Math 6A (periods 2 &4)

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 6 High ( Period 3)

Solving Percent Problems 7.4

You can use a proportion to find what percent one number is of another number.
The statement:

“ a is p percent of b” is expressed by the proportion shown below where

a is the  part of the base      b is the base… and    p is the percent

a/b = p/100

Part of the base / base = percent/100

What percent of 40 is 15?

First think which is the base? 40

Which number is the part?  15

Now write the proportion

a/b = p/100

15/40 =  p/100

Use cross products to multiply… simplify first if you can…

15(100) = 40p

Divide both sides by 40

15(100)/40 = 40p/40

37.5 = p

Note: there are a number of ways to simplify first! You should still arrive at p = 37.5 which means 37.5%

Finding a Percent

Your team won 19 of its 25 softball games. What percent did it win?

Games won/games played = percent/100

It really is still      a/b = p/100

19/25 = p/100

19(100) = 25p

Divide both sides by 25

19(100)/25 = 25p/25

76= p

Again there are a variety of ways to simply BEFORE you multiply! You could even utilize the equivalent fraction method to determine the p percent.

Therefore your team won 76% of its games!

You can use the proportion to find a base or a part of the base.

Finding a part of the base

You buy a pair of pants on sale. The price is 80% of the regular price of $24.50, WOW! That’s a bargain! What is this incredible sales price?

Sale price / regular price = Percent/100

a/24.50 = 80/100

Multiply and divide carefully

100a = 24.50(80)

divide both sides by 100

100a/100 = (24.50)(80)/100

a = 19.6

The sale price is $19.60

Now, you could use the writing an equation method

80% of 24.50 = sale price  or

(.80) (24.50) = p

Multiply carefully

19.60 = p

$19.60 is the sale price.

Finding a Base

8 is 32% of what number?

Method 1: Write and solve a proportion

a/b = p/100

8/b = 32/100
Using cross products
(8)(100) = 32b
Divide both sides by 32

(8)(100)/32 = 32b/32

25 = b

Method 2: Write and solve an equation

8 is 32% of what number?
8 = .32(x)

8/.32 = .32x/.32
Divide carefully

25= b

8 is 32% of 25

Solving Percent Problems

a/b = p/100   means a is p percent of b

Out of 80 students taking Algebra, 95% passed the final exam. About 80% of those who passed the exam got a grade better than a C. How many students got an A or a B?

First find out how many passed the final exam

Using the proportion method for this:

a/80 = 95/100

100a = 80(95)

100a/100 = (80)(95)/100

a = 76

76 students passed the final exam

About 80% of 76 got a grade better than a C

Using the writing an equation method

(.80)(76)  =60.8

Since you can’t have a part of a student your answer would be about 61 students got an A or a B in Algebra

Math 6High ( Period 3)

Large & Small  Percent 7.3

The methods you learned from Lesson 7.2 can be used to find large and small percents.

Rewrite Numbers

Rewrite 1.4 as a percent
1.4 = 1  4/10 = 14/10 = 140/100 = 140%

Rewrite   ½%  as a decimal
½ %= 0.5% = 0.5/100 = 5/1000 = 0.005

Finding a Large Percent of a Number

You can use either of two methods to find 130% of 250

Method 1: Set up a proportion      percent/100 = is/of

130/100 = x/250

Cross multiply carefully

(130)(250) = 100x  Then divide by 100 on both sides

(130)(250)/100 = 100x/100

325 = x

There were a number of different ways to simplify BEFORE you multiply but do them carefully . You still need to get x = 325

Method 2: Rewrite 130% as a decimal and set up a written equation

Find 130% of 250.   Think “ Find”  means “What is…” so you have

What is 130% of 250
x = (1.3 )(250)
x = 325

Find a Small Percent of a Number

You can use the same two methods to find 0.5% of 40

Method 1: Set up a proportion.

