This is going to be somewhat of a general post since the topic is a tad bit wide. Anyways, I think (and hope) that you are all aware of the fact that we have finals this week. After searching a couple of articles about how you can study for your finals and what makes studying easier, I think I have somewhat of a gist of it.
DOs
1. SLEEP
a. You perform better when properly rested
2. Reread your notes, practice on whatever subject you struggle on.
3. Have confidence
4. If you’re studying for a new concept (Chapter 6) then practice on a single section and repeat it until you get all the answers correct.
5. Prepare yourself mentally.
6. Eat healthy
a. Junk food can damage your concentration and memory
7. Choose where you study wisely
a. A quiet, orderly place is most likely the best
b. A peaceful environment will help your concentration
DON'TS
1. Do not just re-read the entire text book.
a. It’s a waste of time
b. You’ll most likely not retain that information
c. Read the chapter summaries
d. Re-read notes
e. Look at charts, pictures, diagrams, and other visual tools.
2. Do not procrastinate.
a. Start studying early and do it often.
3. Do not stress yourself.
a. Take time to rest.
b. You’ll learn more efficiently with breaks in between your studying.
I also have two other things to share~
the first one is a step-by-step how to that shows how it’s possible to pull an all-nighter before a test but still do well.
http://media.tumblr.com/tumblr_liw1rb3Lzj1qbolbn.jpg
~I don't recommend this, I just thought it was interesting.
And the second one is a list of common mistakes that can lower your test scores.
http://homeworktips.about.com/od/schooltests/a/testmistakes.htm
I don’t pick a scribe right? Darn it >//<.
~The dos and the don'ts were in separate charts that were...easier to read, but it all disappeared when I moved it away from the Word document D: ~
Tuesday, December 13, 2011
Monday, December 12, 2011
6.4 Inequalities for one triangle
THEOREM THAT COVERS ALL THE CONCEPTS IN 6.4:
-If the angles of a triangle are distinct, then you can draw conclusions based on the side lengths.
CONCLUSIONS SHOWN BELOW
(diagram is not to scale)
GIVEN: m^A=40; m^B=75; m^ C=65
CONCLUSIONS:
~m^A < m^C < m^B
~a < c < b ->because of the theorem If one angle of a triangle is larger than a second, then the side opposite the first angle is longer than the side opposite the second angle.
BASICALLY:
~the smallest angle is opposite the shortest side
~the "middle" angle is opposite the "middle" side
~the largest angle is opposite the longest side
REMEMBER!!!!!!!!!!!!!!!!!!!!
In order for a triangle to exist, the sum of any 2 sides MUST always be GREATER than the 3rd side. (also called the Triangle Inequality Theorem)
Example:
WHY???
-7+7= 14 < 16
-7+16=23 > 7
The sum of the second pair of side lengths IS greater than the third side, BUT the sum of the first pair of side lengths is NOT greater than the third side, therefore, the "triangle" is NOT a triangle.
ANOTHER EXAMPLE:
If two sides of a "triangle" are 8 cm long and the third side is 14 cm long, then the "triangle" IS a triangle.
WHY???
-8+8=16 > 14
-8+14=22 > 8
The sum of the first pair of side lengths IS greater than the third side, and the sum of the second pair of side lenths IS greater than the third side, therefore, the "triangle" IS a triangle.
COROLLARIES:
-The perpendicular segment from a point to a line is the shortest segment from the point to the line.
-The perpendicular segment from a point to a plane is the shortest segment from the point to the plane.
HELPFUL VIDEO:
-http://www.winpossible.com/lessons/Geometry_Triangle_Inequality.html
-http://www.youtube.com/watch?v=e1prq6UFxJE&noredirect=1 (this is a YouTube video so you gotta watch it at home, but its about Indirect proofs and all of 6.4)
-If the angles of a triangle are distinct, then you can draw conclusions based on the side lengths.
CONCLUSIONS SHOWN BELOW
(diagram is not to scale)
GIVEN: m^A=40; m^B=75; m^ C=65
CONCLUSIONS:
~m^A < m^C < m^B
~a < c < b ->because of the theorem If one angle of a triangle is larger than a second, then the side opposite the first angle is longer than the side opposite the second angle.
BASICALLY:
~the smallest angle is opposite the shortest side
~the "middle" angle is opposite the "middle" side
~the largest angle is opposite the longest side
REMEMBER!!!!!!!!!!!!!!!!!!!!
