Algebra1

hw: 4/23

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hw: ws 11.1

notes 11.1

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11.1 Ratio and Proportion
Proportion- 2 equal ratios or fractions
a/b = c/d a and d are the extremes and b and c are the means

• If 2 ratios are equal, then their reciprocals are equal too
• If you cross multiply, the products will also be equal
Try: 3/y = 5/8

x/3 = 12/x

2007-04-10T12:17:00.001-07:00

Practice for Algebra 1 Test Ch 10!!!!!!!!

10.1 Adding and subtracting Polynomials

Find the sum or difference:

1) (-6x³+5x²-3)-(2x³+4x²-3x+1) 2) (6x²-x+3) + (-2x+x²-7)

3) (12x-8x²+6)-(-8x²-3x+4) 4) -(2x³+3x – x +5) – (x³+2x -1)

10.2 Multiplying Polynomials

Multiply:

4) (3t+5)(t-3) 5) (2x+10)( ½ x + 2 ) 6) (3x+4)²

7) (4x²-3x-1)(2x-5) 8) (2x+3)(8-x-3x²)

9) Write a polynomial expression for the area of a rectangle with a length of 2x-1 and a width of 3x+2.

10.4 Solving Polynomial Equations in Factored Form (Using Zero Product Property)

Solve:

10) y=(x-3)(x+2) 11) y=(x-4)(x+2)

12) (2x+1)(3x-2)(x-1)=0 13) (x+5)²=0

10.5 Factoring x²+bx+c

Solve by factoring:

14) x²+11x+10=0 15) x²-17x+60=0 16) x²+6x+9=0

17) x²-2x=24 18) x²-15x= -54 19) –x²-4x=3

10.6 Factoring ax²+bx+c

Solve by factoring :

20) 2x²+x-6=0 21) 9x²+24x=-16 22) 2x²+7x= -3

23) 20x²+23x+6=0 24) 4x²-5x= 6 25) 2x²+3x+9

10.8 Factoring Using the Distributive Property

Solve by factoring:

26) 2x³-10x²+8x=0 27) x⁴-4x³+x²-4x=0 28) -2x⁵-2x⁴+4x³=0

29) 4x²+40x+100 30) 4x²-16x-48 31) 2x²+28x+98

32) The width of a box is 1 cm less than its length. The height of the box is 9cm greater than the length. The box has a volume of 72 cm³. What are the dimensions of the box?

Chapter Wrap Up- Solve the following using any method you choose. Your choices are: Graphing by hand, “Square Rooting”, Quadratic formula, Factoring, **Factoring by Grouping, **Distributive Property + Factoring. You should then choose a SECOND method to check your answer. Lastly, check with the graphing calculator!!!!!!!!! The bold indicates methods that your test will be focusing on. **= you may only use these methods when applicable.

33) x²-60= -11 34) x²-16=0 35) x²-13x= -40

36) (x+9)² 37) 20x²+23x+6 = 0 38) 4x²+12x+9=0

39) 4x²+4x -5=0 40) -2x²+4x+6=0 41) 3x²+14x=5

notice: Ch 10 Test

2007-04-10T12:13:00.000-07:00

notice: We will be having a test on Chapter 10 this Friday 4/13. The plan is to review tomorrow and Thursday, however if there is a delay or no school due to weather- WE WILL STILL HAVE THE TEST FRIDAY!!!!!!

hw: 4/10

2007-04-10T12:12:00.000-07:00

hw: ws 10-8 DO THE PUZZLE ON THE BACK FOR EXTRA CREDIT!!!!!! You must show work to get extra credit points!

notes 10.7 and 10.8

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10.7 Factoring Special Products
Difference of 2 squares: a2-b2=(a+b)(a-b)
Ex. 16x2-4 = (4x+2)(4x-2)

Perfect Square Trinomial:
a2+2ab+b2= (a+b)2 Ex: x2+8x+16=(x+4)2

a2-2ab+b2= (a-b)2 Ex: x2-12x+36=(x-6)2

10.8 Factoring Using the Distributive Property
Before you begin to factor a polynomial, it is useful to factor out any common factor among all the terms first.
9x2-15 From this, I can factor out a 3 to get 3(3x2-5)

Once you have factored a polynomial completely, it is called prime

5x2-10x-40 –Factor out a 5 first to get:
5(x2-2x-8) since this it equal to 0, divide both sides by 5 to “get rid” of it
Then factor like normal

Try: 45x4-20x2

5x3+30x2+40x

3x3-15x2-6m+30

ws 10-8 //// puzzle

notes 10.6

2007-04-09T13:46:00.001-07:00

10.6 Factoring ax2+bx+c
1) Write the trinomial in standard form
4x2-21x+5=0
2) Multiply a by c
4•5=20
3) Find factors of ac that will add up to “b”
factors of 20: -1,- 20 -2,- 10 -4,- 5
4)Rewrite the quadratic replacing “bx” with the two new factors
4x2-1x-20x +5=0
4) Split this into two groups
4x2-1x -20x+5
5)”Pull out” common factors
x(4x-1) -5(4x-1)
6) You should now have both binomials in the parentheses identical. If you do not, check your work and adjust your negatives.
7)Whatever is inside the parentheses becomes one factor, the second factor is the terms outside the parentheses
(4x-1)(x-5)
8) FOIL to check

try: 7x2-4x-3

6x2-5x-21

10x2+25x-90

21n2+14n+7=6n+11

ws 10-6

notes 10.5

2007-04-09T13:44:00.000-07:00

10.5 Factoring a Quadratic Trinomial
1) Get the quadratic in the form x^2+bx+c
x^2+8x+12
2) look at the c value
12
3) find two factors of c that add up to your b value
Factors of 12: 1, 12 2, 6 3,4 2 and 6 add up to 8
4) start your factors off with (x )(x )
5) Insert in the blanks the two factors you came up with
(x+2)(x+6)
6) FOIL it back together to check
x^2+6x+2x+12 = x^2+8x+12

try: x^2+3x+2

x^2+9x+20

What if b is negative? x^2-6x+8
This means your two factors will be negative
Negative factors of 8 : -1, -8, -2, -4 -2 and -4 add up to be -6
(x-2)(x-4) FOIL to check
x^2-4x-2x+8 = x^2-6x+8

try: x^2-5x+6 x^2-11x+10

What if b and c are negative? This means one of the numbers will have to be negative.
x2-7x-30 Since I will have 1 positive number and 1 negative, I will look for 2 factors of 30 whose DIFFERENCE is 7. Since it is a negative 7, the larger factor will get the negative sign.

