Fold the typing paper in half to produce 2 sections.
Fold the paper in half again to produce 4 sections.
Continue to fold the paper until 6 folds have been made.
Determine the number of sections produced by doubling the section number after each folding.
Open the paper after the sixth folding and count the sections to check your calculated answer.
Refold the typing paper and determine how many times it can be folded in half.
Use the newspaper sheet and determine how many times it can be folded in half.

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.

What is Multiplication?

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

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

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

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

2. 9 + 2 =11 (carry the 1) 892

% 11

*12

3. 8 + 9 =17, 892

17+1 =18 % 11

(carry the 1) 812

4. 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.

2. 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.

3. 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.

4. 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 ---

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)

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

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

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.

An Informal Method of Multiplying

This method is based on the idea of splitting up both the numbers that are being multiplied into their tens units. So 26 becomes 20 + 6, and 34 becomes 30 + 4. Then we have to multiply 20 by, 4, 6, by 30, and 6 by 4.

You can visualize the multiplication as a problem of finding the numbers of counters in a rectangular array of 26 by 34. Thinking of 26 and the 34 as 20 + 6 and 30 +4 respectively suggests that we can split the array into four separate arrays, representing 20 % 30, 20 % 4, 6 % 30, and 6 % 4.

20 % 30

34 30 4

20 kkjhk 20 20%4

6`` 6%4

6 % 30

Figure 1: Using area to interpret 26 % 34

The answer to the multiplication calculation is obtained by working out the areas of the four separate rectangles and adding them up. The actual calculation can be written out as follows:

20 % 30 = 600

6 % 30 = 180

20 % 4 = 80

6 % 4 = 24

26 % 34 = 884

This method extends quite easily to multiplication calculations involving three-digit numbers.

Curriculum Outcomes:

SCOB2

SCOB1

Relating Multiplication To Repeated Addition

This activity is appropriate for large group learning. Activity 2 and 3 are suited to small groups or pairs of students.

This activity has been altered so that the teacher does not require the student handbook. The activity idea comes from the Interaction Math Binder, Unit 2.

1. Counting Wheels:

Display a variety of objects with wheels, such as a pair of in-line skates. Distribute counters or cubes to pairs of children. Focus the children’s attention on the objects with wheels. Ask them to model the number of wheels on 1 skate, then 2 skates, to 6 skates. Do this with different objects (such as a bicycle). As the children are working go from group to group helping and encouraging where necessary. Bring the groups together to share and talk about the problems.

2. Discuss which situations show equal groups even when the objects are not identical. Ask the children to explain the meaning of repeated addition.

3. The children might also find the total number of wheels in a picture provided to them.

Encourage them to do this mentally.

Curriculum Outcomes:

SCOB1 (grade one)

SCOA2 (grade two)

SCOB1 (grade two)

SCOB1 (grade three)

SCOC4 (grade three)

Making Arrays:

Materials: counters, overhead projector

Provide each pair of children with a handful of counters. Invite a child to make a rectangular arrangement of counters on the overhead projector and have the class copy it using their counters. Ask how to describe the array as a number of equal rows and as a repeated addition sentence.

For example: 3 (number of rows) % 4 (number in each row) = 12 (total number of counters)

Discuss how the repeated addition sentence can be rewritten as a multiplication sentence, and the meaning of the word product.

Write a repeated addition sentence on the board. Have the children use their counters to show the array and write the corresponding multiplication sentence. Then write a multiplication sentence on the board and have the pairs show the array and write the corresponding addition sentence.

Have pairs of children do activities such as the following:

One child could make an array or write a repeated addition sentence or a multiplication sentence, and the partner completes the other two descriptions for the one given.
Partners could take turns choosing the total number and making a corresponding array.
The children could make arrays using the last two digits of their first and last names.

Curriculum outcomes:

SCOA2 (grade two)

SCOA5 (grade two)

SCOB1 (grade two)

SCOB5 (grade three)

SCOC4 (grade three)

Counting Money

Materials: coins, items or pictures of items with price tags ending in 5 or 0.

