Unit Plan

Adolescent Exceptional Learners

EDUC 352

December 8, 2010

Jesse Steffen

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A. TEXTBOOK/COURSE INFORMATION

NAME OF COURSE/GRADE LEVEL: Algebra II honors/sophomores and juniors primarily

DESCRIPTION OF COURSE: This course is designed to study algebra in more depth as well as teach the students more about algebra. This course is algebra in probability, geometry, statistics, and discrete math to help the students prepare for future classes.

NAME OF CHAPTER/UNIT: Probability and Statistics

DESCRIPTION OF CHAPTER/UNIT: In this chapter students will learn probability, statistics, distribution, deviation, and be able to analyze the data

TITLE OF TEXTBOOK: Prentice Hall Mathematics Algebra 2

NAME(S) OF AUTHOR(S)/EDITOR(S):Allan E. Bellman, Sadie Chavis Bragg, Randall I. Charles, William G. Handlin Sr, Dan Kennedy

NAME OF PUBLISHING COMPANY: Peasron Prentice Hall

COPYRIGHT DATE: 2004

READING LEVEL OF TEXTBOOK: 9th-12th grade

B. PHILOSOPHY OF READING IN THE CONTENT

STANDARDS: A 2.9.2: Use the basic counting principles, combinations, and permutations to compute probabilities. A2.10.1: Use a variety of problem-solving strategies, such as drawing a diagram, guess-and-check, solving a simpler problem, writing an equation, and working backwards. A2.10.2: Decide whether a solution is reasonable in the context of the original situation.

IMPORTANCE: This chapter is over probability and statistics. We study probability to show the students

how to make smart choices in real life situations. They will know what their chances are on winning the lottery, carnival games, etc. In statistics they will learn how to interpret the different statistics around them as well as calculate statistics themselves. Statistics are everywhere from sports to the weather. You can not read a newspaper without seeing a good amount of statistics.

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PHILOSOPHY: Reading in math is a very important skill to attain to help open up your possibilities

later in life. Many students that are strong readers in their other classes often struggle with math. Why is this the case? Math is a different kind of reading. It sometimes takes a reader two or three times over the same paragraph to understand what is being read. The reason for this (at least for me) is because while you are trying to understand what they are trying to say, you are also trying to figure out the math part of how they went from step to step. The words and the numbers are supposed to help each other but it is still hard to mentally jump back and forth. Its unlike reading in any other subject.

Why would you need this later in life? First off college. I know not everyone will go to college but I feel like we should prepare all the students for college to help the kids that are going to college as well as encourage the kids that do not plan to. Almost every college requires you to take some sort of math class. In these math classes it is not uncommon for a teacher to assign you reading so they do not have to go over it in class. If you are not good at reading math texts, you will struggle in this type of situation. Secondly there are math journals as well as math used in newspapers and magazines frequently. A lot of people would get flustered and have to reread these sections, but a student that has had the practice will go by it like it is any regular text. Just the other day I read an article that was in the New York Times about expected values in football. So even if the students do not feel like it is necessary to practice reading math texts, no matter what you do with your life you will find opportunities to use this skill.

C. READBILITY

FRY READBILITY

Archimedes considered the perimeters of inscribed and circumscribed regular polygons of 96 sides. Seven hundred years later, the Chinese mathematicians Tsu Ch'ung-Chi calculated a more precise value. English mathematicians introduced the notation π about 1700. π is the first letter of the Greek word for perimeter. In 1761, the German mathematician Johann Lambert proved that pi is an irrational number. Although there is no finite formula for calculating pi, mathematicians have found many formulas that involve infinite numbers of additions, subtractions, multiplications, or divisions. More than 500 years ago, the Indian mathematician Madhava discovered another formula. This formula has an (page 11)Number of sentences: 7.5Number of syllables: 191

