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°
Thursday, December 2, 2010
Tuesday, November 30, 2010
Algebra (Period 1)
Multiplying Polynomials 5-11
To multiply two polynomials, multiply each term of one polynomial by every term of the other. THEN ADD the results.
The textbook shows a column approach, please see page 249 for instructions.
In class we used the BOX method... and then combined terms.
When you multiply a trinomial by a binomial or two trinomials, it gets really tricky!
2 ways:
1) box method
2)column method (double or triple distributive with columns to combine like terms)
If you have a trinomial times a binomial, it's easier to use the Commutative Property
and make it a binomial times a trinomial:
(x2 + x - 1) (x - 1)
switch it to
(x - 1) (x2 + x - 1)
Remember the following rules
(A + B)(A + B) = (A + B) 2 = A2 + 2AB + B2
(A - B)(A - B) = (A - B)2 = A2 -2AB + B2
(A + B)(A - B)= A2- B2
You can use FOIL to multiply two binomials
remember FOIL is First Terms, Outside Terms, Inside Terms, Last Terms
You can always use FOIL-- or the BOX method but knowing these rules will make computation quicker if you know the above rules!!
To multiply two polynomials, multiply each term of one polynomial by every term of the other. THEN ADD the results.
The textbook shows a column approach, please see page 249 for instructions.
In class we used the BOX method... and then combined terms.
When you multiply a trinomial by a binomial or two trinomials, it gets really tricky!
2 ways:
1) box method
2)column method (double or triple distributive with columns to combine like terms)
If you have a trinomial times a binomial, it's easier to use the Commutative Property
and make it a binomial times a trinomial:
(x2 + x - 1) (x - 1)
switch it to
(x - 1) (x2 + x - 1)
Remember the following rules
(A + B)(A + B) = (A + B) 2 = A2 + 2AB + B2
(A - B)(A - B) = (A - B)2 = A2 -2AB + B2
(A + B)(A - B)= A2- B2
You can use FOIL to multiply two binomials
remember FOIL is First Terms, Outside Terms, Inside Terms, Last Terms
You can always use FOIL-- or the BOX method but knowing these rules will make computation quicker if you know the above rules!!
Math 6 Honors (Period 6 and 7)
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.
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.
Pre Algebra (Period 2 & 4)
Graphing & Writing Inequalities Review
Some words to really know:
At least--> means greater than or EQUAL TO
I want at least $150 to go shopping for holiday gifts!!
m ≥ 150
AT MOST --> less than or EQuAL TO
You might say that you want at most 15 minutes of homework tonight. That means you will take 15 minutes but you sure would like less....
m ≤ 15
Review of some of the most missed problems from our recent races:
y is at most 5
y ≤ 5
x is no more than 7
x ≤ 7
y is at least 20
y ≥ 20
y is less than 8
y < 8. Now 5 < y reads " Five is less than than y" --> if that is true then y must be greater than 5 or
y > 5 so now you can graph easily because you know exactly which way the arrow ( solution set) must be pointing... in the same direction as the inequality symbol. THis ONLY works when you have the variable on the left!! as in y > 5.
ALWAYS CHANGE so the variable is on the left!!
Solve one step inequalities just like you do for equations.. just make sure to end with the variable on the left-- to be able to graph easily
y - 7 > -3
add 7 to BOTH Sides
y > 4
Now you can graph easily... It's an open dot on your graph Because it DOES NOT inlcude 4 as part of the solution
FORMAL CHECK
(1) REWRITE the inequality
(2) Substitute in for the variable--using ANY number that fits the solution (except the boundary number) In this example y > 4,
4 is the boundary number. We showed in class why you can't use the boundary number so if y > 4 then 5, 10, 100, etc would work...
In class we used 10
10 -7 > -3 (Remember to put a "?" over the inequality)
(3) REALLY DO THE MATH. that is, in this case,
do 10-7.
Well 10-7 = 3 so
3 > -3 That's true-- it works!!
If you get something that is not true, chances are you did your problem wrong. re work the problem and see if you can discover your error!!
WHen dividing or multiplying by a negative coefficient (that's the number attached to the variable) you must remember to
REWRITE the inequality AS YOU "FLIP THE SWITCH" (change the inequality symbol)
y/-5 ≥ -3
We need to multiply both sides by a -5 so we must REWRITE and FLIP
(-5)(y/-5) ≤ (-3)(-5)
y ≤ 15
FORMAL CHECK
(1) REWRITE THE INEQUALITY
(2) Substitute in one of the SOLUTIONS-- any one that really fits and is easy to work with!!
