Algebra Honors (Periods 5 & 6)

Chapter 2 Highlights Continued

Substitution Principle-- an expression may be replaced by another expression that has the same value.

JUSTIFY THE FOLLOWING:
b + (a -b)
= b + [a +(-b)] Definition of subtraction ( we used def-)
= b + [-b + a] C+
=(b + -b) + a A+
= 0 + a Property of Opposites ( How about Prop of Op for this one!!)
= a Id+

Check out the 5 step plan on Page 27... use the index in your textbook to find many concepts and explanations...

Multiplication Property of One a(1) = a and 1(a) = a IDx 1 is the identity element for multiplication

Multiplication Property of Zero A90) = 0 and 0(a) = 0 Ox

Multiplicative Prop of -1
a(-1) = -a and (-1)a = -a

Justify the following
a(-1) + a
= a(-1) + a(1) ID x
= a(-1 +1) DP
=a(0) Prop of Op
=0 Ox

Property of Opposites in Products or the Prop of Opposites in Products or just POP
(-a)b= -ab
a(-b) = -ab
(-a)(-b) = ab

JUSTIFY
a(b-c)
=a[b + (-c)] Def-=ab + a(-c) DP
ab + (-ac) POP
ab -ac Def -

-3(7c + d) - 2(10d-c)
-21c -3d -20d + 2c
-19c -23d

Math 6A (Periods 2 & 4)

Mathematical Expressions 1-1

A variable is a symbol used to represent one or more numbers. The numbers are called the values of the variable.


An expression, such as 3 x n, that involves a variable is called a variable expression.


Expressions, such as 3 x 2, that name a certain number are called numerical expressions


When we write a product that involves a variable, we usually omit the multiplication symbol (whether that be written as x or as ∙ or even with parentheses). Thus, 3 x n is written as 3n
and 2 x a x b is written as 2ab

In numerical expressions for products a multiplication symbol must be used to avoid confusion.

9 x 7 may be written as 9 ∙ 7 or even 9(7)

When a mathematical sentence uses an equal sign, it is called an equation. An equation tells us that two expressions name the same number. The expression to the left of the equals sign is called the left side of the equation and the expression to the right of the equals sign is called the right side.
expression = expression

When a number is substituted for a variable in the variable expression and the indicated operation is carried out, we say that the variable expression has been evaluated. For example, if n has the value 6 in the variable expression 3 x n, then 3 x n has the value 3 x 6, or 18
Example: Evaluate the expression 6a when the variable has the following values:
6a; 2, 4, 6, 8
You would substitute in each value for the variable a
6(2) = 12
6(4) = 24
6(6) = 36
6(8) = 48

148 ÷ 4 =
148/4
37


if m = 3 and n = 18
n ÷ m
substitute in
n/m or 18/3 = 6


If y = 18 and x = 8
4y ÷ 3x immediately set this up as
4y/3x

Now substitute in your values
4(18) / 3(8)
72/24 = 3

Algebra Honors (Periods 5 & 6)

Chapter 1 Highlights
Variable→ symbol used to represent one or more numbers
Variable expression contains a variable
Numerical expression names a particular number
Simplifying the expression→Replacing a numerical expression by the simplest name for its value
Grouping symbols→ parentheses, brackets, fraction bar, etc used to enclose an expression that should be simplified first.
Equation→ two numerical or variable expressions that are equal. Represented by an equals sign placed between the two sides of the equation.
Open Sentences→ contain variables, such as 5x – 1 =9
or y + 2 = 2 + y

The given set of numbers that a variable represents is called the domain of the variable.
Use brackets { } to show a set of numbers. A short way to write “the set whose members are 1, 2, and 3” is {1, 2, 3}
Any value of a variable that turns an open sentence into a true sentence is a solution or root of the sentence.
The set of all solutions of an open sentence is called the solution set of the sentence.
Some equations have only one solution, and some equation have no solutions. The sentence y + 2 = 2 + y has an infinite number. The solution set is the set of all numbers. If, however, you are asked to solve of the domain {0,1,2,3}, you state that the solution set is the domain itself {0, 1, 2, 3}.

Another way to express this is

Read the above as
“y belongs to the set whose members are 0, 1, 2, 3”

Real Numbers
Natural Numbers or Counting Numbers→ 1, 2, 3, 4, 5….
Whole Numbers (Natural Numbers + 0) → 0, 1,2, 3, 4, 5 …
Integers (Natural Numbers , their opposites, + 0) → …-4, -3, -2, -1, 0, 1, 2, 3, 4, …
Rational Numbers (quotient of two integers)
So any number that is either positive, negative, or zero is called a real number.

