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Carlson Home Instructional Program and Hospital School

Division of Special EducationLos Angeles Unified School District

Buy Back Day 2006 In-Service PresentationDate: Tuesday, August 29, 2006

Location: Forest Lawn: Hall of LibertyPresenter: Lydia A. Saxton

Topic: Pre-Algebra Strategies

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Goals and Outcomes To model pedagogical techniques used to

teach pre-algebra concepts and skills including:

integersabsolute value

squares and powersvariables, terms, and expressions

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Goals and Outcomes (con’t.)

To engage teachers in cooperative group activities in order to reinforce and practice the techniques presented

To provide references to resource materials available in the Carlson Resource Room and on the internet.

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Lydia's Rock 'em, Sock 'em, Sure FireBag 'o Tricks to Teach Basic Pre-

Algebra Concepts and Skills

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1.

FIRST THINGS FIRST!

A word or two about Math Vocabulary, Integers, and the Four

Basic Operations of Mathematics

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ABOUT MATH VOCABULARY

Mathematics utilizes a very precise language.....

much more precise (many would argue) than spoken English.

It’s important for students to develop their understanding and use of the language

of mathematics

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I like to call it:

MATHSPEAK

Here are some examples of essential pre-algebra mathspeak vocabulary:

Can you name the ‘parts’ of the following equations?

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12 + 3 = 15addend, addend, sum

3 + 12 = 15(Commutative Property of Addition)

12 - 3 = 9minuend, subtrahend, difference

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12 x 3 = 36factor, factor, product

3 x 12 = 36(Commutative Property of Multiplication)

12 ÷ 3 = 4dividend, divisor, quotient

4 12 3)¯12¯¯ = 12/3 = 3

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SO, IF I ASK:

“WHAT IS THE SUM OF 12 AND 3?”

YOU WOULD SAY:

15

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IF I ASK,

“WHAT IS THE DIFFERENCE OF 12 AND 3?”

YOU WOULD SAY:

9

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IF I ASK,

“WHAT IS THE PRODUCT OF 12 AND 3?”

YOU WOULD SAY:

36

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IF I ASK:

“WHAT IS THE QUOTIENT OF 12 AND 3, IF 12 IS THE DIVIDEND AND 3 IS THE

DIVISOR?”

YOU WOULD SAY:

4

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ABOUT INTEGERS

What are integers?(Take a moment to discuss this

with your group.)

A good definition might be:

The integers consist of the positive natural numbers (1, 2, 3, …), their negatives (−1, −2, −3, ...) and the

number zero.

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What we want students to understand is:

When you hear the word “integer,” think about When you hear the word “integer,” think about thermometers…thermometers…

because integers are basically the numbers above because integers are basically the numbers above and below zero.and below zero.

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So when you ask a student what an integer is…

make sure the answer includes the words:

“positive”“negative”“numbers”

and“zero”

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ABOUT THE FOUR MATHEMATICAL OPERATIONS

What are the four basic operations of mathematics?

Addition

Subtraction

Multiplication

Division

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But when students (even in secondary) need to use these words,

they often say:

“Plusses”

“Take-aways”

“Times”

“Gazintas”

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What we want students to

understand is:

Each of those four symbols means that an operation is being performed!

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There are four basic surgical “procedures” or things you can do to a

number:

You can:

Add to it (addition)

Take away from it (subtraction)

Add the same number to it many times, referred to as ‘repeated addition’

(multiplication)

Take the same number away from it many times or divvy it up into smaller, equal chunks

(division)

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Think of each of these ‘procedures’ as an

OPERATION!

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“What operation are you using?”

and with Word Problems:

“Which operation do you need to use?”

So when you regularly ask the student (and you should ask it

regularly!) the perennial question:

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The student should think ofThe student should think of

a surgical operation…a surgical operation…

employing one of four possible procedures:

Addition

Subtraction

Multiplication

Division

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2.

THE FOUR FACE

A Trick for Multiplying and Dividing Integers

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++ +

When both eyes are open, When both eyes are open, you’re awake, you’re awake,

so you’re happy.so you’re happy.

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+- -

When both eyes are closed, When both eyes are closed, you’re asleep, you’re asleep,

so you’re still happy.so you’re still happy.

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-+ -

But when one eye is openBut when one eye is openAnd one eye is closed, And one eye is closed,

you’re not happy.you’re not happy.

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-- +

And when one eye is closed,And when one eye is closed,And one eye is open, And one eye is open, you’re still not happy.you’re still not happy.

