Week of September 3, 2019

P a r e n t U n i v e r s i t y S e r i e s

  

Understanding the IEP Process

Presented by the

CCSD Special Education Department

MONDAY, SEPTEMBER 16, 2019

6:30 - 8:30 PM @ Pine Mountain Middle School

2720 Pine Mountain Cir, Kennesaw, GA 30152

P l e a s e  r e s e r v e  y o u r  s p o t  f o r  t h e  p r o g r a m  b y  c a l l i n g  t h e  S p e c i a l  E d u c a t i o n P a r e n t  M e n t o r  o f f i c e  a t  7 7 0 - 5 2 9 - 0 0 4 6  o r  b y  r e g i s t e r i n g  a t  t h i s  l i n k :

www.signup.com/go/DXVwPjz

How to Tell If Two Functions Are Inverses

 

If or , then f(x) and g(x) are inverse functions.

So, you really only need to check one of them!

Check it out:

Are these inverse functions?

and

       

       

       

You can double-check it by crunching

.

---

YOUR TURN:

Are these inverse functions?

and

Are these inverse functions?

and

Source:http://www.coolmath.com/algebra/16-inverse-functions/04-how-to-tell-if-two-functions-are-inverses-03

Week of August 26, 2019

Open House

Tomorrow 8/17/19 @ 6.p.m.

Pascal's Triangle

One of the most interesting Number Patterns is Pascal's Triangle (named after Blaise Pascal, a famous French Mathematician and Philosopher).

To build the triangle, start with "1" at the top, then continue placing numbers below it in a triangular pattern. 

Each number is the numbers directly above it added together.

(Here I have highlighted that 1+3 = 4)

Patterns Within the Triangle

Diagonals

The first diagonal is, of course, just "1"s

The next diagonal has the Counting Numbers(1,2,3, etc).

The third diagonal has the triangular numbers

(The fourth diagonal, not highlighted, has the tetrahedral numbers.)

Symmetrical

The triangle is also symmetrical. The numbers on the left side have identical matching numbers on the right side, like a mirror image.

Horizontal Sums

What do you notice about the horizontal sums?

Is there a pattern?

They double each time (powers of 2).

Exponents of 11

Each line is also the powers (exponents) of 11:

• 11⁰=1 (the first line is just a "1")
• 11¹=11 (the second line is "1" and "1")
• 11²=121 (the third line is "1", "2", "1")
• etc!

But what happens with 11⁵ ? Simple! The digits just overlap, like this:

The same thing happens with 11⁶ etc.

Squares

For the second diagonal, the square of a number is equal to the sum of the numbers next to it and below both of those.

Examples:

• 3² = 3 + 6 = 9,
• 4² = 6 + 10 = 16,
• 5² = 10 + 15 = 25,
• ...

There is a good reason, too ... can you think of it? (Hint: 4²=6+10, 6=3+2+1, and 10=4+3+2+1)

Fibonacci Sequence

Try this: make a pattern by going up and then along, then add up the values (as illustrated) ... you will get the Fibonacci Sequence.

(The Fibonacci Sequence starts "0, 1" and then continues by adding the two previous numbers, for example 3+5=8, then 5+8=13, etc)

Odds and Evens

If you color the Odd and Even numbers, you end up with a pattern the same as the Sierpinski Triangle

Using Pascal's Triangle

Heads and Tails

Pascal's Triangle can show you how many ways heads and tails can combine. This can then show you the probability of any combination.

For example, if you toss a coin three times, there is only one combination that will give you three heads (HHH), but there are three that will give two heads and one tail (HHT, HTH, THH), also three that give one head and two tails (HTT, THT, TTH) and one for all Tails (TTT). This is the pattern "1,3,3,1" in Pascal's Triangle.

Tosses	Possible Results (Grouped)	Pascal's Triangle
1	H T	1, 1
2	HH HT TH TT	1, 2, 1
3	HHH HHT, HTH, THH HTT, THT, TTH TTT	1, 3, 3, 1
4	HHHH HHHT, HHTH, HTHH, THHH HHTT, HTHT, HTTH, THHT, THTH, TTHH HTTT, THTT, TTHT, TTTH TTTT	1, 4, 6, 4, 1
	... etc ...

Example: What is the probability of getting exactly two heads with 4 coin tosses?

There are 1+4+6+4+1 = 16 (or 2⁴=16) possible results, and 6 of them give exactly two heads. So the probability is 6/16, or 37.5%

Combinations

The triangle also shows you how many Combinations of objects are possible.

