ELEMENTARY MATH 1

First we want to get your doubles down, these will help you in multiplication. Adding Doubles is the same as Multiplying by 2.

3+3 = &&& + &&& = &&&&&&

Multiplying is read as NUMBER OF ITEMS IN EACH GROUP and NUMBER OF GROUPS

so 6X2 means TWO GROUPS OF SIX or &&&&&& + &&&&&&

In other words DOUBLE the number.

 Double Addition Fact Answer Multiplication Fact 1+1 2 1X2 2+2 4 2X2 3+3 6 3X2 4+4 8 4X2 5+5 10 5X2 6+6 12 6X2 7+7 14 7X2 8+8 16 8X2 9+9 18 9X2 10+10 20 10X2

Match your DOUBLES and then match the Addition problem to the Multiplication problem  COMMUTATIVE PROPERTY

Commute is a word that means TO MOVE. The Commutative property means the numbers can MOVE around and swap places with each other but the answer doesn't change:

For example: 2+3 = 3+2; this is an example of COMMUTATIVE ADDITION

Another example: 6X2 = 2X6; this is an example of COMMUTATIVE MULTIPLICATION

Test yourself using the COMMUTATIVE PROPERTY: Now, you have been studying your doubles, you know that MULTIPLYING by 2 means to double, and you know that the order of the numbers for multiplication and addition can switch and still give you the same answer, so test yourself with the following quiz: BREAKING DOWN NUMBERS

Math is easier if you break down numbers into pieces that you know. Imagine your two hands with 10 fingers. We want to be able to break down a number to 5 + ?

In other words... if you have the number 6, what does that mean for your fingers? It means 1 WHOLE HAND (5 fingers) and 1 FINGER on the other hand: 5 + 1

What about 8? That means 1 WHOLE HAND (5 fingers) and 3 fingers on the other hand.

See you can match the fingers for each number between 6 and 9. Adding a number to 10 is very easy. Notice that 10 has 2 digits - the 1 is in the TENS place and the 0 is in the ONES place. If you add a single digit number to 10, it is added to the 0 part of the ten. A number plus zero is just the number so it is easy:

 10 +5 15 10 +3 13 10 +7 17 10 +9 19 10 +2 12

Now you try it: In the last two exercises you learned to break down a number to a 5 + another number and to add 10 + a number. Now we will combine these two activities.

Step 1: Re-write 6: 6 = 5 + 1

Step 2: Add to 5: 5 + 1 + 5

Step 3: Locate your two 5's, 5+5 = 10 so now we have: 10 + 1

Step 4: Add 10 + number: 10 + 1 = 11

Step 1: Rewrite 7: 7 = 5 + 2

Step 2: Rewrite: 6: = 6 = 5 + 1

Step 3: Combine: 5 + 2 + 5 + 1

Step 4: Find your two 5's, combine to be 10: 10 + 2 + 1

Step 5: Combine the 2 and the 1: 2 + 1 = 3

Step 6: Add 10 + number = 10 + 3 = 13

Right now this may seem like a lot of steps, but once you get good at it, you will be able to do this in your head quickly.

Step 1: Rewrite 9: 5 + 4

Step 2: Add to 5: 5 + 5 + 4

Step 3: Combine the 2 5's: 10 + 4

You should start to VISUALIZE the non-5 number as it's sum of 5 + NUMBER and SEE IT IN YOUR HEAD.

SEE 7 as 5 + 2

NOW SEE 10 + 2 = 12

SEE 6 as 5 + 1

NOW SEE 10 + 1 = 11

Try some matching: + 0 Anything + 0 is itself: 6 + 0 = 6 + 1 Anything + 1 is just the next number: 8 + 1 = 9 + 2 Count on 2 units: 4 + 2 = 6 + 3 Count on 3 units: 8 + 3 = 11 + 5 Break the other number into it's 5 + number parts: 5 + 8 = 5 + 5 + 2 = 12 + 6 Break both numbers into their 5 + number parts: 6 + 7 = (5 + 1) + (5 + 2)= 10 + 3 = 13 + 10 Just change the 0 to the number: 10 + 4 = 14

9's Strategy

Adding with 9 is ALMOST like adding with 10, 9 is ONE LESS than 10 - so if you have to add with a 9, pretend it is a 10 for easy adding and then take-away 1 since it was really a 9.