In order for a triangle to exist, the sum of any 2 sides MUST always be GREATER than the 3rd side. (also called the Triangle Inequality Theorem)
Example:
If two sides of a "triangle" are 7 cm long and the third is 16 cm long, then the "triangle" is NOT a triangle.
WHY???
-7+7= 14 < 16
-7+16=23 > 7
The sum of the second pair of side lengths IS greater than the third side, BUT the sum of the first pair of side lengths is NOT greater than the third side, therefore, the "triangle" is NOT a triangle.
ANOTHER EXAMPLE:
If two sides of a "triangle" are 8 cm long and the third side is 14 cm long, then the "triangle" IS a triangle.
WHY???
-8+8=16 > 14
-8+14=22 > 8
The sum of the first pair of side lengths IS greater than the third side, and the sum of the second pair of side lenths IS greater than the third side, therefore, the "triangle" IS a triangle.
COROLLARIES:
-The perpendicular segment from a point to a line is the shortest segment from the point to the line.
-The perpendicular segment from a point to a plane is the shortest segment from the point to the plane.
HELPFUL VIDEO:
-http://www.winpossible.com/lessons/Geometry_Triangle_Inequality.html
-http://www.youtube.com/watch?v=e1prq6UFxJE&noredirect=1 (this is a YouTube video so you gotta watch it at home, but its about Indirect proofs and all of 6.4)
Chapter 6.3 Indirect Proofs(You might want to read this before reading David's scribe because he spoils the..... next...... Scribe!!!!!!!!!!!!! So read this first!!!!This is possibly the longest title ever, of all time)
Indirect Proofs!!!!!!!!
A long time ago, (since September, 2011) in this class, we have been using proofs. Now they turn out to be direct proofs. At times, they are difficult to find a direct proof so it's possible to reason indirectly. To use a indirect proof, you assume temporarily that the desired conclusion is not true. Since I don't want to explain how to write a indirect prof, I'll write it in steps so... everyone... is... happy!!!
How to write Indirect Proofs!!!!!!!!!!!
Steps
1. Assume temporarily that the conlusion is not true!
2.Reason logically until you reach a contradition of a known fact!
3.Point out that the temporary assumption must be false, and that the conclusion must then be true!
Here is a example of using indirect proofs from the book.
Given: n is an interger and n squared (I don't have microsoft word on my computer so i wrote squared) is even.
Proof: n is even
Assume temporarily that n isnot even. Then n is odd, and
n squared = n x n
= odd x odd x = odd
But this contradits the given information that n squared is even. Therefore the temporarily assumption that n is not false. It follows that n is even.
Now is time for the epic question of the day!!!
Are you ready?
Did you know that provig the contrapositive of a statement is related to indirect proofs?
Confused? Here's an example from the book!
If you want to prove the statement " If p, then q," you can prove the contrapositive " if not q, then p." Or you could write an indirect proof- assume that q is false and show that this assumption implies that p is false.
Now comes the epic videos of the day!!
http://www.5min.com/Video/How-to-Write-an-Indirect-Proof-516909808
This is the only video that I could find that was not on youtube. Somehow, searching for a video on indirect proofs can lead you to Call of Duty Black Ops?
So that is chapter 6.3!!!!!! Prepare for more epic questions and videos of the day!!! Also, expect a epic picture of the day if there is any picture to show.
Now the next scribe is...
Not Jimmy Bob
Not John Bob
Not me
But it's
Whoever David picked!!!!!!!!!!
A long time ago, (since September, 2011) in this class, we have been using proofs. Now they turn out to be direct proofs. At times, they are difficult to find a direct proof so it's possible to reason indirectly. To use a indirect proof, you assume temporarily that the desired conclusion is not true. Since I don't want to explain how to write a indirect prof, I'll write it in steps so... everyone... is... happy!!!
How to write Indirect Proofs!!!!!!!!!!!
Steps
1. Assume temporarily that the conlusion is not true!
2.Reason logically until you reach a contradition of a known fact!
3.Point out that the temporary assumption must be false, and that the conclusion must then be true!
Here is a example of using indirect proofs from the book.
Given: n is an interger and n squared (I don't have microsoft word on my computer so i wrote squared) is even.
Proof: n is even
Assume temporarily that n isnot even. Then n is odd, and
n squared = n x n
= odd x odd x = odd
But this contradits the given information that n squared is even. Therefore the temporarily assumption that n is not false. It follows that n is even.
Now is time for the epic question of the day!!!
Are you ready?
Did you know that provig the contrapositive of a statement is related to indirect proofs?
Confused? Here's an example from the book!