Each number is a pair, such as 4 and -4 is called the opposite of the other number. The opposite of a is written –a.
The numerals -4(lowered minus sign) and ^-4 (raised minus sign) name the same number.
Caution –a read “the opposite of a” is NOT necessarily a negative number.
For example, if a= -2 then –a would be –(-2) = 2

Absolute value is a distance concept. It my be thought of as the distance between the graph of a number and the origin on a number line. The graphs of -4 and 4 are both 4 units from the origin.

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 6H ( Periods 1, 2, & 3)

Congruent Figures 4-7

Two figures are congruent if they have the same size and same shape. If we could lift on of the figures and place it directly on top of the other... all three vertices would match up with the others. In the example in our book ( page 132) we have two congruent triangles ∆ABC and ∆XYZ A would fall on X, B would fall on Y and C would fall on Z. These matching vertices are called corresponding vertices. Angles at corresponding vertices are corresponding angles and the sides joining corresponding vertices are corresponding sides.
The book states Corresponding angles of congruent figures are congruent.
and
Corresponding sides of congruent figures are congruent.
In class we discussed how to abbreviate the above -- when dealing with triangles.
CPCTC
Corresponding PARTS of congruent triangles are congruent!!

When we name two congruent figures we list corresponding vertices in the same order.

∆ABC ≅ ∆XYZ or ∆CAB ≅ ∆ZXY or ∆BCA ≅ ∆YZX

we know that
∠A ≅ ∠X and ∠B ≅ ∠Y and ∠C ≅ ∠Z
and the segments ( which are denoted with a line (but w/o arrows)above each of the two letters
AB ≅ XY and BC ≅ YZ and CA ≅ ZX

If two figures are congruent, we can make the coincide -- occupy the same place-- by using one or more of the following basic rigid motions:

Translation or slide
Rotation
Reflection or flip or mirror
Check the book on page 133 for good examples of these three rigid motions... I like to think of Tetris moves!!

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

 Geometric Constructions  4-8

There is a difference between making a drawing and a geometric construction.
For drawings, we measure segments & angles using a ruler and a protractor to create our shapes or angles.
With geometric constructions, however, we only use a compass and a straight edge. Although we may use the ruler as our straight edge, we ignore the markings.
Construction 1: Bisect a Segment which also creates a 90 degree angle
Construction 2:  Bisect an angle
Construction 3: Construct a 60 degree angle
Construction 4:  Construct an angle congruent to a given angle.

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

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.

Circles 4-6 continued

We continued our study of circles by examining irregular shapes and determined their perimeters.
To see each of the irregular shapes turned to page 131. The numbers we used in class were all different that those of 24-27, but use the shapes to help solve the following:

The circles in the diagrams are parts of circles and the angles are right angles. We found the perimeter of each figure.
The first figure was a semicircle ( see #24 with a diameter of 4).
We noticed that we needed to start with
C =∏d but then we only need half of that
so
∏d/2 or 4∏/2 = 2∏
Using ∏≈ 3.14
we found
≈3.14(2) = 6.26
BUT.. that only was the upper part we needed to add the diameter of 4 to make sure we had all we needed in our perimeter.
6.28 + 4 = 10.28 units... but then we needed to round to 3 digits-- according to our textbook so
10.3 units would be a good approximation for the first perimeter.

Our 2nd irregular shape was a quarter of a circle with a radius of 6
Again, look to our textbook, page 131 # 25 for the shape. Use 6 as the radius.
This time
C = 2∏r
BUT... we only need 1/4 so
(2⋅∏⋅6)/4 = 12∏/4 = 3∏
≈ 3.14(3) = 9.42
BUT... wait.. we aren't finished... we have to sides of this quarter circle that we need to include in our perimeter.
so the perimeter is approximately 9.42 + 6 + 6 = 9.42 + 12
≈ 21.42 which round to ≈ 21.4 units


The next irregular shape looks like something from Griffith Park observatory. Make sure to use the diagram for # 26 but use a radius of 2 as we did in class.