.5/100 = x/40  or

5/1000 = x/40

Cross multiply and divide carefully
(5)(40)/1000 = 1000x/1000

0.2 = x

There were a number of different ways to simplify BEFORE you multiply but do them carefully . You still need to get x = 0.2

Method 2: Rewrite 0.5% as a decimal and use the writing an equation method

Find 0.5% of 40   Think “ Find”  means “What is…” so you have
What is 0.5% of 40.
x = (0.005)(40)
x = 0.2

Estimating:

In many real-life situations—such as figuring the amount for a tip, it is helpful to use mental math to estimate the percent of a number.

Use mental math to estimate 123% of 84. You know it will be MORE than 84… because the percent is over 100% and we know 123%  is about 125%   100% + 25% so all we need to figure is 25% o 84… and that is just ¼ so 21.. 84 + 21 = 105. So 123% is just a little under 105.  

Use mental math to find ½ % of 140

Wait… ½ % is just half of 1% so  figure 1% of 140. Just move the decimal over two places…1.4 would be 1% so half of that is 0.7

estimate 15% tip for a restaurant bill of $43.76

First round the restaurant bill up to $45.00. Then take 10% … that’s 4.50

5% is half of ten percent so half of 4.50  is 2.25

4.50 + 2.25 = 6.75  $6.75 is about 15% of43.76

Algebra Honors ( Periods 5 & 6)

 Solving Problems Involving Quadratic Equations 12-6

You can use quadratic equations to solve problems...
We used the examples in the book to start with:

The park commission wants a new rectangular sign with an area of 25 m² for the visitor center. The length of the sign is to be 4 m longer than the width . To the nearest tenth of a meter, what will be the length and the width of the sign?
Always make sure you check before AND after-- to see what the problem is really asking for...
Let x = the width in meters
then x + 4 = the length in meters

Use the formula for the area of a rectangle to write an equation

x(x+4) = 25
solve it
x(x +4) = 25 becomes
x² + 4x = 25
You can use two methods: the quadratic formula would be my second choice since completing the square works easily here

x² + 4x + 4 = 25 + 4
(x + 2)² = 29
x + 2 = ±√29
You can use your calculator, or the table of square roots... or approximately easily using the method taught in class earlier this year
but to the nearest tenth you get
-2 + √29 ≈3.4
-2 - √29 ≈-7.4

Since you can't have a negative root since a negative length has no meaning... you know the width must be about 3.4 meters and therefore the length is 3.4 +4 or approximately 7.4 meters

Problem 2:
The sum of a number and its square is 156. Find the number

Let x = the number
then
x² + x = 156

x² + x - 156 =0
Using the skills you have for factoring
(x +13)(x-12) = 0
so
x = 13 and x = 12
You have two solutions to this question!!

Problem 3:
The altitude of a triangle is 9 cm less than the base. The area is 143 cm²
What are the altitude and base?

Remember the formula for the area of a triangle is A = ½bh

Let b = the length of the base
then the altitude ( the height) is b-9
so
½(b)(b-9) = 143
b² -9b = 286
b² -9b - 286 = 0
(b -22)(b +13) = 0
b = 22 and b= -13
You can't have a negative length so
the base is 22 cm and the altitude is 13 cm


An object that moves through the air and is solely under the influence of gravity is called a projectile. The approximate height (h) in meters of a projectile at t seconds after it begins its flight from the ground with initial upward velocity v₀ is given by the formula
h = -5t² + v₀t
We can find when such a projectile is at ground level (h=0) by solving
0 = -5t² + v₀t.
If a projectile begins its flight at height c, its approximate height at time t is
h = -5t² + v₀t + c .
We can find when it hits the ground by solving h=0 or
0 = -5t² + v₀t +c.

When a projectile is thrown into the air with an initial vertical velocity of r feet per second, its distance (d) in feet above the starting point t seconds after it is thrown is approximately
d = rt – 16t²

Math 6A (Periods 2 & 4)

Products of Integers 11-4 & 11-5


3 ⋅ -2 = -6
Its really repeated addition
or
-2 + -2 + -2 which we learned a few sections ago was equal to -6.