Factors of 30: 1,30 2,15 3,10 5,6 3 and 10 have a difference of 7 and since 7 is negative I will use -10 and +3

(x-10)(x+3) FOIL to check

Remember that not all quadratics have x- intercepts. You can use the discriminant to tell you quickly. If your discrim is NEGATIVE the parabola never crosses the x-axis and therefore has no solutions!!!!!!

Solve: x^2-3x = 10 x^2+5x-14=0 x^2-9x+18=2x

hw: 4/4

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hw: ws 10-4

notes 10.3 and 10.4

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10.3 Special Products of Polynomials
FOIL always works.
Shortcuts:
(a+b)(a-b) = a2-b2

(a+b)2= a2+ab+b2

(a-b)2= a2-ab+b2

10.4 Solving Polynomial Equations in Factored Form
Quad in standard form are solved by the quadratic or graphing
Factored form is a quad written as 2 or more linear factors

Ex. x 2+3x-10=0 is (x+5)(x-2) = 0

so (x+5) and (x-2) are factors

Because the quad is = to zero, that means either one of these factors can be zero (zero product property)

(x+5)=0 or (x-2)=0

Solve for x these are your x-intercepts!!!!!!!!!!!!!!
Solve: (x-2)(x+3)=0

(x-4)(x+2)=y (just replace y with 0!!!!!!!!)

(5x+2)(2x-1)=y

repeated –factor equation
(x+5)2=0 is just x+5=0 or x= -5 you will get the same answer twice!
Cubic equation:
(2x+1)(3x-2)(x-1)=0 here you will get 3 answers so how many x-intercepts does that mean it has?????

Ws 10-4

hw: 4/3

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hw: ws 10-2

notes 10.2

2007-04-09T13:41:00.000-07:00

10.2 Multiplying Polynomials
When you distribute, you take the term outside the parentheses and multiply that by each term inside.

What happens when you have more than one term?

Binomial x binomial
We use FOIL.
It means multiply the First terms together. The Outside terms, Inside terms, and Last terms.
Then combine like terms and simplify:
(x-3)(x+2)= Firsts:x•x= x2 Outsides: x•2= 2x Insides: -3•x = -3x Lasts: -3•2 = -6
x2+2x-3x-6
x2-x-6
(6x+1)(2x+9)

Binomial x Trinomial
Take the first term in the binomial and multiply it by EACH term in the trinomial. Repeat for the second term. Combine like terms and simplify

(2x-3)(-x2+x-5)

Try:
1. (x + 2)(x + 2)
2. (x - 3)(x + 3)
3. (3x + 2)(x - 4)
4. (1 + x)(3 + 2x)
5. (-1+x)(x2-6x+2)
6. (x+9)(2x2-4x+1)

foil worksheet

ws 10-2

hw: 4/2

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hw: ws 10-1

notes 10.1

2007-04-02T12:53:00.001-07:00

10.1 Adding and subtracting Polynomials
Polynomials are expressions that are in the form axk .
-Standard form is when the expression is written with the terms in descending degree order.
-the degree is the exponent of the variable
-the degree of the entire polynomial is determined by the highest exponent.

Monomial- poly with only 1 term
Binomial- poly with 2 terms
Trinomial- poly with 3 terms

Ex. Polynomial Degree Classified by degree classified by # of terms
6 0 constant monomial
-2x 1 linear monomial
3x+1 1 linear binomial
-x2+2x-5 2 quadratic trinomial
4x3-8x 3 cubic binomial
2x4-7x3-5x-1 4 quartic polynomial

****************You can only combine LIKE TERMS**********************

Ex: (5x3-x+2x2+7)+(3x2+7-4x)+(4x2-8-x3)

!Subtracting!
When you subtract a polynomial you MUST “distribute” the negative sign throughout

Ex: (-2x3+5x2-x+8)-(-2x3+3x-4) becomes : (-2x3+5x2-x+8)+2x3-3x+4

Ws 10-1

hw:3/29

2007-03-30T05:20:00.001-07:00

hw:STUDY!!!!!!!!!!! TEST CH 9 TOMORROW!!!!!!!!

hw:3/28

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hw: Finish rounds sheet

notes 9-8

2007-03-30T05:18:00.000-07:00

9.8 Comparing Linear, Exponential, and Quadratic Models
Types of models that fit a collection of data:
Linear model: y=mx+b
Exponential Model: y=C(1±r)t c= initial amount r= percent t=time period
Quadratic model: y= ax2+bx+c

To determine which model fits best:
1) Draw the scatter plot
a- If it is linear (same rise/run throughout) a lineat model fits it best
b- if you see a curve:
-test for an exponential model by finding the ratio of consecutive entries. If the ratios are consistent, use an exponential model (see ex 2 pg 555)
- test for quadratic model by finding the ratio of consecutive entries again. If the ratios are NOT consistent, the data follows the quadratic model.

