Math 6A (Periods 1 & 2)

Graphs of Equations 11-9

An equation in two variables y = x + 2      produces an infinite number of ordered pairs
If we give x the value of 1, a corresponding value of y is determined
y = (1) + 2 = 3
The ordered pair is (1, 3)

If we let x = 4
y = (4) + 2 = 6
and we get the ordered pair (4, 6)

What happens if x = 0
y = (0) + 2 = 2 ( 0, 2)
or x = -1
y = (-1) + 2 = 1 ( -1, 1)

I like to remember ordered pairs---> ( ordered, pairs)

We graphed the line on a mini graph stickie.
The line is graphed using a ruler and connecting all the points we plotted. Put arrows at each end (since a line continues with out end) and write the line's equation right above the line.


For each value of x there is EXACTLY 1 value of y.
set of ordered pairs in which no two ordered pairs have the same x is called a FUNCTION


y = x + 2
in the future you will see it written as
f(x) = x + 2
so if x = 2
f(2) = (2) + 2 = 4
if x = 5
f(5) = (5) +4 = 9


We used a three column chart to compute our ordered pairs.

Algebra Honors ( Periods 6 & 7)

Percents 7-5

You have been doing THIS since 6^th grade ( or even  before!)

The word percent means per cent or hundredths or divided by 100.

29 percent = 29%= 29/100 = 0.29

2.6 percent = 2.6% = 2.6/100 = 26/1000 = 0.026

637 percent = 637% = 637/100 = 6  37/100 = 6.37

0.02 percent = 0.02% = 0.02/100 =  2/10,000 = 0.0002

¼ percent = ¼% = 0.25% = 0.25/100 = 25/10,000 = 0.0025

33 ⅓ percent = 33 ⅓% = 100/3% = 100/3 ÷ 100 = 1/3

It is easy to translate word problem questions about percents if you use the methods reviewed in class:

Translating into AN EQUATION

Remember  when translating into an equation “is” means =  and “of” means multiplication
Change the % into a decimal ( or fraction)

Example 1

15%   of   180    is    what number?

0.15 (180)   =   x

or as a fraction

(15/100) (180) = x

In both cases you ge

x = 27

So 15% of 180 is 27.

Example 2

23 is 25% of what number?

23 = .25(x)

or as a fraction

23 = (25/100)(x)

Again in both cases you will arrive at

x = 92

23 is 25% of 92

Example 3

What percent  of 64  is 48?

x(64) = 48

This time you need to remember you found

x = 48/64 which simplifies to ¾

You need to remember to change this into a % 

 ¾ = 75%

This is more easily done using a fraction

(x/100)(64) = 48

x = 75

You just need to remember to add the percent symbol!

Let’s use those three examples but using the proportion method;

Translating into A PROPORTION

Remember  when translating into a proportion “is” means part  and “of” means whole
Change the % into a fraction  with 100 as the denominator. You will have 3 out of 4 numbers. The one missing is the one you are solving for—in each problem

  

That’s all you need to remember

Example 1

15%   of   180    is    what number?

You definitely want to simplify before using cross products

but you will get

x = 27

Example 2

23 is 25% of what number?

You definitely want to simplify before using cross products

but you will get

x = 92

Example 3

What percent  of 64  is 48?

You definitely want to simplify before using cross products

but you will get

x = 75

so it is 75%

Always remember to check what the question is asking for—so you label it appropriately.

When you solve an equation with decimal coefficients, you can multiply both sides of the equation by a power of 10 (10, 100, and so on)  to get an equivalent equation with integral coefficients.

Solve    1.2x = 36 + 0.4x

Multiply both sides by 10 when the coefficients are tenths

10(1.2x)  = 10(36 + 0.4x)

12x = 360 + 4x
8x = 360
x= 45
{45}

Solve     94 = 0.15x + 0.08(1000 - x)

Multiply BOTH SIDES by 100 when the coefficients are hundredths

100(94) = 100[0.15x + 0.08(1000 - x)]

9400 = 15x +8(1000 –x)
9400 = 15x + 8000 -8x
1400 = 7x
200 = x
{200}

Word Problem Example:
During a sale, a sporting goods store gave a 40% discount on sleeping bags. How much did Ross pay for a sleeping bag with an original price of $75?