Example: 9 + 4

THINK: 10 + 4 = 14

THINK ONE LESS: 13

Example: 8 + 9

THINK 8 + 10 = 18

THINK ONE LESS: 17

Try some: ALMOST A DOUBLE

Another strategy is noticing that the pair of numbers is just ONE away from being a DOUBLE:

Example: 5+4

This is almost 5+5 but 4 is ONE LESS than 5.

We add it like 5+5 but take one less: 10 -1 =9

Example: 7+6

This is almost 7+7 but 6 is ONE LESS than 7.

We add it like 7+7 but take one less: 14-1= 13

This strategy is a little harder but with practice, you can do it!

Try finding the match: Let's try putting all these strategies together: Get a piece of paper and write the answers. Try to use as many strategies as you can.

 9 + 3 6 + 1 8 + 2 5 + 6 10 + 3 5 + 9 1 + 4 7 + 2 5 + 7 10 + 7 7 + 9 8 +1 2 + 6 5 + 9 4 + 10 6 + 9 1 + 7 2 + 4 5 + 8 10 + 8 9 + 4 5 + 0 3 + 9 6 + 6 5 + 10 8 + 9 0 + 3 7 + 3 7 + 6 7 + 7 9 + 9 8 + 0 3 + 4 8 + 7 8 + 8

 9 + 3 =12 6+1 = 7 8 + 2 = 10 5 + 6 = 11 10 + 3 = 13 5 + 9 = 14 1 + 4 = 5 7 + 2 = 9 5 + 7 = 12 10 + 7 = 17 7 + 9 = 16 8 +1 = 9 2 + 6 = 8 5 + 9 = 14 4 + 10 = 14 6 + 9 = 15 1 + 7 = 8 2 + 4 = 6 5 + 8 = 13 10 + 8 = 18 9 + 4 = 13 5 + 0 = 5 3 + 9 = 12 6 + 6 = 12 5 + 10 = 15 8 + 9 = 17 0 + 3 = 3 7 + 3 = 10 7 + 6 = 13 7 + 7 = 14 9 + 9 = 18 8 + 0 = 8 3 + 4 = 7 8 + 7 = 15 8 + 8 = 16

QUESTIONS:

1. Which is your favorite strategy you learned? Why?

2. Which strategy is the most challenging? Why?

SUBTRACTION

Subtraction is the same as TAKE-AWAY. When the number you are subtracting is small, you can just count backwards:

10-2: Think 10, 9, 8

7-3: Think 7, 6, 5, 4

12 - 1: Think 12, 11

When the number being subtracted is big, you want to count the other way.

10-8: Think how many MORE do I need to get from 8 to 10: 2 more

9-6: Think how many MORE do I need to get from 6 to 9: 3 more

When the top number is bigger than 10, we will use the BUILD TO 10 strategy.

14

-7

For each number, we need to figure out how far each number is away from 10.

For our 14, the answer is EASY - as we did in the previous section we know that 10 + 4 = 14, in fact it will just be the second digit of the number: 4.

For our 7, we will use a quick and easy number chart to find out:

 1 2 3 4 5 9 8 7 6 5

To make this table, write the numbers 1 through 5. Below the 5, write another 5 and then writing to the LEFT, continue to 9. Each pair should add to 10 if you did it right. So, if we want to find out how far 7 is from 10, we look on the chart for it's partner: 3.

Once we find the distance from 10 for both numbers, we add those two numbers together: 4 + 3 = 7 and that is our answers.

Sometimes it is easy to remember: Copy (the ones digit), Partner (use the table), and Add.

Examples: Note that the first number is just copied from the ones place, the second number is the partner from the table, then I add.

 13 -7 3 +3 6 15 -8 5 +2 7 14 -6 4 +4 8 14 -5 4 +5 9

Let's do it one step at a time together:

First how far from 10 are these numbers:

Use the chart to figure out how far these numbers are from 10?

Now to convert from a subtraction problem to an addition problem - build each to 10 and add the two numbers:

Multiplication

Multiplication by 0

Anything multiplied by 0 is 0.

5 X 0 = 0

Multiplication by 1

Anything multiplied by 1 is itself.

7 X1 = 7

Multiplication by 2

Double the number that is multiplied by 2.

8 X 2 = 16 (Double 8)

Multiplication by 10

Add a zero after the number multiplied by 10.

4 X 10 = 40 (Add a zero after the 4)

Multiplication by 100.

Add two zeros after the number multiplied by 100.

8 X 100 = 800 (Add two zeros after the 8)

Multiplication by 11.

Write the number multiplied by 11 twice.