Example: You have 16 pool balls. How many different ways could you choose just 3 of them (ignoring the order that you select them)?

Answer: go down to the start of row 16 (the top row is 0), and then along 3 places (the first place is 0) and the value there is your answer, 560.
Here is an extract at row 16:

1    14    91    364  ...
1    15    105   455   1365  ...
1    16   120   560   1820  4368  ...

 

A Formula for Any Entry in The Triangle

In fact there is a formula from Combinations for working out the value at any place in Pascal's triangle:

It is commonly called "n choose k" and written like this:

Notation: "n choose k" can also be written C(n,k), ⁿC_k or even _nC_k.

The "!" is "factorial" and means to multiply a series of descending natural numbers. Examples: 4! = 4 × 3 × 2 × 1 = 24 7! = 7 × 6 × 5 × 4 × 3 × 2 × 1 = 5040 1! = 1

So Pascal's Triangle could also be
an "n choose k" triangle like this one.

(Note how the top row is row zero
and also the leftmost column is zero)

Example: Row 4, term 2 in Pascal's Triangle is "6" ...

... let's see if the formula works:

Yes, it works! Try another value for yourself.

This can be very useful ... you can now work out any value in Pascal's Triangle directly (without calculating the whole triangle above it).

Polynomials

Pascal's Triangle can also show you the coefficients in binomial expansion:

Power	Binomial Expansion	Pascal's Triangle
2	(x + 1)2 = 1x2 + 2x + 1	1, 2, 1
3	(x + 1)3 = 1x3 + 3x2 + 3x + 1	1, 3, 3, 1
4	(x + 1)4 = 1x4 + 4x3 + 6x2 + 4x + 1	1, 4, 6, 4, 1
	... etc ...

The First 15 Lines

For reference, I have included row 0 to 14 of Pascal's Triangle

1

1

1

1

2

1

1

3

3

1

1

4

6

4

1

1

5

10

10

5

1

1

6

15

20

15

6

1

1

7

21

35

35

21

7

1

1

8

28

56

70

56

28

8

1

1

9

36

84

126

126

84

36

9

1

1

10

45

120

210

252

210

120

45

10

1

1

11

55

165

330

462

462

330

165

55

11

1

1

12

66

220

495

792

924

792

495

220

66

12

1

1

13

78

286

715

1287

1716

1716

1287

715

286

78

13

1

1

14

91

364

1001

2002

3003

3432

3003

2002

1001

364

91

14

1

Week of August 18,2019

Adding and Subtracting Polynomials

A polynomial looks like this:

example of a polynomial this one has 3 terms

To add polynomials we simply add any like terms together ... so what is a like term?

Like Terms

Like Terms are terms whose variables (and their exponents such as the 2 in x²) are the same.

In other words, terms that are "like" each other.

Note: the coefficients (the numbers you multiply by, such as "5" in 5x) can be different.

Example:

7x

x

-2x

πx

are all like terms because the variables are all x

Example:

(1/3)xy2

-2xy2

6xy2

xy2/2

are all like terms because the variables are all xy²

Example: These are NOT like terms because the variables and/or their exponents are different:

2x

2x2

2y

2xy

Adding Polynomials

Two Steps:

• Place like terms together
• Add the like terms

Example: Add   2x² + 6x + 5   and   3x² - 2x - 1

Start with:2x² + 6x + 5   +   3x² − 2x − 1

Place like terms together:2x²+3x²   +   6x−2x   +   5−1

Which is:(2+3)x²  +   (6−2)x   +   (5−1)

Add the like terms:5x²  +   4x   +   4

Here is an animated example:

(Note: there was no "like term" for the -7 in the other polynomial, so we didn't have to add anything to it.)

Adding in Columns

We can also add them in columns like this:

Adding Several Polynomials

We can add several polynomials together like that.

Example: Add     (2x² + 6y + 3xy)  ,   (3x² - 5xy - x)   and   (6xy + 5)
Line them up in columns and add:

2x² + 6y + 3xy
3x²      - 5xy - x
           6xy     + 5

5x² + 6y + 4xy - x + 5

Using columns helps us to match the correct terms together in a complicated sum.

Subtracting Polynomials

To subtract Polynomials, first reverse the sign of each term we are subtracting (in other words turn "+" into "-", and "-" into "+"), then add as usual.

Like this:

Note: After subtracting 2xy from 2xy we ended up with 0, so there is no need to mention the "xy" term any more.

