Algebra Honors (Periods 5 & 6)

Dividing Fractions 6-3

Use the same rule for dividing fraction that you use for dividing real numbers—multiply by its reciprocal.    
a/b  ÷ c/d = a/b ∙ d/c

Divide           x/2y  ÷ xy/4   so 

  x/2y  ∙ 4/xy

Simplify to 2/y²       

Divide  18/(x²-25) ÷   [24/(x+5)]

That becomes

18/(x²-25) ∙   [(x+5)/24]

     =  [3 ∙ 6/(x+5)(x-5)] ∙   [(x+5)/4 ∙ 6]

 Simplify

= 3/4(x+5)

Divide:

[x²+3x-10/(2x+6)]    ÷ [(x²-4)/ (x²-x-12)]

Which becomes

[x²+3x-10/(2x+6)]  ∙ [(x²-x-12)/ (x²-4)]

Factor

 [(x+5)(x-2)/2(x+3)]  ∙ [(x+3)(x-4)/ (x+2)(x-2)]

 [(x+5)(x-4)]/[2(x+2)]   

Make sure to use the O³ when simplifying an expression that involves more than one operation.   For Example:

   (2x/y)³÷(y²/x) ∙  x/4

= (8x³/y³⁾∙ (x/y²) ∙  x/4

2x³ /4

Math 6High (Period 3)

Rates 3.8 

If two quantities a and b have different units of measure, then the rate of a per b is a/b. The units for a rate tell you which numbers goes in the numerator and which units goes in the denominator .

For example

Miles per hour = miles/hour

miles

hour

When a rate is simplified so that it has a denominator of 1, it is a unit rate.

You are traveling from San Jose,  CA to Santa Rosa, CA You travel a distance of 101 miles in 2 hours. What is your average speed in miles per hour?

Speed is one type of a unit rate. 
To find the average speed in miles per hourà divide the distance traveled by the time

Average speed = distance/time

101miles/2hours =   50.5miles/hour

You can always estimate to check that your result is reasonable.

100miles in 2 hours is 50 miles/hour so your answer makes sense

In order to compare prices at the supermarket, you can calculate unit prices. A 12 ounce box of cereal costs $ 3.72  What is the unit price of the cereal?

A unit price is another type of unit rate. Divide the cost by the weight

Unit price = Cost/weight

$3.72/12 oz

DIVIDE CAREFULLY

$0.31/1 oz

The unit price of the cereal is $0.31 per ounce.

Then we talked about a scenario involving babysitting. Suppose you babysat for one family on Saturday for 2 ½ hours and were paid $10. Then on Sunday, you babysit for another family for 3 hours and were paid $10.50. 

Find the hourly rate that you were paid on each day. Which family do you want to work for again? (That is which one paid your more per hour?)

We used a table to organize the information

Day	Total	Time	Rate
Saturday	$10	2  ½ hours	$__/hour
Sunday	$10.50	3 hours	$__/hour

Saturday:  Amount paid/time = 10/2.5 = $ 4/hour

Sunday: Amount paid/ time = 10/3 = $3.50/hour

Your hourly rate was greater on Saturday.

Two rates for answering questions on a test are given below. 
We wrote each as a unit rate

60 minutes/ 75 questions  and   75 questions/ 60 minutes

for  60 minutes/ 75 questions  we found it was 0.8 min/question

and

for  75 questions/ 60 minutes , we found it was 1.25 questions/ min

Both rates are equivalent and would allow you to complete the test on time. However, it would be easier to use the rate of 0.8 min/question to figure out how long it took you per question!

You drive 1350 miles in 3 days Each day you drive for 9 hours. What is your speed in miles per hour?  

First you need to calculate how many hours you drove. 3(9)= 27à so you drove 27 hours

1350miles/27 hours = 50miles/hour

Algebra Honors ( Periods 5 & 6)

Multiplying Fractions 6-2
You know from previous years that
ac/bd = a/b ⋅ c/d

and you know the converse is also true
a/b ⋅ c/d = ac/bd
That means you could solve
8/9⋅3/10 by either multiplying first and then simplify or you could simplify first and then multiply.
I find it works so much better to simplify first
8/9⋅3/10 = 4/15

6x/y³⋅y²/15 = 2x/5y where y ≠0

Which simplifies to

This textbook wants us to keep the factored form as our answers--> so let's continue to do that. In addition it states, " ...from now on, assume that the domains of the variables do not include values for which any denominator is ZERO. Therefore it will NOT be necessary to show the excluded [or restrictions] values of the variables."
I know everyone is jumping for joy!!

Rule of Exponents for a Power of a Quotient
(a/b)^m = a^m/b^m
(x/3)³ = x³/27

(-c/2)²⋅4/3c
you must do the exponent portion first!!
c²/4⋅(4/3c)
c/3

Find the volume of a cube if each edge has length 6n/7 in
You just need to cube each factor
(6n/7)³ = 216n³/343 inches cubed

If you traveled for 7t/60 hours at 80r/9 mi/h, how far have you gone?
Just multiply
7t/60⋅80r/9
but simplify first and you get
28rt/27 miles

Math 6A (Periods 2 & 4)

Prime Numbers & Composite Numbers 5-4

A prime number is one that has only two factors: 1 and the number itself, such as 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31...
A counting number that has more than two factors is called a composite number, such as 4, 6, 8, 9, 10...