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++ +

+

+ +-

- -

---

The Four Faces

A Trick for Multiplying and Dividing Integers

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Cooperative Activities with The Four Faces

1. Work with your colleagues to fill in the four faces on Handout #1 with the appropriate positive and negative signs.

2. Work cooperatively to solve the equations on Handout #2.

(The Four Faces Trick for Multiplying and Dividing Integers)

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The Four Faces

A Trick for Multiplying and Dividing Integers

Handout #1

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(The Four Faces Trick for Multiplying and Dividing Integers)

1. -3 (9) =

2. -6 (-6) =

3. 2 (-21) =

4. -15 / 5 =

5. 0 (-8) =

6. 5 · 9 =

7. -5 · -9 =

8. -1 · 16 =

Cooperative Activities with The Four Faces

Handout #2

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Answers: (The Four Faces Trick for Multiplying and Dividing

Integers)

1. -3 (9) =

2. -6 (-6) =

3. 2 (-21) =

4. -15/5 =

5. 0 (-8) =

6. 5 · 9 =

7. -5 · -9 =

8. -1 · 16 =

-27

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-42

-3

0

45

45

-16

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3.

HOW MANY LICKS DOES IT TAKE TO GET TO THE CENTER OF A

TOOTSIE® POP?

Some Tricks for Understanding

Absolute Value

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Imagine that the following is a giant oval-shaped lollipop

with a candy center.

Yummmm!

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How many hops does it take for the bunny to get to 0?

|-5| = 5| 5| = 5

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So the absolute value of any real number (whose value is either positive or negative), is always expressed as a

positive number.

|-63| =

63

|-527| = 527|12| = 12|-328| = 328

|-9| = 9

|100| = 100 |49| = 49|1| = 1

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I like to use a variety of learning modalities

when I teach.

So, for my visual/kinesthetic

learners, I use hand signals.

The hand signal for absolute value is:

Two arms, bent straight

up at the elbow to form two

parallel lines.

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But absolute value signs also appear in equations with variables, like:

|x + 1| = 6

So now what?

The total value of what’s inside the | | sign could be either 6 or -6,

because, either way, the absolute value of whatever’s

inside the | | sign will always be expressed as a

positive number.

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So that means, that for:

|x + 1| = 6

x + 1 = 6 x + 1 = -6x = 5 or x = -7

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Huh?

|x + 4| = 23

x + 4 = 23 x + 4 = -23x = 19 or x = -27

One more time…

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Work with your colleagues to find solutions for the equations on Handout

#3.

Cooperative Activity with Absolute Value

(Some tricks for Understanding Absolute Value)

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Can you find solutions for these equations?

|x| = 5; so x =

|x + 2| = 4; so x =

|3x| = 27; so x =

Cooperative Activities with Absolute Value

Handout #3

(Some Tricks for Understanding Absolute Value)

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Answers:(Some tricks for Understanding Absolute Value)

|x| = 5; so x =

|x + 2| = 4; so x =

|3x| = 27; so x =

5, -5

2, -6

9, -9

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4.

HOTS AND COLDS and THE SUN AND THE ICE CAULDRON

Tricks for Adding and Subtracting Integers

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+7

+6

+5

+4

+3

+2

+1 0-1

-2

-3

-4

-5

-6

-7

+

-

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+7

+6

+5

+4

+3

+2

+1 0-1

-2

-3

-4

-5

-6

-7

3 + 4 = 7

3 + 4 = 7

+ + +

- - -

If you’re adding “HOTS,” you go

up on the thermometer!

If you’re adding

“COLDS” you go down!

+

-

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But what if the signs

in an equation involving

addition of integers are not the same?

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+7

+6

+5

+4

+3

+2

+1 0-1

-2

-3

-4

-5

-6

-7

6 + 4 = ?

+ -

Absolute Valueto the rescue!

| 6 |+ -

6 4- = 2+

| 4 |

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+7

+6

+5

+4

+3

+2

+1 0-1

-2

-3

-4

-5

-6

-7

3 + 7 =+ -

LET’S TRY ANOTHER ONE!

| 3 |+ -

3 7

| 7 |

- = 4-

?4-

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+7

+6

+5

+4

+3

+2

+1 0-1

-2

-3

-4

-5

-6

-7

Cooperative Activity with

Adding Integers

Work with your

colleagues to solve the addition

equations on Handout #4.

Use the thermometer to assist you…and ask yourself the question:

“Am I addingHOTS

or

COLDS?”