If you want to prove the statement " If p, then q," you can prove the contrapositive " if not q, then p." Or you could write an indirect proof- assume that q is false and show that this assumption implies that p is false.
Now comes the epic videos of the day!!
http://www.5min.com/Video/How-to-Write-an-Indirect-Proof-516909808
This is the only video that I could find that was not on youtube. Somehow, searching for a video on indirect proofs can lead you to Call of Duty Black Ops?
So that is chapter 6.3!!!!!! Prepare for more epic questions and videos of the day!!! Also, expect a epic picture of the day if there is any picture to show.
Now the next scribe is...
Not Jimmy Bob
Not John Bob
Not me
But it's
Whoever David picked!!!!!!!!!!
6.5 Inequalities For Two Triangles
The Next Publisher is Phil J
If two sides of one triangle are congruent to two sides of another triangle, but the included angle of the first triangle is larger than the included angle of the second, then the third side of the first triangle is longer than the third side of the second triangle.
Given: Seg. HA is congruent to Seg. KC; Seg. HB is congruent to Seg. KD; m Angle K is greater than m Angle H.
Prove: Seg. CD is greater than Seg. AB
Proof:
Draw Ray KL so that m Angle LKD = m Angle H. On Ray KL take point M so that KM = HA. Then either M is on Seg. CD or M is not on Seg. CD.
In both cases Triangle MKD = Triangle AHB by Hinge Theorem, and MD = AB.
Case 1: M is on Seg. CD
CD is greater than MD (Seg. Add. Post. and Prop. of Ineq.)
CD is greater than AB (Substitution Prop.)
Converse of Hinge Theorem
If two sides of one triangle are congruent to two sides of another triangle, but the third side of the first triangle is longer than the third side of the second triangle, then the included angle of the first triangle is larger than the included angle of the second.
Same diagram as shown above
Given: Seg. KC is congruent to Seg. HA; Seg. KD is congruent to HB; CD is greater than AB.
Prove: m Angle K is greater than m Angle H.
Proof:
Assume temporarily that m Angle K is not greater than m Angle H.
Then either m Angle K = m Angle H or m Angle K id less than m Angle H.
Case 1: If m Angle K = m Angle H, then Triangle CKD is congruent to Triangle AHB by the Hinge Theorem. (CD = AB)
Case 2: If m Angle K = m Angle H, then CD is less than AB by the Hinge Theorem.
http://www.schooltube.com/video/c8315510b6fb421f9b0b/L23T3
Thursday, December 8, 2011
6.2 Inverses and Contraposistives
Class Blog 6-2 By Matthew Jones
If Then Statements!!! AGAIN!!!!
The Statement - If P then Q
Converse or P and Q flipped - If Q then P
NEW INFO ALERT
Inverse or the Not statement – If NOT P Then NOT Q.
Contropositive or the Not Converse – If NOT Q Then NOT P
Logical Equivalence
- A statement and it’s Contrapositive are Logically Equivalent
- A statement is NOT Logically Equivalent to it’s Converse and Inverse.
- Similar to an = and the biconditional
- Results of either
Both True
Both False
Examples of everything above :
True conditional : If two lines are not coplanar, then they do not intersect.
Inverse of it : If two lines are coplanar, then they intersect. (false)
Contropositive : If two lines intersect, then they are coplanar. (True)
Venn Diagrams!!!
You use Venn diagrams to represent conditionals. You Use Venn Diagrams/Truth Tables to determine Logical Equivalence.
BUT HOW!!?! You might ask. Well I’m getting to that. It’s easiest to use
EXAMPLES :
G: all runners are athletic.
P: Leroy is a runner.
C: Leroy is an athlete.
HOW DID I JUST DO THAT!
Well it is quite simple you use the Venn Diagram ( Since Ms.Hunt doesn’t want us using the Truth tables since they will confuse us).
The P is runners and the Q is Athletic. So then what is Leroy? Leroy is a runner as given so he goes in the Runner section or Q. and Q is inside of P. Since in the given.
Then it is quite obvious that Leroy is an Athlete since he is also in the P section.
Or you can look at the given info and see that runners are athletes and Leroy is a runner making him an athlete. TRANSITIVE PROPERTY!!
MORE EXAMPLES : With INVERSES
P: Lucia is NOT an Athlete
C: Lucia is NOT a runner
The conclusion is true. We are still using the same given as the previous Question/ Example.
The reason Lucia is not an athlete is because She is neither in the runner circle nor the Athlete circle so she is a lame-o.
