Worked Example 6
Problem: Create a simple Grade 4 example about Inverse operations using the numbers 2 and 16.
- Identify what the topic means.
- Use the numbers in a way that follows the rule.
- Check that the result or representation makes sense.
Very beginner explanation: Meaning comes first; the numbers should illustrate the concept clearly.
Answer: A correct example that demonstrates Inverse operations.
Worked Example 7
Problem: Create a simple Grade 4 example about Inverse operations using the numbers 4 and 5.
- Identify what the topic means.
- Use the numbers in a way that follows the rule.
- Check that the result or representation makes sense.
Very beginner explanation: Meaning comes first; the numbers should illustrate the concept clearly.
Answer: A correct example that demonstrates Inverse operations.
Worked Example 8
Problem: Create a simple Grade 4 example about Inverse operations using the numbers 20 and 16.
- Identify what the topic means.
- Use the numbers in a way that follows the rule.
- Check that the result or representation makes sense.
Very beginner explanation: Meaning comes first; the numbers should illustrate the concept clearly.
Answer: A correct example that demonstrates Inverse operations.
Worked Example 9
Problem: Create a simple Grade 4 example about Inverse operations using the numbers 2 and 20.
- Identify what the topic means.
- Use the numbers in a way that follows the rule.
- Check that the result or representation makes sense.
Very beginner explanation: Meaning comes first; the numbers should illustrate the concept clearly.
Answer: A correct example that demonstrates Inverse operations.
Worked Example 10
Problem: Create a simple Grade 4 example about Inverse operations using the numbers 2 and 4.
- Identify what the topic means.
- Use the numbers in a way that follows the rule.
- Check that the result or representation makes sense.
Very beginner explanation: Meaning comes first; the numbers should illustrate the concept clearly.
Answer: A correct example that demonstrates Inverse operations.
Practice Exercise
Create and solve one new example of Inverse operations. Show the steps and explain how you checked it.
Common Mistake
Do not rush into a calculation before understanding the question. Check place values, labels, units, and whether your answer fits the situation.
13.11 Choosing an operation
Choosing an operation is an important Grade 4 mathematics idea in Properties and Relationships of Operations. Understand the meaning first, then use a clear representation or strategy and check whether the answer makes sense. Work one step at a time, use a model or number sentence when useful, and check whether the answer is reasonable.
Beginner Explanation
First identify what is given and what must be found. Then choose a strategy, show the steps clearly, and check the final answer.
10 Worked Examples with Answers
Worked Example 1
Problem: Create a simple Grade 4 example about Choosing an operation using the numbers 17 and 6.
- Identify what the topic means.
- Use the numbers in a way that follows the rule.
- Check that the result or representation makes sense.
Very beginner explanation: Meaning comes first; the numbers should illustrate the concept clearly.
Answer: A correct example that demonstrates Choosing an operation.
Worked Example 2
Problem: Create a simple Grade 4 example about Choosing an operation using the numbers 19 and 9.
- Identify what the topic means.
- Use the numbers in a way that follows the rule.
- Check that the result or representation makes sense.
Very beginner explanation: Meaning comes first; the numbers should illustrate the concept clearly.
Answer: A correct example that demonstrates Choosing an operation.
Worked Example 3
Problem: Create a simple Grade 4 example about Choosing an operation using the numbers 17 and 5.
- Identify what the topic means.
- Use the numbers in a way that follows the rule.
- Check that the result or representation makes sense.
Very beginner explanation: Meaning comes first; the numbers should illustrate the concept clearly.
Answer: A correct example that demonstrates Choosing an operation.
Worked Example 4
Problem: Create a simple Grade 4 example about Choosing an operation using the numbers 16 and 4.
- Identify what the topic means.
- Use the numbers in a way that follows the rule.
- Check that the result or representation makes sense.
Very beginner explanation: Meaning comes first; the numbers should illustrate the concept clearly.
Answer: A correct example that demonstrates Choosing an operation.
Worked Example 5
Problem: Create a simple Grade 4 example about Choosing an operation using the numbers 14 and 12.
- Identify what the topic means.
- Use the numbers in a way that follows the rule.
