Multiplication and Division Activities

 

Multiplication Activities

 

 

 

Division Activities

 

 

Combination of both: multiplication and division

 

 

Multiplication Activities:

 

What is Multiplication?

Kelly, K. & Zeman, A, (1994). Everything You Need To Know About Math Homework: a desk reference for students and parents. Scholastic Inc., New York.

 

What is Multiplication?

Multiplication is quick form of addition. By multiplying numbers together, you are really adding a series of one number to itself.

For example: if you wanted to calculate the number of days in five weeks, you could add 7 days + 7 days + 7 days + 7 days + 7 days, or you could multiply 7 days times 5. Either way you arrive at 35, the number of days in five weeks.

 

7 + 7 + 7 + 7 + 7= 35

7 % 5 =35

 

The multiplication sentence is different from the addition sentence.

   2 ----- multiplicand

% 2 ----- multiplier

  4 ----- product

 

Curriculum outcomes:

Grade one: SCOB1

Grade three: SCOB1

 

Multiplication Table:

 

X

  0

1

2

3

4

5

6

7

8

9

10

11

12

1

  0

  1

2

3

4

5

6

7

8

9

10

11

12

2

  0

2

4

6

8

10

12

14

16

18

20

22

24

3

  0

3

6

9

12

15

18

21

24

27

30

33

36

4

  0

4

8

12

16

20

24

28

32

36

40

44

48

5

  0

5

10

15

20

25

30

35

40

45

50

55

60

6

  0

6

12

18

24

30

36

42

48

54

60

66

62

7

  0

7

14

21

28

  35

42

49

56

63

70

77

84

8

  0

8

16

24

32

40

48

56

64

72

80

88

96

9

  0

9

18

27

36

45

54

63

72

81

90

99

108

10

  0

10

20

30

40

50

60

70

80

90

100

110

120

11

  0

11

22

33

44

55

66

77

88

99

110

121

132

12

  0

12

24

36

48

60

72

84

96

108

120

132

144

Have students explore the following patterns in the multiplication table and explain why they work:

-         The numbers in each row and column increase by the same amount

-         The square numbers are found on the left-right diagonal and the numbers on the left-right diagonal increase by 1, 3, and 5…

-         The row for products of 4 is double the row for products of 2,

The row for products of 6 is double the row for products of 3.

-         When you add the corresponding products for 2 and 3, you get the products for 5, e.g. 2%4 (8) plus 3%4 (12) is the same as 5%4 (20).

-         When you “cross multiply” any 4 numbers that form a square on the grid, the product is the same, e.g.

     2%6 = 3%4

-         When you “cross add” these numbers and subtract, you always get 1.

-         The grid is symmetrical, i.e., numbers under the left-right diagonal are reflections of the numbers over this diagonal.

 

Curriculum outcomes:

SCOC3

 

Doubling Activity

Taken from:

VanCleave’s J. (1991). Math for Every Kid. John Wiley & Sons, Inc., New York.

 

Purpose: To determine the number of sections formed by folding a sheet of paper a specific number of times.

 

Materials: Typing paper

               Newspaper

 

Procedure:

 

Results:

Six folds produce 64 sections. It is difficult to fold any size paper more than sic times because of the thickness of the paper. The seventh fold produces 128 sections and an eighth fold would again double the number of sections, forming 256 sections.

 

 

 

 

Speedo Multiplication

Burns, M. (1982). Math For Smarty Pants. Page 33-37. Little, Brown and Company, Toronto.

 

This activity is dedicated to teaching children how to multiply the “speedo” way. It shows some of the patterns involved in multiplication so that students can multiply faster then they would if they memorized their times tables.

 

Speedo way to multiply by 11:

When you multiply a number by 11 using this method, you get your answer one digit at a time, starting in the ones place and moving to the left. Here’s an example: 523 % 11.

