Algebra Period 3

Coordinate Plane
The coordinate system is a way of locating points in a plane in relation to two perpendicular number lines. These lines intersect at the origin (0, 0) and divide the plane into four quadrants. Points are designed by ordered pairs (x, y) which indicate their location on the coordinate plane. The horizontal axis line is called the x-axis-- the first number in the ordered pair, the x-coordinate tells how far to the right (if positive) or to the left (if negative) the point (x, y) is located. The vertical axis is called the y-axis-- the second number, the y- coordinate-- tells how far up-- if it is positive or down (if it is negative) the point (x, y) is located.
In the first quadrant both the x- coordinate and the y-coordinate are positive. In the second quadrant, the x is negative and the y is positive; in the third quadrant BOTH are negative; and in the fourth, the x is positive and the y is negative.

Now to make sure you have the correct answers:

A = 6
B = 8
C = -11
D = -4
E = 10 (2 + 10 - 2)
F = 3 (1 is neither prime nor composite!)
G = -14
H = -13
J = 11




Remember: that just as x comes before y in the alphabet AND h (for horizontal) comes before v (vertical) -- first move the x-distance in the horizontal direction and then the y-distance in the vertical direction... HAVE FUN and make your graph PIZZAZZY!!

Pre Algebra Period 2

Coordinate Plane
The coordinate system is a way of locating points in a plane in relation to two perpendicular number lines. These lines intersect at the origin (0, 0) and divide the plane into four quadrants. Points are designed by ordered pairs (x, y) which indicate their location on the coordinate plane. The horizontal axis line is called the x-axis-- the first number in the ordered pair, the x-coordinate tells how far to the right ( if positive) or to the left ( if negative) the point (x, y) is located. The vertical axis is called the y-axis-- the second number, the y- coordinate-- tells how far up-- if it is positive or down (if it is negative) the point (x, y) is located.
In the first quadrant both the x- coordinate and the y-coordinate are positive. In the second quadrant, the x is negative and the y is positive; in the third quadrant BOTH are negative; and in the fourth, the x is positive and the y is negative.



Now to make sure you have the correct answers to the questions about mean:
A = 6
B = 2
C = 5
D = -9
E = 4
F = 7
G = -1
H = 0
J = 8
K = 10
M = -3
N = 1
P = -8

Remember: that just as x comes before y in the alphabet AND h (for horizontal) comes before v (vertical) -- first move the x-distance in the horizontal direction and then the y-distance in the vertical direction... HAVE FUN and make your graph PIZZAZZY!!

Math 6 Honors Periods 1, 6 & 7

Coordinate Plane
The coordinate system is a way of locating points in a plane in relation to two perpendicular number lines. These lines intersect at the origin (0, 0) and divide the plane into four quadrants. Points are designed by ordered pairs (x, y) which indicate their location on the coordinate plane. The horizontal axis line is called the x-axis-- the first number in the ordered pair, the x-coordinate thells how far to the right ( if positive) or to the left ( if negative) the point ( x, y) is located. The vertical axis is called the y-axis-- the second number, the y- coordinate-- tells how far up-- if it is positive or down ( if it is negative) the point ( x, y) is located.
In the first quadrant both the x- coordinate and the y-coordinate are positive. This is the quadrant we are graphing for our first assignment. All ordered pairs are positive.

Remember: that just as x comes before y in the alphabet AND h ( for horizontal) comes before v ( vertical) -- first move the x-distance in the horizontal direction and then the y-distance in the vertical direction... HAVE FUN and make your graph PIZZAZZY!!

Math 6 Honors Periods 1, 6 & 7

Our Class Blog
Welcome to our class blog... where you can earn extra credit by adding your own relevant comments about our class notes for the day.. or where you can find answers from others in your class. Check here often, especially if you have been absent. You might just find out the math strategy that works for you!!

Email me if you are interested in adding notes and/or comments to this blog-- for extra credit!!

Pre Algebra Period 2

Our Class Blog
Welcome to our class blog... where you can earn extra credit by adding your own relevant comments about our class notes for the day.. or where you can find answers from others in your class. Check here often, especially if you have been absent. You might just find out the math strategy that works for you!!

Email me if you are interested in adding notes and/or comments to this blog-- for extra credit!!

Algebra Period 3

Our Class Blog
Welcome to our class blog... where you can earn extra credit by adding your own relevant comments about our class notes for the day.. or where you can find answers from others in your class. Check here often, especially if you have been absent. You might just find out the math strategy that works for you!!