- Check that the result or representation makes sense.
Very beginner explanation: Meaning comes first; the numbers should illustrate the concept clearly.
Answer: A correct example that demonstrates Choosing an operation.
Worked Example 6
Problem: Create a simple Grade 4 example about Choosing an operation using the numbers 13 and 13.
- Identify what the topic means.
- Use the numbers in a way that follows the rule.
- Check that the result or representation makes sense.
Very beginner explanation: Meaning comes first; the numbers should illustrate the concept clearly.
Answer: A correct example that demonstrates Choosing an operation.
Worked Example 7
Problem: Create a simple Grade 4 example about Choosing an operation using the numbers 18 and 5.
- Identify what the topic means.
- Use the numbers in a way that follows the rule.
- Check that the result or representation makes sense.
Very beginner explanation: Meaning comes first; the numbers should illustrate the concept clearly.
Answer: A correct example that demonstrates Choosing an operation.
Worked Example 8
Problem: Create a simple Grade 4 example about Choosing an operation using the numbers 10 and 12.
- Identify what the topic means.
- Use the numbers in a way that follows the rule.
- Check that the result or representation makes sense.
Very beginner explanation: Meaning comes first; the numbers should illustrate the concept clearly.
Answer: A correct example that demonstrates Choosing an operation.
Worked Example 9
Problem: Create a simple Grade 4 example about Choosing an operation using the numbers 19 and 18.
- Identify what the topic means.
- Use the numbers in a way that follows the rule.
- Check that the result or representation makes sense.
Very beginner explanation: Meaning comes first; the numbers should illustrate the concept clearly.
Answer: A correct example that demonstrates Choosing an operation.
Worked Example 10
Problem: Create a simple Grade 4 example about Choosing an operation using the numbers 13 and 3.
- Identify what the topic means.
- Use the numbers in a way that follows the rule.
- Check that the result or representation makes sense.
Very beginner explanation: Meaning comes first; the numbers should illustrate the concept clearly.
Answer: A correct example that demonstrates Choosing an operation.
Practice Exercise
Create and solve one new example of Choosing an operation. Show the steps and explain how you checked it.
Common Mistake
Do not rush into a calculation before understanding the question. Check place values, labels, units, and whether your answer fits the situation.
13.12 Checking calculations with inverse operations
Checking calculations with inverse operations is an important Grade 4 mathematics idea in Properties and Relationships of Operations. Understand the meaning first, then use a clear representation or strategy and check whether the answer makes sense. Work one step at a time, use a model or number sentence when useful, and check whether the answer is reasonable.
Beginner Explanation
First identify what is given and what must be found. Then choose a strategy, show the steps clearly, and check the final answer.
10 Worked Examples with Answers
Worked Example 1
Problem: Create a simple Grade 4 example about Checking calculations with inverse operations using the numbers 13 and 19.
- Identify what the topic means.
- Use the numbers in a way that follows the rule.
- Check that the result or representation makes sense.
Very beginner explanation: Meaning comes first; the numbers should illustrate the concept clearly.
Answer: A correct example that demonstrates Checking calculations with inverse operations.
Worked Example 2
Problem: Create a simple Grade 4 example about Checking calculations with inverse operations using the numbers 4 and 7.
- Identify what the topic means.
- Use the numbers in a way that follows the rule.
- Check that the result or representation makes sense.
Very beginner explanation: Meaning comes first; the numbers should illustrate the concept clearly.
Answer: A correct example that demonstrates Checking calculations with inverse operations.
Worked Example 3
Problem: Create a simple Grade 4 example about Checking calculations with inverse operations using the numbers 10 and 17.
- Identify what the topic means.
- Use the numbers in a way that follows the rule.
- Check that the result or representation makes sense.
Very beginner explanation: Meaning comes first; the numbers should illustrate the concept clearly.
Answer: A correct example that demonstrates Checking calculations with inverse operations.
Worked Example 4
Problem: Create a simple Grade 4 example about Checking calculations with inverse operations using the numbers 8 and 20.
- Identify what the topic means.
- Use the numbers in a way that follows the rule.