 

  1. The ones digit of the answer is the same as the ones digit on the number you are multiplying by 11:3.
  2. To get the tens digit of the answer look at the tens digit in the number: 2. Add that to its right-hand neighbor (the 3 in the ones place): 2 + 3 = 5. That’s the tens digit in the answer. Now you have 53.
  3. Continue the same way. To find the next digit of the answer, find the digit in the same place in the number and add it to its right-hand neighbor.

The answer so far: 753.

4.     The farthest left-hand digit in the answer is the left-hand digit in the number. The final                                 answer using this method is 5,753. Do you agree?

 

To practice here is another example:

145 % 11

Step one: the ones digit in the answer is 5.

Step two: now add the 4 to the 5 and get 9 for the tens place.

Step three: Now add the 1 to the 4 and that’s 5. The answer so far is 595.

Step four: last of all the 1 goes in front to finish the answer.

1,595!!!

Give your students some more problems to work on, they will eventually become really fast at multiplying by 11.

Exception!!!

Sometimes when you add a number to its neighbor, that sum is more than 9, so you have to carry 1, just like in regular addition. For example: 892 % 11.

Here’s how to do it:

  1. 892 % 11 The ones digit is 2. 892

                                                %  11

                                                **2

  1. 9 + 2 =11 (carry the 1)  892

                                        %  11

                                        *12

 

  1. 8 + 9 =17,           892

17+1 =18          %  11

(carry the 1)       812

 

  1. 8 + 1 =9       892

                            %  11

                          9,812 

 

In step three you add the 1 carried from 9 + 2 = 11.

In step from you add the 1 carried from 17 + 1 = 18.

 

Speedo Multiplication by 12:

 

To multiply by 12, the method is almost the same as for multiplying by 11. The difference is that you double the digit before adding its right-hand neighbor to it (except for the final digit). So to multiply 564 % 12, follow these steps:

 

1.     To get the ones digit, double the ones digit in the number you’re multiplying by 12:8.

  1. For the next digit in the answer, take the 6 and double it, then add it to its right-hand neighbor: 12 + 4= 16. Write the 6 and carry the 1; now you have 68.
  2. For the next digit, take the 5 in the number, double it, and then add it to its right-hand neighbor, which is 6: 10 + 6 = 16. Then add the 1 you carried from the previous addition to get 17. Write the 7 and carry the 1. This gives you 768.
  3. The left hand digit of the answer is the same as the left hand digit in the problem, but you have to add the 1 you’ve carried, so it’s 6. The final answer is 6,768.

 

Curriculum Outcomes:

Grade five: SCOC3

            SCOB9

                SCOB13

 

Making Multiplication Books:

 

This lesson plan describes an activity during which the children create a set of personalized mini multiplication books. The process of creating these books reinforces the concept of multiplication and provides a tool to help the children internalize basic multiplication facts.

 

I recommend that the children make one book per week so that they are not overwhelmed. Furthermore, exposure over an extended period of time assists in the internalization process.

 

 This activity is suitable for children (early elementary grades) who have a good recall of basic addition facts and have been introduced to the concept of multiplication.

 

You will find that the children are proud of their individualized creations. They are generally enthusiastic about showing their books to others and about using their books to solve multiplication problems.

 

 

Materials:

 

- 12 3"x3" pieces of construction paper for each book

 

- 2 4"x4" pieces of construction paper for each book

 

- yarn

 

- hole punch

 

- markers

 

 

 

Method:

 

It is a good idea to make a sample set of these books to show the students what it is they will be making.

 

1. Explain to the students that they will be making a set of mini multiplication books that will be used in subsequent lessons to help them solve multiplication problems.

 

2. For the first book, the 2's multiplication facts, give each student 12 pieces of 3" x 3" construction paper of assorted colors.

 

3. For the first page, instruct the students to draw two stars in the center of the book, one on top of the other. Then tell the students to write the multiplication fact under the diagram: 2 x 1 = 2

 

4. For the second page, tell the students to draw two sets of two stars and to write the multiplication fact under the diagram: 2 x 2 = 4

 

The contents of the first two pages should look similar to the following:

 

* **

* **

2 x 1 = 2 2 x 2 = 4

 

5. Continue to create the pages for each fact up to 2 x 12.

 

6. Use the larger pieces of construction paper for the front and back covers of the book. Have each student write the title, author, and date on the front cover (something along the lines of My Multiplication book of 2's by Student X, March 17, 1996).