Two different methods to find the same solution:
Method 1
Find 40% of $75? 0.40 × 75 = 30
Subtract the amount of discount from the original price
75-30 = $45
Ross paid $45 for the sleeping bag

Method 2
If the sleeping bag was discounted 40%, it then cost 100% -40% or 60% of its original price
Find 60% of 75 0.60 × 75 = 45
Ross paid $45 for the sleeping bag.

Algebra Honors ( Periods 6 & 7)

Fractional Equations 7.4

The total resistance R of an electrical circuit with two resistors R₁ and R₂, that are connected in parallel is given by the formula
1/ R₁ +1/ R₂ = 1/R
What do you notice about the difference between this equation and those with fractional coefficients?

This formula is an example of a fractional equation.
An equation with a variable in the denominator of one or more terms is called a fractional equation. To solve a fractional equation, you can multiply BOTH sides of the equation by the LCD or you could use the method of solving a proportion when the equation consists of one fraction equal to another fraction.
3/x -1/4 = 1/12
The LCD of the fractions is 12x
Multiply BOTH sides of the equation by the LCD, 12x
Notice that x ≠ 0 because in this case 3/0 is undefined
12x(3/x – ¼) = (1/12)(12x)
36 -3x = x
36 = 4x
x = 9
{9}

(2-x)/(3-x) = 4/9
There are two different ways to solve this
First by finding the LCD, which is 9(3-x) Notice that x ≠3 Why?
9(3-x)[(2-x)/(3-x) = (4/9)[9(3-x)]
18-9x =12-4x
6 = 5x
6/5 = x
{6/5}
OR solve as proportion
(2-x)/(3-x) = 4/9

(2-x)(9) = (4)(3-x)
18- 9x = 12- 4x
we are at the same spot as with the first method and we arrive at the same solution
6 = 5x
6/5 = x
{6/5}

The following gets a little more complicated to do and to display here...
Solve
(2/b² - b) – 2/(b-1) = 1
Find the LCD by first factoring the denominators first
b² - b = b(b-1) so the LCD of the two fractions in this equation is in fact
b² - b BUT use it in factored form b(b-1) Notice: b ≠ 1 why?
(2/b² - b) – 2/(b-1) = 1
[b(b-1][ (2/b² - b) – 2/(b-1) ]= 1[b(b-1)]
which separates to
[b(b-1) (2/b² - b)] – [b(b-1)2/(b-1)] = 1
[b(b-1) (2/b(b - 1)] – [b(b-1)2/(b-1)] = 1
2 -2b=b(b-1)
or
2 – 2b = b² - b
solve for b now
0 = b² - b + 2b -2
0 = b² + b – 2
0 = (b-1)(b+2)
b = 1 and b = -2
Remember in this case b ≠ 1 because of the ORIGINAL EQUATION
the solution set is only
{-2)

Multiplying both sides of an equation by a variable expression sometimes results in an equation that has an extra root. You must check each root of the transformed equation to see if it satisfies the original equation.

Math 6A (Periods 1 & 2)

Graphs of Ordered Pairs 11-8

A PAIR of numbers whose ORDER is important is called an
ordered pair!!
(ordered, pair)

(2,3) is not the same as (3,2)

The two perpendicular lines are called axes.
The x-axis deals with the 1st number of the ordered pair and the y-axis deals with the 2nd number of the ordered pair.
The AXES meet at a point called the Origin (0,0)
The plane is called the coordinate plane

There are 4 quadrants, Use Roman Numerals to name them!!
Quadrant I ---> both the x and y coordinates are positive
(x,y) (+,+)

Quadrant II --> the x coordinate is negative but the y is positive
(-x,y) (-,+)

Quadrant III -->. both the x and y coordinates are negative
(-x,-y) (-,-)