The product of a positive integer and a negative integer is a negative integer.

The product of ZERO and any integer is ALWAYS ZERO!!
a⋅0 = 0

Math imitates life...and Karma(?)
What was the story I told in class... it applies to
Multiplication & Division ...
+ ⋅ + = +
- ⋅ + = -
+ ⋅ - = -
- ⋅ - = +



The product of -1 and any integer equals the opposite of that integer.
(-1)(a) = -a

The product of two negative integers is a positive integer

For a product with NO ZERO factors:
-->if the number of NEGATIVE factors is odd, the product is negative
-->if the number of NEGATIVE factors is even, then the product is positive

Every integer and its opposite have equal squares!!

Remember-- if its all multiplication use the Associative & Commutative Properties of Multiplication to make your work EASIER!!

Math 6 High ( Period 3)

 Finding a Percent of a Number 7.2

or… it’s just proportions!

Percent/100 = Percent of a number/number

Find 30% of 400
30/100 = x/400
Its a proportion!
30(400) = 100(x)
30(400)/100 = x
120 = x
Therefore 120 is 30% of 400

You drive 300 miles. the first 20% of the road is under construction. How many miles of the road are under construction?
20/100 = x/300
Its a Proportion!!
20(300) = 100(x)
20(300)/100 = x
60= x
therefore 60 miles of the road are under construction.

You can also change the percent to a fraction and multiply!
For example,
Find 50% of 46. 
[Remember: the word "OF" in a mathematical sentence means to multiply]

50% is just 1/2 
so
1/2(46) = 23

75% of 120
75% = 3/4
so 3/4 (120) = 90

You can also change the percent into a decimal and multiply
Find 35% of 150
Read this as
What is 35% of 150?
x= 0.35(150)
x = 52.5
Find 92% of 75
Read this as
What is 92% of 75?
x = 0.92(75)
x = 69

Are these reasonable?
Well 92% is pretty close to 90% and !0% of 75 would be 7.5 so 90% would be 75-7.5 or 67.5.   Since  69 is very close to 67.5  the answer is reasonable.

Math 6 H ( Period 3)

Percents, Fractions, and Decimals 7.1

The word percent means “per hundred.” The symbol is %.
A percent is a ratio whose denominator is 100.  

25% = 25/100 = 0.25

To write a fraction or a decimal as a percent or to write a percent as a fraction or a decimal, the first step is to rewrite the given form as a fraction with a denominator of 100.  

¼ = 25/100 = 25%

2/5 = 40/100 = 40%

0.75 = 75/100 =75%

0.8 = 8/10 = 80/100 = 8%

0.104 = 104/1000 = 10.4/100 = 10.4%

86% = 86/100 = 0.86

15% = 15/100 = 0.15

Some percents need to be written exact – or rounded to a specific decimal place.

For example

1/3 = 0.3333333...

to write as an exact percent you would write 33 1/3 %

You would round it to the nearest tenth as 33.3%

1/8 = 12.5% but if it asked to round to the nearest tenth you would write 13%

Algebra Honors (period 5 &6)

 Methods of Solutions 12-5

You have learned five different methods of solving quadratics.. picking the best one is the challenge
Here are some tips on when to use each method:
Quadratic Formula--> when to use: ax² +bx + c = 0 (It always works) Great when you have a calculator

Factoring--> when to use: ax² +bx + c = 0 you see the factors easily or you have ax² +bx = 0

Using the Property of SQRT's --> when to use: ax² + c = 0

Completing the square --> when to use: x² +bx + c = 0 especially when b is even
Graphing... this really does takes time but if done correctly... it really shows what are  the roots... and.... the x-intercepts, the solutions, the zeroes...

Which method would you use for the following?
x² +5x - 6 = 0
Answer: Factoring

x² -2x = 1
Answer: Completing the square

11x² = 44
Answer: SQRTing

3x² -5x = 4
Answer: Quadratic Formula