- Check that the result or representation makes sense.
Very beginner explanation: Meaning comes first; the numbers should illustrate the concept clearly.
Answer: A correct example that demonstrates Checking calculations with inverse operations.
Worked Example 5
Problem: Create a simple Grade 4 example about Checking calculations with inverse operations using the numbers 16 and 7.
- Identify what the topic means.
- Use the numbers in a way that follows the rule.
- Check that the result or representation makes sense.
Very beginner explanation: Meaning comes first; the numbers should illustrate the concept clearly.
Answer: A correct example that demonstrates Checking calculations with inverse operations.
Worked Example 6
Problem: Create a simple Grade 4 example about Checking calculations with inverse operations using the numbers 3 and 20.
- Identify what the topic means.
- Use the numbers in a way that follows the rule.
- Check that the result or representation makes sense.
Very beginner explanation: Meaning comes first; the numbers should illustrate the concept clearly.
Answer: A correct example that demonstrates Checking calculations with inverse operations.
Worked Example 7
Problem: Create a simple Grade 4 example about Checking calculations with inverse operations using the numbers 7 and 15.
- Identify what the topic means.
- Use the numbers in a way that follows the rule.
- Check that the result or representation makes sense.
Very beginner explanation: Meaning comes first; the numbers should illustrate the concept clearly.
Answer: A correct example that demonstrates Checking calculations with inverse operations.
Worked Example 8
Problem: Create a simple Grade 4 example about Checking calculations with inverse operations using the numbers 18 and 11.
- Identify what the topic means.
- Use the numbers in a way that follows the rule.
- Check that the result or representation makes sense.
Very beginner explanation: Meaning comes first; the numbers should illustrate the concept clearly.
Answer: A correct example that demonstrates Checking calculations with inverse operations.
Worked Example 9
Problem: Create a simple Grade 4 example about Checking calculations with inverse operations using the numbers 3 and 11.
- Identify what the topic means.
- Use the numbers in a way that follows the rule.
- Check that the result or representation makes sense.
Very beginner explanation: Meaning comes first; the numbers should illustrate the concept clearly.
Answer: A correct example that demonstrates Checking calculations with inverse operations.
Worked Example 10
Problem: Create a simple Grade 4 example about Checking calculations with inverse operations using the numbers 20 and 17.
- Identify what the topic means.
- Use the numbers in a way that follows the rule.
- Check that the result or representation makes sense.
Very beginner explanation: Meaning comes first; the numbers should illustrate the concept clearly.
Answer: A correct example that demonstrates Checking calculations with inverse operations.
Practice Exercise
Create and solve one new example of Checking calculations with inverse operations. Show the steps and explain how you checked it.
Common Mistake
Do not rush into a calculation before understanding the question. Check place values, labels, units, and whether your answer fits the situation.
30 Review Questions and Answers
Q1. What is important to understand about Addition and subtraction are related?
Answer: Addition combines quantities. Align place values, regroup when needed, and estimate first to predict a reasonable answer.
Q2. What is important to understand about Multiplication and division are related?
Answer: Multiplication combines equal groups. Arrays, place value, known facts, and breaking apart numbers can all help.
Q3. What is important to understand about Commutative property of addition?
Answer: Addition combines quantities. Align place values, regroup when needed, and estimate first to predict a reasonable answer.
Q4. What is important to understand about Commutative property of multiplication?
Answer: Multiplication combines equal groups. Arrays, place value, known facts, and breaking apart numbers can all help.
Q5. What is important to understand about Associative property of addition?
Answer: Addition combines quantities. Align place values, regroup when needed, and estimate first to predict a reasonable answer.
Q6. What is important to understand about Associative property of multiplication?
Answer: Multiplication combines equal groups. Arrays, place value, known facts, and breaking apart numbers can all help.
Q7. What is important to understand about Distributive thinking?
Answer: Distributive thinking is an important Grade 4 mathematics idea in Properties and Relationships of Operations. Understand the meaning first, then use a clear representation or strategy and check whether the answer makes sense.
Q8. What is important to understand about Identity in addition?
Answer: Addition combines quantities. Align place values, regroup when needed, and estimate first to predict a reasonable answer.
