4-21 3rd and 4th Block

 

Chapter 5 Triangles Overview w ans

Chapter 5 Triangles      Name __________________

Classifying Triangles by their Angles

Acute Triangles – All angles are less than 90°

 

 

 

Obtuse Triangles – Has an angle greater than 90°

 

 

Right Triangles – Has a 90° angle 

 

Equiangular Triangles – All angles are equal(60°) all

Angles are equal, all side lengths are equal.

 

 

Classifying Triangles by their Sides 

 

Scalene Triangles – All sides are different lengths.

 

 

Isosceles Triangles – At least two sides are congruent (at least two angles will also be congruent)

 

 

Equilateral Triangles – All sides are equal lengths.

 

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In a triangle, the largest angle is across from the __longest______ side of a triangle.  List the angles from smallest to largest.

 

 

 

 

 

 

 

 

 

The smallest angle is across (opposite) from the ___shortest____ side of a triangle. List the sides from smallest to largest.

   

 

 

 

 

 

Triangle Inequality Theorem states that the sum of the lengths of any two sides of a triangle is greater than the length of the third side of the triangle.

 

 

 

AB + BC > AC

AC + AB > BC

BC + AC > AB

 

 

 

 

 

 

 

 

Exterior Angles of a Triangle

 

 

 

Remote interior angles

 

 

A + B + C = 180

C + BCD = 180

 

The measure of an exterior angle of a triangle is equal to the _measure___ of the measures of the two __remote____ _interior____ ___angles____.

 

 

 

 

The Exterior Angle Inequality Theorem states that the measure of an exterior angle of a triangle is greater than the measure of either of its remote interior angles.

 

 

 

 

 

 

 

 For more detail information on constructions, see yesterday's post and view the websites listed.

 

                                                                                            Point of Concurrency

 

Angle Bisectors of a Triangle- Divides the Angles in half       

_Incenter_

 

Medians of a Triangle - From the vertex to the middle of the      ___Centroid______

opposite side      

 

Altitudes of a Triangle – Right angle with the opposite side,      ___Orthocenter___

through the vertex      

 

Perpendicular Bisectors of the sides of a Triangle – Bisect the      ___Circumcenter___

Sides of the triangle at a right angle 

 

 

 

Triangle Congruence Postulates

Angle-Side-Angle

 

  

 

Side-Side-Side

 

    

 

Side-Angle-Side

 

 

  

 

Angle-Angle-Side

   

 

Hypotenuse-Leg

 

   

4-19 3rd and 4th Block

4/19 3^rd and 4^th Block

 

Triangle Types

class="Normal__Char" style=" text-decoration: none;">By Angle	By Side
Acute – all angles less than 90°	Scalene – all three sides are different
Right – has a 90° angle	Isosceles – at least two sides are the same
Obtuse – has an angle greater than 90°	Equilateral – all three sides are equal

 

Remember any triangle can be categorized by both their sides and their
angles.

 

Below are links to constructions of the four centers of circles we
looked at today.

 

Median – http://www.mathopenref.com/constmedian.html

 

Perpendicular Bisector – class="Hyperlink__Char">http://www.onlinemathlearning.com/perpendicular-bisector.html

 

Angle Bisector – class="Hyperlink__Char">http://strader.cehd.tamu.edu/geometry/bisectangle1.0/bisectangle.html

 

Altitudes – class="Hyperlink__Char">http://www.mathopenref.com/constorthocenter.html

 

Vocabulary to know…

 

1. Acute Triangle


2. Obtuse Triangle


3. Right Triangle


4. Exterior Angle


5. Interior Angles


6. Scalene Triangle


7. Isosceles Triangle


8. Equilateral Triangle


9. Angle Bisector


10. Points of Concurrency


11. Incenter


12. Bisect


13. Circumcenter


14. Median


15. Centroid


16. Altitude


17. Orthocenter

 

 

4-13 1st Block

 

4-12 3rd and 4th Block

Mathematics I              Unit 4                                                                       2^nd Edition

 

Spinner Learning Task 1

 

 

Make an arrow for the spinner above so that the spinner is useable.  You can use brads and a paper clip for an arrow.  Is this homemade spinner fair?  How can you tell?