(3) DO THE MATH-- look at your results-- does it make sense?... don't just put a happy face!!
(1) y/-5 ≥ -3
well, we found out the y ≤ 15 that means any number less than 15 could be used to check I would pick something that was easy to work with, something that was divisible by 5... like 0, or 5 or even 10
we used 0 in class
(2) 0/-5 ≥ -3
(3) 0 ≥ -3 That's true!! IT WORKED
But what if I used 5?
well step 2 would be
(2) 5/-5 ≥ -3
(3) -1 ≥ -3 and that's still true !!
5y ≤ -35
I just divide by 5. I don't need to switch anything because I am NOT dividing by a negative number
5y/5 ≤ -35/5
y ≤ -7
But what about
-15 ≤ -3m
In this case I must rewrite as I "FLIP THE SWITCH" (Remember the poster from class)
-15/-3 ≥ -3m/-3
5 ≥ m
Before we continue we need to realize that if five is greater than or equal to m... that means m must be less than or equal to 5 or m ≤ 5. Now its easy to graph.
FORMAL CHECK
-15 ≤ -3m
-15 ≤ -3(4)
-15 ≤ -12 which is true!!
In class we used a number that was not correct to show what happens.
When working with Inequalities with 2 steps and variables on both sides:
(1) Do Distributive Property (DP) carefully-- if necessary
(2) combine like terms on each side
(3) "Jump" the variables to one side of the equation
(4) Add or subtract
(5) Multiply or Divide-- being careful if have negative coefficients to rewrite the inequality and "flip the Switch"
(6) MAKE SURE the variable is on the LEFT Side
Some words to really know:
At least--> means greater than or EQUAL TO
I want at least $150 to go shopping for holiday gifts!!
m ≥ 150
AT MOST --> less than or EQuAL TO
You might say that you want at most 15 minutes of homework tonight. That means you will take 15 minutes but you sure would like less....
m ≤ 15
Review of some of the most missed problems from our recent races:
y is at most 5
y ≤ 5
x is no more than 7
x ≤ 7
y is at least 20
y ≥ 20
y is less than 8
y < 8. Now 5 < y reads " Five is less than than y" --> if that is true then y must be greater than 5 or
y > 5 so now you can graph easily because you know exactly which way the arrow ( solution set) must be pointing... in the same direction as the inequality symbol. THis ONLY works when you have the variable on the left!! as in y > 5.
ALWAYS CHANGE so the variable is on the left!!
Solve one step inequalities just like you do for equations.. just make sure to end with the variable on the left-- to be able to graph easily
y - 7 > -3
add 7 to BOTH Sides
y > 4
Now you can graph easily... It's an open dot on your graph Because it DOES NOT inlcude 4 as part of the solution
FORMAL CHECK
(1) REWRITE the inequality
(2) Substitute in for the variable--using ANY number that fits the solution (except the boundary number) In this example y > 4,
4 is the boundary number. We showed in class why you can't use the boundary number so if y > 4 then 5, 10, 100, etc would work...
In class we used 10
10 -7 > -3 (Remember to put a "?" over the inequality)
(3) REALLY DO THE MATH. that is, in this case,
do 10-7.
Well 10-7 = 3 so
3 > -3 That's true-- it works!!
If you get something that is not true, chances are you did your problem wrong. re work the problem and see if you can discover your error!!
WHen dividing or multiplying by a negative coefficient (that's the number attached to the variable) you must remember to
REWRITE the inequality AS YOU "FLIP THE SWITCH" (change the inequality symbol)
y/-5 ≥ -3
We need to multiply both sides by a -5 so we must REWRITE and FLIP
(-5)(y/-5) ≤ (-3)(-5)
y ≤ 15
FORMAL CHECK
(1) REWRITE THE INEQUALITY
(2) Substitute in one of the SOLUTIONS-- any one that really fits and is easy to work with!!
(3) DO THE MATH-- look at your results-- does it make sense?... don't just put a happy face!!
(1) y/-5 ≥ -3
well, we found out the y ≤ 15 that means any number less than 15 could be used to check I would pick something that was easy to work with, something that was divisible by 5... like 0, or 5 or even 10
we used 0 in class
(2) 0/-5 ≥ -3
(3) 0 ≥ -3 That's true!! IT WORKED
But what if I used 5?
well step 2 would be
(2) 5/-5 ≥ -3
(3) -1 ≥ -3 and that's still true !!