7 X 11 = 77 (Write the 7 twice)

1. 7 X 0

2. 9 X 0

3. 4 X 1

4. 8 X 1

5. 2 X 3

6. 5 X 2

7. 9 X 2

8. 4 X 2

9. 9 X 0

10. 7 X 2

11. 2 X 6

12. 4 X 10

13. 8 X 10

14. 5 X 10

15. 9 X 10

16. 10 X 3

17. 5 X 100

18. 100 X 4

19. 100 X 9

20. 4 X 11

21. 11 X 7

22. 7 X 10

23. 8 X 11

24. 9 X 11

25. 3 X 100

TAKING HALF

It will be helpful for multiplication and division if we can quickly take half of a number.

Take half of 20. You get 10.

Take half of 10. You get 5.

Take half of 40. You get 20.

Take half of 70. You get 35 - this one is a little trickier.

Take half of 30. You get 15.

Take half of 50. You get 25 (think about money, 50 cents is 2 quarters, half of 50 is one quarter or 25 cents).

Take half of 60. You get 30.

Take half of 80. You get 40. MULTIPLYING BY 5.

One way to multiply by 5 is to count by 5's. If you had 5 X 6 - you would count by 5's six times: 5, 10, 15, 20, 25, 30. This is a good strategy but if you know how to take HALF, you can get your answer quicker. Since 5 is HALF of 10, multiply your number by 10 and then take HALF.

So - 5 X 6 - THINK 6 X 10 = 60, take half, 30.

Try another one:

8X5 --> THINK 8 X 10,= 80, take half, 40.

Try some:

1. 5 X 3

2. 6 X 5

3. 5 X 5

4. 5 X 4

5. 8 X 5

6. 7 X 5

1. 5 X 3

2. 5 X 9

3. 5 X 2

4. 5 X 4

5. 5 X 8

6. 5 X 1

7. 5 X 5

8. 5 X 6

9. 5 X 0

10. 5 X 7

11. 5 X 10

12. 5 X 11

MULTIPLYING BY 9

To multiply by 9, there are a couple of tricks. First, the first (tens) digit of your answer is ONE LESS than the number you are multiply by. If you have 8 X 9, then your tens digit will be ONE LESS than 8 which is 7. If multiplying 6 X 9, then your tens digit will be 5; if doing 3 X 9, your tens digit will be 2. To find your ones digit, figure out what you need to add to the tens digit to get a sum of 9.

8 X 9

The tens digit will be 7.

Two more is needed to get from 7 to 9, so the answer is 72.

4 X 9

The tends digit will be 3.

Six more is needed to get from 3 to 9, so the the answer is 36.

FINGER TRICK: Another way to do it is to hold out both hands. Count from the finger on the farthest left hand side. Fold down the finger that corresponds to the number being multiplied by 9. So, if you have 9 X 5, fold down the fifth finger, this would be your thumb on your left hand. Count the number of finger (both hands) that fall to the LEFT of the folded finger, that is your TENS digit, then count the number of fingers to the RIGHT side of the folded finger, that is your ONES digit. So in 9 X 5, I have FOUR fingers to the LEFT of my folded down thumb and 5 fingers to the RIGHT of the folded down thumb, therefore the answer is 45.

If I have 9 X 8. I will count 8 fingers from the starting left-hand. This would have me fold down the middle finger on my RIGHT hand. I count the number of fingers to the LEFT of that finger and I get 7, to the RIGHT of the finger, I get 2 therefore my answer is 72.

1. 9 X 2

2. 9 X 9

3. 9 X 6

4. 4 X 9

5. 5 X 9

6. 1 X 9

7. 9 X 0

8. 9 X 8

9. 7 X 9

10. 9 X 10

11. 11 X 9

MULTIPLYING BY 4

DOUBLE AND DOUBLE AGAIN

Let's practice doubling a number twice.

If you have 4 and double it, you get 8. If you then double 8, you get 16.

If you have 9 and double it, you get 18. To double it again, you may have to write out the problem sometime if it cannot be done in your head:

18

+18

36

So, when you start with 9, double to 18, double again to 36.

If you have 6 and double, you get 12. Doubling 12 in your head is easier, just line up the 2's and add to 4. Line up the 1's and add to get 2. Your solution is 24. Practice the easier ones in your head until you get good at it.

Now, when you multiply by 4, you circle the number that is multiplied by the 4, double that number and double it again and you have your answer.

3 X 4: Double 3 and get 6, double 6 and get 12. 3 X 4 = 12

7 X 4: Double 7 and get 14, double 14 and get 28. 7 X 4 = 28

8 X 4: Double 8 and get 16, double 16 (write it out on paper if you can't do this one in your head) and get 32. 8 X 4 = 32.

