Pre Algebra Periods 1, 2, & 4

AREA Sections 10-1, 10-2 & 10-3

All these formulas are related to the basic concept of A = bh
Where the perimeter/circumference fenced in my puppy, the area of the yard will tell me how much sod (grass) I should buy to stop the puppy's paws from getting muddy!

Area of rectangles and parallelograms 10-1

A = (base)(height)
In a parallelogram, whether it's a rectangle, rhombus, square or other parallelogram
A = bh with the height being a line perpendicular to both bases (not the slanted side!)
You have learned the area of a rectangle as A = lw, but the l = b and the w = h
You may have learned the area of a square as A = s² , but that's because the b = h


Area of triangles 10-2
A = (1/2)(base)(height or altitude)
Any parallelogram can be split into 2 triangles using a diagonal.
Because of this, the area of a triangle is half that of a parallelogram.

A = ½bh

Area of trapezoid = (average of the 2 bases)(height)
A = ½(b₁+ b₂)h

Just add the two bases together and divide by 2—that’s the average!!

Then multiply by the height

A trapezoid has 2 bases that ARE NOT EQUAL. So which base is THE base?
If you use the smaller base, you won't have enough sod for your yard and the puppy's paws are still getting muddy.
If you use the larger base, you'll have too much sod for your yard and the extra will rot.
Sooooooooo..... you actually need to take the average of the two bases times the height
A = (average of the 2 bases)(height)
A = (b₁ + b₂) h
theitheis2

Area of Circles 10-3
Area of circle = (Pi)(r²)

If you are given the diameter remember to take half of the diameter to find the radius and make sure to square the radius!!

Use 3.14 or 22/7 as a good approximation of pi. Remember pi is irrational—never ending and never repeating

Algebra Period 3 (Friday)

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 <>if it's negative , there are no real roots

Pre- Algebra Periods: 1, 2, & 4

Transforming geometry, 9-8, 9-9, & 9-10

Rigid Motions:
Translations (slides)
Rotations (turns)
Reflections (flips)

Translations 9-8

You can move pattern blocks by sliding them, flipping them, or turning them. Each of these moves is a transformation.
Sometimes called a slide. Moves a figure - Doesn't change the shape or size
A translation is a transformation that moves points in the same distance and in the same direction. A figure and its translated image are congruent. You can see examples of translations in wallpaper, fabric and wrapping paper.
The figure you get after a transformation is called the image.
To name the image of a point, you use the prime notation, Point A and its image Point A^’
After the figure is translated (moved), it is then called the image and you use ' (the prime notation)
So triangle ABC once it's moved up 2 spaces is now named ABC'

You can describe a transformation using arrow notation, which describes the mapping of a figure onto its image. A --> A^’
You can also use arrow notation to write a general rule that describes a transformation.

So... If point A is (3,5)----> A' is (5,1)
(A prime)
Write an equation.
Ask yourself, what do you have to do to X as 3 to get X' as 5?
And what do you have to do to Y as 5 to get Y' as 1

(X,Y)---> (X + 2, Y - 4) then you have A' as (5,1) and you proved it.


Symmetry 9-9

Symmetry means that if you divide a figure in half, both halves look exactly the same
Line of symmetry: where you can "fold" the figure and have it exactly the same
Figures can have 0 or more lines of symmetry
A random drawing would probably have no lines of symmetry
A butterfly has one line of symmetry
A rectangle has 2 lines of symmetry
A square has 4 lines of symmetry
A circle has infinite lines of symmetry (you can rotate it infinitely and divide it in half)

A figure has reflective symmetry when one half is a mirror image of the other half. A line of symmetry divides a figure with reflectional symmetry into two congruent halves.

Again, Reflections (flips)

Line of symmetry---> Fold Figure
Reflection Reflection is sometimes called a flip of the figure
The figure flips over a LINE OF REFLECTION
For example, if a triangle is in Quadrant I, it may flip over the y axis into Quadrant II
(the right side of the triangle is now the left side)
Or it may flip over the x axis into Quadrant IV
(the top of the triangle is now the bottom)
Or it may flip over any other line like y = x. This is more complicated to understand!





Rotation 9-10

This transformation takes a figure and TURNS IT
about a point called the CENTER OF ROTATION.
The angle of turning is called the ANGLE OF ROTATION.
The two most common angles: 90 degrees and 180 degrees

If you are going to rotate 90 degrees the rule is this

The Value of X Becomes the opposite of the Y Value & the Y value becomes what the value of x was.

EX: (X,Y)--->(-Y, X)

(5,4) rotated 90 degrees would be (-4,5)
(5,4)---> (-4,5)

If you rotate 90 degrees a figure in quadrant I it would rotate into quadrant II
I-->II
II-->III
III-->IV
IV-->I

The rotation is one quadrant over-- counterclockwise

If you are going to rotate 180 degrees the rule is this
(X,Y)--->(-X,-Y)
180 degrees: ends up in the opposite quadrant (I goes to III , II goes to IV and so on)
The image is: (x, y) goes to (-x, -y) SAME NUMBERS...OPPOSITE SIGNS!



submitted by Lorenzo