Pg 559 #30, 31, 32 tell which type of graph for each

Bridge ws

rounds

hw: 3/27

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hw: ws 9-7

notes 9.7

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9.7 Graphing Quadratic Inequalities
1) Sketch the graph of the related equation
2) Draw a - - - - line if there is a > or < sign. __________ line if there is a <= or >= or= sign.
3) Pick a point not on the parabola and test it in your equation
4) if it works  shade the graph so that point is included
5) if it doesn’t shade the graph so that point is not included.

Try: y<2x2-3x

Y<= -x2-5x+4

hw: 3/26

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hw: ws 9-6

notes 9.6

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9.6 Applications of the Discriminant
Quad form:
The expression inside the radical is called the discriminant b2-4ac

If the discriminant is:
-Positive the equation has 2 solutions
-Zero the equation has 1 solution
-Negative the equation has no real solutions (can’t take the sq. root of a – number)

Try: tell me how many solutions each has:

x2-3x-4=0 -x2+2x-1=0 2x2-2x+3=0

guided prac. Pg 544 #1-6

ws 9-6

hw: 3/23

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hw:ws 9-5

notes 9.5

2007-03-26T08:22:00.000-07:00

*Many of the symbols needed on this page will not show up properly, please refer to your book or see me for a printout*
9.5 Solving Quadratic Equations by the quadratic formula
a cannot be 0
1) write formula x2+9x+14
2) label a, b, and c a=1 b=9 c=14
3) write quadratic formula
4) plug in a b and c- *WATCH NEGATIVES*
5) simplify
6) split into 2 problems + and –
7) solve for 2 answers (usually) x=-2 x=-7
8) check!

Try: 2x2-3x=8

-x2-2x+5=0

Vertical Motion: When an object is dropped, you can follow its height by using the formula:
h= -16t2+s where h=final height (ft) , t=time(sec) s=initial height (ft)

Try: If a ball is dropped from an initial height of 20 ft. how long will it take to hit the ground?
1.11s
(do actual example if time allows- if not do tomorrow)

What if a ball is thrown? Then we use the formula:
h=-16t2+vt+s where h=height (ft) t=time(sec) s=initial height(ft)
v=initial velocity (ft/sec) *if thrown up the velocity is +; if thrown down the velocity is - *

Try: A baseball is thrown up from a height of 6ft at 30ft/sec. How long until it hits the ground?
X= -.18 and x= 2.06. We”reject” the -.18 answer because we cannot have negative time in this situation.

Guided prac pg 536 #1-22

Ws 9-5

How-To Graphing Calc

2007-03-26T08:12:00.000-07:00

The upcoming test on Ch 9 (likely this Friday 3/30) will be part graphing calculator.
You can either use my handwritten sheet to practice or the website:
http://www.richland.edu/james/ti82/
Is really good too. You would be interested in the following on that page:

# Finding zeros, roots, solutions, or x-intercepts
# Finding minimums and maximums

The calc we use in class is the TI-83 so be sure to follow the directions for that one

**Remember if you do not have a graphing calc at home, PLEASE STAY AFTER TO PRACTICE WITH ONE I will be here this Tuesday, Weds, AND Thursday!!!!!!!

notes 9.4

2007-03-20T11:41:00.000-07:00

9.4 Solving Quadratic Equations by Graphing
1- Write the equation in the form: ax2+bx+c=0
2- Graph the related function y= ax2+bx+c
3- Sketch the graph. The solutions, or roots are the x- intercepts

try: ½ x2=8

If you solve algebraically you get +- 4.

Solve it by sketching the graph.

½ x2-8=0 graph y=½ x2-8  you will find that your graph crosses the x axis twice at 4 and -4. WOW!

Try:
x2-x=2

x2+4x-12=0

pg 529 #18-32 evens, then check them with the graphing calc

ws 9-4

hw: 3/20

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hw: ws 9-4

notes 9-4

2007-03-19T14:52:00.001-07:00

notes 9-3

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9.3 Graphing Quadratic Functions

quadratic: function written in the STANDARD form:
y=ax2+bx+c where a is not 0
When you graph this function, you get a parabola which is a U-shaped graph
If a your leading coefficient is POSITIVE the graph opens up like a cup
If a is NEGATIVE the graph opens down like a frown.

The vertex is the highest or lowest point on a parabola, depending on if it opens up or down.
The axis of symmetry runs through the vertex and divides the parabola into 2 congruent parts

y= ax2+bx+c
-If a is + the graph is up like a cup
- If a is – the graph is down like a frown
The X- coordinate of the vertex is always: -b/2a
The axis of symmetry is the line x=-b/2a

To graph a quadratic:
1) find the x- coordinate using the formula -b/2a
2) make a table of values (input output) using x values to the left and right of the vertex
3) plot the points and connect with a smooth curve

ex: sketch: y=x2-2x-3 so a=1, b= -2 and c= -3 up like a cup? Or down like a frown?
x- coord of vertex: -b/2a = -(-2) / 2*1 = 2/2 = 1 so pick some points to the left and right

input output: x: -2 -1 0 1 2 3 4
y: 5 0 -3 -4 -3 0 5

you try: y= -2x2-x+2

y= -3x2 y= x2-6x+8 y=2x2-8x+3

pg 521 # 30-42 every 3 : 30,33,36,39,42
pg 522 #44-62 every 3: 44, 47, 50, 53, 56, 59, 62

ws 9-3

notes 9.2

2007-03-19T14:49:00.002-07:00

9.2 Simplifying Radicals
Product property: the sq rt of a product equals the product of the sq rts of the factors:
√ab =√a * √b
√400 = √4 * √100= 2*10 = 20

Quotient prop: the sq rt of a quotient equals the quotient of the sqare roots of the numerator and denominator.
√(a/b) = √a / √b
√(9/25) = √9 / √25 = 3/5

simplest form:
-there are no more perfect square factors other than 1 in the radicand (in the “house”)
- there are no fractions in the radicand
- there are no radicals in the denominator of a fraction

try √50 √20 /4 √(32/50) 1/√4 8√(13/9)