Source:https://www.mathsisfun.com/algebra/polynomials-adding-subtracting.html

Week of August 12, 2019

Factoring

Factors

Numbers have factors:

And expressions (like x²+4x+3) also have factors:

Factoring

Factoring (called "Factorising" in the UK) is the process of finding the factors:

Factoring: Finding what to multiply together to get an expression.

It is like "splitting" an expression into a multiplication of simpler expressions.

Example: factor 2y+6

Both 2y and 6 have a common factor of 2:

• 2y is 2 × y
• 6 is 2 × 3

So we can factor the whole expression into:

2y+6 = 2(y+3)

So 2y+6 has been "factored into" 2 and y+3

Factoring is also the opposite of Expanding:

Common Factor

In the previous example we saw that 2y and 6 had a common factor of 2

But to do the job properly we need the highest common factor, including any variables

Example: factor 3y²+12y

Firstly, 3 and 12 have a common factor of 3.
So we could have:

3y²+12y = 3(y²+4y)

But we can do better!
3y² and 12y also share the variable y.
Together that makes 3y:

• 3y² is 3y × y
• 12y is 3y × 4

So we can factor the whole expression into:

3y²+12y = 3y(y+4)

Check: 3y(y+4) = 3y × y + 3y × 4 = 3y²+12y

More Complicated Factoring

Factoring Can Be Hard !

The examples have been simple so far, but factoring can be very tricky.

Because we have to figure what got multiplied to produce the expression we are given!

It is like trying to find which ingredients
went into a cake to make it so delicious.
It can be hard to figure out!

Experience Helps

With more experience factoring becomes easier.

Example: Factor 4x² − 9

Hmmm... there don't seem to be any common factors.
But knowing the Special Binomial Products gives us a clue called the "difference of squares":

Because 4x² is (2x)², and 9 is (3)²,

So we have:

4x² − 9 = (2x)² − (3)²

And that can be produced by the difference of squares formula:

(a+b)(a−b) = a² − b²

Where a is 2x, and b is 3.

So let us try doing that:

(2x+3)(2x−3) = (2x)² − (3)² = 4x² − 9

Yes!

So the factors of 4x² − 9 are (2x+3) and (2x−3):

Answer: 4x² − 9 = (2x+3)(2x−3)

How can you learn to do that? By getting lots of practice, and knowing "Identities"!

Remember these Identities

Here is a list of common "Identities" (including the "difference of squares" used above).

It is worth remembering these, as they can make factoring easier.

a2 − b2	=	(a+b)(a−b)
a2 + 2ab + b2	=	(a+b)(a+b)
a2 − 2ab + b2	=	(a−b)(a−b)
a3 + b3	=	(a+b)(a2−ab+b2)
a3 − b3	=	(a−b)(a2+ab+b2)
a3+3a2b+3ab2+b3	=	(a+b)3
a3−3a2b+3ab2−b3	=	(a−b)3

There are many more like those, but those are the most useful ones.

Advice

The factored form is usually best.

When trying to factor, follow these steps:

"Factor out" any common terms
See if it fits any of the identities, plus any more you may know
Keep going till you can't factor any more

There are also Computer Algebra Systems (called "CAS") such as Axiom, Derive, Macsyma, Maple, Mathematica, MuPAD, Reduce and many more that are good at factoring.

More Examples

Experience does help, so here are more examples to help you on the way:

Example: w⁴ − 16

An exponent of 4? Maybe we could try an exponent of 2:

w⁴ − 16 = (w²)² − 4²

Yes, it is the difference of squares

w⁴ − 16 = (w² + 4)(w² − 4)

And "(w² − 4)" is another difference of squares

w⁴ − 16 = (w² + 4)(w + 2)(w − 2)

That is as far as I can go (unless I use imaginary numbers)

Example: 3u⁴ − 24uv³

Remove common factor "3u":

3u⁴ − 24uv³ = 3u(u³ − 8v³)

Then a difference of cubes:

3u⁴ − 24uv³ = 3u(u³ − (2v)³)

= 3u(u−2v)(u²+2uv+4v²)

That is as far as I can go.

Example: z³ − z² − 9z + 9

Try factoring the first two and second two separately:

z²(z−1) − 9(z−1)

Wow, (z-1) is on both, so let us use that:

(z²−9)(z−1)

And z²−9 is a difference of squares

(z−3)(z+3)(z−1)

source: https://www.mathsisfun.com/algebra/factoring.html