 

 

 

 

 

 

 

 

 

 

 

 

 

Calculate the following probabilities for the spinner (assuming the spinner is fair).

 

 

1) What is the probability of obtaining $800 on the first spin?

 

 

2) What is the probability of obtaining $400 on the first spin?

 

 

3)   Is it just as likely to land on $100 as it is on $800?

 

 

4)   What is the probability of obtaining at least $500 on the first spin?

 

 

5)   What is the probability of obtaining less than $200 on the first spin?

 

 

6)   What is the probability of obtaining at most $500 on the first spin?

 

 

7)   If you spin the spinner twice, what is the probability that you will have a sum of $200?

 

 

8)   If you spin the spinner twice, what is the probability that you will have a sum of at most $400?

 

 

9)   If you spin the spinner twice, what is the probability that you will have a sum of at least $1500?

 

 

 

10)         If you spin the spinner twice, what is the probability that you will have a sum of at least 300?

 

 

 

11)         Given that you landed on $100 on the first spin, what is the probability that the sum of your two spins will be $200?

 

 

 

12)         Given that you landed on $800 on the first spin, what is the probability that the sum of your two spins will be at least $1000?

 

 

 

 

Spinner Learning Task 2 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1)  If you spin the spinner once, how much money, on average, would you expect to receive?

 

 

 

2)  If you spin the spinner twice, how much money, on average, would you expect to receive?

 

 

 

3)  If you spin the spinner 10 times, how much money, on average, would you expect to receive?

 

 

 

 

4)  On average what is the fewest number of spins it will take to accumulate $1000 or more? 

 

 

Spinner Learning Task 3

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1)   What is the probability of obtaining $800 on the first spin?

 

 

 

2)   What is the probability of obtaining $500 on the first spin?

 

 

 

3)   Is it just as likely to land on $100 as it is on $800?

 

 

 

4)   What is the probability of obtaining at least $500 on the first spin?

 

 

5)   What is the probability of obtaining less than $200 on the first spin?

 

 

 

6)   What is the probability of obtaining at most $500 on the first spin?

 

 

 

The following questions will be a little more difficult because there are two spins involved.  You may want to draw a tree diagram and label the branches with the associated probabilities.  Since each spin should be independent of each other, you can multiply the probabilities.

 

7)   If you spin the spinner twice, what is the probability that you will have a sum of $200?

 

 

 

8)   If you spin the spinner twice, what is the probability that you will have a sum of at most $400?

 

 

 

9)   If you spin the spinner twice, what is the probability that you will have a sum of at least $1500?

 

 

 

10)         Given that you landed on $100 on the first spin, what is the probability that the sum of your two spins will be $200?

 

 

 

 

11)         Given that you landed on $800 on the first spin, what is the probability that the sum of your two spins will be at least $1500?

 

 

 

12)         If you spin the spinner once, how much money, on average, would you expect to receive?

 

 

 

13)         If you spin the spinner twice, how much money, on average, would you expect to receive?

 

 

 

14)         If you spin the spinner 10 times, how much money, on average, would you expect to receive?

 

 

Spinner Learning Task 4

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

This spinner is different than yesterday’s spinner because the sections are not of the same area. 

 

 

1)   What is the probability of obtaining $800 on the first spin?

 

 

 

2)   What is the probability of obtaining $500 on the first spin? 

 

 

 

3)   Is it just as likely to land on $100 as it is on $800?

 

 

 

4)   What is the probability of obtaining at least $500 on the first spin?

 

 

 

5)   What is the probability of obtaining less than $200 on the first spin?

 

6)   What is the probability of obtaining at most $500 on the first spin?

 

 

 

7)   If you spin the spinner twice, what is the probability that you will have a sum of $200?

 

 

 

8)   If you spin the spinner twice, what is the probability that you will have a sum of at most $200?

