Introduction
Properties of Operations is an important Grade 3 math skill because students are moving from simple answers toward explaining how the math works.
In this lesson, students use models, real questions, worked examples, practice problems, and two online quizzes to build confidence with properties of operations.
What Is Properties of Operations?
Properties of Operations means choosing a model, naming what each number means, and explaining the strategy.
The goal is not only to get the answer. Students should be able to show the idea, explain the strategy, and check whether the answer makes sense.
Understanding Properties of Operations
Before solving, students should slow down and decide what each number, shape, unit, or label represents.
- Read the question carefully and identify what is being asked.
- Choose a model, equation, table, or diagram that matches the situation.
- Solve one step at a time and keep units or labels attached.
- Use the answer explanation to check that the result makes sense.
Visual Models
Visual Model 1
Question: Which array shows \(4\times3\)?
Why it works: An array for \(4\times3\) has 4 columns and 3 rows. Option C shows the correct array.
Answer: The \(4\times3\) array
Visual Model 2
Question: Ava has 5 bags with 3 pencils in each. Which array shows \(5\times3\) pencils?
Why it works: 5 bags with 3 pencils in each is shown by an array with 5 columns and 3 rows: \(5\times3=15\) pencils.
Answer: \(5\times3\)
Worked Examples
Example 1
Question: Mia splits a rectangle to show \(3\times8=3\times(5+3)\). What is \(3\times5+3\times3\)?
- A. \(14\)
- B. \(20\)
- C. \(27\)
- D. \(24\)
- \(3\times5=15\) and \(3\times3=9\), so \(15+9=24\).
- The distributive property shows \(3\times(5+3)=24\).
Answer: \(24\)
Example 2
Question: Look at this area model. Which equation does it show?
- A. \(2\times3=6\)
- B. \(5+2=7\)
- C. \(3\times(2+2)=12\)
- D. \(2\times(3+2)=10\)
- The area model shows a rectangle split into two parts: one part is \(3\times2=6\) and the other is \(2\times2=4\).
- Using the distributive property: \(2\times(3+2)=2\times3+2\times2=6+4=10\).
Answer: \(2\times(3+2)=2\times5=10\)
Example 3
Question: Which statement about these two arrays shows the commutative property?
- A. The arrays show the same property as \(9-2\).
- B. \(2+9=11\) and \(9+2=11\)
- C. Array B has more squares.
- D. \(2\times9=9\times2\)
- These arrays show the commutative property.
- One has 2 rows and 9 columns, and the other has 9 rows and 2 columns, but both have 18 squares total.
Answer: \(2\times9=9\times2=18\)
Real-World Word Problems
Problem 1
Question: Lily arranged 6 rows with 2 stickers in each row. How many stickers does she have?
- A. \(8\)
- B. \(10\)
- C. \(14\)
- D. \(12\)
Why it works: \(6\times2=12\). She has 12 stickers.
Answer: \(12\)
Problem 2
Question: Ben has 4 boxes with 5 markers in each. Which number sentence shows how many markers he has?
- A. \(4+5=9\)
- B. \(5+5+5=15\)
- C. \(4\times4=16\)
- D. \(4\times5=20\)
Why it works: 4 boxes with 5 markers in each is \(4\times5=20\) markers total.
Answer: \(4\times5=20\)
Common Mistakes
- Rushing before identifying what the numbers represent.
- Choosing an operation that does not match the situation.
- Dropping labels, units, or context from the answer.
- Skipping the estimate or reasonableness check.
Strategy Tips
- Underline the question being asked.
- Use a model before jumping to computation.
- Write an equation that matches the story or picture.
- Explain the final answer in a sentence.
Practice Questions
Question 1
If \(8\times3=24\), what is \(3\times8\)?
- A. \(11\)
- B. \(21\)
- C. \(27\)
- D. \(24\)
Question 2
Which number sentence shows the commutative property of multiplication?
- A. \((2\times3)\times2=2\times(3\times2)\)
- B. \(5\times4=20\)
- C. \(2\times5=10\)
- D. \(5\times4=4\times5\)
Question 3
Noah knows that \(7\times2=14\). What is \(2\times7\)?
- A. \(9\)
- B. \(12\)
- C. \(16\)
- D. \(14\)
Question 4
Which pair of equations shows the commutative property with \(3\), \(4\), and \(12\)?
- A. \(3+4=7\) and \(4+3=7\)
- B. \(3\times3=9\) and \(4\times3=12\)
- C. \(12-3=9\) and \(12-4=8\)
- D. \(3\times4=12\) and \(4\times3=12\)
Question 5
Look at this problem. Which property of multiplication lets you group the factors differently?
\((2\times3)\times4=2\times(3\times4)\)
- A. Distributive property
- B. Commutative property
- C. Identity property
- D. Associative property
Question 6
Which expression uses the associative property to regroup \(4\times(3\times2)\) without changing the order of the factors?
- A. \(4+(3\times2)\)
- B. \((4\times3)+2\)
- C. \((4\times2)\times3\)
- D. \((4\times3)\times2\)
Full Answer Explanations Click to show all answers and explanations
Question 1
Answer: \(24\)
The commutative property of multiplication says we can switch the order of the factors and get the same product. So \(3\times8=8\times3=24\).
Question 2
Answer: \(5\times4=4\times5\)
Commutative property means we can switch the order of the factors. Both sides have the same factors in a different order.
Question 3
Answer: \(14\)
The commutative property says you can switch the order of the factors and still get the same answer. So \(2\times7=7\times2=14\).
Question 4
Answer: \(3\times4=12\) and \(4\times3=12\)
The commutative property shows that switching the factors in a multiplication gives the same product: \(3\times4=4\times3=12\).
Question 5
Answer: Associative property
The associative property says we can group factors in different ways and still get the same product. The groups change, like \((2\times3)\times4 = 6\times4 = 24\) or \(2\times(3\times4) = 2\times12 = 24\), but the product stays the same.
Question 6
Answer: \((4\times3)\times2\)
The associative property changes the grouping but keeps the factors in the same order. So \(4\times(3\times2)\) can be regrouped as \((4\times3)\times2\).
Connection to Standards
This lesson supports Grade 3 math expectations for reasoning, modeling, problem solving, and explaining answers clearly. It connects classroom skills to the kind of questions students see on state math assessments.
Summary
Properties of Operations becomes easier when students connect the question to a model, use clear steps, and explain why the answer fits.
GOLDEN RULE
Understand the model before choosing the operation.