You have learned to multiply binomials using the FOIL method and the Distributive Property. If you are raising a single binomial to a power, you have another option for finding the product. Consider the expansion of several binomials. To expand a binomial being raised to a power, first multiply; then write the result as a polynomial in standard form. In the first case, the

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coefficients of the product are 1, 2, 1. In the second case, they are 1, 3, 3, 1. Notice that each set of coefficients matches a row of Pascal's Triangle below. Pascal's Triangle is a triangular array of numbers formed by (page 347)Number of sentences: 6.3Number of syllables: 167

In the past, you have used degrees to measure angles. When angles are used in periodic functions, they are often measured in larger units called radians. Measure the diameter of a cylinder and calculate its radius. On a piece of string, mark off a “number line” with each unit equal to the radius. Mark at least seven units. Wrap the string around the cylinder. How many radius units are needed to go around the cylinder one more time? Use the end of the cylinder to draw a circle on a sheet of paper. Keep the cylinder in place and wrap the (page 711)Number of sentences: 8.6Number of syllables: 148

AVERAGE NUMBER OF SENTENCES: 7.5 AVERAGE NUMBER OF SYLLABLES: 169READING LEVEL OF THE TEXTBOOK: 17 years old

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Reflection Over Fry Readability Test:I think this is pretty accurate. I feel like any math text will be a lot harder to comprehend than other textbooks so this number might be somewhat inaccurate. For most book you are able to turn to the first page, count out the first 100 words there and go from there. For math books it is hard to find 100 words that are together and do not have a bunch of numbers and questions in them. That was the hardest part about finding this number. This is one of the reasons we have to get students to read more of their math textbooks. It is a different kind of reading than over classes and the only way to get good at it is practice. Also most students taking algebra 2 are freshman and sophomores and the reading level shows more for juniors. I feel like this is still ok because algebra 2 is designed for sophomore or junior level students and the ones that are taking it as a freshman are usually the higher level students. This means they should still be able to read it just fine, but teachers should still be aware of students that can not.

D. TRADE BOOKS/ANNOTATED BIBLIOGRAPHY

Annotated Bibliography:

Abbott, E. A. (2010). Flatland. Pittsburgh: Bibliopolis Books. In this book the characters are two-dimensional geometric shapes. They all think that their planar world is the only one that exists. These characters are known as “Flatlanders.” One Flatlander is a square who discovers a third dimension. In discovering this he realizes the confusing problems in having a higher dimension. This book also has funny satire dealing with the society of Victorian England.

Egan, G. (1995). Permutation City. New York: Eos; Book Club (BCE/BOMC).In this book the main character is an electronic code that is living in a virtual reality program. In this program you can not die, and you live on forever. You are a copy and you know it. The only way to escape this fate is to terminate yourself and go back into a flesh and blood human. The only problem for the main character is that he can not terminate himself because the program to do so is blocked. The irony is that he knows the person who blocked it, it is the person he is a copy of. The real version of him, he wants to keep his copy in the virtual world forever.

Enzensberger, H. M. (2000). The Number Devil. New York:Holt Paperbacks.The main character Robert has a dream of a place of the number devil. He is the host and asks Robert math questions that have number tricks. They use giant furry calculators, coconuts, and much more to solve problems. If Robert tries to use an easy way out, the devil will lose his temper or simply say that the math is boring and useless. The humor and adventure in this book is so captivating that you do not realize the mathematical principles you are learning.

Stephenson, N. (2002). Cryptonomicon. New York: Avon Books. This book jumps back and forth between two times periods, present-day and back to World War II. There are two people the book focuses on in the 1940's era. One of them is

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Lawrence Watherhouse who is a cryptanalyst extraordinaire. The other one is Bobby Shaftoe who is a morphine-addicted marine. They are part of a group that works to break enemy communication codes as well as make their own codes that are too difficult for the enemy to break. All of the things happening in the past are directly relating to things that happen in the present in the lives of the two grandchildren of Waterhouse and Shaftoe. One of them is a programming geek while the other is a lovely and powerful girl. The two are working together to create an off shore data haven in Southeast Asia. This book has a math problem, humor, or an interesting idea on almost every page.