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-16 + 7 =

8 + -3 =

-7 + -7 =

-5 + 5 =

Cooperative Activities with Adding Integers

Handout #4

(A Trick for Adding Integers)

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Answers:(A Trick for Adding Integers)

-16 + 7 =

8 + -3 =

-7 + -7 =

-5 + 5 =

-9

5

-14

0

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Subtracting integers, on the other hand, requires several

steps.

Careful, though! You’ve got to be really comfortable with addition of

integers before you tackle subtraction!!

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Here are the basic steps:

1. First, keep the value of the minuend and its sign exactly as it is.

2. Next, change the “operation” symbol from subtraction to addition.

3. Then, change the sign of the subtrahend to its opposite sign. (ADD THE OPPOSITE!)

4. Finally , treat the whole thing as an additional problem and follow the steps for addition of integers.

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How’s that again???

+ -6 4- = ?

+ +6 4+ =

6 + 4 =

+ +

?

10+

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One more time!

+ -3 4- = ?

+ +3 4+ =

3 + 4 =

+ +

?

7+

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But what if the minuend is negative?

- -3 4- = ?

- +3 4+ = ?

? 3 + 4 =

- +Now you’ve got an addition equation.

You know what to do!

Use Absolute Value and subtract. Then take the sign of the larger number.

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So here’s your newly created addition

equation.

? 3 + 4 =

- +

| 3 |- | 4 |+

4 3- = 1++ +

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+7

+6

+5

+4

+3

+2

+1 0-1

-2

-3

-4

-5

-6

-7

3 - 4 =- ?- -

Let’s see how this looks on our

thermometer.Remember!

You are taking away COLDS.

So that means you’re getting warmer,

so you must be going up! 3 - 4 = 1- - +

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+7

+6

+5

+4

+3

+2

+1 0-1

-2

-3

-4

-5

-6

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Cooperative Activity with Subtracting

Integers

Work with your

colleagues to solve the

subtraction equations on Handout #5.

Use the thermometer

to assist you…

and remember

to

add the

opposite!

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7 - ˉ5 =

ˉ6 - ˉ8 =

3 - 9 =

-5 - 5 =

Cooperative Activities with Subtracting Integers

Handout #5

(A Trick for Subtracting Integers)

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+7

+6

+5

+4

+3

+2

+1 0-1

-2

-3

-4

-5

-6

-7

+

-

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Answers:(A Trick for Subtracting Integers)

7 - ˉ5 =

ˉ6 - ˉ8 =

3 - 9 =

-5 - 5 =

12

2

-6

-10

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5.

THE MAGIC BOX – PART 1

(An Introduction to Squares and Powers)

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When you were in elementary school, you learned your

multiplication tables, right?

So you all know the answers to the following:

7 x 7 =8 x 8 = 9 x 9 =

etc.

496481

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So how about:

11 x 11 =

12 x 12 =

13 x 13 =

121

144

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14 x 14 =

15 x 15 =

16 x 16 =

Having trouble?

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Don’t worry. You’re not alone. But there’s hope!

It’s called:

The Magic Box

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10 3+

103

100 30

30 9+

100 + 30 + 30 + 9 = 169

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10 4+

104

100 40

40 16

+

100 + 40 + 40 + 16 = 196

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10 5+

105

100 50

50 25

+

100 + 50 + 50 + 25 = 225

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10 6+

106

100 60

60 36

+

100 + 60 + 60 + 36 = 256

Now you try it!

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This is also a good way to introduce the concept of

“squares” which is also referred to as

powers of two

x²

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So:

12 x 12 = 12²= 144

13 x 13 = 13² =

Hint:Visualize The Magic Box in your

head!

17 x 17 = 17² =

18 x 18 = 18² =

169

289

324

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Cooperative Activity with The Magic

Box (Part 1)Work with

your colleagues to

solve the equations on

Handout #6. Use The Magic

Boxes on Handout #7 to help you.

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Cooperative Activity with The Magic Box

16² =

17² =

18² =

19² =

(m + p)² =

(Use the boxes on Handout #7 or try to do these in your head.)

Handout #6

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Cooperative Activities with The Magic Box

Handout #7

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Answers:

16² =

17² =

18² =

19² =

(m + p)² =

Cooperative Activity with The Magic Box

256

289

324361

m²+2mp+p²

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6.

THE MAGIC BOX – PART 2

(Squaring Binomials and FOIL, too!)

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The Magic Box has some interesting qualities.

When you use it, you are

actually using

the FOIL technique.

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You remember

FOIL.

It’s an acronym

for:

FIRST

OUTER

INNER

LAST.

