Algebra Honors (Period 6 & 7)

Star Prep

Conjunctions & Disjunctions
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

Solve the equation twice - Once with the solution positive and once with it negative
I2x - 4I = 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

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 up between them
I3xI < 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

I3xI >15 is equal to 3x < -15 and 3x > 15

Math 6 Honors ( Periods 1, 2, & 3)

Polygons 4-5

A polygon is a closed figure formed by joining segments ( the sides of the polygon) at their endpoints ( the vertices of the polygon). Polygons are named for the number of sides they have.
Triangle- 3 sides
Quadrilateral- 4 sides
Pentagon- 5 sides
Hexagon- 6 sides
Octagon- 8 sides
Decagon- 10 sides

A polygon is regular if ALL of its sides are congruent and ALL of its angles are congruent.

To name the polygon we name its consecutive vertices IN ORDER.

A diagonal of a polygon is a segment joining two nonconsecutive vertices.
Look at the quadrilateral on Page 123 and notice the two segments that represent the diagonals of the quadrilateral PQRS.

Certain quadrilaterals have special names.
A parallelogram has it opposite sides parallel and congruent.

A trapezoid has just one pair of parallel sides.

Certain parallelograms also have special names

A rhombus ( rhombii plural) has all it sides congruent.
A square has congruent sides and congruent angles
A rectangle has all its angles congruent.

Thus a square is a rectangle.. but a rectangle isn't necessarily a square!!

TH\he perimeter of a figure is the distance (think fence) around it. Thus the perimeter of a polygon is the sum of the lengths of its sides.

Always label your perimeters. If the figure provides a specific measure, such as meters (m), centimeters (cm), feet (ft), inches (in.)-- make sure to use that label.

If no unit of measure is given, always include "units"
Circles 4-6

A circle is the set of all points in a plan at a given distance from a given point O (called the center).
A segment joining the center to a point on the circle is called a radius ( plural: radii) of the circle. All radii of a given circle have the same length and the length is called the radius of the circle.

A segment joining two points on a circle is called a chord... and a chord passing through the center is a diameter of the circle. the ends of the diameter divide the circle into two semicircles. The length of a diameter is called the diameter of the circle.
Two radii equal one diameter-- a fact we will use in the formulas below

The perimeter of a circle is called the circumference and the quotient

circumference ÷ diameter is the same for all circles--> regardless of size
This quotient is denoted by the Greek letter ∏ ( pronounced "pie")
No decimal gives ∏ exactly
No fraction gives ∏ exactly, either

A fairly good approximation is either 3.14 or 22/7

If we denote the circumference by C and the diameter by d we can write

C ÷ d = ∏
This formula can be put into several useful forms.

Let C = circumference d = diameter and r = radius
Then:

C = ∏d
d = C/∏

C = 2∏r
and
r = C/(2∏)

We tried a few examples.
Using ∏≈ 3.14 and rounding to three digits, as described by our textbook.
The diameter of a circle is 6 cm. Find the circumference.
WE are given d and are asked to find C.
WE use the formula
C = ∏d
C ≈ 3.14(6) = 18.84
C ≈ 18.8
So, the circumference is approximately 18.8 cm


The circumference of a circle is 20 feet. Find the radius.
To find the radius, use the formula
r = C/(2∏)
r = 20/2∏
Simplify first
r = 10/∏
r ≈ 10/3.14
r ≈ 3.1847
Since the third digit from the left is in the hundredths' place, round to the nearest hundredth.
r ≈ 3.18
The radius is approximately 3.18 feet

A polygon is inscribed in a circle if all of its vertices are on the circle. Check on the diagram in our textbook on page 129-- we added that to our notes as well.

Three noncollinear points (not on a line) determine one and only one circle that passes through the three given points.

Preparing for the STAR

This was made over two years ago... but it is still fun to watch..
Larry & Sarah.. and the CA STAR

Algebra Honors (Period 6 & 7)

Puzzle Problems 9-7

This includes digit, age, and fraction problems

Example 1: Digit problem
The sum of the digits in a 2 digit number is 12. The new number obtained when the digits are reversed is 36 more than the original number. Find the original number.

Let t = the ten's digit of the original number
Let u = the unit's digit of the original number

then the quantity relationship is t + u = 12

The value expression of the original number is 10t +u so when the digits are reversed the value would be 10u+ t

SO, the value equation becomes
10u + t = 10t + u + 36 or
9u - 9t = 36 Now, divide by 9

u - t = 4
So the two equations are
u - t = 4
u + t = 12

2u = 16
u = 8
If u = 8 then t= 4
and the original number is {48}

EXAMPLE 2: Age Problem

Mini is 4 years older than Ronald
5 years ago she was twice as old as he was. Find their ages now.