-√4 * √(81/36) √200

try to take the simplest form and turn it back into 1 number under the radicand:
ex. 2√5 would be √4*5= √20
4√2 =√32
4√7
2√6

pg 514 #10-52

ws 9-2

notes 9-1

2007-03-19T14:49:00.001-07:00

9.1 Solving Quadratic Equations by Finding Square Roots
Square root one of 2 equal factors of a number
Factor?  Any numbers being multiplied are factors
4= (4*1) or (2*2)
10=(10*1) or (2*5)
81= (81*1) or (3*27) or (9*9)
If a number has 2 factors that are the same, like 4 and 81, we call them perfect squares
Square roots of non-perfect squares are irrational numbers
So the square root would be one of the factors

Symbol for square root is “√” which is called a radical sign
√64 = 8 √49= 7 √(9/16)= √9/√16= ¾

Must know:

12=1 √1=1
22=4 √4=2
32=9 √9=3
42=16 √16=4
52=25 √25=5
62=36 √36=6
72=49 √49=7
82=64 √64=8
92=81 √81=9
102=100 √100=10
112=121 √121=11
122=144 √144=12

There are also negative square roots
-2 * (-2) is still 4
√36 indicates the POSITIVE, or principal root = 6
-√36 indicates the NEGATIVE root = -6
±√36 indicates BOTH roots = 6 and -6

How about the square root of a negative number? That you cannot do, it falls under the category of imaginary numbers because by definition, we cannot have two negative numbers that are multiplied by themselves to leave us with a negative number, it will always be positive.

Quadratic:is an equation that can be written in the form:
ax2+bx+c = 0
where a is not 0 and is called the leading coefficient

pg 507 #12-14 together
pg 508 # 45-77 odds

ws 9-1

hw: 3/19 GRADED!

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hw: To be completed on a separate sheet to be turned in and graded:
pg 521 # 30-42 every 3
pg 522 #44-62 every 3

hw: 3/15

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hw: ws 9-3

hw: 3/13

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hw: ws 9-2

hw: 3/12

2007-03-19T14:45:00.001-07:00

hw: ws 9-1

hw: 3/7

2007-03-07T12:21:00.002-08:00

hw: finish study guide

hw:3/6

2007-03-07T12:21:00.001-08:00

hw: finish volume and surface area sheet

hw: 3/5

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hw: circumference and area worksheet

notes Geometry unit 3/5

2007-03-05T11:09:00.000-08:00

Circles:- can be named by their center point
Arc- a portion of a circles edge
Major arc: a portion of the circle that is larger than 180 degrees
Minor arc: a portion of the circle that is less than 180 degrees
Semicircle: an arc that measures 180 degrees
Chord: a segment that has its endpoints ON the circle
Diameter: a chord that runs through the center of the circle, divides the circle into two halves
Radius: half the length of the diameter- runs from the center of the circle to the edge
Secant: intersects two or more points on the curve
Tangent: intersects a circle at only 1 point

G

F

A

Z E

B l

m
C D

Circumference of a circle is the distance around the edge- like perimeter but for circles
C=2*pi*r where pi= the constant 3.14 and r= the radius
Area of a circle is found by : A=pi*r2

Circle drawing sheet

hw: 3/2

2007-03-02T14:18:00.001-08:00

hw: "books never written" puzzle on SOHCAHTOA

hw: 3/1

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hw: puzzle worksheet on SOHCAHTOA

notes: Geometry Unit 3/2

2007-03-02T14:16:00.000-08:00

More on SOHCAHTOA

If given the SIDES of a triangle and asked to solve for the ANGLE we use what we call the inverse ratios.
If I have a right triangle and I have to solve for the angle I will still set it up like before: sin=opp/hyp cos=adj/hyp tan=opp/adj

1) Set up the equation
2) Divide the fraction
3) use chart to find original angle, OR use INVERSE on your calc.

notes: Geometry Unit 3/2

2007-03-02T14:13:00.000-08:00

Trigonometric Ratios
Trig ratio is the ratio of 2 sides of a RIGHT triangle
Trigonometry means measurement of triangles (Tri= 3)
3 basic trigonometric ratios:
-Sine of angle A= sinA=

-Cosine of angle A= cosA=

-Tangent of angle A= tanA=

SOHCAHTOA, SOHCAHTOA, SOHCAHTOA
*Use a trig table- available here: http://argyll.epsb.ca/jreed/math9/strand3/tables.htm

Find an angle with only the sides given: Check on calc’s mode should be set on DEGREE

Pg 755-756 #10-22 go over

hw: 2/27

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hw: puzzle sheet on symmetry

notes: Geometry Unit 2/27

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Symmetry
Line symmetry: If a shape or figure can be folded along a line so that one half exactly matches the other half. We say that one side is a reflection of the other.
-Figures can have more than one line of symmetry
Name things in nature that have symmetry (sunflowers, jellyfish, octopus, humans (externally))
Write your name with vertical and horizontal symmetry

Rotational symmetry: If you spin a figure and there are positions where it looks exactly like the original.
Show ex’s- 2 headed arrow, X shape, star
We can specify how many degrees a rotationally symmetric shape has
(2 headed arrow- 180 degree symm. ; X is 90 degree symm; star 72)

Ws 5-4 both sides

hw: 2/29

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hw: ws 6-2

HW: 2/15

2007-02-15T12:29:00.001-08:00

hw: study for TEST ON GEOMETRY UNIT (test #1)

notes: Geometry Unit 2/13

2007-02-13T10:57:00.000-08:00

Midpoint formula: (x1+x2 / 2 , y1+y2 / 2)

RIGHT TRIANGLES ONLY:
Pythagorean Theorem: a2+b2=c2 where a and b are lengths of the legs and c is ALWAYS the length of the hypotenuse