Q9. What is important to understand about Identity in multiplication?
Answer: Multiplication combines equal groups. Arrays, place value, known facts, and breaking apart numbers can all help.
Q10. What is important to understand about Inverse operations?
Answer: Inverse operations is an important Grade 4 mathematics idea in Properties and Relationships of Operations. Understand the meaning first, then use a clear representation or strategy and check whether the answer makes sense.
Q11. What is important to understand about Choosing an operation?
Answer: Choosing an operation is an important Grade 4 mathematics idea in Properties and Relationships of Operations. Understand the meaning first, then use a clear representation or strategy and check whether the answer makes sense.
Q12. What is important to understand about Checking calculations with inverse operations?
Answer: Checking calculations with inverse operations is an important Grade 4 mathematics idea in Properties and Relationships of Operations. Understand the meaning first, then use a clear representation or strategy and check whether the answer makes sense.
Q13. How would you explain Addition and subtraction are related?
Answer: Addition combines quantities. Align place values, regroup when needed, and estimate first to predict a reasonable answer.
Q14. How would you explain Multiplication and division are related?
Answer: Multiplication combines equal groups. Arrays, place value, known facts, and breaking apart numbers can all help.
Q15. How would you explain Commutative property of addition?
Answer: Addition combines quantities. Align place values, regroup when needed, and estimate first to predict a reasonable answer.
Q16. How would you explain Commutative property of multiplication?
Answer: Multiplication combines equal groups. Arrays, place value, known facts, and breaking apart numbers can all help.
Q17. How would you explain Associative property of addition?
Answer: Addition combines quantities. Align place values, regroup when needed, and estimate first to predict a reasonable answer.
Q18. How would you explain Associative property of multiplication?
Answer: Multiplication combines equal groups. Arrays, place value, known facts, and breaking apart numbers can all help.
Q19. How would you explain Distributive thinking?
Answer: Distributive thinking is an important Grade 4 mathematics idea in Properties and Relationships of Operations. Understand the meaning first, then use a clear representation or strategy and check whether the answer makes sense.
Q20. How would you explain Identity in addition?
Answer: Addition combines quantities. Align place values, regroup when needed, and estimate first to predict a reasonable answer.
Q21. How would you explain Identity in multiplication?
Answer: Multiplication combines equal groups. Arrays, place value, known facts, and breaking apart numbers can all help.
Q22. How would you explain Inverse operations?
Answer: Inverse operations is an important Grade 4 mathematics idea in Properties and Relationships of Operations. Understand the meaning first, then use a clear representation or strategy and check whether the answer makes sense.
Q23. How would you explain Choosing an operation?
Answer: Choosing an operation is an important Grade 4 mathematics idea in Properties and Relationships of Operations. Understand the meaning first, then use a clear representation or strategy and check whether the answer makes sense.
Q24. How would you explain Checking calculations with inverse operations?
Answer: Checking calculations with inverse operations is an important Grade 4 mathematics idea in Properties and Relationships of Operations. Understand the meaning first, then use a clear representation or strategy and check whether the answer makes sense.
Q25. What should a beginner check when working with Addition and subtraction are related?
Answer: Addition combines quantities. Align place values, regroup when needed, and estimate first to predict a reasonable answer.
Q26. What should a beginner check when working with Multiplication and division are related?
Answer: Multiplication combines equal groups. Arrays, place value, known facts, and breaking apart numbers can all help.
Q27. What should a beginner check when working with Commutative property of addition?
Answer: Addition combines quantities. Align place values, regroup when needed, and estimate first to predict a reasonable answer.
Q28. What should a beginner check when working with Commutative property of multiplication?
Answer: Multiplication combines equal groups. Arrays, place value, known facts, and breaking apart numbers can all help.
Q29. What should a beginner check when working with Associative property of addition?
Answer: Addition combines quantities. Align place values, regroup when needed, and estimate first to predict a reasonable answer.
Q30. What should a beginner check when working with Associative property of multiplication?
Answer: Multiplication combines equal groups. Arrays, place value, known facts, and breaking apart numbers can all help.