Tahan, M. (1993). The Man Who Counted. New York City: W. W. Norton & Company. This book is about an Arabian man that is very good at math. As he travels he uses his skills to help settle conflicts. His adventures involve solving math puzzles and gives a look into some of the mathematical minds of history. This book combines math as well as good storytelling with an “Arabian Nights” style that makes it fun to read. The puzzles fit in the story so well that you do not even realize they are there.

Trade books in mathematics:These trade books will greatly enhance my classroom content. With math more than

other subjects, it is way to easy to get caught in the textbook learning. It is very hard to make a math textbook fun. These books teach some of the same concepts that are in those textbooks, but they do it in a way that is fun. They are well written enough that they can either teach the student without them knowing they are learning, or spark the interest of the student so he wants to learn how to do something. Any book that makes math interesting and fun while teaching the student is a success for mathematics. I wish that I would have known about these books when I was growing up.

E. LESSON PLANS

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Lesson Plan #1

Lesson: Introduction to probability and statistics (12-1) Length: 45 minutes Age or Grade Level Intended: Algebra 2 Honors

Academic Standard(s): A 2.9.2: Use the basic counting principles, combinations, and permutations to compute probabilities.

Performance Objective(s): Given 5 probability problems, students will answer them using counting and combinations with 60% accuracy. The reason this is low is because I want to know what the students already know, what they learned from my presentation, and what I have to go over again in my next lesson

Assessment: A worksheet with 5 probability problems will be given as homework and will be handed in tomorrow.

Advance Preparation by Teacher: A deck of cards, 2 dice, 1 spinner (#2-12 in equal sections), a Vegas style dealers outfit, a 2 liter of coke, sprite and Dr. Pepper with cups, and a table in front of the class with a sign on front saying “Welcome to Vegas”

Procedure:Introduction/Motivation: The students come into the class and are met with you

standing behind a table in the front of the room with your dealer attire. When the students walk into class you ask them if they would like a drink. When the bell rings you walk behind the table in the table and say, “Welcome to Vegas! You all have $50 to spend. There are 3 separate games you can use this on: Face Plant, Roll of the Dice, and The Spinning Wheel. The only guidelines is you can only pick one game to play and you have to bet all your money on one game. You win, you walk away with $100; you lose, you walk away $50 in the hole.” You then explain the 3 games to them.Face Plant: You are given 2 cards for a standard 52 deck. If both those cards are face cards you win.Roll of the Dice: 2 dice are rolled, if the sum of the dice are 7 you win.The Spinning Wheel: There is a wheel with equal sections labeled 2-12, you spin the spinner in the middle and if it lands on 7 you win.

Step-by-Step Plan:1. Have all the students get up and go to the right of the room if they would play Face Plant,

the left of the room if they would play Roll of the Dice, and to the back of the room for the Spinning wheel. Pick a student or two from each group and ask why they made their choice. (when they are done tell them to go back to their seats) (Gardner:

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Bodily/Kinesthetic) (Gardner: Verbal/Linguistic) (Bloom: Analysis)2. Explain the probability of winning Face Plant: Out of the 52 cards your first card has to

be a face card. A face card includes a jack, queen, or king in any suit. There are 4 suits and 3 face cards each so you have 12 cards you could draw. So the probability is 12/52 (or 3/13). If you do not choose a face card in the first pick you already loose. For your second card there are still 11 face cards out there (since you already drew one), there are also only 51 cards (for the same reason) left in the deck. So the chances of you picking out a face card is 11/51. To find the probability of winning you take both of these probabilities and multiply them. So the probability of you winning is 33/663 which reduces to 11/221. So you have a 4.98% chance of winning.