Arrange the class in groups and distribute coins or play money. Have the children separate 30 pennies into groups of 5. Then ask them to write the repeated addition sentence and multiplication sentence for their model.

6 groups of 5 pennies.

5 + 5 + 5 + 5 + 5 + 5 = 30

6 % 5 = 30

Have the children replace each group of 5 pennies with a nickel. Discuss how this model can be written as a multiplication sentence, where each nickel represents 5 pennies.

6 (number of nickels) % 5 (number of cents in each nickel) = 30 (total number of cents)

Repeat the activity, this time having the children separate 30 pennies into groups of 10, and replacing each group of 10 pennies with a dime.

3 (number of dimes) % 10 (number of cents in each dime)= 30 (total number of cents)

Ask the children such as:

How much are 4 nickels worth? 4 dimes? Write the multiplication sentences.
How can using coins help find 5 % 5? Find 4 % 10? Show 3 % 10 = 6 % 5?
Why can’t a bunch of nickels be worth .64 cents?

Display several items with price tags ending in 5 or 0. Have the children use their coins to show the number of nickels required to pay for each item, then if possible, the number of dimes. Ask them to write the addition and corresponding multiplication sentences.

Display items that have a price tag of 5 cents and 10 cents. Discuss their multiplication sentences. 1 % 5 = 5 and 1 % 10 = 10. Use these examples to help the children realize that a number multiplied by 1 is the number itself.

Curriculum Outcomes:

SCOA2 (grade two)

SCOB1 (grade two)

SCOB1 (grade three)

Measuring Containers

Materials: litre containers graduated in units of 100ml, pourable material (salt, rice, water), and jars marked at 100ml or 200ml.

Distribute to each group of children a jar marked at 100ml and another jar marked at 200ml, a litre container marked at 100ml intervals, and a supply of pourable material. Ask the groups to predict, then find out how many times the 100ml jar needs to be filled to make 200ml in the other jar. Help them write a repeated addition sentence and a multiplication sentence to describe the results.

100ml + 100ml = 200ml

2 % 100ml = 200ml

Have the children use the 100ml jar to fill the litre container to different levels (such as 300ml, 400ml, 500ml, ... , 1000ml), and then write the addition and multiplication sentences to show the results.

100ml + 100ml + 100ml + 100ml + 100ml = 500ml

5 % 100ml =500ml

Then have the groups use the 200ml jar to fill the litre container to different levels (such as 200ml, 400ml, 600ml, . . ., 100ml), and write the number sentence to show the results.

Invite one child from each group to write on the board, the results of their pouring experiments. Discuss with the class any patterns they see in their number sentences.

Curriculum outcomes:

SCOB1 (grade one)

SCOA2 (grade two)

SCOA5 (grade two)

SCOB1 (grade two)

SCOB5 (grade two)

SCOC4 (grade two)

SCOB2 (grade four)

SCOB9 (grade five)

Grouping For Fun…Division activity

Materials: Counters or connecting cubes

Involve children in creating a display chart similar to the following:

Activities for 1: solitaire, reading, puzzles.

Activities for 2: badminton, tennis, and video games.

Activities for 3: marbles, jacks, skipping.

Activities for 4: doubles tennis, table tennis, relay teams

Activities for 5: basketball

Activities for 6: hockey, volleyball

Challenge small groups to choose 1 activity from each and find out how many equally groups are possible for the entire class. Have the groups use counters or cubes to represent the children in their class.

Bring the class together to share solutions for each grouping. Check several solutions by having the children arrange themselves in groups. Use the terms dividing, equal, remainder, and leftover as you help the children summarize each grouping in a division statement. Talk about how leftovers or remainders could be handled:

For example: 26 divided into groups of 3 is 8 equal groups with 2 remaining.

The children can go with other groups and take turns.

The two left over children could jump rope on their own.

Curriculum outcomes:

SCOB2 (grade three)

SCOB5 (grade three)

SCOB9 (grade four)