Converse: if a2+b2=c2 is true, then you have a right triangle
If a2+b2 is GREATER than c2 you have yourself an acute triangle
If a2+b2 is LESS than c2 you have yourself an obtuse triangle

A Pythagorean triple is when a, b, and c are all whole numbers

Pg 741 #13-33

Special Right Triangles
: In a 45,45,90 tri, the hypot is √2 times as long as each leg
: in a 30-60-90 the shortest leg is x, the hypot is 2x and the longer leg is √3x

hw: 2/13

2007-02-13T10:56:00.000-08:00

hw: ws "practice right triangles"
valentine's day extra credit assignment also given due Thursday!

hw: 2/12

2007-02-12T12:26:00.001-08:00

hw: ws 5-6 and monster mysteries puzzle

notes Geometry unit 2/9

2007-02-09T08:15:00.000-08:00

Similar triangles: have the same shape but different sizes. The symbol”~” means similar.

If the measures of the 3 angles in 2 different triangles are the same, the two are ~

The corresponding sides of similar triangles form proportions

hw: 2/9

2007-02-09T08:14:00.001-08:00

hw: triangle puzzle sheets both sides

notes: Geometry Unit 2/6

2007-02-06T11:40:00.002-08:00

TRIANGLES: Classification by sides: Equilateral Triangle
An equilateral triangle has 3 sides of equal length and 3 angles that are equal.
Since ALL the angles in a triangle add up to 180º then 180 divided by 3 must be 60º.
Isosceles Triangle An Isosceles triangle has two sides of equal length and 2 angles equal.
Scalene Triangle A scalene triangle is the easiest of them all. The scalene triangle has NO sides of equal length and NO angles the same.
Classification by angles
Acute Triangle Right Triangle Obtuse Triangle
All angles are less than 90 Has a right angle (90) Has an angle more than 90°

°
Triangles can fit into more than one category:
Be as specific as you can: angles of 58, 65, and 57 make an acute scalene triangle
< 130 and 2 congruents sides make and obtuse isolsceles
Each “corner” of the triangle is called a vertex (pl. vertices)
Two sides sharing a common vertex are called adjacent sides

Right : Sides that form the rt angle are legs, the opposite side, or the longest side of the triangle is the hypotenuse
and Isosceles: if only 2 sides are congruent, they are called legs, the other side is the base

Triangle Sum Theorem: The sum of the measures of the interior angles of a triangle= 180
Exterior angle Theorem: The m of an ext. angle of a triangle is = to the sum of the m of the 2 nonadjacent interior angles

WS 4-1 FROM GEO BOOK

hw: 2/5

2007-02-06T11:40:00.001-08:00

hw: study for quiz on Geometry unit

hw: 2/1

2007-02-01T11:39:00.001-08:00

hw: ws 3-1

notes: Geometry Unit 2/1

2007-02-01T11:36:00.000-08:00

Lines and Angles
parallel lines are coplanar and do not intersect
skew lines do not intersect and are not coplanar
parallel planes do not intersect

transversal- line that intersects 2 or more coplanar lines
The angles a transversal creates have special names:
Corresponding angles: occupy corresponding positions
Alternate Exterior: lie outside the 2 lines on opposite sides
Alternate Interior: ;lie between the 2 lines on opposite sides
Consecutive Interior: between the 2 lines and on the same side (aka same side interior)
Consecutive Exterior: lie outside the 2 lines but on the same side of the transversal

hw: 1/31

2007-01-31T08:30:00.000-08:00

hw: ws 1-6

notes: Geometry unit 1/31

2007-01-31T08:29:00.000-08:00

ANGLE RELATIONSHIPS:
vertical angles- have sides that form 2 pairs of opposite rays (opposite angles will be congruent)
linear pairs- adjacent angles whose non-common sides are opposite rays (the measures of linear pairs must add up to 180
complementary angles have measurements that add up to 90 degrees (can be adjacent or not)
supplementary angles: have measurements that add up to 180 degrees (can be adjacent or not)
adjacent angles: Two angles in a plane which share a common vertex and a common side but do not overlap.

pairs of angles //// ws “practice” 1-7

ws 1-6

notes: Geometry unit 1/30

2007-01-30T08:29:00.001-08:00

Segment Addition Postulate:
If B is between C and A, then AB+BC=AC
If AB+ BC=AC, then B is between A and C

Angle addition postulate pg 27:
If P is in the interior of ∟RST (meaning it is between points that lie on each side of the triangle) then m∟RSP + m∟PST= m∟RST

Distance Formula- used for computing the distance between two points in a coordinate plane
Length of AB= which is just rearranging the Pythagorean theorem (a2+b2=c2)
Plot the points (1,4) and (10, 8) and find the distance using the formula
Try (-9,14) and (5, -10)
Segments that have the same length are called congruent

notes: Geometry unit 1/29

2007-01-30T08:26:00.002-08:00

• POINT - any given spot, points are named with capitol letters; Point A
• LINE - a straight line that never ends, named with lower case Script letters
• PLANE - a flat surface that reaches out forever, named by any three points that are on that same plane; plane ABC
• CONGRUENT - two lines or geometric figures that are exactly the same
• LINE SEGMENT - an area of a line defined by 2 points on that line, they are named for those 2 points; segment AB
• MIDPOINT - a point that is in the exact middle of a line segment, this point divides the segment into equal halves
• ENDPOINT - a point on the end of a line segment or ray
• RAY - a line extending from one endpoint and continues forever, named by listing the endpoint and then another point on the ray; ray AB
• OPPOSITE RAY- two rays that share a common endpoint and that form a line
• ANGLE - an angle is made by two rays extending from the same endpoint, named by listing a point on one ray, the endpoint, and the a point on the other ray; angle ABC
• DEGREE - type of measurement used to show amount of opening between the two rays of an angle
• ACUTE ANGLE - angle measures less than 90 degrees
• RIGHT ANGLE - an angle measuring exactly 90 degrees
• OBTUSE ANGLE - an angle measuring greater than 90 degrees
• BISECT - when an line or point divides a geometric figure into exact halves.
• COLLINEAR- two or more points that are on the same line
• COPLANAR- two or more points that are on the same plane