C = 2∏r
but you notice we only need half of the full circle so
2∏r/2 or just ∏r
Now substitute int he radius-- which is also 2
2∏ ≈ 2(3.14) = 6.28
But... we still need the bottom perimeter
so this shape ≈ 6.28 + 2 + 4 + 2 or 6.28 + 8
≈14.28
≈14.3 units

Period 7 said the next shape ( # 27 in our textbook) looks like a bandaid... What do you think?
(At least.. with the way I drew it.. having a radius of 10)
Again we need the formula
C = 2∏r
This time we realized we had two semicircles.. but that is one whole circles so we kept the formula and substituted in our radius of 10
C =2(10)∏
=20∏
≈20(3.14)= 62.8
Then we added the two sides of 10 and found the perimeter to be approximately
62.8 + 10 + 10
≈82.8 units


I described the last shape to be a teardrop. Make sure to check #28 in our textbook. I actually used the same radius as the book so that drawing is exactly what we did.

C =2∏r
C =2⋅6⋅∏ = 12∏
But... we only need 3/4 of the circle so what is 3/4 of 12... in class everyone knew it was 9 so
3/4 of 12∏ is 9∏
≈9(3.14) = 28.26
But then we need to make sure we include the two sides of 6 each
≈28.26 + 6 + 6 = 28.26 + 12
≈40.26 units
and rounding to three digits
≈ 40.3 units.

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

Polygons 4-5

A polygon is a closed figure formed by joining segments—the sides of the polygons at that endpoints—the vertices of the polygon. Polygons are names according to 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 its sides and all its angles are congruent.
A regular triangle is the equilateral triangle
A regular quadrilateral is the square.

To name a polygon we name its consecutive vertices in order.
A diagonal of a polygon is a segment joining two non consecutive vertices.

To find the perimeter of a polygon add all the lengths of its sides. The perimeter is the distance around the figure. Finding the perimeter of a parallelogram can be done by computing the sum of the lengths or by using the distributive property to obtain
For instance a parallelogram with sides 9 cm and 6 cm
has a perimeter of 9 + 6 + 9 + 6 = 30 cm
but you could calculate that by 2(9) + 2(6) = 18 + 12 = 30 cm or using the distributive property, even 2(9+6) = 2(15) = 30 cm

If you have a regular polygon you can simple multiple the side by the number of sides in the polygon
For example,
a quadrilateral with side 16.5 m has a perimeter of 4(16.5) = 66 m

The sum of the measures of the angles of any pentagon is 540 degrees. If it is a regular pentagon, what must be the measure of each angle of the regular pentagon? 540/5 = 108 degrees.

The sum of the measures of the angles of any pentagon is 540 degrees. How can you prove that? Draw your pentagon and then draw all the diagonals from ONE of the vertices. Count the number of triangles created. Three. How many degrees does a triangle have? 180. Multiply the number of triangles created by 180… 540 is your answer. IT works every time. So How could you create a general rule or formula for the sum of the measures of the angles of any polygon with n sides?

Practice drawing various polygons—now practice drawing all the diagonals for each of them. Can you determine a general rule for the number of diagonals that can be drawn for any polygon?

Math 6H ( Period 1, 2, & 3)

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.

Algebra Honors (Period 6 & 7)

Inequalities in One Variable
Solving Problems Involving  Inequalities 10-3

For practice, we went through the examples in our textbook on Page 469
We discovered that reading and re-reading the problem was critical to make sure we answered the exact question.  As noted in Example 1: the question asked what is the minimum total distance, to the nearest mile, that she will have to travel...?"  The critical part, was "to the nearest mile."  Please read the problem and then realize  why re arrived at the following:
Let d = the distance fro the sign to home
d - 16 > 25
solving that open sentence we get
d > 41
Since the distance needs to be greater than 41, the next whole number is 42 so the answer is
The minimum distance she will travel is 42 miles.

To translate phrases such as "is at least" and "is no less than"  you will need   ≥ 
...think I want at least $200 when going to Disneyland. I obviously want more.. but I will be happy with $200.

To translate phrases such as "is at most" and "is no more than"  you will need   ≤
...think I want at most 7 problems of homework. I really want fewer than 7 but I'll be okay with 7.

Algebra Honors (Period 6 & 7)

Inequalities in One Variable
Solving Inequalities 10-2


Property of Comparison
For all real numbers a and b, one and only one of the following statements is true:

a < b,            a = b,       or a >; b

Transitive Property of Order
For all real  numbers a, b, and c :

1. If a < b and b < c , then  a <; c
2. If a > b, and b > c, then a > c

Addition Property of Order
For all real  numbers a, b, and c :

1. If  a < b, then a + c < b + c
2. If a > b, then a + c > b + c

Multiplication Property of Order


For all real  numbers a, b, and c such that
c > 0 ( c is positive)

1. If a < b, then ac < bc
2. If a > b then ac > bc

c < 0 ( c is negative)

1. If a < b, then ac > bc
2. If a > b, then ac < bc

Multiplying each side of an inequality by a negative number reverses the direction or order of the inequality.