 

 

 

9)   If you spin the spinner twice, what is the probability that you will have a sum of at least $1500?

 

 

 

10)                     Given that you landed on $100 on the first spin, what is the probability that the sum of your two spins will be $200?

 

 

 

11)                     Given that you landed on $800 on the first spin, what is the probability that the sum of your two spins will be at least $1500?

 

 

 

12)                     If you spin the spinner once, how much money, on average, would you expect to receive?

 

 

 

13)                     If you spin the spinner twice, how much money, on average, would you expect to receive?

 

 

 

14)                     If you spin the spinner three times, how much money, on average, would you expect to receive?

 

 

 

15)                     How many times on average would you have to spin until you land on “bankrupt?”

 

 

 

 

Spinner Learning Task 5:

A student created a spinner and records the outcomes of 100 spins in the table and bar graph below.

Amount of Money on each section of the spinner	$0	$100	$200	$300	$400	$500	$600
Number of times the student lands on that section	30	10	15	20	10	10	5

 

 

1.                                                 Based on the table and graph above, calculate the experimental probabilities of landing on each section of the spinner.  Use these probabilities to draw what the spinner would look like below:

 

 

 

 

2.                                                 Calculate the average amount of money a person would expect to receive on each spin of the spinner.

 

 

 

 

3.                                                 Calculate the probability that you will receive at least $400 on your first spin.

 

 

 

 

4.                                                 Given you land on $300 the first time, what is the probability that the sum of your first two spins is at least $600?

 

 

5.                                                 What is the probability that the sum of two spins is $400 or less?

Spinner Learning Task 7

 

 

Suppose you are playing “Wheel of Fortune” and you are the first player to spin the wheel.  Is it likely that you will solve the puzzle (provided you never land on “bankrupt” or “lose a turn”)? In order to answer this question, calculate the following probabilities.  Use those probabilities in your explanation.

 

1.       Suppose you are the first player to spin the wheel, and you do not know the phrase, what is the probability that you guess a correct letter?

 

 

 

2.       If it turns out that no letters in the word are repeated, what is the probability that the 2^nd letter you guess is correct (if you still do not know the phrase)?

 

 

 

3.       So, if you have a 5 letter word, and no letters in the word are repeated, what is the probability that you guess all 5 correct (provided that you still have no idea what the word is even after the 4^th letter)?

 

 

 

4.       Suppose that you are the 3^rd player to spin the wheel.  You know that “S” and “T” are not in the phrase since those were the guesses of the first two players.  What is the probability that you guess all 5 letters correctly if you never know what the word is?

 

 

 

Spinner Learning Task 8

 

Suppose you are playing the bonus round on the game show, “Wheel of Fortune.” 

 

1.    If you were allowed to pick any 8 consonants and any 2 vowels, which letters would you pick? 

 

2.    In a small group, play the “bonus round” from “Wheel a Fortune.”  This is a modified version of hangman.  Let one member of your group come up with a phrase.  You are to use only the 8 consonants and 2 vowels that you pick.  Record the time it takes you to guess the phrase (do not take more than 3 minutes per phrase).  Perform this simulation within your group 5 times.  Record the length of time it took you to guess each of the 5 phrases.

 

 

 

3.    Record the class data below. 

 

 

 

 

 

4.    Calculate the 5 number summaries and draw a box plot.

1. Are there outliers? 

 

2. What is the interquartile range?

 

3. Calculate the mean and mean deviation of the class data.

 

4. Which is a better measure of center to use, the mean or the median?  Why?

 

5. Which is a better measure of spread to use, the interquartile range (IQR) or the mean deviation?  Why?

 

 

5.    Now, use statistics to determine which letters and consonants are used the most in the English language.

·       Prior to beginning this simulation, make a tally sheet.  Write down all 26 letters to the alphabet in a vertical column on your notebook paper.