Quadrant IV --> the x coordinate is positive but the y coordinate is negative
(x,-y) (+,-)

Algebra Honors ( Periods 6 & 7)

Ratios 7-1

The ratio of one number to another is the quotient when the first number is divided by the second number (and the 2nd number does not equal 0) 

Ratios = fractions with meaning 

A ratio is the comparison of a number a and a non zero number b using division. The ratio a to b can be written three ways-- and you read them all the same

1) as a quotient using the division sign ÷   1 ÷ 3

32) as a fraction  1/3

3) as a ratio using a colon  1:3

A ratio of 7 to 4 can be written 7:4 or 7/4
A Ratio needs two numbers. DO NOT make it into a Mixed number!

32:48 becomes 2:3

   = 3x/2y

You can use ratios to compare 1 quantities of the SAME KIND

To write the ratio of two quantities of the same kind

1) First express the measures in the same unit

2) Then write their ratio

Write each ratio in simplest form

3h: 15 min

or 12:1

That’s the same ratio whether you changed the numerator to minutes or the denominator to hours

9in: 5 ft

Write a ratio of the height of a tree 4m tall to the height of a sapling 50cm tall

1) Express both heights in centimeters

2) Express both heights in meters

When you solve a word problem, you may need to express a ratio in a different form. If two numbers are in the ratio 3:5 you can use 3x and 5x to represent them, because

The lengths of the sides of a triangle are in the ratio 3:4:5. The perimeter of the triangle is 24 in. Find the lengths of each side.

Let the lengths of the sides be 3x, 4x, and 5x

3x + 4x + 5x = 24

12x= 24

x = 2

So the sides of the triangle are 6in, 8 in, and 10 in

Find the ratio of x to y 
Collect x-terms on one side and y terms on the other. Then factor

3x = 7y

Divide both sides by 3 and then divide both sides by y

cx –ay = aby - bcx

collecting  the x terms on one side and the y terms on the other you have

cx + bcx = aby + ay

Now factor

x(c + bc) = y( ab + a)

divide both sides by c + bc

then divide both sides by y

you have

BUT… you can still factor

which can simplify to

 

Math 7 ( Period 4)

Solving  Percent Problems 7.5

There are 2 methods: Equations and Proportions

For both methods, there are THREE parts of any percent problem:

1) the BASE à the number you are taking a percent of

2) the PERCENT rateà the percent

3) the PERCENTAGE AMOUNT à (base)(%)

EQUATION METHOD:

Just translate and solve for the missing number.

Remember:

isà =

of à multiply

EXAMPLES:

Finding the PERCENTAGE AMOUNT

Find 60% of 15

First realize that “Find” means “What is”

and change the percent to a decimal or a fraction

so

What is 60% of 15

n = (.60)(15)

n = 9

Using fractions we found

first that 60/100 simplifies to 3/5

so

n =(3/5)(15)

That simplifies to

n = 9

Finding the BASE:

35% of what number  is 52.5

.35n = 52.5

simple one step equation

.35n/.35 = 52.5/.35

n = 150

Again, what if we used fractions

35% = 35/100 = 7/20

so

(7/20)n = 53.5

n = 150

Finding the PERCENT rate

What percent of 25 is 13?

25p = 13

p = 13/25

Now you need to make this a percent and that can be accomplished by dividing

but you can also use equivalent fractions   that is,

13/25 = 52/100 so that is 52%

PROPORTION METHOD

%/100 = is/of

Set this up each time and then substitute in what you know

Firstà look for the 5 rate and put that in ( if you are given it)

Second à find  either the “of” or the “is”

Lastà  Whatever is missing is what you are finding…

We did the three examples above as proportions

EXAMPLES:

Find 60% of 15

60/100 = x/15

Simplify and you get

3/5 = x/15

you can simplify even more and get

3/1 =x/3

so x = 3(3) = 9

35% of what number  is 52.5

35/100 = 52.5/x

Simplifying again we get

7/20 = 52.5/x

Using cross products

7x = 20(52.5)

7x = 1050

so x = 150

What percent of 25 is 13?