Remember that if you have 9 X4 - you get to choose, do you want to use your 4's trick (double and double again) or use the 9's trick (fingers) - so do whatever is easier for you.

MEMORIZING FACTS

We now have strategies for many different facts, but there are just a few that have to be memorized.

3's: There is no good trick for the 3's. You can double it and then add one more if you like:

3 X 6 = double 6 to 12 and add a 6 to get 18.

You can just use the other tricks for many of the facts.

For 6X6, 6X8 and 8X8, I try to remember either a little song or rhyme.

6 times 6 is 36 and 6 times 8 is 48.

For 8 X 8, you can either remember: Bend down touch the floor, 8 X 8 is 64.

Or Skate X Skate is Sticky Floor can help too.

If you need to with 8's you can double 3 times too.

8 X 7: Doulbe 7 to get 14, double 14 to get 28 and finally double 28 to get 56.

6 X 8 is a little easier for the Triple Double: Double once, 12, Double again 24, Double again, 48. (Those are easy enough to do in your head).

Learning 12 FACTS

I don't believe in memorizing the 12 facts when it is so easy to multiply by 10 and by 2. You just need to do both and add together.

6 X 12: (6 X 10 ) + (6 X 2) = 60 + 12 = 72.

5 X12 = 50 + 10 = 60

8 X 12 = 80 + 16 = 96

Even for 11 X 12, you can multiply 11 by 10 - you get 110 and then add the double of 11, 22: 110 + 22 = 132.

1. 2 X 12

2. 8 X 12

3. 4 X 12

4. 9 X 12

5. 3 X 12

6. 0 X 12

7. 5 X 12

8. 6 X 12

9. 1 X 12

10. 7 X 12.

11. 10 X 12

12. 11 X 12

13. 12 X 12

MULTIPLICATION CONCEPT

Don't forget as you work through learning your facts that you need to know the CONCEPT behind multiplication. There are 2 ways to think of multiplication:

1) Number of Groups and Number of Objects in each group.

One of the numbers tells you how many groups you have, the other number tells you how many objects are in that group. The total number of objects in the answer.

You can think about how many things in the real world come in groups. Each person has 2 eyes. Each dog has 4 legs. Each dozen eggs has 12 eggs. Each basketball team has 5 people playing.

Multiplication problems will state: There are 10 people, each person has 2 eyes, how many eyes in all? 10 "groups" with 2 "items" in each group: 10 X 2 = 20.

There are 6 dogs, each dog has 4 legs, how many legs in all? 6 "groups" with 4 "items" in each group: 6 X 4 = 24.

There are 8 baskets, each basket has 3 apples in it, how many apples in all? 8 groups of 3 or 8 X 3 = 24.

2) An Array - Imagine a rectangle shape with a certain number of objects across and then a certain number of objects down and filling in to form a rectangle. It you multiply the number across by the number down, you will have the total number in the array.

There are 6 row with 10 trees in each row, how many trees in all? This is an array with 6 across and 10 down, for a total of 6 X 10 = 60 trees.

There are 8 rows of cars with 8 cars in each row, how many cars in all? Again this makes an array (this time it is a square): 8 X 8 = 64.

It is easy to get a list of problems that are all multiplication, so to check that you understand the difference between the concept of multiplication, addition, and subtraction - here are a mixed set. Read each carefully and determine which operation to use and then write the number sentence and find the answer.

Concept Problems

1. There are 5 apples in one basket and 9 apples in another basket, how many apples in all?

2. There are 5 baskets and each basket has 9 apples in it, how many apples in all?

3. There are 9 apples in a basket, I eat 3 apples, how many apples are left?

4. There are 7 candy bars on the table and 8 candy bars on the counter, how many candy bars in all?

5. There are 5 friends, each friend is carrying 4 candy bars, how many candy bars in all?

6. I have 15 candy bars and give 7 away to my friends, how many candy bars are left?

7. There are 9 rows of marbles and 6 marbles in each row, how many marbles in all?

8. Mike has 9 marbles and Sue has 6 marbles, how many more marbles does Mike have than Sue?

9. Lara has 9 marbles and jeff has 6 marbles, how many marbles are there in all?

10. Wes had 16 cards and givens 9 to his friend, how many does he have left?

11. Wes has 16 cards and his friend has 9 cards, how many more cards does Wes have than his friend?

12. Wes lines up his cards, he has 5 rows of cards and 8 cards in each row, how many cards does he have in all?

Before starting to work on division, it is helpful to think about FACTORS. Factors are the things that multiply to give a PRODUCT (Product is what we call the ANSWER in a multiplication problem).

