Apply properties of operations as strategies to multiply and divide. Examples: If 6 × 4 = 24 is known, then 4 × 6 = 24 is also known. (Commutative property of multiplication.) 3 × 5 × 2 can be found by 3 × 5 = 15, then 15 × 2 = 30, or by 5 × 2 = 10, then 3 × 10 = 30. (Associative property of multiplication.) Knowing that 8 × 5 = 40 and 8 × 2 = 16, one can find 8 × 7 as 8 × (5 + 2) = (8 × 5) + (8 × 2) = 40 + 16 = 56. (Distributive property.)
Official wording from the Common Core State Standards for Mathematics (© 2010 National Governors Association Center for Best Practices and Council of Chief State School Officers). View on thecorestandards.org
Third graders do not need to memorize property names to benefit from them; they need to use them as shortcuts. The commutative property says the order of the factors does not change the product, so knowing 6 × 4 = 24 means 4 × 6 = 24 comes free, which roughly halves the number of facts to learn. The associative property lets students regroup three factors, finding 3 × 5 × 2 as 3 × 10 instead of 15 × 2.
The distributive property is the most powerful of the three. It lets a child break a tricky fact into two friendly ones: 8 × 7 is 8 × 5 plus 8 × 2, which is 40 + 16 = 56. Arrays make this visible, because a 8-by-7 array can be split with one line into an 8-by-5 part and an 8-by-2 part. Students who can explain these moves with pictures are reasoning about the structure of multiplication, not just recalling answers, and they will use the same ideas for multi-digit multiplication and for algebra later on.
Some students turn 8 × 7 into 8 × 5 + 2 and get 42. Each part of the split factor must be multiplied by 8, which an array picture makes clear.
After learning that 4 × 6 = 6 × 4, children may think 12 ÷ 3 equals 3 ÷ 12. Show that sharing 12 among 3 is not the same as sharing 3 among 12.
A student who knows 3 × 8 but treats 8 × 3 as unknown is missing a free fact. Rotate an array 90 degrees to show it is the same total.
Regrouping 2 + 3 × 4 as (2 + 3) × 4 changes the value. The associative property only applies when every operation is multiplication.
Use facts you know to find 6 × 8.
Answer: 6 × 8 = 48, found as (6 × 5) + (6 × 3) = 30 + 18.
Grid paper is ideal here. Students draw an array for a fact, cut it into two rectangles along a line, and write the two smaller facts and their sum. Rotating an array shows the commutative property, and stacking layers of cubes shows why three factors can be grouped in any order.
Test questions often show a strategy and ask which equation describes it, or ask which expression is equal to a given product. Students who use the property names are fine, but explaining the move in words is what the standard values.
Original questions written for this standard. Choose an option or type your answer, then press Check. Every question has a worked explanation.
Answer: C) 4 × 7 = 28
Changing the order of the factors does not change the product (commutative property), so 4 × 7 = 28.
Answer: B) 9 × 5 + 9 × 2
Split 7 into 5 + 2 and multiply each part by 9: 45 + 18 = 63, the same as 9 × 7.
Answer: 80
2 × 5 = 10, and 10 × 8 = 80. Grouping the factors in a different order does not change the product.
Answer: 32
8 = 5 + 3, so 4 × 8 = 20 + 12 = 32.
Answer: C) (2 × 3) × 4 = 2 × (3 × 4)
The associative property is about how three factors are grouped. Both sides equal 24.
Answer: 42 (also accepted: 42 dots)
7 × 5 = 35 and 7 × 1 = 7. 35 + 7 = 42, which is 7 × 6.
Use addition to find the total number of objects arranged in rectangular arrays with up to 5 rows and up to 5 columns; write an equation to express the total as a sum of equal addends.
A full lesson with slides, activities and an exit ticket on properties of multiplication, pitched to grade 3 and editable in PowerPoint or Google Slides.
Make a lesson →A printable, differentiated worksheet on 3.OA.B.5 with an answer key, ready in about a minute.
Make a worksheet →Turn properties of multiplication into a quiz students answer online that marks itself, with a class summary for you.
Build a test →No. The standard says students apply the properties as strategies. Using them correctly and explaining why they work matters more than the vocabulary.
It lets students build hard facts from easy ones and later powers multi-digit multiplication (4.NBT.B.5), area models and expanding expressions in middle school.