And that’s all folks! Here is one helpful video one funny video and an amazing link to tons of awesome info!
http://www.blinkx.com/watch-video/g1d-converse-inverse-and-contrapositive/z_3OtHw6mD_ALGPXFjtUZQ
http://vimeo.com/15367677
http://accgeo435.blogspot.com/
And the NEXT SCRIBE IS!!!!!!!!!!!
Well uh....
Almost there....
If Not Phil J, then it is not Ms.Hunt
If not Ms.Hunt, then is is not David.
If not David then it is not Francis
If not any other person but one, then it's..............
KATIE HAHAHAHAHAH!!!!
If Then Statements!!! AGAIN!!!!
The Statement - If P then Q
Converse or P and Q flipped - If Q then P
NEW INFO ALERT
Inverse or the Not statement – If NOT P Then NOT Q.
Contropositive or the Not Converse – If NOT Q Then NOT P
Logical Equivalence
- A statement and it’s Contrapositive are Logically Equivalent
- A statement is NOT Logically Equivalent to it’s Converse and Inverse.
- Similar to an = and the biconditional
- Results of either
Both True
Both False
Examples of everything above :
True conditional : If two lines are not coplanar, then they do not intersect.
Inverse of it : If two lines are coplanar, then they intersect. (false)
Contropositive : If two lines intersect, then they are coplanar. (True)
Venn Diagrams!!!
You use Venn diagrams to represent conditionals. You Use Venn Diagrams/Truth Tables to determine Logical Equivalence.
BUT HOW!!?! You might ask. Well I’m getting to that. It’s easiest to use
EXAMPLES :
G: all runners are athletic.
P: Leroy is a runner.
C: Leroy is an athlete.
HOW DID I JUST DO THAT!
Well it is quite simple you use the Venn Diagram ( Since Ms.Hunt doesn’t want us using the Truth tables since they will confuse us).
The P is runners and the Q is Athletic. So then what is Leroy? Leroy is a runner as given so he goes in the Runner section or Q. and Q is inside of P. Since in the given.
Then it is quite obvious that Leroy is an Athlete since he is also in the P section.
Or you can look at the given info and see that runners are athletes and Leroy is a runner making him an athlete. TRANSITIVE PROPERTY!!
MORE EXAMPLES : With INVERSES
P: Lucia is NOT an Athlete
C: Lucia is NOT a runner
The conclusion is true. We are still using the same given as the previous Question/ Example.
The reason Lucia is not an athlete is because She is neither in the runner circle nor the Athlete circle so she is a lame-o.
And that’s all folks! Here is one helpful video one funny video and an amazing link to tons of awesome info!
http://www.blinkx.com/watch-video/g1d-converse-inverse-and-contrapositive/z_3OtHw6mD_ALGPXFjtUZQ
http://vimeo.com/15367677
http://accgeo435.blogspot.com/
And the NEXT SCRIBE IS!!!!!!!!!!!
Well uh....
Almost there....
If Not Phil J, then it is not Ms.Hunt
If not Ms.Hunt, then is is not David.
If not David then it is not Francis
If not any other person but one, then it's..............
KATIE HAHAHAHAHAH!!!!
Wednesday, December 7, 2011
6 - 1: Inequalities
It's a new chapter and we're getting to the inequalities.
First things first, Ms. Hunt wasn't in class today and we had an assignment that I presume needed to be finished at home. We both had to take notes on chapter 6-1 and we had an in-book assignment to do. The notes are self-explanatory, but if you forgot the in-book assignment:
Page 206: #1-13.
Ok, getting back to the chapter . . .
Example 1:
Given: AC > AB; AB > BC
Conclusion: AC ? BC
AC > BC because of the Properties of Inequality.
What are the properties of inequality? Well, they're a group of properties of . . . inequality. You can use these for proofs, and you would just say Prop. of Inequality as the reason.
Properties of Inequality:
If a > b and c >/= d, then a + c > b + d.
If a > b and c > 0, then ac > bc and (a/c) > (b/c).
If a > b and c < 0, then ac < bc and (a/c) < (b/c).
If a > b and b > c, then a > c.
If a = b + c and c > 0, then a > b.
Example 2:
Given: AC > BC; CE > CD
Prove: AE > BD
Statements: Reasons:
1. AC > BC; CE > CD 1. Given
2. AC + CE > BC + CD 2. Prop. of Inequality
3. AC + CE = AE; BC + CD = BD 3. Segment Addition Postulate
4. AE > BD 4. Substitution Prop.
Example 3:
Given: Angle 1 is an exterior angle of Triangle DEF.
