OPPOSITES AND EXPONENTS

It was 1st period, and the class was Algebra I.  I put he following on the board:

                         – 3² =       A) –6  B)  6   C)  –9   D)  9

Sadly of the 30 students 28 chose D, and 1 chose C, and 1 chose A.
I then asked 3 of them to read the problem out loud.  They all said,

                            "negative 3 squared"


This is what they were saying  (–3)², or –3 the quantity squared. 


This is why 28 of them were incorrect by sign and another has no idea what 
an exponent does.  One of them was correct, because we know the answer is
C.  This is why I strongly suggest that we teach our students to call this sign
"–", "the opposite of".  Now read the problem.


             "the opposite of",  3 squared


The value of 3 squared is 9, and its opposite is –9.

This was a lead in to the question what is;

                             – x²; x = – 5  or x = 7

the opposite of (–5)²  or   the opposite of (7)²

                                x = – 25  or  x = – 49 


I hope these comments will help.

SQUARING A 3–DIGIT NUMBER

In my last blog I talked about squaring a 2–digit number. I know that most

teachers, even Middle School, will not be interested in my last blog, however

why stop there. Let's look at squaring a 3–digit number.

Last time it was: (a + b)(a + b) = a² + 2ab + b²

This time we have: (a + b + c)(a + b + c) =

a² + ab + ac + ba + b² + bc + ca + cb + c² =

a² + 2ab + b² + 2bc + c² + 2ac

So, when we look a number like 426, we need to look at it as

400 + 20 + 6 = 426

Square it: (400 + 20 + 6)(400 + 20 + 6)

We only need to use our counting digits, 4, 2, 6, with the help

of our zero to answer our question.

a = 400, b = 20, and c = 6

Thus, 400² + 2(400)(20) + 20² + 2(20)(6) + 6² +

2(400)(6)

1. 400² = 160000

2. 2(400)(20) = 16000

3. 20² = 400

4. 2(20)(6) = 240

5. 6² = 36

6. 2(400)(6) = 4800

Grand total is 181476

You may not be able to do something like this in your head

without a great deal of practice, but what if the number you

were squaring had a zero in it, like 403, or 260.

In 403 the b value would be 0, and the only therms in the formula

you would need would be a², 2ac, and c²

400² = 160000

2(400)(3) = 2400

3² = 9 TOTAL = 162409

260² = 200² + 2(200)(60) + 60²

= 40000 + 24000 + 3600

TOTAL = 67600

These last two examples, with practice, you can do in you

head and will show you how to use zeros correctly

when multiplying. I find that these kinds of mental

practices improve the mind. These will work well as

warm–up problems in the morning, or at the beginning

of any Math class.

SQUARING A 2-DIGIT NUMBER

Remember those first days of multiplying? We memorized our

2's, 3's, 5's, 10's, and struggled with our our 7's. If you do, you

are much younger than I, but then again, except for the baby–

boomers, everybody else is.

After memorizing everything from 1 x 1 = 1 to 12 x 12 = 144

we began to multiply things like

345 x 8 = 2760

Or did we think

(345 x 10) – (345 x 2) =

3450 – 690 = 2760

I know the second method was never mentioned, however I

thought it might be a good lead into what I am about to tell

you. That chart you memorized had a diagonal. These were

the perfect squares.

Why stop at 12 x 12 = 144. What if we could square any

2–digit number in your head. You might not think that it

would come in handy, but it gives you ideas about how you

can manipulate numbers to make what you are doing easier.

What is 93² or 93 x 93.

Well it is 8649, and I did not need a calculator to know it.

I simply added 8100 + 540 + 9 in my head.

Say what?

Well, you need to look at the problem as;

(90 + 3) (90 + 3)

and multiply every number in the 1st parenthesis by each

in the 2nd parenthesis.

90 x 90 = 8100

90 x 3 = 270 (twice) which makes 540

3 x 3 = 9

Thus, 8100 + 540 + 9 = 8649

You have to know how zeros work, but otherwise you are

just multiplying 9 x 9, 3 x 9, and 3 x 3.

It all comes from Algebra, and the F–O–I–L system.

Lets try 67 x 67, or (60 + 7)(60 + 7)

60 x 60 = 3600

60 x 7 = 420

7 x 60 = 420

7 x 7 = 49/ 3600 + 840 + 49 = 4449

With practice you can do these in your head.

THE NEW YEAR

Its January 1st, 2012. My dad died four years ago today, and he was the

last of his generation in our family. I miss him, like I miss my mom who

died 3 years before him. But as we all march towards our own end, we are

looking at a year that for many brings big questions.