These properties guarantee that the following transformations of a given inequality will always produce an equivalent inequality

1. Substituting for either side of the inequality an expression equivalent to that side
2. Adding to (or subtracting from) each side of the inequality by the same real number
3. Multiplying (or dividing) each side of the inequality by the same positive number.
4. Multiplying  or dividing) each side of the inequality by the same negative number AND  reversing the direction of the inequality. 

To solve an inequality you use the same steps used to solve equations

1. Simplify each side of the inequality as needed
2. Use the inverse operations to undo any additions or subtractions
3. Use the inverse operations to undo any multiplications or divisions

Remember when graphing these inequalities a closed dot indicates that the number is included while an open dot indicates a less than or a greater than.

When working with inequalities it is important to read both the symbols and the words slowly and carefully!!

Word problems involving inequalities also require very careful reading. An important step to solving this type of problem is determining which inequality symbol to use. Before you write your inequality you should be certain that you will use the correct symbol.

Math 6H ( Period 1, 2, & 3)

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°

Algebra Honors (Period 6 & 7)

Inequalities in One Variable
Order of Real Numbers 10-1

This sections should be review... you have been working with less than and greater than symbols since 6th grade. A number line shows order relationships among all real numbers. The value of a variable may be unknown but you may know that is either greater than or equal to another number.
For example,
x ≥ 5 is read "x is greater than or equal to 5."
x ≥ 5 is another way of writing " x >5 or x = 5"

Translating statements into symbols is a critical concept. Practice these as review
-3 is greater than -5.    -3 > -5
and
x is less than or equal to 8  x    ≤   8

To show that x is between -4 and 2 you write
-4 < x < 2
which is read 
"-4 is less than x and x is less than 2."
or you could read it as
" x is greater than -4 AND less than 2."

The same comparisons are stated in the sentence  2 > x  > -4

When all the numbers are know you can classify the statement as true or false.
Thus,
-4 < 1 < 2   is true
but
-4 < 8 < 2 is false

An inequality is formed by placing an inequality symbol (  > ,  < ,     ≤ , or    ≥) between numerical pr variable expressions-- called the SIDES of the inequality
You solve an inequality by finding the values from the domain of the variable which make the inequality a true statement.   Such values are called the solutions of the inequality. All the solutions make up the solution set of the inequality.

Math 6H ( Period 1, 2, & 3)

Points, Lines, Planes 4-1
We can describe but CANNOT DEFINE point, line or plane in Geometry

We use a single small dot to represent a point and in class we labeled with a P and we called it Point P
A straight line in Geometry is usually just called a line...
Two points DETERMINE exactly ONE LINE

we connected Point P with Point Q and created Line PQ
We placed this type of arrow ↔ over PQ to show a line
↔
PQ

that would represent the line PQ but we found we could write
↔
QP
and mean the SAME line!!

You can name ANY line with ANY TWO points that fall on that line!! USE only TWO points to name a line!!


Three or more points on the same line are called collinear. Notice the word "line" in collinear.
collinear

Points NOT on a same line are called noncollinear.

We have a RAY if we have an endpoint and it extends through other points. We name the rame by Naming the endpoint FIRST
→
PQ is RAY PQ and it begins at P and goes through Q. It is NOT the SAME as

→
QP which is Ray QP, which begins at Q and goes through P

Segments are parts of lines with TWO ENDPOINTS.
⎯
PQ is a segment with endpoints Point P and POint Q

We then looked at the drawing from page 105 and the class named all of the names for the line in the drawing, all of the rays that existed in the figure as well as the segments. We found that there were just 3 collinear points: A, X, B but that we could name 3 sets of non collinear points
X, Y, B and A, X, Y, AND A, B, Y

Three non-collinear points determine a flat surface called a plane.
We name a Plane by using three of its non collinear points!! Plane ABC was out example.

Lines in the same plane that do not intersect are PARALLEL lines. Two segments or rays are parallel if they are parts of parallel lines.
↔
AB is parallel to

↔
CD

may be written
↔ ↔
ABllCD
using two straight lines to indicate parallel

Parallel lines DO NOT intersect.

Intersecting lines intersect in a single point!!



Planes that do not intersect are called parallel planes... we looked around the room and found examples of parts of planes.. noticing which ones were parallel!! (the floor and ceiling were a great example)
Then we drew the box from PAge 106 and identified parallel segments and lines from that box.

Two non parallel lines that do not intersect are called SKEW LINES.