 

·       Next, open a book/novel, close your eyes, and put your finger somewhere on the page.  Begin at that spot and count how many A’s, B’s, C’s, etc. occur in the first 150 letters that they see.  Tally on the notebook paper

 

 

 

 

 

 

6.               Based on your simulation, answer the following questions:

·       Which 8 consonants and 2 vowels are used the most often in the English language?

·       Compare your answer with your classmate.  Did you pick the same letters?

·       Write the class data below.  Compute the percent of A’s, percent of B’s, percent of C’s, etc. from the class data.

·       Compare your individual answer to the class answer.

Answer the question, “which 8 consonants and 2 vowels are used most often in the English language?”

 

7.               Using the 8 consonants and 2 vowels that are used most frequently in the English language, play the bonus round of wheel of fortune again within your group five times. Record the time it takes you to guess the phrase (do not take more than 3 minutes per phrase). 

 

The teacher will collect the class data.  Using the class data, calculate the 5 number summary (minimum, 1^st quartile, median, 3^rd quartile, and maximum) and draw the associated box plot.

 

 

Compare and contrast the two box plots (old box plot before we knew the most frequently used letters with this box plot).  In your explanation, you should compare the centers (medians), the IQR’s (interquartile range), and the shapes (skewed or symmetric).  You should then use these values to answer the following questions:

 

a.               Did you save time today by using the letters we found to be used most often?  Explain.

 

b.               It took more than ______ seconds to answer 25% of the puzzles when we randomly provided the letters.  It took more than _____ seconds to answer 25% of the puzzles when we used the most frequently used letters.

 

c.                We answered 25% of the puzzles in less than _______seconds when we randomly guessed the letters.  We answered 25% of the puzzles in less than _________ seconds when we used the most frequently used letters.

 

d.               It took more than ________ seconds to answer half of the problems when we randomly provided the letters.  It took more than ______ seconds to answer half of the problems when we used the most frequently used letters.

 

e.               The bonus round only allows the player about 10 seconds to guess the phrase.  Based on that, would we win more often or less often by randomly guessing or by using the frequently chosen letters?

 

 

Spinner Learning Task 9  

 

Susan played the bonus round of wheel of fortune 30 times.  She recorded how long it took her to guess the phrase to the nearest second.  The following are the lengths of time it took her to guess each phrase correctly: 

10, 11, 11, 12, 12, 12, 13, 13, 14, 14, 15, 15, 15, 15, 17, 18, 19, 21, 24, 24, 24, 26, 28, 31, 33, 34, 35, 35, 37, 40

 

 

Monique also played to bonus round of wheel of fortune 25 times.  She recorded how long it took her to guess the phrase to the nearest second.  The following are the lengths of time it took her to guess each phrase correctly:

12, 13, 13, 13, 13, 14, 14, 14, 14, 14, 15, 15, 15, 15, 15, 15, 16, 16, 16, 17, 17, 17, 18, 18, 55

 

 

1. Graph the two distributions below.  Which measure of center (mean or median) is more appropriate to use and why?  Calculate that measure of center.

 

 

2. Comment on any similarities and any differences in Susan’s and Monique’s times. Make sure that you comment on the variability of the two distributions.

 

 

3. If you are only allowed 15 seconds or less to guess the phrase correctly in order to win, which girl was more likely to win and why?

 

4. If Susan found that she could have guessed each phrase 3 seconds faster if she had chosen a different set of letters, would that have made any difference in your answer to part c?  Why/why not?

 

 

 

 

Georgia Department of Education

Kathy Cox, State Superintendent of Schools

May 5, 2008

Copyright 2008 © All Rights Reserved

Unit 4, Page 8 of 12

4-12 1st Block

 

Math 1 Review Combined 1

This review is due April 14th and I am only counting 90 questions. Any questions you get right greater than 90 is a bonus to you. This is a BIG grade and is designed only to help you study for your final exam which is May 12 and 13. I am aware that we have yet to cover all of this material, that is why only 90 of 144 questions count. Please do your very best work. Remember that if the question is multiple choice you must show your work and the answers were included in the packet you received in class to help you. There may be errors for which I apologize.