PRINCIPLE 1: Multiplication of Any Number by Zero Equals Zero. a X 0 = 0

Therefore, prior to applying my Seven Day system, I strongly suggest that teachers be certain their students have been given the appropriate background so they understand what multiplication is all about. To do this, I recommend Marilyn Burns’ wonderful book entitled Math By All Means, Multiplication.

Depending on the background of the student, you may not have to do every lesson but can pick and choose from the delightful menu of lessons and activities. Also, before beginning my Seven Day program, there are three math principles of which one needs to make students aware:

PRINCIPLE 1: Multiplication of any number by zero equals zero. A x 0 = 0

PRINCIPLE 2: Multiplying any number by one yields the number multiplied. A x 1 = A

PRINCIPLE 3: The order in which we multiply does not change the result. A x B = B x A (The Commutative Property of Multiplication)

To begin, I recommend that students be given a multiplication table (see Table 1) without the products and have them complete the table.

To complete the table they may use any prior knowledge, manipulatives, or strategy they choose. Completing the table will help them learn how to “read the table” and will also show them some of the many patterns.

Next, give each student Table 2 and discuss how the three principles above allow us to black out certain facts (see Table 3). The next step is to look at Table 3 and observe any patterns. Students may suggest many patterns, but there are three that need to be found.

· First, notice that the product of any number and zero equals zero (a x 0 = 0). So we need not memorize any facts involving zero as a factor. (see Table 3)

· Secondly, it should be observed that one times any number, a, has a product of a (1 x a = a). Therefore we need not memorize any products involving one, since 1 x a = a for example, 1 x 3 = 3, 1 x 7 = 7, etc. Mark out those products as shown in Table 3.

· Thirdly, it should be observed that the products of numbers where the order is reversed is the same.

For example, notice 3 x 4 = 4 x 3, 6 x 5 = 5 x 6, etc. Therefore, since a x b = b x a, only half of the table needs to be memorized, since when you know 6 x 9 you already know 9 x 6, etc.

Mark out all products that “commute”, are reverses of a set of products (see Table 3).

At this point, you have only 36 products that have not been marked out. These are the facts we need to memorize.

Note that we have eliminated a x 0 because the answer is always zero. Nothing to memorize.

Note that we have eliminated a x 1 because multiplication by one does not change anything. 1 x a = a

Note that since a x b = b x a (order does not matter), only part of the facts need to be memorized because when you know 5 x 6 = 30 you know that 6 x 5 = 30, for example.

Now the process. (Refer to Table 3)

v Special note: Each day send the Parent Letter home to ask parents to help.

Day 1: Learn the Squares (see Lesson 1)

2 x 2 = 4 4 x 4 = 16 6 x 6 = 36 8 x 8 = 64

3 x 3 = 9 5 x 5 = 25 7 x 7 = 49 9 x 9 = 81

Why squares first? Actually the answer is “we don’t know”, but for some reason the squares are easiest to learn. This encourages students because it builds confidence. I also like to solicit the parents’ help and send home Letter A to the parent or guardian. Parents are eager to help and this saves classroom time normally used for more drill. Use Lesson 1 in class.

Day 2: Learn the Twos (see Lesson 2)

Actually, there are only 7 facts to learn:

2 x 3 = 6 2 x 5 = 10 2 x 7 = 14 2 x 9 = 18

2 x 4 = 8 2 x 6 = 12 2 x 8 = 16

Why twos next? Two reasons:

1. They are easy to learn (probably because we frequently count by 2’s)

2. Again, we build confidence that this is an easy task.

Do Lesson 2 and send home Letter B. Review squares.

Day 3: Learn the Fives

Again, we learn fives next because they are easily learned and there are only 4.

5 x 6 = 30 5 x 7 = 35 5 x 8 = 40 5 x 9 = 45

Do Lesson 3 and send home Letter C. Review squares and twos.

Day 4: Learn the Threes

3 x 4 = 12 3 x 6 = 18 3 x 8 = 24

3 x 5 = 15 3 x 7 = 21 3 x 9 = 27

Do Lesson 4 and send home Letter D. Review squares, twos, and fives.

Day 5: Learn the Fours

4 x 5 = 20 4 x 7 = 28 4 x 8 = 32

4 x 6 = 24 4 x 9 = 36

Do Lesson 5 and send home Letter E. Review squares, twos, fives and threes.

Day 6: Learn Sixes, Sevens and Eights

6 x 7 = 42 6 x 9 = 54 7 x 9 = 63

6 x 8 = 48 7 x 8 = 56 8 x 9 = 72

Do Lesson 6 and send home Letter F. Review squares, twos, fives, threes, and fours.

Day 7: All Facts, Review and Test

Focus on those facts the children still have trouble with and individualize Letter G to parents.

That’s it! Now the only other thing you need to do is to periodically review. I strongly suggest using games. For a good set of games I recommend Multi-Level Math Fun from Creative Teaching Press.