 

7. Assemble the book and punch two holes along the left-hand edge. Have the students thread the piece of yarn through the holes and tie in a knot to hold the book together.

 

8. As an alternative to drawing stars, have each student use the first letter of his/her name as the symbol. This way the book is truly personalized.

 

9. Show the students that the books can be used as a reference tool. Some activities with the books

include:

 

- Fact lookup. Tell the students to take out a particular book, for instance the fours multiplication book. Tell the students to find the multiplication fact 4 x 7. After each student has found the particular fact, give them several more facts to find from that book.

 

- Multiplication riddles: Make up riddles such as "I'm thinking of a number which when multiplied by 3 equals 21. What number am I?" and let the students use a particular book to find the answer.

 

Encourage the children to find patterns in their books and to talk about those patterns.

 

Curriculum Outcomes:

SCOB1

SCOB7

 

Multiplication Properties:

 

Repeated Aggregation

Repeated aggregation is an elementary concept of describing multiplication as so many sets of. For example, if we have 10 sets of 3 counters, then multiplication as repeated aggregation is considered as 3 % 10. If 10 sets of 3 counters is 3 % 10, how many counters are there altogether?

 

This multiplication formula is simply an extension of the aggregation structure of addition, in which 3 % 10 can be displayed as the repeated addition of 3 + 3 + 3 + 3 + 3 + 3 + 3+ 3 + 3 + 3.

 

The Commutative Property of Multiplication

 

The basic notion of this theory is that any two numbers can be multiplied together, regardless of the order in which they are written and always result in the same answer. Addition also displays this quality. We can recognize this commutative property formally by the following two generalizations:

a+ b = b + a

and

b % a = a % b.

It is important to note that subtraction and division do not have the same property.

In multiplication, the use of the commutative property enables us to simplify some calculations. For example, 14 sets of 5, would be evaluated by many of us as 5 sets of 14, because we have more experience multiplying by 5 than we do multiplying by 14.

Teachers should realize that the commutative property of multiplication is by no means obvious. Aside from counting the squares in each picture, we would not immediately recognize that panel a and panel b of figure 1 have the same number of counters.

 

x x x           ---        

x x x           ---

x x x           ---

x x x           ---

x x x           ---

 

Figure 1: examples of rectangular arrays for 3 % 5

 

Associative Property: When three or more numbers are multiplied, the product is the same regardless of the order of the multiplication. For example (2 % 3) % 4 = 2 % (3 % 4)

 

Multiplicative Identity Property: The product of any number and one is that number. For

example 5 * 1 = 5.

 

Distributive property: The sum of two numbers times a third number is equal to the sum

of each addend times the third number. For example 4 * (6 + 3) = 4*6 + 4*3

 

Curriculum Outcomes:

SCOB1(grade three)

SCOB7

SCOB2 (grade four)

 

Division Activities:

 

What is Division?

Division is the process of finding out how many times one number, the divisor, will fit into another number, the dividend. The division sentence results in a quotient. You can think of division as a series of repeated subtractions. For example, 40 + 10 could also be solved by subtracting 10 four times:

40 –10 – 10 – 10 – 10 = 0

Because 10 can be subtracted four times, you can say that 40 can be divided by 10 four times, or 40 + 10 = 4.

 

40 (dividend) + 10 (divisor) = 4 (quotient)

 

Curriculum outcomes:

Grade one: SCOB1

Grade two: SCOB2

Grade three: SCOB2

 

 

Children’s Literature

Froman, R. (1978). The Greatest Guessing Game: A book about division. New York.

 

This is a wonderful children’s book that focuses on division. It shows children that all division involves guessing (or estimating). Whether you are dividing the contents of a bottle of pop, a pile of peanuts, money a group of friends has earned, or the books in a second-hand library, the important thing you do is guess how many or how much each person should get of what you are dividing.