Let's start with 6. What multiplication facts multiply to give you 6? With any number, you can always take the number and multiply it by one to get self. In other words: 6 X 1 = 6. What other fact gives 6 as an answer? Our next check should be if the number is an EVEN number. If it is an EVEN number then you know that 2 will be a factor. In the case of 6, we get 2 X 3 = 6. These are our factors of 6.

6 = 1X6

6 = 2X3

Let's test ourselves to see if we know which number will have a factor of 2. If the number is EVEN, 2 is a factor. Divide it in half to find the other factor.

Write down the numbers that have 2 as a factor, then write the other factor. If 2 is not a factor, write ODD

To determine if 5 is a factor, we only have to look to the last digit of the number, if it is a 5 or a 0, then 5 will be a factor. Think about counting by 5's: 5, 10, 15, 20, 25, 30, 35, 40, 45, 50. All the numbers end in a 5 or a 0.

Let's find the factors of 20:

First, we can see if 2 is a factor. Yes, 20 is even, so 2 is a factor. If we divide 20 in half we get 10, therefore 2 X 10 is one way to get 20.

Next, we see that 5 must be a factor because 20 ends in a zero. Do you remember what times 5 gave us 20? If not, you can count by 5's until you get there and see how many numbers you had to count: 5, 10, 15, 20. You had to count 4 times to get to 20, so 4 X 5 = 20.

Our factors of 20 are going to be: 1 X 20, 2 X 10, and 4 X 5.

To deterine if 10 is a factor, think about what the pattern is when you count by 10: 10, 20, 30, 40, 50, ... You can see that all the numbers end in a 0. Remember how you learned to multiply by 10? You just added a zero to the number. 4 X 10 = 40. (Add a zero after the 4).

Let's find the factors of 30:

First, 30 is even so 2 must be a factor. Divide 30 in half and you can see that 15 X 2 = 30.

Since 30 ends in a 0, it must also have 5 as a factor. Count by 5's to see how many 5's are in 30: 5, 10, 15, 20, 25, 30. There are 6 numbers so 5 X 6 = 30.

Now we also notice that 30 ends in a zero, so 10 must be a factor. We can see that 3 X 10 = 30.

Let's see all our factors: 1 X 30, 2 X 15, 3 X 10, 5 X 6.

Let's just practice with our 2's, 5's and 10's right now.

 Number 2 Factor 5 Factor 10 Factor 10 2 X 5 5 X 2 10 X 1 15 ODD 5 X 3 Doesn't end in 0 20 2 X 10 5 X 4 10 X 2 30 2 X 15 5 X 6 10 X 3 40 2 X 20 5 X 8 10 X 4

Let's now think about 4's for a minute. To find 6 X 4, we doubled 6 (and got 12) and then doubled again (and got 24). To go BACKWARDS, we can take HALF and then take HALF AGAIN.

Let's start with 24. Take half, you get 12, take half again, you get 6. So we know that 6 X 4 = 24.

Let try 32. Taking half is harder but we get 16. Half of 16 is 8. So we can see that 4 is a factor of 32.

What happens if 4 is not a factor. Let's see what happens when we have 30. Take half, we get 15, when we go to take half again, we can't 15 is ODD. So 4 cannot be a factor of 30.

Multiplication are OPPOSITES of each other. In math we call then INVERSES.

If 6 X 4 = 24; then writing it backwards we get 24 ÷ 4 = 6.

For each FACT, we can write the FACT FAMILY. It uses the commutative property of multiplication that we used earlier. If 6 X 4 = 24 then 4 X 6 = 24. In division, we can swithc out the divisors: We can write:

24 ÷ 4 = 6 but can also switch the 4 and the 6 to write: 24 ÷ 6 = 4. So our FACT FAMILY WOULD LOOK LIKE THIS:

 6 X 4 = 24 24 ÷ 4 = 6 4 X 6 = 24 24 ÷ 4 = 6

Let's work on another FACT FAMILY:

 3 X 4 = 12 12 ÷ 3 = 4 4 X 3 = 12 12 ÷ 4 = 3

Now I will give you 1 piece of the family and you practice writing the other 3 pieces:

1. 5X4 = 20

2. 6X3 = 18

3. 9X5 = 45

4. 7X8 = 56

5. 42÷7 = 6

6. 15÷3 = 5

7. 72÷8 = 9

8. 48÷6 = 8