Table 1

x / 0 / 1 / 2 / 3 / 4 / 5 / 6 / 7 / 8 / 9
0
1
2
3
4
5
6
7
8
9

Table 2

x / 0 / 1 / 2 / 3 / 4 / 5 / 6 / 7 / 8 / 9
0 / 0 / 0 / 0 / 0 / 0 / 0 / 0 / 0 / 0 / 0
1 / 0 / 1 / 2 / 3 / 4 / 5 / 6 / 7 / 8 / 9
2 / 0 / 2 / 4 / 6 / 8 / 10 / 12 / 14 / 16 / 18
3 / 0 / 3 / 6 / 9 / 12 / 15 / 18 / 21 / 24 / 27
4 / 0 / 4 / 8 / 12 / 16 / 20 / 24 / 28 / 32 / 36
5 / 0 / 5 / 10 / 15 / 20 / 25 / 30 / 35 / 40 / 45
6 / 0 / 6 / 12 / 18 / 24 / 30 / 36 / 42 / 48 / 54
7 / 0 / 7 / 14 / 21 / 28 / 35 / 42 / 49 / 56 / 63
8 / 0 / 8 / 16 / 24 / 32 / 40 / 48 / 56 / 64 / 72
9 / 0 / 9 / 18 / 27 / 36 / 45 / 54 / 63 / 72 / 81

Table 3

Letter A

Dear Parent,

Over the next seven days, we will be learning the multiplication facts. We will be using a system called Seven Days to Multiplication Mastery. Each day I will send you some problems that I want you to help your child master. Have your child drill these facts and these facts only so that your child will master these by the next day. How you choose to do this is really up to you.

TODAY’S FACTS

2 x 2 = 4 3 x 3 = 9 4 x 4 = 16

5 x 5 = 25 6 x 6 = 36 7 x 7 = 49

8 x 8 = 64 9 x 9 = 81

It is very important to learn the facts only today and only these facts. I really appreciate your help.

______

Teacher

Letter B

Dear Parent,

This is the second day in our series of lessons designed to teach your child all of the multiplication facts in seven days. Today I would like for you to help your child learn the following facts:

TODAY’S FACTS

2 x 3 = 6 2 x 7 = 14

2 x 4 = 8 2 x 8 = 16

2 x 5 = 10 2 x 9 = 18

2 x 6 = 12

Review the squares (below):

2 x 2 = 4 3 x 3 = 9 4 x 4 = 16

5 x 5 = 25 6 x 6 = 36 7 x 7 = 49

8 x 8 = 64 9 x 9 = 81

Thank you for your help.

______

Teacher

Letter C

Dear Parent,

Today we continue having your child learn the multiplication facts. On this third day we are focusing on the multiplication facts below. Please drill these facts and review the previous facts (squares and twos).

TODAY’S FACTS

5 x 6 = 30 5 x 8 = 40

5 x 7 = 35 5 x 9 = 45

Review:

2 x 2 = 4 7 x 7 = 49 2 x 5 = 10

3 x 3 = 9 8 x 8 = 64 2 x 6 = 12

4 x 4 = 16 9 x 9 = 81 2 x 7 = 14

5 x 5 = 25 2 x 3 = 6 2 x 8 = 16

6 x 6 = 36 2 x 4 = 8 2 x 9 = 18

Thank you for your help.

______

Teacher

Letter D

Dear Parent,

On this fourth day of our “Seven Days to Learning Multiplication Facts”, please drill the following facts:

TODAY’S FACTS

3 x 4 = 12 3 x 6 = 18 3 x 8 = 24

3 x 5 = 15 3 x 7 = 21 3 x 9 = 27

Review these facts:

2 x 2 = 4 2 x 3 = 6 5 x 6 = 30

3 x 3 = 9 2 x 4 = 8 5 x 7 = 35

4 x 4 = 16 2 x 5 = 10 5 x 8 = 40

5 x 5 = 25 2 x 6 = 12 5 x 9 = 45

6 x 6 = 36 2 x 7 = 14

7 x 7 = 49 2 x 8 = 16

8 x 8 = 64 2 x 9 = 18

9 x 9 = 81

Thanks for your help.

______

Teacher

Letter E

Dear Parent,

This is the fifth day your child is learning the multiplication facts. Please drill your child on the following facts:

TODAY’S FACTS

4 x 5 = 20 4 x 7 = 28 4 x 8 = 32

4 x 6 = 24 4 x 9 = 36

Review the following:

2 x 2 = 4 2 x 3 = 6 5 x 6 = 30 3 x 4 = 12

3 x 3 = 9 2 x 4 = 8 5 x 7 = 35 3 x 5 = 15

4 x 4 = 16 2 x 5 = 10 5 x 8 = 40 3 x 6 = 18

5 x 5 = 25 2 x 6 = 12 5 x 9 = 45 3 x 7 = 21

6 x 6 = 36 2 x 7 = 14 3 x 8 = 24

7 x 7 = 49 2 x 8 = 16 3 x 9 = 27

8 x 8 = 64 2 x 9 = 18

9 x 9 = 81

Thank you for your help.