hw: 1/20

2007-01-30T08:26:00.001-08:00

hw: Distance Formula worksheet

hw: 1/29

2007-01-30T08:25:00.001-08:00

hw: ws 1-2

hw: 1/24

2007-01-24T13:16:00.001-08:00

hw: both puzzle sheets

hw: 1/23

2007-01-23T11:52:00.000-08:00

finish ws for bingo and on a seperate sheet of paper do your test corrections!
Keep studying for your "retest" on FRIDAY!

hw: 1/22

2007-01-22T12:31:00.001-08:00

hw: finish exponent packet

notes 8-5

2007-01-11T11:35:00.000-08:00

8.5 Exponential Growth functions
Exponential Growth: when a quantity increases by the same percent, r, in each time period, t.
y= C(1+r)t
where C= initial amount
(1+r) = the growth factor
r= the growth rate
t=time

100r= percent of increase

try: deposit $500 in an account that pays 8% annual interest compounded yearly. What is the account balance after 6 years?
y= C(1+r)t
y=500 ( 1+ .08)6

try: channel catfish has a mass of .3g at birth and grows 10% each day for 6 weeks. Mass after 10 days? 20? 6 weeks?

Bunny sheet

Pg 480-481 #6-26 skip 23
Ws 8.5

hw: 1/11

2007-01-11T11:32:00.001-08:00

hw: ws 8-5

Exponents Notes

2007-01-10T12:02:00.000-08:00

The thing about exponents…

If you have a number like 5³, we call the 3 an exponent (or whatever number is written in superscript like that) and the 5 (or the large regular number) the base.

When you have an exponent with a base it means to multiply the BASE by ITSELF as many times as the EXPONENT says.

Ex: 5³really means 5·5·5 7² means 7·7

When you have two numbers being MULTIPLIED by each other that have THE SAME base, you ADD the exponents. Watch what happens if we do it out longways:

Ex. 3⁴ ·3² this is really: 3·3·3·3 · 3·3 which is 3⁶

We get the same thing if we follow the rule and ADD the exponents:

3⁴ ·3²= 3 ⁴⁺²=3⁶

When we have what we call a “power to a power” we have to MULTIPLY the exponents. Watch here in longhand:

Ex: (6³)⁵

Something to the 5^th power is telling me to multiply whatever it is by itself 5 times like so:

(6³)· (6³)· (6³)· (6³)· (6³)

Now if I expand what is really inside each parenthesis, I get something that looks like:

(6·6·6)· (6·6·6)· (6·6·6)· (6·6·6)· (6·6·6)

Condense all this and I end up with 6¹⁵

Much easier if I remember the rule: (6³)⁵= 6^3·5=6¹⁵

***Any number to the power of zero is ONE except zero to the power of zero is UNDEFINED***

Now things get tricky…

A NEGATIVE exponent means you must do the following:

1) Take the RECIPROCAL of the base

2) Raise the numerator and denominator to the positive power given

Ex: 5^-3 reciprocal of base= raise both to the power of 3= =

Ex: reciprocal of base= raise both to the power of 9= =

Now, when you have to find the power of a negative number, it is CRUCIAL how it is written

(-3)² and -3² are very different.

The first one tell us to do: (-3)·(-3) = 9

The second one, however: -3²if you remember your PEMDAS order of operations, you must evaluate the EXPONENT before you multiply (by -1 in this case). -3²= - 3·3 = -9

Two VERY different answers. Make sure you stay alert!!

DIVIDING powers you should SUBTRACT exponents; again in longform:

Ex: = cross out what you can and you are left with n·n or n²

Again the rule is much faster: = n^5-3 = n²

Some basic math concepts that you NEED to remember to make this all go smoothly:

-A negative PLUS a negative is still a negative : -5 + -9 = -14

-When you SUBTRACT a negative, you ADD: 4 – (-8) = 4+(+8) = 4+8 = 12

- When you have a whole number times a fraction, multiply the whole number by the numerator:

4· = = or y·=

-When you divide by a fraction you are really MULTIPLYING THE NUMERATOR BY THE RECIPROCAL OF THE DENOMINATOR:

= which as you just learned is :

Ok lets practice: Please do this on another piece of paper!

hw: 1/10

2007-01-10T11:58:00.000-08:00

hw: ws practice exam front & back

hw: 1/8

2007-01-08T12:43:00.002-08:00

hw: pg 497 #1-28 QUIZ TOMORROW ON 8.1-8.4

notes 8-4

2007-01-08T12:43:00.001-08:00

8.4 Scientific Notation
To write a number in scientific notation:
Put the decimal after the first digit and drop the zeroes.
In the number 123,000,000,000 The coefficient will be 1.23
To find the exponent count the number of places from the decimal to the end of the number.
In 123,000,000,000 there are 11 places. Therefore we write 123,000,000,000 as:
Try: 2,300,000 54,300,000 1200 453,700,000,000

For small numbers we use a similar approach. Numbers less than 1 will have a negative exponent. A millionth of a second is:
0.000001 sec. or
Put the decimal after the first number, this is now your coefficient.
Write “x 10”
Count how many spaces there are from your new decimal places.
Write that number as a negative exponent

Try: 0.00000000234 0.0033 0.000987 0.0000002

To change a # from sci not. Back into standard form:
-Look at the exponent. Move the decimal that many times putting in 0s where needed
*If the exponent is +, move the decimal to the RIGHT to get a BIG number
* If the exponent is -, move the decimal to the LEFT to get a SMALL number