3. Roll of the Dice: You have 2 dice. Here is a table of all the outcomes possible when the dice are rolled. The left is the number that the first die landed on, and the top is the number of the second die. Where they meet in the chart is what their sum would be.

(Draw these on the board quickly) The graph shows there are 6 different ways to roll a 7. There are 36 total possibilities, so there is a 6/36 (or 1/6) chance of winning, so there is a 16.67% chance of winning. (Gardner: Visual/Spatial)

4. The Spinning Wheel. There are 11 equal sections, therefore every section of the wheel has an equal chance of being landed on. Since there is only one that is a 7, you have a 1/11 chance of landing on it. So you have a 9.1% chance in winning.

5. When we compare these probabilities (Face Plant= 4.98%, Roll of the Dice=16.67%, and Spinning the Wheel= 9.1%) we see that we have the best chance in winning in Roll of the Dice.

6. Ask the Students why they think the probability was different in Roll of the Dice and The Spinning Wheel. Tells them to turn to their neighbors and briefly discuss. (if they do not figure it out explain that its because there are more than one way for the sum to be 7 on the die but only 1 way to spin a 7) (Gardner: interpersonal, Bloom: synthesis)

7. Tell them to get out their books and turn to page 636. choose someone to read what a frequency table is (a list of outcomes in a sample space and the number of times each outcome occurs)

8. Ask them what we did in our activity that relates to this. (the rolling 2 dice graph)(Bloom: Knowledge)

9. Ask another student to read what probability distribution is on the next page (a function that gives the probability of each event in a sample space)

10. Ask the students what we could change on the graph to make it into a graph showing the

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probability distribution. (put the numbers in the Y axis over 36 [total number of outcomes for 2 dice)(Bloom: Evaluation)

Closure: Tell the students to think about the game “Rock, Paper, Scissors.” Tell them to think about how a frequency table would look. What would be the probability of winning 2 in a row? 3 in a row? What about best 2 out of 3? By the end of this unit the students will be able to give a survey, graph the results, and be able to predict probabilities from their surveys.

Reflection: The worksheet will let me know if the students are able to solve some simple probability problems. If the students are able to answer 3 out of 5 correctly (60%) it will be ok to move on to the chapter. If they can not we will go over what they got wrong and I will give them a few more to walk through in class until I am confident they are ready for the chapter.

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Introduction WorksheetAnswer these on a separate sheet of paper

1) You have 20 pairs socks in your drawer. You have 5 black pairs, 4 blue pairs, 8 white pairs, and 3 green pairs. Draw a frequency table OR graph when you pick out a pair of socks at random to wear to school.

2) What is the probability of choosing a BLACK pair?

3) What is the probability of choosing a WHITE pair

4) Draw a probability distribution graph for your answer in problem 1.

5) What is the probability of picking a GREEN pair, and then drawing a WHITE pair since

green did not match your outfit? (The question is not asking for the probability green

matches your outfit) (Hint: Since you do not like the green pair you do not put them back

in the drawer before you pick out another pair)

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Lesson Plan #2

Lesson: Probability Distribution/ Vocab (12-1) Length: 45 min Age or Grade Level Intended: Algebra 2 Honors

Academic Standard(s): A 2.9.2: Use the basic counting principles, combinations, and permutations to compute probabilities.

Performance Objective(s): Given 15 problems from the Probability Distribution section in the text, students will answer them with 80% accuracy.

Assessment: 15 problems from the text will be given as homework and will be handed in tomorrow.

Advance Preparation by Teacher: Answers to the worksheet due for today, number 3 and 10 on page 639-640 done to use as examples, and a vocab hand out with the sections 1, 2, and 3 vocab and definitions.

Procedure:Introduction/Motivation: Hand out the vocab sheets. Point out a few of the

vocab words we are already used (frequency table, probability etc). Many of these words they will see again because they are continuing the same lesson they did yesterday. It is just about the same with a little bit of extra steps with it. Tell the students you are going to separate them into groups and give each group a different set of the vocab list. Explain to the students to work with their group to think of ways to remember the definition of the words. Make sure they know it can be anything to help them remember, not just math related. It could be a song they know, it could rhyme with something, or it could even seem totally out the the blue.

