3rd Grade - Multiply Numbers In Any Order

 
     
 
     
 
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3rd
Multiplication and Division
Multiply Numbers in Any Order
Recognize and use the commutative and associative properties of multiplication (e.g., if 5 x 7 = 35, then what is 7 x 5? and if 5 x 7 x 3 = 105, then what is 7 x 3 x 5?).
Algebra and Functions: commutative and associative properties, multiplication The ability to understand that the order and grouping of multiplication factors does not change the product.
 

Sample Problems

(1)

Define commutative property of multiplication. (When numbers are multiplied, the order of the factors can be changed without changing the product.)

(2)

Define associative property of multiplication (When numbers are multiplied, the factors can be grouped in different ways without changing the product.)

(3)

Give an example of the commutative property of multiplication. (e.g. 3 x 4 = 4 x 3)

(4)

Give an example of the associative property of multiplication. (e.g. (5 x 3) x 4 = 5 x (3 x 4).

(5)

What do the parentheses mean in mathematical problems? (Do that operation first.)

Learning Tips

(1)

If you have them, use base ten blocks to model multiplication problems: 6 x 8 and 8 x 6, for example.


Use coins, buttons, or other small objects to help your child discover associative and commutative properties for him/herself. For addition, lay a number of objects (perhaps 8-10 to start with) on the table and ask your child to arrange them in 2 groups. Write down the number sentence that you get (for example, with 8 items, the number sentences that are possible are 0+8=8, 1+7=8, 2 +6+8, 3+5=8, and 4+4=8. After that, we are repeating ourselves, because 3+5 is the same as 5+3, but children don’t automatically know this. Allow your child to reverse the order of the groups he/she laid out (the group of 3 and the group of 5 in this example change places) and see that the total is still 8. The child can then discover, with this same grouping that 8-5 is 3 and 8-3 is 5.


Make up a little story about number families. Each family has a grown-up and (for our purposes, for now) 2 children. If the family happens to only have 1 child, we’ll hold a place for the child that is “away visiting” and name that child “zero.” A family has to “prove” that it gets to be a family in this way: The number names of the children have to be equal to the sum (or product) of the grown-up. All families are named “the Add family” or “the Multiply family.” Now, let’s play. “Here are the children in the Add family. Their names are three (3) and (5). What’s the mommy’s (or daddy’s or grown-up’s) name?” Expect the answer “Eight.” When the child can do this confidently, change the question: “Here is the Add family. The mommy’s name is Eight and one of the children’s names is Five. What’s the name of the other child?” (It’s best if you say this in a sort of playful-“we’re pretending” fashion because while it’s a game that makes sense mathematically, it is just a game!) Soon your child will automatically think of these 3 numbers as having this special relationship. For the Multiply families, you could say, “Here is a Multiply family. The children are named Three and Five. What’s the daddy’s name?” Expect the child to answer “Twenty-four.” Eventually you can make this game harder with this question: “I see a Multiply Mommy! Her name is Twenty-four! Hi, Ms. Twenty-four? Where are your children today? Wait! Who ARE your children? The children, in a Multiply family, could be Twenty-four and One, they could be Twelve and Two, they could be Six and Four . . . you get the idea. This is a valuable game to play because when children begin to work with fractions later in their schoolwork they will need to find common denominators—a number that will allow several selected numbers to have a common division factor. When your child sees denominators of, say, 3, 4, and 6, a child who is good at playing “Number Families” will quickly think “12 is my common factor for these three numbers.”

(2)

Use flashcards with the multiplication problems written on one side and the product (answer) on the back. Make sure to write the multiplication problems both ways 9 x 8 and 8 x 9. Draw children’s attention to the fact that no matter the order of the numerals, the answer is the same.

(3)

Remind students that these properties also work for addition and that remembering their times tables is easier because if the child knows 2 x 6 they will know 6 x 2 when they get to the 6s.

(4)

Mental math (mental math is math you “do in your head” without writing anything down) is easier if children realize addition and multiplication use the commutative and associative properties. For example, if a student is trying to add 3 + 25 + 37, the student can add 37 and 3 first to get 40 and then add in the 25 to get 65.

(5)

Order matters in subtraction and division. Use a number line to help children explore addition and subtraction (multiplication and division) in terms of commutative properties. Addition and multiplication work, subtraction and division don’t.

Extra Help Problems

(1)

Write the order that you would do these calculations to make them as quick and easy as possible: 10 x 3 x 2

[(10 x 3) x 2 = 60]

(2)

2 x 10 x 4 [(10 x 4) x 2 = 80]

(3)

4 x 10 x 3 [(4 x 10) x 3 = 120]

(4)

3 x 2 x 5 [(5 x 3) x 2 = 30]

(5)

5 x 4 x 2 [(5 x 4) x 2 = 40]

(6)

50 x 14 x 2 [(50 x 2) x 14=100 x 14=1400]

(7)

4 x 9 x 25 [(25 x 4) = 100 x 9 = 900]

(8)

4 x 8 x 5 [(8 x 5) = 40 x 4 = 160]

(9)

20 x 6 x 5 [(20 x 5) = 100 x 6 = 600]

(10)

15 x 7 x 2 [(15 x 2 ) = 30 x 7 = 210]

(11)

5 x 7 is the same as 7 x ? (5)

(12)

8 x 9 is the same as ? x 8 (9)

(13)

20 x 5 is the same as ? x ? Use the commutative property. (5 x 20)

(14)

9 x 5 = 45 ? x 9 = 45

(15)

Change the parentheses to show the associative property of multiplication: (4 x 3) x 2 = 4 x 3 x 2 (3 x 2)

(16)

8 x (6 x 10) = 8 x 6 x 10 (8 x 6)

(17)

5 x (3 x 7) = 5 x 3 x 7 (5 x 3)

(18)

9 x 8 = ? x ? Show the commutative property. ( 8 x 9)

(19)

? x ? = 6 x 4 Show the commutative property. (4 x 6)

(20)

Your neighbor, Billy, can’t seem to remember his 8 times tables. He always seems to get stuck on 8 x 2. He knows his 2 times tables. Can you give him a clue that could help him remember his 8s? (Tell Billy to think of 2 x 8 instead of 8 x 2)

(21)

Gina sells lemonade outside of her house. She needs to add $5 + $6 + $10 to figure out how much money she’s made, but it’s taking her a long time. Is there a way you could help Gina solve the problem quickly? (Tell her to add $5 + $10 first and then add the $6)

(22)

4 x 5 = 5 x ? (4)

(23)

20,000 x 9 = 9 x ? (20,000)

(24)

Is 10 – 7 the same as 7 – 10? (No, the commutative property doesn’t work for subtraction)

(25)

Is 120 divided by 6 the same as 6 divided by 120? (No, the commutative property doesn’t work for division)

 

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