Prove: m Angle 1 > m Angle D; m Angle 1 > m Angle E.
Statements: Reasons:
1. . . . 1. Given
2. m Angle 1 = m Angle D + m Angle E. 2. The measure if an ext. angle of a triangle = the sum of the measures of the two remote int. angles.
3. m Angle 1 > m Angle D; 3. Prop. of Inequality
m Angle 1 > m Angle E
This specific example proves the following theorem:
Theorem 6-1: Exterior Angle Inequality Theorem
The measure of an exterior angle of a triangle is greater than the measure of either remote interior angle.
OKAY, so I couldn't find a video about Inequalities in Geometry because it's too broad of a term and all that showed up were different types of inequalities - not even from geometry. So I specified my search to "Properties of Inequalities Geometry" and that came up with nothing except for a 30-minute video of this one guy teaching his class about inequalities, but half of them weren't related to this lesson. Then I just tried to find ANYTHING having to do with this specific lesson and nothing came up. So, what I decided to do, was to give a link to main points of EACH section in Chapter 6. It's basically a link to notes for the whole chapter. I figured this would be helpful to understand mainly what the chapter is about.
TURN OFF YOUR SOUND BEFORE GOING TO THIS LINK, THERE ARE ADDS IN THE SIDEBAR AND THEY'RE LOUD.
http://www.scribd.com/doc/522464/Geometry-Notes-Chapter-Six-Inequalities-in-Geometry
I REALLY hope this link is aloud past the filter.
So, it's that time of the day . . . who will be the next scribe.....?
Yeah, Matthew, you should have known it was going to be you from the beginning. HAVE FUN!
First things first, Ms. Hunt wasn't in class today and we had an assignment that I presume needed to be finished at home. We both had to take notes on chapter 6-1 and we had an in-book assignment to do. The notes are self-explanatory, but if you forgot the in-book assignment:
Page 206: #1-13.
Ok, getting back to the chapter . . .
Example 1:
Given: AC > AB; AB > BC
Conclusion: AC ? BC
AC > BC because of the Properties of Inequality.
What are the properties of inequality? Well, they're a group of properties of . . . inequality. You can use these for proofs, and you would just say Prop. of Inequality as the reason.
Properties of Inequality:
If a > b and c >/= d, then a + c > b + d.
If a > b and c > 0, then ac > bc and (a/c) > (b/c).
If a > b and c < 0, then ac < bc and (a/c) < (b/c).
If a > b and b > c, then a > c.
If a = b + c and c > 0, then a > b.
Example 2:
Given: AC > BC; CE > CD
Prove: AE > BD
Statements: Reasons:
1. AC > BC; CE > CD 1. Given
2. AC + CE > BC + CD 2. Prop. of Inequality
3. AC + CE = AE; BC + CD = BD 3. Segment Addition Postulate
4. AE > BD 4. Substitution Prop.
Example 3:
Given: Angle 1 is an exterior angle of Triangle DEF.
Prove: m Angle 1 > m Angle D; m Angle 1 > m Angle E.
Statements: Reasons:
1. . . . 1. Given
2. m Angle 1 = m Angle D + m Angle E. 2. The measure if an ext. angle of a triangle = the sum of the measures of the two remote int. angles.
3. m Angle 1 > m Angle D; 3. Prop. of Inequality
m Angle 1 > m Angle E
This specific example proves the following theorem:
Theorem 6-1: Exterior Angle Inequality Theorem
The measure of an exterior angle of a triangle is greater than the measure of either remote interior angle.
OKAY, so I couldn't find a video about Inequalities in Geometry because it's too broad of a term and all that showed up were different types of inequalities - not even from geometry. So I specified my search to "Properties of Inequalities Geometry" and that came up with nothing except for a 30-minute video of this one guy teaching his class about inequalities, but half of them weren't related to this lesson. Then I just tried to find ANYTHING having to do with this specific lesson and nothing came up. So, what I decided to do, was to give a link to main points of EACH section in Chapter 6. It's basically a link to notes for the whole chapter. I figured this would be helpful to understand mainly what the chapter is about.
TURN OFF YOUR SOUND BEFORE GOING TO THIS LINK, THERE ARE ADDS IN THE SIDEBAR AND THEY'RE LOUD.
http://www.scribd.com/doc/522464/Geometry-Notes-Chapter-Six-Inequalities-in-Geometry
I REALLY hope this link is aloud past the filter.
So, it's that time of the day . . . who will be the next scribe.....?
Yeah, Matthew, you should have known it was going to be you from the beginning. HAVE FUN!
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