This is the only one where we do not have the %

n%/100 = 13/25

Can we simplify first?

Yes

n%/4 = 13/1

so n = 4(13)

n = 52

Remember to add the percent sign

52%

Math 7 ( Period 4)

Scale Drawings & Models  7.3

A scale drawing is a diagram of an object in which the length and the width are proportional to the actual length and width of the object

The scale for the drawing gives the relationship between the drawing’s measurement and the actual measurements, such as “ 1 in : 2ft.” 

This scale means that 1 inch on the drawing represents the actual distance of 2 feet.

The scale factor for a scale drawing is the ratio of the length and width in the drawing to the corresponding actual length and width (in the same units of measurements)  for instance a sale of “.25 in : 2 ft” has a scale factor of  1/96 because you need to change either the inches to feet in the numerator  or the feet to inches in the denominator to have the same units of measurement. 

2 ft is equivalent to 24 inches so you have  .25/24 when you simplify that you get 1/96

Scale ratios are usually rates… They have two different labels.
1.5 in : 1000 miles.

Scale rates have NO LABELS at all because both the model and the actual are expressed in the same label.  The scale rate no longer has any labels and simplifies.

Using cross products you find that the missing  side is 12 inches

ALWAYS SET UP THE PROPORTION WITH 2 KNOWN MEASUREMENTS.

We looked at the scale drawing of the theater stage on Page 328 and found the length of the back wall. 
In the scale drawing .25 inches  = 2 ft  . 

The back wall in the drawing measures 1.5 inches  Let x = the actual length of the back wall

We set up a proportion

 

You could simplify BEFORE you cross multiply
but without you get  .25x = 2(1.5)

divide both sides by .25

x = 12
12 feet

What if you had changed the scale to a scale  with the same units of measures first? That is changed the “.25 inches = 2 feet” to “.25 inches = 24 inches”   So

Notice the difference – now everything is in inches and when you solve it you get    x = 144

BUT THAT IS IN INCHES… 

and 144 inches = 12 feet… the same answer you arrived at—without having to change to the same units of measure!!

In Geometry the definition of similar polygons is that they have proportional sides and congruent (equal) angles. The congruent angels keep the shape the same. The proportional sides make the polygons smaller or larger. 

So if you know that 2 triangles are similar, you can find a missing measurement of a side by setting up a proportion.

Example: You know that 2 triangles are similar. One is larger than the other You know the measurement of 2 sides of the smaller  are 3 inches and 4 inches. You know that the larger triangles corresponding shorter side is 9 inches.

COMPARE PERIMETER AND AREA

Perimeter of a rectangle is 2l + 2w

So if a photo is 8 inches by 10 inches, the perimeter is
2(8) + 2(10) = 36 inches

If you reduce the size of the picture 50% on the copier
P = 2(4) + 2(5) = 18. So the perimeter is 50% smaller

Area of a rectangle = lw = 8(10) = 80 inches squared
Is the reduced photo’s area 40 inches squared?

NO 4(5) = 20 inches squares.  WHY?

each of the sides is ½ the original so (1/2)(1/2) = ¼ of the original area..

 20 is 1/4 of the original area of 80

Math 7 (Period 4)

Writing & Solving Proportions 7.2

An equation that states tow ratios are equal is called a proportion.

The proportion  

 is read “ a is to b as c is to d.”

It has two cross
products ad and bc.

In words: In a proportion the cross products are equal

In Algebra: if 

 then ad = bc

With examples: because 2/5 = 4/10 you know that 2(10) = 4(5)

When you know three numbers in a proportion, you can find the missing value by using cross products

Use cross products property to solve the proportion  

3(15) = 5(m)
Now it is a one step equation
Divide both sides by 5

3(15)/5 = 5m/5
9 = m

Be sure to compare quantities in the same order in a proportion.

You are reading a novel—we talked about The Giver, one day you read 30 pages in 50 minutes. The next day you read 24  more pages in 40 minutes. Did you read at the same rate?