Email me if you are interested in adding notes and/or comments to this blog-- for extra credit!!

Math 6 Honors Periods 6 & 7 (Wednesday)

Volumes of Cylinders 10-7

A cylinder is a space figure that has two circular bases and one curved surface. The perpendicular distance between the bases is the height (h) of the cylinder.
If the base radius is r, then the base area, B is

πr²

Volume of cylinder = Base area X height
V = BH
V = πr² h
Find the volume of a cylinder having a base radius of 6 cm and a height of 8 cm.
B = πr² so B = π6²

36π

≈

36 X 3.14 113 ( rounded to three digits as our book requests)

V = Bh

113 X 8 = 904 904 cubic cm.

The volume of a cylinder or box is often called its capacity. For containers of liquids, capacity is usually measured in liters (L) or milliliters (mL)

Note: 1 L = 1000 cm³ and 1 mL = 1 cm³.


Note: Certain mathematics may have different meanings than they have in ordinary usage. For example, in everyday language, base refers to the bottom of an object and height refers to how tall an object is. In mathematical usage these terms have special meanings,

Math 6 Honors Periods 6 & 7 (Tuesday )

Polyhedrons 10-5

A polyhedron is a figure formed of polygonal parts of planes that enclose a region of space.

A prism is a polyhedron that has two congruent regions called bases that are parallel. Prisms are named according to their bases. Thus, a box is a rectangular prism. Take a look at Page 339 for some more examples of prisms.

A pyramid has only one base and a vertex. It is also named by the shape of its base. A triangular pyramid is also called a tetrahedron.
A regular polyhedron has all of its faces bounded by congruent regular polygons. There are only five such polyhedrons having 4, 6, 8, 12, and 20 faces.
Regular tetrahedron, cube, regular octahedron, regular dodecahedron, and regular icosahedron.

Check out Page 340 in our textbook for a picture of these.



Volumes Of Prisms 10-6

A polyhedron together with the region inside it is called a solid. The measure of the space occupied by a solid is called the colume ofthe solid.
Volume of prism = Base area X height
V = Bh where B = the Base area or the area of the base

Math 6 Honors Periods 6 & 7

Area of Circles 10-3

Recall that there are two formulas for the circumference C of a circle. If the diameter of the circle is denoted by d and the radius by r, then

C =πd and C = 2πr

Two approximations for the number π are 3.14 and

The part of the plane enclosed by a circle is called the area of the circle.

Formula

Area of circle = π · (radius)²

A = πr²

Find the area of the shaded regions. Use π ≈ 3.14

Recall that we give answers to only three digits when we use the approximation π≈ 3.14. In fact, sometimes to avoid approximation we give the answer in terms of π.

Find the area of the shaded region. Leave your answer in terms of π

(Look at your textbook for examples)

Area of shaded region = Area of large circle – Area of small circle.

We must first find the area of the small circle.

A = πr² = π2² = 4π

We then find the area of the large circle. Since the radius of the large circle is the same as the diameter of the small circle we know that the radius of the circle must be 4 m so A = πr² = π4²= 16π

Thus, the area of the shaded region is equal to 16π - 4π = 12π

Math 6 Honors Periods 6 & 7 (Wednesday)

Areas of Triangles and Trapezoids 10-2

Any side of a triangle can be considered to be the base. The height is then the perpendicular distance from the opposite vertex to the base line.

Let us find the area of a triangle having base b and height h. The triangle and a congruent copy of it can be put together to form a parallelogram

Since the area of the parallelogram is bh—from yesterday’s lesson--the area of the triangle is half the area of the parallelogram so we have the following

Formula

Area of triangle = ½ · base · height

A = ½ bh

Find the area of each triangle

Look at the pictures in your textbook Page 325 and practice a few

Note that in the second example, the lengths of the sides of the right angle of the triangle were used as the base and the height. this can be done for any right triangle.

Trapezoids

The height of a trapezoid is the perpendicular distance between the parallel sides. These parallel sides are called the bases of the trapezoid. The method used to find the formula for the area of a triangle can be used to find the formula for the area of a trapezoid having bases b₁ and b₂ and height h. The trapezoid and a congruent copy of it can be put together to form a parallelogram. The area of the parallelogram is (b₁ + b₂)h and the area of the original trapezoid is half the area of the parallelogram.