Try: multiplying 2 #’s or a sci. notation raised to a power, or dividing

Practice ws 8-5

hw: 1/4

2007-01-04T11:24:00.002-08:00

hw: ws 8-3

notes 8.3

2007-01-04T11:24:00.001-08:00

8.3: Division Property of Exponents
To divide two powers with the same base, SUBTRACT the exponents
3^7/3^5 = 3^7-5= 3^2
If a fraction has an exponent you must apply it to the numerator AND denominator.
(2/3)^2 = 2^2/3^2

guided prac pg 466

pg 466 #19-48, go over, put on board

ws 8-3

hw: 1/3

2007-01-03T14:22:00.000-08:00

hw: ws titled "Practice"

notes 8-2

2007-01-03T14:20:00.000-08:00

8.2 Zero and Negative Exponents
Any number to the power of zero is 1. Zero to the zero is undefinded

Any number with a negative exponent is the reciprocal a-n= 1/an

2-3 5x-2 5(2-x) 2x-2y-3

Exponential Function: a function in the form y=abx (y=amount after x periods, a= initial value, and b= growth factor)
I put $5 into an account this year. If my money doubles every year, how much will I have in 20 years? Y= 5*2x sketch graph

“practice 8.4” ws

pg 459-460 #14-48, 53-57

ws 8-2

notes 8-1

2007-01-02T14:13:00.001-08:00

***********Exponents may not properly show see me for copy**********
Review: 32=9 (-3)2= 9 -32= -9
Ch 8.1- Multiplication Properties of Exponents:
In the example 53, 5 is called the base and 3 is called the exponent
-To multiply 2 #’s with the same base, ADD the exponents
43 * 42 is really (4*4*4)*(4*4) or 45

try: c*c*c 22*24 x4*x5

-Power of a power property: to find a power of a power, multiply the exponents
(33)4 is 33*4 or 312

try: (53)3 (-24)3 = -212

-Power of a product:
(2*7)3 is 23*73

try: (2m2)3 (ab2)2 (5x)2 (-2x3)3= (-2)3x9

pg 453 #22-68
go over
ws 8-1

hw: 1/2

2007-01-02T14:12:00.001-08:00

hw: ws 8-1

WELCOME BACK!!!!!!!!!!

hw: 12/20

2006-12-20T11:35:00.000-08:00

hw: review packet (both sheets)

hw: 12/19

2006-12-20T04:54:00.001-08:00

hw: ch 4-6 tests in packet

hw: 12/18

2006-12-20T04:53:00.000-08:00

hw: ch 1-3 tests done in packet

Notice: Midterm

2006-12-20T04:51:00.000-08:00

The week of 12/19-12/21 we will be doing review for the Algebra 1 midterm to occur on Friday, 12/22. Review session will be held today, Wednesday 12/20 and possibly Thursday 12/21.

hw: 12/15

2006-12-15T06:04:00.001-08:00

hw: None!

hw: 12/14

2006-12-15T06:03:00.002-08:00

hw: extra credit: solve 4x4 puzzle STUDY FOR TEST ON CH 7 TOMORROW!

hw: 12/13

2006-12-15T06:03:00.001-08:00

hw: pg 440-442 #1-32

hw: 12/12

2006-12-12T12:35:00.000-08:00

hw: ws 7-6 and Take home quiz

notes 7-6

2006-12-12T12:34:00.004-08:00

FF#29

7.6 Solving Systmes of Linear Inequalities
When graphing sys of ineq, shade like normal. The areas that are shaded for all ineq are the solution areas.
Ex: y<2, x>= -1, y> x-2
Select a point inside that shaded region to check

Ex 2 pg 433 ex 3

6-8 pg 435 on board

pg 436 34 and 35

ws 7-6

practice quiz

hw: 12/11

2006-12-12T12:34:00.003-08:00

hw: pg 429 12-23

notes 7-5

2006-12-12T12:34:00.001-08:00

7.5 Special types of Linear Systems

solve: 2x + y = 5 and 2x + y = 1 (sub)

no solution? Weird!

Parallel lines have the same slope and have no solution as they never intersect

Solve: -2x + y = 3 -4x + 2y = 6 (line comb)

Lines that coincide (lie on top of each other) have infinite many solutions, so you will get BOTH variables eliminated and get 2 values = to each other, regardless of what x and y are.

Try- in nbs and at the board
3x+y= -1 -9x – 3y =3

x-2y=5 -2x+4y=2

2x +y = 4 4x -2y = 0

probs 6-11 pg 429 on the board together

pg 429 #12-23

ws 7-5
quiz-for practice

notes 7-4

2006-12-12T12:33:00.001-08:00

7.4 Applications of Linear Systems
When to use which method:
-Graphing: When approximating, checking, or for visual solution
-Substitution: Useful when 1 variable has a coefficient of 1 or -1
-Linear Combination: when NO variable has coeff. Of 1 or -1

ex 1 pg 418
ex 3 pg 420

guided prac pg 421

pg 422 #31-51

hw: 12/8

2006-12-12T12:32:00.001-08:00

hw: ws 7-4

hw: 12/7

2006-12-07T12:22:00.000-08:00

hw: ws that says "extra practice 31"

notes 7-3

2006-12-06T11:28:00.002-08:00

7.3 Solving Linear Systems by Linear Combinations
1) Arragne the equations with like terms in columns
2) Multiply 1 or both of the equations by a number to get opposite coefficients for one of the variables (so they will cancel out when you add them)
3) Add the equations by combining like terms (one of your variables should be eliminated)
- Solve for the remaining variable
4) Substitute that answer into one of the originals to find what the other variable is.
5) CHECK in each original equation

try: 3x + 5y = 6 -4x +2y= 5

4x + 3y = 16 2x- 3y= 8

3x + 2y= 8 2y= 12-5x

hw: 12/6

2006-12-06T11:28:00.001-08:00

hw:ws 7-3

notes 7-2

2006-12-05T11:59:00.000-08:00

7.2 Solving Linear Systems by Substitution
1) Solve one of the equations for one of the variables
2) Substitute your answer in for the variable in the next equation
3) Solve for the only variable left and get a numerical answer
4) substitute the answer in to find the other variable to make an ordered pair (x,y)
5) CHECK!