          The next step is to revise your original guess so that the division comes out as evenly as possible, and finally you decide what to do with the left overs (if there is anything left over).

          This book provides lifelike examples so that children can grasp this often-difficult concept of dividing. It is a wonderful place to start when introducing division to children.

 

Suggested Activity: After reading the book once, you and your students can bring it to life. Take each example provided in the book and re-enact it through drama. In doing this the children will manipulate money, peanuts, and pop and will be able to do the divisions on their own.

 

Which Numbers are Divisible?

 

2… For a number to be divisible by 2, for example, it has to be even. Numbers such as 8,10, 64, and 2, 368 are divisible by 2; 5, 67, and 103 aren’t. So you can tell whether a number is divisible by 2 just by looking to see if it’s even. You don’t actually have to do the division.

 

3…Deciding if a number is divisible by 3, however, isn’t so obvious. Take 144, for example. You can’t tell just by looking whether 3 will go into 144 with no remainder or not. You could find out by dividing, but there’s another way. To test if a number is divisible by 3, add up the digits in the number you’re testing. Is that sum divisible by 3? Take 144, for example: If you add 1+4+4, you get 9. Since 9 is divisible by 3, so is 144.  This test is useful for large numbers such as 273, 645. Add the digits: 2+7+3+6+4+5. The sum is 27. Still not sure? Then add the 2 and the 7. That should convince you.

 

4…For divisibility by 4, you can test by looking at the last two digits of the number you’re testing. If that number is divisible by 4, so is the entire number. This works well for large numbers such as 2,365,716. The last two digits are 16, which is divisible by 4. Check with your calculator to see if the larger number is also.

 

5…You can go back to the “look” method to test for divisibility by 5. Any number that ends in 0 or 5 is divisible by 5. That’s easy!

 

6…See if you can figure a divisibility test for dividing by 6. Hint: you have to combine two of the tests given so far.

 

7…This is a tricky one. A weird 3—2—1 pattern helps here. For example, to test an enormous number, such as 6,124,314, you have to figure like this: 3% the ones digit + (2 % the tens digit) – (1 % the hundreds digit)- (3 % the thousands digit) – (2 % the ten thousands digit) + (1 % the hundred thousands digit) + (3 % the millions digit). Try the test on 6, 124, 314, then check your test by actually dividing it out.

 

 

8…If the last three digits form a number divisible by 8,

               then so is the whole number.

           

9… If the sum of the digits is divisible by 9, the number is also.

          

10… If the number ends in 0, it is divisible by 10.

 

11…Alternately add and subtract the digits from left to right.

         If the result (including 0) is divisible by 11, the number is also.

         Example: to see whether 365167484 is divisible by 11, start by subtracting:

        3-6+5-1+6-7+4-8+4 = 0; therefore 365167484 is divisible by 11.

       

12…If the number is divisible by both 3 and 4, it is also divisible by 12.

          

13…Delete the last digit from the number, then subtract 9 times the deleted

          digit from the remaining number. If what is left is divisible by 13,

          then so is the original number.

 

Curriculum Outcomes:

Grade four: SCOB5

Grade five: SCOB14

                 SCOC3

 

Combination Activities:

 

Multiplication and Division Bonds:

 

Foster L. (1985). Mathematics Encyclopedia. Rand McNally & Company, United States of America.

 

This activity will help students understand the relationship between division and multiplication.

 

Explanation: Division is the opposite of multiplication. They are related to each other in the same way as addition and subtraction. Multiplication and division bonds can be learned together after understanding how they work.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

4%3=12

 

 

 

 

3%4=12

 

 

12+3=4

 

 

 

 

 

 

 

 

 

 

 

 

 

12+4-3

 

Through using tables like this one, students will get a visual picture of the relationship between multiplication and division.

 

Curriculum Outcomes:

Grade three: SCOB3

                   SCOB8