Since one has exactly ONE factor, it is NEITHER PRIME NOR COMPOSITE!!
Zero is also NEITHER PRIME NOR COMPOSITE!!
Sieve of Eratosthenes - We did it!! :)

Every counting number greater than 1 has at least one prime factor -- which may be the number itself.
You can factor a number into PRIME FACTORS by using a factor tree or the inverted division, as shown in class.

Using the inverted division, you also start with the smallest prime number that is a factor... and work down
give the prime factors of 42
2⎣42
3⎣21
7

When we write 42 as 2⋅3⋅7 this product of prime factors is called the prime factorization of 42.

Two is the only even prime number because all the other even numbers have two as a factor.

Explain how you know that each of the following numbers must be composite...
111; 111,111; 111,111,111; and so on....
Using your divisibility rules you notice that the sums of the digits are multiples of 3.

List all the possible digits that can be the last digit of a prime number that is greater than 10.
1, 3, 7, 9.

Choose any six digit number such that the last three digits are a repeat of the first three digits. For example
652,652. You will find that 7, 11, and 13 are all factors of that number... no matter what number you choose... why is that???? email me your response.

Math 6A (Periods 2 & 4)

Square Numbers and Square Roots 5-3

Numbers such as 1, 4, 9, 16, 25, 36, 49... are called square numbers or PERFECT SQUARES.

One of two EQUAL factors of a square is called the square root of the number. To denote a square root of a number we use a radical sign (looks like a check mark with an extension) See our textbook page 157.

Although we use a radical sign to denote cube roots, fourth roots and more, without a small number on the radical sign, we have come to call that the square root.
SQRT = stands for square root, since this blog will not let me use the proper symbol) √ is the closest to the symbol

so the SQRT of 25 is 5. Actually 5 is the principal square root. Since 5 X 5 = 25
There is another root because
(-5)(-5) = 25 but in this class we are primarily interested in the principal square root or the positive square root.

Evaluate the following:
SQRT 36 + SQRT 64 = 6 + 8 = 14
SQRT 100 = 10
Is it true that SQRT 36 + SQRT 64 = SQRT 100? No
You cannot add square roots in that manner.
However look at the following:
Evaluate
SQRT 225 = 15
(SQRT 9)(SQRT 25)= (3)(5) = 15
so
SQRT 225 = (SQRT 9)(SQRT 25)

Also notice that the SQRT 1600 = 40
But notice that SQRT 1600 = SQRT (16)(100) = 4(10) = 40

Try this:
Take an odd perfect square, such as 9. Square the largest whole number that is less than half of it. ( For 9 this would be 4). If you add this square to the original number what kind of number do you get? Try it with other odd perfect squares...

In this case, 9 + 16 = 25... hmmm... what's 25???

Algebra Honors (Periods 5 & 6)

Simplifying Fractions 6-1

When the numerator and the denominator of an algebraic fraction have no common factor other than 1 or -1, the fraction is said to be in simplest form. To simplify a fraction, first factor the numerator and the denominator.

Simplify:

(3a + 6)/ (3a + 3b)

3(a + 2)/3(a + b)

=(a  + 2)/(a +b)    (where a ≠ -b)

REMEMBER : YOU CANNOT DIVIDE BY ZERO. You must restrict the variables in a denominator by excluding any values that would make the denominator equal to ZERO.

So with the above example, a  CANNOT EQUAL –b

Simplify

(x²-9)/(2x+1)(3+x)

(x+3)(x-3)

(2x+1)(3+x)

since x + 3 = 3 +x

you can simplify both the numerator and denominator  to

x-3

2x+1

  (where x ≠-1/2, x≠ -3)

To see which values of the variable to exclude look at the denominator of the original fraction as well. Neither 2x + 1 nor 3 +x can be equal to zero.  Since 2x + 1 ≠ 0  x ≠ =1/2  and since 3 + x≠ 0  x ≠ -3

Simplify

2x² + x - 3

2- x- x²

First factor the numerator and the denominator, using  the skills you developed from the last chapter. If you do not see any common factors, look for opposites—as in this case

(x - 1)(2x+3)

(1- x)(2 + x)

Notice that (x-1) and (1-x) are opposites.

(1 -  x) = -( x - 1)

So change the sign on the fractions and use the opposites.

That is 

(x - 1)(2x+3)

-(x- 1)(2 + x)

and that can simplify to

(2x+3)

-(2 + x)

  or

-    (2x+3)

        x+2

 (where  x≠1, x≠-2)

Solve for x

ax - a²=bx –b²  

Collect all terms with x on one side of the equation

ax – bx = a² –b²

Factor BOTH sides of the equation

x(a –b) = (a+b)(a –b)

Divide BOTH sides of the equation by the coefficient of x ( which is    a-b)

x = a + b     ( where a≠ b)