Formula

Area of trapezoid = ½ · (sum of bases) · height

A = ½ (b₁ +b₂)h

Algebra Period 3 (Tuesday)

Quadratic Equations Review: Chapter 13


Finding the x intercepts of a parabola:
x intercepts = roots = solutions = zeros of a quadratic
You can find these one of two ways:
1) Read them from the graph

2) Set y or f(x) = 0 and then solve

Where the graph crosses the x axis is/are the x intercepts.

(Remember, y = 0 here!)
The x intercepts are the two solutions or roots of the quadratic.
When we factored in Chapter 6 and set each piece equal to zero,
We were finding the x value when y was zero.
That means we were finding these two roots!

You can find these roots (solutions, x intercepts, zeros) by several different methods.
You already know the following 2 methods:

1) Graphing and see where the graph crosses the x axis
2) Factoring, then using the zero products property to solve for x for each factor

ANOTHER METHOD:
Chapter 13-2 Solving Quadratics Using Square Roots -
You did this in Ch 11 for Pythagorean Theorem!
If there is no middle x term, it's easiest to just square root both sides to solve!

BUT THE DIFFERENCE FROM PYTHAGOREAN--NOT LOOKING FOR JUST THE PRINCIPAL SQUARE ROOT ANY MORE.

YOU NEED THE + OR - SYMBOL!!

EXAMPLE:
3x² = 18
divide both sides by 3 and get: x² = 6
square root each side and get x = + or - SQRT of 6

HARDER EXAMPLE:
(x - 5)² = 9
SQRT each side and get: x - 5 = + or - 3
+ 5 to both sides: x = 5 + 3 or x = 5 - 3
So, the 2 roots are x = 8 or x = 2

HARDEST EXAMPLE:
(x + 2)² = 7
SQRT each side and get: x + 2 = + or - SQRT of 7
-2 to both sides: x = -2+ SQRT 7 or x = -2 - SQRT 7

THE QUADRATIC FORMULA 13-4

-b plus or minus the square root of b squared minus 4ac all over 2a

Notice how the first part is the x value of the vertex -b/2a
The plus or minus square root of b squared minus 4ac represents
how far away the two x intercepts (or roots) are from the vertex!!!!

Very few real world quadratics can be solved by factoring or square rooting each side.
And completing the square always works, but it long and cumbersome!

All quadratics can be solved by using the QUADRATIC FORMULA.

(you will find out that some quadratics have NO REAL solutions, which means that there are no x intercepts - the parabola does not cross the x axis! Think about what kinds of parabolas would do this....ones that are smiles that have a vertex above the x or ones that are frowns that have a vertex below the x axis. You will find out in Algebra II that these parabolas have IMAGINARY roots)

So now you know 5 ways that you know to find the roots:

1. graph
2. factor if possible
3. square root each side
4. complete the square - that's what the quadratic formula is based on!
5. plug and chug in the Quadratic Formula -
This method always works if there's a REAL solution!

DON'T FORGET TO PUT THE QUADRATIC IN STANDARD FORM BEFORE PLUGGING THE VALUES INTO THE QUADRATIC FORMULA!

ax² + bx + c = 0

DISCRIMINANTS - a part of the Quadratic Formula that helps you to understand the graph of the parabola even before you graph it!
the discriminant is b² - 4ac

(the radicand in the Quadratic Formula, but without the SQRT)

Depending on the value of the radicand, you will know
HOW MANY REAL ROOTS IT HAS

1) Some quadratics have 2 real roots (x intercepts or solutions) - Graph crosses x axis twice
2) Some have 1 real root (x intercept or solution) - Vertex is sitting on the x axis
3) Some have NO real roots (no x intercepts or solutions) - vertex either is above the x axis and is a smiley face (a coefficient is positive) or
the vertex is below the x axis and is a frown face (a coefficient is negative)

In both of these cases, the parabola will NEVER CROSS (intercept) the x axis!

b² -4ac > 0 That is, if it's positive, then there are 2 roots
b² -4ac = 0 That is, if it's zero , then there is 1 root
b² -4ac <0>

Algebra Period 3 (Tuesday)

Review of how to graph a quadratic

Find the vertex ( x = -b/2a, then plug in to the equation to find the y coordinate)

Find the axis (line) of symmetry. Just use the x value of the vertex that is
x = -b/2a

Draw the line of symmetry using a dotted or dashed line

Find 2 more points in an x y table (go either left or right of the vertex and use 2 points close to the line of symmetry for the x, then plug into the equation to find the y values)

Find the 2 shadow (mirror) points by counting over from the axis at the exact same y value

Draw a U (not a V) through the points. Extend it past and put arrows at the end
Remember to label the x and y axes and put arrows on the axes.

