Grade 3 · Operations & Algebraic Thinking · MAT-03-OA-006
Use Multiplication Properties
Represent use multiplication properties with words, symbols, diagrams, or models appropriate to Grade 3.
Learning objectives
What you should be able to do
- Represent use multiplication properties with words, symbols, diagrams, or models appropriate to Grade 3.
- Solve problems involving use multiplication properties accurately and explain why the method works.
- Check results for reasonableness using a second representation, inverse relationship, benchmark, or property when appropriate.
Prerequisite check
Make sure the foundation is ready.
Checks conceptual readiness before new Grade 3 reasoning.
Checks whether prior skills are available for use, not only recalled by name.
Learn
Build the idea from meaning, not memorization.
Meaning of Use Multiplication Properties
The commutative, associative, and distributive properties let multiplication be rearranged or decomposed without changing value.
A reliable reasoning strategy
Choose a property that turns a difficult product into easier known facts, then recombine the parts.
Representations that should agree
Properties explain why array rotations, regrouped factors, and split rectangles preserve total quantity. A correct model, equation, verbal explanation, or diagram should preserve the same quantities and relationships.
How to check the result
Check whether the answer fits the quantities, units, and structure of the problem. Use a second representation or inverse relationship when available, and explain what would make an answer unreasonable.
Math Box · Interactive lesson
Touch the math. Change it. See what stays true.
Use the interactive model before and after the worked examples. Change the inputs, make a prediction, then use the model to test whether your reasoning holds.
Array Builder
Build equal rows and columns, then connect the grid to repeated addition.
- build a model
- change one quantity
- predict before calculating
- explain why the result changes
Worked examples
Twenty different ways to see the concept work.
Use the commutative property to rewrite 3 × 8.
3 × 8 = 8 × 3 = 24Changing factor order does not change the product.
Use the commutative property to rewrite 4 × 7.
4 × 7 = 7 × 4 = 28Both arrays contain the same number of cells when rows and columns swap.
Use the identity property to evaluate 1 × 9.
1 × 9 = 9Multiplying by 1 leaves the other factor unchanged.
Use the zero property to evaluate 6 × 0.
6 × 0 = 0Six groups with zero objects contain zero objects altogether.
Break 6 × 7 into 6 × 5 and 6 × 2.
6 × 7 = 30 + 12 = 42The distributive property splits 7 into 5 + 2.
Break 4 × 9 into 4 × 10 minus 4 × 1.
4 × 9 = 40 − 4 = 36Using a nearby ten fact is a distributive strategy.
Break 8 × 6 into 5 × 6 and 3 × 6.
8 × 6 = 30 + 18 = 48Splitting 8 into 5 + 3 preserves the total product.
Regroup (2 × 3) × 4 using the associative property.
(2 × 3) × 4 = 2 × (3 × 4) = 24Changing the grouping of factors does not change their order or product.
Regroup 5 × (2 × 3) using the associative property.
5 × (2 × 3) = (5 × 2) × 3 = 30Both groupings multiply the same three factors.
Use commutativity to make 2 × 9 easier to connect to a known fact.
2 × 9 = 9 × 2 = 18Swapping factors can connect the problem to a familiar doubles fact.
Use the distributive property for 7 × 8 by splitting 7 into 5 + 2.
7 × 8 = 5 × 8 + 2 × 8 = 40 + 16 = 56The two partial products combine to the original product.
Use the distributive property for 9 × 6 by thinking 10 × 6 − 1 × 6.
9 × 6 = 60 − 6 = 54Nine groups are one group fewer than ten groups.
Which property explains 3 × 4 = 4 × 3?
Commutative propertyThe order of the two factors changes while the product stays equal.
Which property explains (3 × 2) × 5 = 3 × (2 × 5)?
Associative propertyThe factors stay in the same order but are grouped differently.
Which property explains 5 × (4 + 2) = 5 × 4 + 5 × 2?
Distributive propertyThe factor 5 is multiplied by each addend and the partial products are combined.
Which property explains 8 × 1 = 8?
Identity property of multiplicationOne group of eight—or eight groups of one—preserves the factor's value.
Which property explains 0 × 7 = 0?
Zero property of multiplicationZero groups contain no objects.
Use 5 × 7 = 35 to find 6 × 7.
6 × 7 = 35 + 7 = 42One additional group of 7 extends the known five-times fact.