Ex: -x + y = 1 2x+y= -2

2x + 2y = 3 x- 4y = -1

guided prac pg 408

You are selling 2 types of cookies Choc Chip for $4 a box and PB for $6. You sell 525 boxes total and collected $2876 How many of each type did you sell????????

Pg 408 #17-34 and #44

Ws 7.2

hw: 12/5

2006-12-05T11:58:00.000-08:00

hw: ws 7-2

notes 7-1

2006-12-04T11:13:00.000-08:00

7.1- Solving Linear Systems by Graphing
If you have 2 equations, the solution of the system is the coordinate where the two lines meet
-To check this point, put in the coordinates (x, y) of the intersection point into the equations

Graph and Check Method:
1) Write each equation in y=mx+b
2) Graph them
3) Estimate the coordinate where they intersect
4) Check the coordinate in BOTH equations

Ex: x + 2y= 5 and 2x-3y=3
X + y = -2 and 2x – 3y =-9
-3x +y = -12 and y= -2x +3
3x + y = 11 and x- 2y =6

ex 3 pg 400

snow shoveling problem

pg 402 20-36 need graph paper

hw: 12/4

2006-12-04T11:12:00.000-08:00

hw: pg 402 #20-36

hw: 11/30

2006-11-30T11:37:00.000-08:00

hw: pg 384-385 #1-25

aaaaaaaaaaaaand

STUDY FOR YOUR TEST TOMORROW!

2006-11-29T11:29:00.000-08:00

Box and Whisker Plots

A box and whisker plot is easy to make if you know how to find the median of a set of numbers. This type of plot shows specific information at a glance.

1) The first step is to find the median of a set of numbers

Example: * Remember to list your numbers in order!!!!

18 27 34 52 54 59 61 68 78 82 85 87 91 93 100

Find the median of this set. This number represents the second quartile.

If you got 68, then you are right!

2) Now look at the numbers to the left of the median which are:

18 27 34 52 54 59 61

Now find the median of this set! You should get the number 52. This number represents the first quartile.

3) Go back to your original number set and look at the numbers to the right of the median. Now, find the median of this number set.

78 82 85 87 91 93 100

You should get the number 87 which represents the third quartile.

4) Finally, you put all the information you found on to a number line to create your box and whisker plot!

To make the boxes:

a. Make a dot above the number representing your original median (SECOND QUARTILE)

b. Make a dot above your first quartile number

c. Make a box through these two points

d. Make a dot at your third quartile number

e. Make another box through your median and now through the third quartile number

To make the whiskers:

f. Make a dot at your lowest number in your original number set

g. Draw a line from your first quartile number to the lowest number

h. Make a dot at your highest number in your original number set

i. Draw a line from your third quartile number to your highest number

Given the number set:

45, 44, 52, 53, 45, 47, 49, 37, 39, 40, 45, 47, 52

Rewrite the list in order:

Find the median:

Copy the numbers that are to the left of your median after you put the numbers in order:

Find the median of this number set. This is the FIRST QUARTILE:

Copy the numbers that are to the right of your median after you put the numbers in order :

Find the median of this number set. This is the THIRD QUARTILE:

Make your box and whisker plot using the instructions on the other side:

Make a box and whisker plot using the following number set:

1, 1, 2, 2, 2, 5, 7, 7, 8, 9, 10, 10, 11, 12, 12, 12

2006-11-29T11:28:00.000-08:00

Stem and Leaf Plots

A stem and leaf plot is a way to display data. There are two parts to the plot a “stem” and a “leaf”

To construct a stem and leaf plot:

Put your data set in order from least to greatest.

Math test scores out of 50 points: 35, 36, 38, 40, 42, 42, 44, 45, 45, 47, 48, 49, 50, 50, 50

Now make a table with 2 columns labeled “stem” and “leaf”.

3) Separate each number into a stem and a leaf. Since these are two digit numbers, the tens digit is the stem and the units digit is the leaf. List the stems in the stem column and the leaves in the leaf column. If you have two numbers with the same stem, just add another leaf (see example)

Math Test Scores (out of 50 pts)
Stem	Leaf
3	5 6 8
4	0 2 2 4 5 5 7 8 9
5	0 0 0

******* If you had a test score of 8 for example, your stem would be “0” and your leaf would be 8 since this is only a one-digit number.********

Now you try! Make a stem and leaf plot of the following data sets:

1) 12, 13, 21, 27, 33, 34, 35, 37, 40, 40, 41

2) 15, 19, 14, 12, 22, 23, 24, 15, 24, 22, 19, 22, 21, 22

notes 6-6

2006-11-29T11:24:00.000-08:00

6.6 Stem and Leaf Plots, Mean, Median, Mode
Measures of Central Tendency
-Mean: finding the average: Take all the data, add it up, divide by the # of pieces of data you had
ex: 3 people got tests back with scores of: 91, 79, and 85. To find the mean, add all the scores up and divide by 3.
Find the mean number of hours the class sleeps each night
-Median: Write the data in order from greatest to least. The MIDDLE number is the median. If there are TWO middle numbers, find the average, or MEAN of the #s.
Do with class sleep data
-Mode: The mode is the number that occurs the most frequently. You can have more than one mode if 2 #s tie for having the highest frequency. You can have no modes if all the numbers have the same frequency.