Firsts

Firsts

Outer

Outer

Inner

Inner

Last

Last

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So 13²

is the same as

13 x 13

which is the same as

(10 + 3) x (10 + 3)

or, as we would write it mathematically,

(10+3)(10+3) = (10+3)²

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(10 3)+

(10

3)

100 30

30 9+

(10 + 3) (10 + 3) =

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Let’s look at that for a moment.

(10 + 3)(10 + 3) =

10 x 10 = 100 First

10 x 3 = 30 Outer

3 x 10 = 30 Inner

3 x 3 = 9 Last

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(10 + 3)(10 + 3) = 169

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So it’s a fairly easy transition from the Magic Box to this:

(a + b)² =

(a + b) (a + b) =

a²+ ab + ab + b² =

a²+ 2ab + b²

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Cooperative Activity with The Magic

Box (Part 2)

No handout for this one. We’ll do this one together!

WATCH!

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(a+b)² = (a+b)(a+b) =

first: a²

outer: ab

inner: ab

last: b²

a²+ ab + ab +b² =

a²+ 2ab + b²Bravo! That wasn’t

so bad, was it!

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7.

A TRIP TO WHITE CASTLE®

Understanding Combining Like Terms

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I went to White Castle® yesterday with my daughter’s Girl Scout troop.

They wanted

5 hamburgers and 2 fries

So I wrote down5h²

(White Castle has ‘square’ burgers – hah, hah! Get it?!)

and2f.

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So our order was

5h² and 2f

or, mathematically-speaking

5h² + 2f

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So far so good!

But then we met girls from another troop.

We decided to pool our money

and eat together.

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The girls from the other troop wanted

3 hamburgers

3h²

and

4 fries

4f

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Now our order included the original

5 hamburgers and 2 fries

that’s 5h² + 2f

to which we added

3 hamburgers and 2 fries

that’s 3h² + 4f

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So all together we had:

5h² + 2f + 3h² + 4f

Now I didn’t think the young woman behind the busy counter would

appreciate it if I gave her the order as:

5 hamburgers plus 2 fries plus 3 more hamburgers

plus 4 more fries!

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So I combined the hamburgers from both orders and the fries from

both orders and told her:

8h² + 6f

This example demonstrates the efficiency of combining like terms.

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Now you try it!

10x + 6y + 5x = ?

15x + 6y

6c² + 2 + 8b³ + 4 + 3c² + 2b³= ?

10b³ + 9c² + 5

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8.

WHY DO WE HAVE TO LEARN THIS STUFF ANYWAY?

A Practical Example You Can Show Your Students to Demonstrate the Role and Relevance of Algebra in Everyday Life

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A Carpenter’s TaleSam wants to brace the back of a bookcase by

nailing a strip of wood from the lower left corner to the upper right corner. The bookcase is 1

meter wide and 2 meters high. How long should the brace be (to the nearest hundredth of a

meter)?

2 m.

1 m.

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The Pythagorean Theorem tells us that for any right triangle:

a² + b² = c²So, if we know the length of the legs of a right triangle, we can find the length

of the opposing side (hypotenuse).

a

b

c

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a² + b² = c²

then:

2² + 1² = c²

so:

4 + 1 = c²

5 = c²

√5 = √c²

2.24m = c

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BUT THAT’S NOT THE REAL STORY HERE!

2 m.

1 m.

The real story is that the boy’s dad, who was a carpenter, knew exactly how to

solve this problem. He just didn’t realize that he was using a formula, let alone, that

it was invented by a famous Greek mathematician!

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Page 10of your handout

includesa brief listing of materialsfrom our resource roomand from the internet

which I have found useful!

THANK YOU, ALL!

Hope you had fun…and maybe learned a thing or two

along the wayto help your students!

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Resource Room Resources

1. The Key to Algebra Series (Key Curriculum Press)

2. Algebra Survival Guide and Algebra Survival Guide Workbook by John Rappaport(Singing Turtle Press).

3. California Standards Key Concepts Book for Algebra I (McDougal Little)

Internet Resources

1. A Math Dictionary for Kids http://www.teachers.ash.org.au/jeather/maths/dictionary.html

2. Harcourt Math (6th Grade) http://www.harcourtschool.com/menus/math2002/ca/menu_ca.html

3. Purple Math http://www.purplemath.com/index.htm

4. SparkNotes for Pre-Algebra http://www.sparknotes.com/math/#algebra1

5. Illuminations – The website of the National Council for Teachers of Mathematics http://illuminations.nctm.org/