3rd Grade - Multiplication And Division Are Opposites

 
     
 
     
 
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3rd
Multiplication and Division
Multiplication and Division Are Opposites
Use the inverse relationship of multiplication and division to compute and check results.
Number Sense: Inverse relationship of multiplication and division The ability to use the information that division is the opposite of multiplication (and visa-versa) to check the results of computations.
 
 

Building Blocks/Prerequisites

 

Sample Problems

(1)

How are division and multiplication related?

Division is the inverse (opposite) operation of multiplication: axb=c so a=c/b.

(2)

How are inverse relationships used?

This principle can be used for a child to check his or her work. For example, after completing a division problem the answer can be multiplied by the divisor to get the third number.

(3)

How are division problems written?

There are two common ways to indicate division: c divided by b can be written as c/b or c÷b.

(4)

What are the names for the numbers in a division problem?

a=c ÷b, a is the quotient, c is the dividend, and b is the divisor. Quotient=dividend÷divisor or

quotient=dividend

divisor

(5)

Is the order of numbers important? When dividing the order of the numbers is important: 40/5 does not equal 5/40.

(6)

Are there any special rules when dividing? Division is special when dividing by 0 and 1. A number divided by zero is not defined. Any number divided by one is equal to that number.

Learning Tips

(1)

Children can use manipulatives to multiply numbers. For example, to use manipulatives to multiply 7 x 8, ask the child to count out either 7 groups of 8 manipulatives in each group, or 8 groups with 7 manipulatives in each. Bring them all together into one pile and count how many there are. After noting that this problem was either 7x8=56 or 8x7=56 on paper (so no one forgets what work was completed already), ask the child to divide the 56 blocks back into groups that show the original problem. Ask your child, “Do you see 2 ways to use the special relationship between these 3 numbers to check that your work is correct?” (If you are sure that 8x7=56, then you can be sure that 7x8 is 56. Similarly, if 56÷7 is 8, you can check it by grouping 8 manipulatives into each of 7 groups and counting to see that it is still 56.

(2)

Use the sheet of blank diagrams (attached to encourage children to write in the number relationships that correspond to multiplication facts. Reproduce the sheet as often as needed to cover all of the possibilities. Then allow children to play a guessing game in which they take turns covering any one number in a number group. Another child should be able to name the covered number, just by seeing what the other two are. Point out that if the covered number is the higher-value one, the child is using his/her multiplication knowledge to name this “mystery number.” Conversely, if one of the smaller numbers is covered, the child is using his/her knowledge of the division combinations to name the mystery number.

(3)

Later in your child’s mathematics education the ability to group these multiplication/division relationships in a way that the child can quickly state what the missing number is will be a critical factor in your child’s ability to manipulate fractional parts quickly. It is a valuable skill to teach your child and well worth your time to make sure that your child can quickly name the other number of one of these relationships very quickly when the first two are presented. Later you can extend these to the game of “The large number in this triad is 12. What other groups also have 12 as the large number? (3,4,12; 2,6,12; 1, 12, 12)

(4)

Encourage your child to take the time, when working multiplication/division problems involving more than basic combination, to use what he/she knows about how numbers are related through multiplication and division. If, for example, the problem is 23 x 16, after your child solves it ask him/her to write 16 x 23 and see if the same answer is obtained. Similarly, if 60 is divided by 20, with the answer calculated as 3, ask your child to multiply 20 x 3 and see if the result obtained is the third number of the set, namely 60.

(5)

Remind your child that 0 has a special circumstance. While you can always multiply by 0 (the answer will always be 0 because if you have either no groups, or nothing in the group, the only possible result is 0) you cannot divide by 0 because, as a little experimentation with manipulatives will show, it is not possible to divide a number by 0 because while you can represent a group with nothing in it, you cannot represent dividing items into no groups. If there are no groups to divide items into, then there can be no division.

Extra Help Problems

(1)

81 divided by 9= (9)

9 x 9= (81)

(2)

1,000 x 8 = (8,000)

8,000 divided by 8 = (1,000)

(3)

25 ÷ 5= (5); 5 x 5= (25)

(4)

78 ÷ 6 = (13); 13 x 6 = (78)

(5)

45 ÷ 9 = (5); 5 x 9= (45)

(6)

35 ÷ 5=(7); 5 x 7 = (35)

(7)

72 ÷4=(18); 18 x 4 = (72)

(8)

91 ÷7=(13); 13 x 7=91

(9)

60 ÷4 = (15); 15 x 4 = (60)

(10)

88 ÷ 4 = (22); 22 x 4 = (88)

(11)

84 ÷ 6 = (14); 14 x 6 = (84)

(12)

70 ÷ 7 = (10); 10 x 7 = (70)

(13)

65 ÷ 5 = (13); 13 x 5 = 65

(14)

30 ÷ 1= (30); 30 x 1 = 30

(15)

60 ÷ 2 = (30); 30 x 2 = 60

(16)

27 ÷ 3 = (9); 9 x 3 = (27)

(17)

48 ÷ 8 = (6); 6 x 8 = (48)

(18)

99 ÷ 9 = (11); 11 x 9 = (99)

(19)

90 ÷ 6 = (15); 15 x 6 = (90)

(20)

64 ÷ 8 = (8); 8 x 8 = (64)

(21)

72 ÷ 9 = 8); 8 x 9 = (72)

(22)

7 x 9 = (63); 63 ÷ 7 = (9)

(23)

63 ÷ 3 = (21); 21 x 3 = (63)

(24)

42 ÷ 3 = (14); 14 x 3 = (42)

(25)

52 ÷ 4 = (13); 13 x 4 = (52)

 

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