5y ≤ -35
I just divide by 5. I don't need to switch anything because I am NOT dividing by a negative number
5y/5 ≤ -35/5
y ≤ -7
But what about
-15 ≤ -3m
In this case I must rewrite as I "FLIP THE SWITCH" (Remember the poster from class)
-15/-3 ≥ -3m/-3
5 ≥ m
Before we continue we need to realize that if five is greater than or equal to m... that means m must be less than or equal to 5 or m ≤ 5. Now its easy to graph.
FORMAL CHECK
-15 ≤ -3m
-15 ≤ -3(4)
-15 ≤ -12 which is true!!
In class we used a number that was not correct to show what happens.
When working with Inequalities with 2 steps and variables on both sides:
(1) Do Distributive Property (DP) carefully-- if necessary
(2) combine like terms on each side
(3) "Jump" the variables to one side of the equation
(4) Add or subtract
(5) Multiply or Divide-- being careful if have negative coefficients to rewrite the inequality and "flip the Switch"
(6) MAKE SURE the variable is on the LEFT Side
Algebra (Period 1)
Multiplying Binomials: Special Products 5-10
LEARN TO RECOGNIZE SOME SPECIAL PRODUCTS - IT MAKES IT EASIER!
Remember: You can FOIL these just like the other products until you remember these special patterns....but when we get to factoring next week, it will really help you to know these patterns by heart.
When you do, you actually don't need to show any work because you do it in your head!
(That should make a lot of you happy! :)
DIFFERENCE OF TWO SQUARES:
You will notice that the two factors are IDENTICAL except they have DIFFERENT SIGNS
(x + 6)(x - 6) =
x2 - 6x + 6x - 36 =
x2 - 36
This will happen every time!
The middle terms are additive inverses so they become zero.
You're left with a difference (subtraction) of two terms that are squared.
SQUARING A BINOMIAL:
When you multiply one binomial by itself (squaring it), you end up with:
First term squared + twice the product of both terms + last term squared
(x + 6)2 =
(x + 6)(x + 6) =
x2 + 2(6x) + 62 =
x2 + 12x + 36
If you foiled you would have:
x2 + 6x + 6x + 36
CAN YOU SEE THAT THE 2 MIDDLE TERMS ARE JUST DOUBLING UP???
WHY???
Another example with subtraction in the middle:
(x - 6)2 =
(x - 6)(x - 6) =
x2 + 2(-6x) + 62 =
x2 - 12x + 36
If you foiled you would have:
x2 - 6x - 6x + 36
CAN YOU SEE THAT THE 2 MIDDLE TERMS ARE JUST DOUBLING UP???
WHY???
PLEASE NOTE:
NOTICING THESE SPECIAL PRODUCTS HELPS YOU DO THESE MULTIPLICATIONS FASTER!
IF YOU EVER FORGET THEM, JUST FOIL!
(but you will need to recognize them for factoring in Chapter 6)
LEARN TO RECOGNIZE SOME SPECIAL PRODUCTS - IT MAKES IT EASIER!
Remember: You can FOIL these just like the other products until you remember these special patterns....but when we get to factoring next week, it will really help you to know these patterns by heart.
When you do, you actually don't need to show any work because you do it in your head!
(That should make a lot of you happy! :)
DIFFERENCE OF TWO SQUARES:
You will notice that the two factors are IDENTICAL except they have DIFFERENT SIGNS
(x + 6)(x - 6) =
x2 - 6x + 6x - 36 =
x2 - 36
This will happen every time!
The middle terms are additive inverses so they become zero.
You're left with a difference (subtraction) of two terms that are squared.
SQUARING A BINOMIAL:
When you multiply one binomial by itself (squaring it), you end up with:
First term squared + twice the product of both terms + last term squared
(x + 6)2 =
(x + 6)(x + 6) =
x2 + 2(6x) + 62 =
x2 + 12x + 36
If you foiled you would have:
x2 + 6x + 6x + 36
CAN YOU SEE THAT THE 2 MIDDLE TERMS ARE JUST DOUBLING UP???
WHY???
Another example with subtraction in the middle:
(x - 6)2 =
(x - 6)(x - 6) =
x2 + 2(-6x) + 62 =
x2 - 12x + 36
If you foiled you would have:
x2 - 6x - 6x + 36
CAN YOU SEE THAT THE 2 MIDDLE TERMS ARE JUST DOUBLING UP???
WHY???
PLEASE NOTE:
NOTICING THESE SPECIAL PRODUCTS HELPS YOU DO THESE MULTIPLICATIONS FASTER!
IF YOU EVER FORGET THEM, JUST FOIL!
(but you will need to recognize them for factoring in Chapter 6)
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