Setup a proportion

Does     

Notices on both the pages are in the numerator and the minutes are in the denominator.

Could we have written it  

 Yes.  Notices on both the pages are in the denominator now and the minutes are in the numerator.

We looked at the CORN BREAD RECIPE on Page 336

Amount	Ingredient	Directions
1  ¼ c	flour	Pre heat oven to 400 º
¾  c	Corn meal	Grease 8 by 8 inch pan
¼ c	sugar
2 t	Baking powder	Combine dry ingredients. Mix in milk,
1 c	Skim milk	oil and egg. Pour into pan. Bake 20
½ t	salt	to 25 minutes
¼ c	oil
1	Egg, beaten	9 servings

If you had 8 cups of corn meal, how many loaves of corn bread could you make?  Set up a proportion

  (3/4)x = 8  solve 

 x = 32/3 which is  10 2/3 so you could make 10 loaves

To make 10 loaves of corn bread, how much sugar would you need?

Again set up a proportion

  Solve 10(¼) = s   10/4 = s or 2½ cups of sugar

Math 6A ( Periods 1 & 2)

Solving Equations 11-7

Now that we have learned about negative integers, we can solve an equation such as
x + 7 = 2
We need to subtract 7 from both sides of the equation
x + 7 = 2
- 7 = - 7

to do this use a side bar and use the rules for adding integers
Notice the signs are different so
ask yourself... Who wins? and By How Much?
stack the winner on top and take the difference
so
x + 7 = 2
- 7 = - 7
x = -5



t - -10 = 19
becomes -- with add the opposite---
t+ + 10 = 19
which is just
t + 10 = 19
so subtract 10 from both sides
t + 10 = 19
- 10 = - 10
t = 9

w - - 26 = -44
"Add the Opposite"
w + + 26 = -44
- 26 = - 26
x = -70

Know your integer rules and it becomes easy!!
Side bars are great, if you need them with difference signs!!


y -^- 6 = 4
add the opposite and you get
y + 6 = 4
now you need to subtract 6 from both sides of the equation
y + 6 = 4
- 6 = - 6
Again the signs are different -- ask your self those all important questions
"Who Wins? and "By How Much?"
Use a side bar, stack the winner on top and take the difference. Make sure to use the winner's sign in your answer!!
y = 2

What about -5u = 125?
Whats happening to u?
It is being multiplied by -5... so you must divide by -5

-5u = 125
-5 -5
u = -25

or written easier to read -5u/-5 = 125/-5
u = -25

(1/-9)c = 33
Need to multiply both sides by the reciprocal of (1/-9) which is (-9/1)

(-9/1)(1/-9)c = 33(-9/1)
c = -297



2- STEP EQUATIONS

What about

3u - 1 = -7
You need to do the reverse of PEMDAS... remember unwrapping the present? We did the exact opposite of what we had done to wrap the present!!
so
3u - 1 = -7
+ 1 = + 1
3u = -6
Now divide by 3 on both sides
3u/3 = -6/3
u = -2


3z - ^- 15 = 9
add the opposite first and you get
3x + 15 = 9
In order to solve this 2 step equation
we need to do the reverse of PEMDAS-- as we did with unwrapping the present so many months ago
3x + 15 = 9
subtract 15 from both sides of the equation
3x + 15 = -9
- 15 = - 15

This time the sides are the same-- so just add them and use their sign
3x + 15 = -9
- 15 = - 15
3x = -24
Now divide both sides by 3
3x = -24
3 3

x = -8

Make sure to BOX your answer!!
What about this one
(1/2)(x) + 3 = 0
subtract 3 from both sides
(1/2)x = -3

Multiple by the reciprocal of 1/2 which is 2/1
(2/1)(1/2)x = -3(2/1)
x = -6
Again box your answer.