What about 12-21-12 and the end of the world. The alignment of our

planets with the center of our galaxy. The Mayans said it, so we should

all believe it. It truly amazes me what we believe, and what we spend

our whole lives trusting in. Two things that I do know, on 12-22-12

1. 40% of 250 will still be 100

2. The firs derivative of 5X³ will be 15X²

Math will not change, 1 will still be less than 3, and ∏ will be a

transcendental number 3.14159265. . .. Here we are on this tiny

little planet, in a universe of billions of galaxies, in a galaxy of

billions of stars, and most of us believe that we are the most

intelligent in the universe, and that one "god" did it all. Many of

us spend our entire life worrying about the next life time, when

we need to get along in this life time, and are not doing such

a great job.

I am not saying that we should not have our beliefs, but they

should not destroy our ability to get along with each other.

So as this year begins, let us strive to get along better, and

recognize what must be true, and spend less time debating

what might be true.

ANOTHER LOOK AT INTEGERS

I think what sparked my reason to look at this again was a

resent visit to my bank. I asked them for our balance after

a deposit, and then came home. I gave that information to

my incredibly copious spouse, who has been the guardian

of our check book for the last 40 years. The gap between

their amounts was more than I could understand. Somehow,

had a positive become a negative somewhere, and if so, this

could be a big mess.

Imagine you had a deposit of $500, but in your check book

recording you subtracted. You would be off by $1000. The

simple idea of adding integers seams to be such a problem

for so many. It is all about getting back to zero. If I earn

$200, but have $200 in bills to pay, when I am done I have

no money at all.

All we need ask now is which came first, my earnings or

my bills.

$200 + (–$200) = 0 or

(–$200) + $200 = 0

Why is this confusing?

Students know that 5 + 4 = 4 + 5, don't they?

What are we not being clear about?

Perhaps we just do not explain ourselves. Let me show

you what I mean.

We start with our digits 0 – 9 and we have hundreds

of worksheets with 7 + 2 = __ and 5 + 1 = __ .

Most of our students fill in the blanks, with little or

no retention, because all they do is stare and write

a number. Where it goes in those fine minds is

anyone's guess.

Try having them write the entire problem and the

answer with the = sign. Now their minds have the

whole story to place somewhere. Even better,

7 + (+ 2 ) = __ Read as: 7 + a positive 2 = 9

Then when they get to SUBTRACTION

7 + (–2 ) = __ Read as 7 + "the opposite of" 2 = 5

On the other side of zero

–7 + (– 2) = __ Read as "the opposite of" 7

+ "the opposite of" 2 = – 9

and with SUBTRACTION

–7 – (– 2) = __ Read as "the opposite of" 7

"the opposite of the opposite of 2 = – 5

When we add we go away fro zero, and when we

subtract we head back to zero, from either side.

I believe it is truly important that students see

integers from Kindergarten. They need not be

a surprise. We do not have to work with them,

but letting them know they are there will save

a great deal of anxiety.

SUBTRACTION WITH ZEROS

We have all been in class when the following problem was put on the board.

5003

– 1468

We remember that we cannot take 8 away from 3, so we have to borrow, or

REGROUP in todays vernacular. Thus, our 0's become a 9, our 5 is reduced

to 4 and off we go. But, why do we do that? Our 0's become 9's, really?

That's cool, we get 9 out of nothing. Oh, hold on there cowboy!

Look at this a different way.

5000 + 0 + 0 + 3

– 1000 + 400 + 60 + 8

Now the 3 needs an extra 10 to be 13, but 5000 can only hand out

1000's. Thus, we will drop off 900 in the hundreds place, 90 in the

tens place and the ones place only gets 10 leaving the following.

4000 + 900 + 90 + 13

– 1000 + 400 + 60 + 8

As we see, by expanding the numbers we get a clearer understanding

of what is happening. I hope you feel better about those 9's now.

Palindromic Numbers and Companions

Many years ago I came across a method for practicing your addition

facts and having fun at the same time. It all has to do with

PALINDROMIC NUMBERS.

A Palindromic number is a numbers that looks the same backwards

and forwards; 101, 232, 5445, 63436.

If a number is Palindromic then it is its own companion, but if its

not then we need to figure it out. We start with 3–digit numbers,

and as you saw above 101 and 232 are already Palindromic. But what

about a number like 117. Sense it is not Palindromic we reverse the

number and add them together.

117

+ 711

828 Thus: 828 is the companion of 117.

What about 536

536

+ 635

1171 Not Palindromic so we do it again.

1171

+ 1711

2882 Thus: 2882 is the companion of 536

Now try 468

468

+ 864

1332

2331

+ 1332

3333 Thus: 3333 is the companion of 468

Last we will look at 579

579

+ 975

1554 Try again

1554

+ 4551

6105 Try again

6501

+ 1056

7557

We found the Companion 7557

If your class works on all of the 3–digit numbers up to 999, you will

find them being able to add faster. As they get into them, they will find

some that take as many as 15 steps. Also, when you see common

companions look back at the numbers and see how they relate.