Function vs. Relations: Functions are special relations where there is a unique x for each y
Therefore, there will never be 2 x’s that will repeat and if you use a vertical line on the graph, it will only hit one point on the graph.

Domain vs. Range: the domain for any quadratic function is ALL REAL NUMBERS
The range depends on where the vertex is and whether the quadratic is a smile or a frown.
Generally, it will be of the form:

y such that y is either greater than or equal to (≥ ) or less than or equal to (≤) the y value of the vertex. {y l y≥ n} or {y l y ≤ n}

f(x) is just a different way of saying y. What is better about it? Without seeing the work before the solution, you can actually tell the x value as well as the y value in the solution.

For example, if the solution is f(-3) = 12 you know that the point is (-3, 12)
Compare that to the solution y = 12. For that solution, you would not know the x value unless you look back in the problem.

Math 6 Honors Periods 6 & 7 (Tuesday )

Areas of Rectangles and Parallelograms 10-1

In Chapter 4 (so long ago) we measured lengths of segments and found perimeters of various polygons. Now we will measure the part of the plane enclosed by a polygon. We call this measure the area of the polygon.

In the metric system the unit area often use is the square centimeter. cm²

Notice that the area of each rectangle is the product of the lengths of the two consecutive sides. These sides are called the length and the width of the rectangle The length names the longer side and the width names the shorter side. These sides are also named the base and height.

Formula

Area of rectangle = length · width

A= lw

The length and the width of the rectangle are called its dimensions

In the case of a parallelogram, we may consider either pair of parallel sides to the bases. (the word base is also used to denote the length of the base) the height is the perpendicular distance between the bases.

Formula

Area of parallelogram = base · height

A= bh

The unit areas used are square meters (m²), square millimeters (mm²) and square centimeters (cm²). For very large regions, such as the State of California, or entire countries (Canada), we could use square kilometers (km²)

Sometimes we work with an unspecified unit of length. Then the unit of area is simply called a square unit. It is vital, however, that you indicate the area is in square units.

Algebra Period 3

Compound Sentences 9-2

Conjunction = and - the graph is an intersection ("yo") and inequality looks like our domains and ranges on our projects EXAMPLE: 5 < x < 10
open dots; between 5 and 10 is colored in

Disjunction = or - graph will go opposite ways ("dorky" dancer) and inequality looks like this:
x < -2 OR x > 4
open dots; one arrow goes right at 4 and the other arrow goes left at -2

Equations and Absolute Value - 1 VARIABLE 9-3
Solve the equation twice - Once with the solution positive and once with it negative in this setting [ ] represents absolute value.
[2x - 4] = 10
Solve it twice:
2x - 4 = 10 or 2x - 4 = -10
x = 7 or x = -3

If there is a term on the same side of the equation as the absolute value, move that to the other side of the equation first (just like we did with radical equations!) Then solve twice.

REMEMBER THAT THE SOLUTION GIVEN CANNOT BE NEGATIVE (the null set)

Inequalities and Absolute Value - 1 VARIABLE 9-4
There are 2 possible types of inequalities - less than and greater than
For less thAND:
These are conjunctions and so you solve it twice and the solution ends us between them
[3x] < 15 is equal to -15<3x<15
so x is greater than -5 and less than 5

For greatOR than:
These are disjunctions and are solved twice with the solution infinitely in different directions
[3x] >15 is equal to 3x < -15 and 3x > 15

INEQUALITIES IN 2 VARIABLES 9-5
You will shade an x y graph to find the side of a linear equation that fits the solution
EXAMPLE: x + y > 5
You graph the line with DOTTED line because it cannot be equal to 5
You pick an easy point on one side of the line and substitute to see if that side is a solution.
If that does not work, pick an easy point on the other side to see if that side checks.
I USUALLY USE (0,0) as my first point!
If there is more than 1 Inequality (a system of inequalities), follow the same procedure and where the 2 shadings overlap each other is called the solution of the system of inequalities.

SYSTEMS OF INEQUALITIES IN TWO VARIABLES 9-6
If there is more than 1 Inequality (a system of inequalities),
1. Follow the same procedure as above for one equation
but you will need to do it for each equation.
2. Use a different type of shading for each so you won't get confused
(Ex: Use slanted lines one way and then slanted lines the other way. Use different colors if possible. Make one set of lines wavy and the other set straight)
3. Where the 2 shadings overlap each other is called the solution of the system of inequalities.
(any point in the overlap should work in BOTH inequalities - make sure you check!!!!!)