Let m = Mimi's age now
Let r = ronald's age now
The two equations created from this information are the following:
m = r + 4
and
m-5 = 2(r-5)

m -5 = 2r - 10 -->I would use substitution with this one so
(r+4) - 5 = 2r - 10
r-1 = 2r - 10
9 = r
Since Ronald is 9 years old
Mimi is 13 years old

Example 3: Fraction Problem

The numerator of a fraction is 3 less than the denominator
If the numerator and the denominator are each increased by 1, the value of the resulting fraction is 3/4 What was the original fraction?

Let n = the numerator
Let d = the denominator

n/d is the fraction

n = d -3
which can be translated to n - d = -3

and
n+1=3
d+1 4

4(n+1) = 3(d+1)
4n + 4 = 3d+ 3

4n-3d = -1
n - d = -3

Using the addition/subtraction method and multiplying the 2nd equation by (-4)

4n-3d = -1
-4n - (-4)d = -3(-4)

4n-3d = -1
-4n+4d = 12
d = 1

Since d = 11
n = 8
and the fraction is 8/11
{8/11}

We then turned to Page 447 and completed several even problems # 8-16 even.
Here are a few of the problems worked out. TUrn to our textbook to read the actual wording of each problem.

#8. Let t = ten's digit
Let u = unit's digit

t = u-5

t + u = (1/3)(10t + u)
3t + 3u = 10t + u
-7t + 2u = 0

Using substitution

-7(u-5) + 2u + 0
-7u + 35 + 2u = 0
-5u = -35
u = 7

Since u = 7 t = 2 and the number is 27
{27}

10.
Let t = ten's digit
Let u = unit's digit

The original number would be 10t + u and the reverse would be 10u + t

so 10u + t = 10t + u -54
9u -9t = -54
divide that by 9
u - t = -6
so
u - t = -6
u + t = 8
2u =2
u = 1
so the number must be 71
{71}

12.
Cecelia is 24 years younger than Joe
Let c = Cecelia's age NOW
:et j = Joe's age now
c + 24 = j
c - 6 = (1/2)(j - 6)

2c - 12 = j- 6

2c - j = 6
c - j = -24

Becomes
2c - j = 6
- c + j = 24
c = 30

So Cecelia's age is 30 years old and Joe is 54 years old

Math 6 Honors ( Periods 1, 2, & 3)

Prep for STAR
Angles and Angle Measure 4-3
An angle is a figure formed by two rays with the same endpoints. The common endpoint is called the vertex. The rays are called the sides.
We may name an angle by giving its vertex letter if this is the only angle with that vertex, or my listing letters for points on the two sides with the vertex letter in the middle. We use the symbol from the textbook.

To measure segments we use a rule to mark off unit lengths. To measure angles, we use a protractor that is marked off in units of angle measure called degrees.

To use a protractor, place its center point at the vertex of the angle to be measured and one of its zero points on the side.


We often label angels with their measures. When angles have equal measures we can write m angle A = m angle B
We say that angle A and angle B are congruent angles


If two lines intersect so that the angles they form are all congruent, the lines are perpendicular. We use the symbol that looks like an upside down capital T to mean “is perpendicular to.”



Angles formed by perpendicular lines each have measure of 90° . A 90° angle is called a right angle. A small square is often used to indicate a right angle in a diagram

An acute angle is an angle with measure less than 90°. An obtuse angle has measure between 90° and 180°



Two angles are complementary if the sum of the measures is 90°
Two angles are supplementary if the sum of their measures is 180°
Triangles 4-4
A triangle is the figure formed when three points, not on a line are jointed by segments.

Triangle ABC ΔABC
Each of the Points A, B, C is called a vertex
(plural: Vertices) of ΔABC

Each of the angles angle A. angle B. and angle C is called an angle of ΔABC

In any triangle-
The sum of the lengths of any two sides is greater than the length of the third side
The sum of the measures of the angles is 180

There are several ways to name triangles. One way is by angles

Acute Triangle
3 acute angles

Right Triangle
1 right angle

Obtuse Triangle
1 obtuse angle

Triangles can be classified by their sides

Scalene Triangle
no 2 sides congruent

Isosceles Triangle
at least 2 sides congruent

Equilateral Triangle
all 3 sides congruent


The longest side of a triangle is opposite the largest angle and the shortest side is opposite the smallest angle. Two angles are congruent if and only if the sides opposite them are congruent.