Step-by-Step Plan:1. Put the students into groups of 4 or 5 and give them a section of the vocab to

create memorization tips for. (Gardner: Interpersonal, Bloom: Synthesis)2. Give them the example of “mean”. When you think of mean, you think of its

relatives median and mode. Well you can remember mean because it's the mean one; it makes you add everything up and then divide by the number of terms to figure it out. Mode and median are much easier to find.

3. Give them about 10 min to discuss this.4. Go to each group and ask them their memory tricks. Tell the other students to take

notes so they have a way to study them. (Gardner: Verbal/Linguistic, Bloom: Comprehension)

5. If a group has a word that they could not think of anything, ask the rest of the

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class if they can think of one for that word. If they, and you, can't, then tell them to try and think of one for tomorrow. Be sure to tell them it's not for a grade.

6. Tell them to get out their worksheet that is due today and ask them to find some vocab words in this homework assignment. Use a highlighter or circle these words.

7. Go over homework on the board. For #1 have one volunteer draw the frequency table and another draw the graph. Once they are done ask the rest of the class how the table and graph on the board compare to their own.(Gardner: Visual/Spatial, Bloom: Analysis)

1) Frequency table(in tally marks):

black=5blue=4white=8green=3

2) 5 black socks, 20 total. 5/20=1/43) 8 white, 20 tot. 8/20=2/54) (see the graph on #1, just put all the numbers over 20)5) (3/20)(8/19)=6/95 first pick there is 20 socks, but since it is not replaced, the second pick there are only 19 pairs left.

8. Have a volunteer go up to the board and go over #3 on page 639 on the board as a quick example. This uses a lot of things the students already know but is a little bit new, so if they struggle ask the class what they should do next. If they do not know then give them help. (Gardner: Bodily/Kinesthetic)Steps for #3:

i. P(5 or more responses), To figure this out you need to know the probability of 5 responses and 6 or more responses.

ii. Refer to the table to find that P(5 responses)=16/200 and P(6+ responses)=6/200.

iii. Since it says “5 OR more responses” you add these 2 numbers. If it said “AND” you would multiply these probabilities.

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iv. SO: P(5 responses)+P(6+ responses)=16/200+6/200=22/200 =11/100=0.11

9. Go over #10 on page 640 as an example. Make sure to tell the students that we have not done this yet so they should pay close attention and take good notes. When you are done tell them to look at the data and the answer we got and ask themselves if it logically makes sense to them. Stress that they should always take time to do this. ( Bloom: Evaluation)

i. Construct a different table using the table in the book:Age Probability Cumulative Prob. Assigned Numbers<20 0.051 0.051 1-5120-29 0.176 0.227 52-27730-39 0.211 0.438 278-43840-49 0.211 0.649 439-64950-59 0.156 0.805 650-80560-69 0.096 0.901 806-90170-79 0.070 0.971 902-971>=80 0.029 1 972-1000

ii. Explain that the probability is the probability that a randomly selected person of that age has their license. (ex: you have a probability of 17.6% of selecting someone age 20-29)

iii. We assign the numbers by assuming we pick 1000 people.iv. From here we pick random numbers from 1-1000 and then see

what category they are in. Once we get 20 of them we stop and count the number of people we have in each group. (ex: 37, 503, 690, 550. Go to assigned number to find what age the numbers are in. 37=<20; 503=40-49; 690=50-59; 550= 40-49; so there are 1 person in <20 and 50-59, but 2 people in 40-49.)