______

Teacher

Letter F

Dear Parent,

This is the sixth day of learning the multiplication facts. Your child is memorizing the following. Please drill your child on the following facts:

TODAY’S FACTS

6 x 7 = 42 7 x 8 = 56 8 x 9 = 72

6 x 8 = 48 7 x 9 = 63

6 x 9 = 54

Review the following:

2 x 2 = 4 2 x 3 = 6 5 x 6 = 30 3 x 8 = 24

3 x 3 = 9 2 x 4 = 8 5 x 7 = 35 3 x 9 = 27

4 x 4 = 16 2 x 5 = 10 5 x 8 = 40 4 x 5 = 20

5 x 5 = 25 2 x 6 = 12 5 x 9 = 45 4 x 6 = 24

6 x 6 = 36 2 x 7 = 14 3 x 4 = 12 4 x 7 = 28

7 x 7 = 49 2 x 8 = 16 3 x 5 = 15 4 x 8 = 32

8 x 8 = 64 2 x 9 = 18 3 x 6 = 18 4 x 9 = 36

9 x 9 = 81 3 x 7 = 21

Thank you for your help.

______

Teacher

Letter G

Dear Parent,

This is the last day in our series of Learning the Multiplication Facts in Seven Days. All of the facts have been covered, but there may be a few your child needs to review and memorize. Below is a set of all the facts. See that your child knows the facts and spend extra time reviewing any that cause your child to hesitate or that the child cannot recall. Occasionally go back and review the facts.

Squares Twos Fives Threes

2 x 2 = 4 2 x 3 = 6 5 x 6 = 30 3 x 4 = 12

3 x 3 = 9 2 x 4 = 8 5 x 7 = 35 3 x 5 = 15

4 x 4 = 16 2 x 5 = 10 5 x 8 = 40 3 x 6 = 18

5 x 5 = 25 2 x 6 = 12 5 x 9 = 45 3 x 7 = 21

6 x 6 = 36 2 x 7 = 14 3 x 8 = 24

7 x 7 = 49 2 x 8 = 16 3 x 9 = 27

8 x 8 = 64 2 x 9 = 18

9 x 9 = 81

Fours Sixes Sevens Eights

4 x 5 = 20 6 x 7 = 42 7 x 8 = 56 8 x 9 = 72

4 x 6 = 24 6 x 8 = 48 7 x 9 = 63

4 x 7 = 28 6 x 9 = 54

Thank you for your help.

______

Teacher

Lesson 1, Squares

Complete the following. If you need help, find the answers in your table.

(1) 2x2 = _____

(2) 3x3 = _____

(3) 4x4 = _____

(4) 5x5 = _____

(5) 6x6 = _____

(6) 7x7 = _____

(7) 8x8 = _____

(8) 9x9 = _____

(9) 8x8 = _____

(10) 7x7 = _____

(11) 6x6 = _____

(12) 5x5 = _____

(13) 4x4 = _____

(14) 3x3 = _____

(15) 2x2 = _____

(16) 2x2 = _____

(17) 4x4 = _____

(18) 6x6 = _____

(19) 8x8 = _____

(20) 3x3 = _____

(21) 5x5 = _____

(22) 7x7 = _____

(23) 9x9 = _____

(24) 2x2 = _____

(25) 8x8 = _____

(26) 9x9 = _____

(27) 7x7 = _____

(28) 3x3 = _____

(29) 5x5 = _____

(30) 6x6 = _____

(31) 4x4 = _____

(32) 6x6 = _____

(33) 9x9 = _____

(34) 5x5 = _____

(35) 7x7 = _____

(36) 8x8 = _____

(37) 6x6 = _____

(38) 4x4 = _____

(39) 5x5 = _____

(40) 7x7 = _____

(41) 3x3 = _____

(42) 7x7 = _____

(43) 8x8 = _____

(44) 4x4 = _____

(45) 2x2 = _____

(46) 9x9 = _____

(47) 8x8 = _____

(48) 3x3 = _____

(49) 6x6 = _____

(50) 9x9 = _____

Lesson 2, Twos

Complete the following.

(1) 2x3 = _____

(2) 2x4 = _____

(3) 2x5 = _____

(4) 2x6 = _____

(5) 2x7 = _____

(6) 2x8 = _____

(7) 2x9 = _____

(8) 2x8 = _____

(9) 2x7 = _____

(10) 2x6 = _____

(11) 2x5 = _____

(12) 2x4 = _____

(13) 2x3 = _____

(14) 2x9 = _____

(15) 2x3 = _____

(16) 2x8 = _____

(17) 2x4 = _____

(18) 2x7 = _____

(19) 2x6 = _____

(20) 2x5 = _____

(21) 2x2 = _____

(22) 2x4 = _____

(23) 2x6 = _____

(24) 2x8 = _____

(25) 2x3 = _____

(26) 2x5 = _____

(27) 2x7 = _____

(28) 2x9 = _____

(29) 9x9 = _____

(30) 7x7 = _____

(31) 2x2 = _____

(32) 8x8 = _____

(33) 4x4 = _____

(34) 9x9 = _____

(35) 3x3 = _____

(36) 6x6 = _____

(37) 8x8 = _____

(38) 6x6 = _____