Use 4 × 8 = 32 to find 8 × 8 by doubling.
8 × 8 = 32 + 32 = 64Eight groups can be split into two sets of four groups.
Explain why 3 × (4 + 5) and 3 × 4 + 3 × 5 must match.
Both equal 27.The distributive property preserves the total when a factor is applied to each part of a sum.
Practice
Guided practice with hints
01Split 3 × 8 using 5+3.
Hint: Choose a property that turns a difficult product into easier known facts, then recombine the parts.
Answer: 3 × 5 + 3 × 3
02Regroup 4 × 7 × 4 without changing factor order.
Hint: Choose a property that turns a difficult product into easier known facts, then recombine the parts.
Answer: 4 × (7 × 4)
03Use commutativity to rewrite 5 × 3.
Hint: Choose a property that turns a difficult product into easier known facts, then recombine the parts.
Answer: 3 × 5
04Split 6 × 8 using 5+3.
Hint: Choose a property that turns a difficult product into easier known facts, then recombine the parts.
Answer: 6 × 5 + 6 × 3
05Regroup 2 × 7 × 4 without changing factor order.
Hint: Choose a property that turns a difficult product into easier known facts, then recombine the parts.
Answer: 2 × (7 × 4)
06Use commutativity to rewrite 3 × 3.
Hint: Choose a property that turns a difficult product into easier known facts, then recombine the parts.
Answer: 3 × 3
Practice
Independent practice
01Use commutativity to rewrite 3 × 3.
Answer: 3 × 3
02Split 4 × 8 using 5+3.
Answer: 4 × 5 + 4 × 3
03Regroup 5 × 7 × 4 without changing factor order.
Answer: 5 × (7 × 4)
04Use commutativity to rewrite 6 × 3.
Answer: 3 × 6
05Split 2 × 8 using 5+3.
Answer: 2 × 5 + 2 × 3
06Regroup 3 × 7 × 4 without changing factor order.
Answer: 3 × (7 × 4)
07Use commutativity to rewrite 4 × 3.
Answer: 3 × 4
08Split 5 × 8 using 5+3.
Answer: 5 × 5 + 5 × 3
Common mistakes
Learn to catch the error, not just the answer.
Track exactly which quantities are reordered, regrouped, or decomposed.
Reconnect the answer to use multiplication properties and verify that the model, calculation, and label describe the same quantity.
Applications & challenge
Use the idea beyond a single exercise.
Applications
- Break harder products into friendly facts during mental math.
- Use use multiplication properties to interpret or solve a Grade 3 situation with quantities, diagrams, measurements, or data.
Challenge problems
Regroup 3 × 7 × 4 without changing factor order.
3 × (7 × 4)
Use commutativity to rewrite 4 × 3.
3 × 4
Split 5 × 8 using 5+3.
5 × 5 + 5 × 3
Regroup 6 × 7 × 4 without changing factor order.
6 × (7 × 4)
Flashcards
Retrieve it from memory.
Flashcards
Card 1 of 4
Game mode
Use Multiplication Properties Challenge
Solve and explain Grade 3 problems involving use multiplication properties without relying on answer guessing.
Practice
Mastery check
01Split 3 × 8 using 5+3.
Answer: 3 × 5 + 3 × 3
02Regroup 4 × 7 × 4 without changing factor order.
Answer: 4 × (7 × 4)
03Use commutativity to rewrite 5 × 3.
Answer: 3 × 5
04Split 6 × 8 using 5+3.
Answer: 6 × 5 + 6 × 3
05Regroup 2 × 7 × 4 without changing factor order.
Answer: 2 × (7 × 4)
06Use commutativity to rewrite 3 × 3.
Answer: 3 × 3
07Split 4 × 8 using 5+3.
Answer: 4 × 5 + 4 × 3
08Regroup 5 × 7 × 4 without changing factor order.
Answer: 5 × (7 × 4)
Terminology
Words to know
- commutative property
- Changing factor order without changing the product.
- distributive property
- Breaking a factor into parts and adding partial products.
Continue learning
Connected concepts
Curriculum references
Standards and instructional references
California Common Core State Standards for Mathematics — Grade 3, 3.OA.5
Primary standards alignment for Use Multiplication Properties.2023 Mathematics Framework for California Public Schools
Instructional emphasis on reasoning, representations, problem solving, and equitable access.