What about x = -6 + 3x
OH dear... we have variables on BOTH sides of the equations... we need to get the variables on one side all the constants on the other.
We need to isolate the variable!!

x = -6 + 3x
What if we add six to both sides
x = -6 + 3x
+6 = + 6
x + 6 = 3x
now we need to subtract x from both sides
x + 6 = 3x
- x - x
6 = 2x
so now divide both sides by 2
6/2 = 2x/2
3 = x

How about this one
3 - r = -5 + r
- 3 = - 3
-r = -8 + r
if subtract r from both sides, I will get rid of the +r on the right side

-r = -8 + r
- r = -r
-2r = -8
Now divide by -2 on both sides

-2r/-2 = -8/-2
r = 4

Math 6A ( Periods 1 & 2)

Quotients of Integers 11-6
Now that we have learned about negative integers, we can solve an equation such as
x + 7 = 2
We need to subtract 7 from both sides of the equation
x + 7 = 2
- 7 = - 7

to do this use a side bar and use the rules for adding integers
Notice the signs are different so
ask yourself... Who wins? and By How Much?
stack the winner on top and take the difference
so
x + 7 = 2
- 7 = - 7
x = -5



t - -10 = 19
becomes -- with add the opposite---
t+ + 10 = 19
which is just
t + 10 = 19
so subtract 10 from both sides
t + 10 = 19
- 10 = - 10
t = 9

w - - 26 = -44
"Add the Opposite"
w + + 26 = -44
- 26 = - 26
x = -70

Know your integer rules and it becomes easy!!
Side bars are great, if you need them with difference signs!!


y -^- 6 = 4
add the opposite and you get
y + 6 = 4
now you need to subtract 6 from both sides of the equation
y + 6 = 4
- 6 = - 6
Again the signs are different -- ask your self those all important questions
"Who Wins? and "By How Much?"
Use a side bar, stack the winner on top and take the difference. Make sure to use the winner's sign in your answer!!
y = 2

What about -5u = 125?
Whats happening to u?
It is being multiplied by -5... so you must divide by -5

-5u = 125
-5 -5
u = -25

or written easier to read -5u/-5 = 125/-5
u = -25

(1/-9)c = 33
Need to multiply both sides by the reciprocal of (1/-9) which is (-9/1)

(-9/1)(1/-9)c = 33(-9/1)
c = -297



2- STEP EQUATIONS

What about

3u - 1 = -7
You need to do the reverse of PEMDAS... remember unwrapping the present? We did the exact opposite of what we had done to wrap the present!!
so
3u - 1 = -7
+ 1 = + 1
3u = -6
Now divide by 3 on both sides
3u/3 = -6/3
u = -2


3z - ^- 15 = 9
add the opposite first and you get
3x + 15 = 9
In order to solve this 2 step equation
we need to do the reverse of PEMDAS-- as we did with unwrapping the present so many months ago
3x + 15 = 9
subtract 15 from both sides of the equation
3x + 15 = -9
- 15 = - 15

This time the sides are the same-- so just add them and use their sign
3x + 15 = -9
- 15 = - 15
3x = -24
Now divide both sides by 3
3x = -24
3 3

x = -8

Make sure to BOX your answer!!
What about this one
(1/2)(x) + 3 = 0
subtract 3 from both sides
(1/2)x = -3

Multiple by the reciprocal of 1/2 which is 2/1
(2/1)(1/2)x = -3(2/1)
x = -6
Again box your answer.



What about x = -6 + 3x
OH dear... we have variables on BOTH sides of the equations... we need to get the variables on one side all the constants on the other.
We need to isolate the variable!!

x = -6 + 3x
What if we add six to both sides
x = -6 + 3x
+6 = + 6
x + 6 = 3x
now we need to subtract x from both sides
x + 6 = 3x
- x - x
6 = 2x
so now divide both sides by 2
6/2 = 2x/2
3 = x

How about this one
3 - r = -5 + r
- 3 = - 3
-r = -8 + r
if subtract r from both sides, I will get rid of the +r on the right side

-r = -8 + r
- r = -r
-2r = -8
Now divide by -2 on both sides

-2r/-2 = -8/-2
r = 4