10. Assign homework problems 1-14 and 16 on page 639-640.

Closure: Go over some of the best tricks they thought of to memorize some vocab words. Point out some of the vocab they will use on the homework tonight (Cumulative Probability and Probability Distribution). Tell the students to think about how the probabilities would change on their worksheet that was due today if #2 was: What is the probability of picking a black sock given that you just picked a white sock? This is what we will learn next lesson.

Reflection: The worksheet handed in today will let me know how well the students know

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this material already. The vocab game will hopefully get the kids to be creative and break out of the math cycle for a little bit. This will also help them remember their vocab words for the first half of a chapter. If I find that the students still do not know their vocab after the first couple lessons I will give them a quiz over it so they study and learn them.

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Chapter 12 Vocab12-1Cumulative Probability= Probability over a continuous range of events.

Frequency Table= A list of the outcomes in a sample space and the number of times each outcome occurs.

Probability Distribution= A function that gives the probability of each event in a sample space.

12-2Conditional Probability= Contains a condition that may limit the sample space for an event.

12-3Mean= sum of the data values divided by the total number of data values (average).

Measure of Central Tendency= The single, central values that help describe a set of data (mean, median, and mode).

Median= The middle value OR mean of the 2 middle values (if there is an even number of values).

Mode= The most frequently occurring value.

Bimodal= Data sets that has 2 modes.

Quartiles= The values separating the 4 parts found by finding the median, and then the median of the 2 parts (the median of the median).

Percentile= a value that divides the range of a data set into 2 parts such that the part below the percentile contains a given percent of the data.

Box-and-Whisker Plot= A method of displaying data that uses quartiles to form the center box and min and max values to form the whiskers.

Outlier= An item of data with a value substantially different from the rest of the items in the data set.

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Lesson Plan #3Jesse Steffen

Lesson: Conditional Probability (12-2) Length: 45 min Age or Grade Level Intended: Algebra II Honors

Academic Standard(s): A2.9.2: Use the basic counting principles, combinations, and permutations to compute probabilities.

Performance Objective(s): Given 10 problems on computing conditional probabilities, students will be able to answer them with 80% accuracy.

Assessment:10 problems from the text will be given as homework and will be handed in tomorrow.

Advance Preparation by Teacher: The hand out “Chances Are” from the New York Times Opinion Page.

Procedure:Introduction/Motivation: As you hand the students the article from the New

York Times Opinion Page, explain that yesterday we were doing the probability of something happening. Today we are going to take that a step further and learn about finding the probability of something happening if we know something else already happened. For example, the probability that it will rain today knowing that it rained yesterday. Once they all have a copy Tell them you are going to read it out loud to them as they follow along. Tell them to circle the part in the paper that defines conditional probability. Write the words “conditional probability” on the board so they do not forget. This not only defines conditional probabilities it also gives some real life examples. This is a good break from the text book and shows them a way to use what they are learning in real life as well as other professions not typically connected to math. (Gardener's Verbal/Linguistic)(Gardener's Naturalist)

Step-by-Step Plan:1. After reading through the article, ask the students what the definition of conditional

probability is (Bloom's Knowledge). Tell them to open their books and see how closely that definitions goes with the one in our book. (Bloom's Analysis)

2. Go over example 1on page 642. This is explained in the text so just go through it in more depth on the board by drawing the table and putting in circle or arrows to emphasize where you get the numbers. (Gardener's Visual/Spatial)

3. Read the property box about the Conditional Probability Formula. Do a quick review of how to do the probability of A and B. Do the following example on the board:

If you have a drawer with 5 red socks, 3 blue socks, and 7 white socks and a standard

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deck of cards, what is the probability of picking a blue sock from the drawer and a diamond from the deck of cards? P(blue sock)= 3/15=0.05, P(diamond)=13/52=0.25. P(blue sock and diamond)= 0.05×0.25=0.0125=1.25%

4. Ask the students to explain how they would do example 2 knowing what they know from example 1. Make sure to involve as many students as possible. If they do something wrong ask the others students to correct them. (Bloom's Comprehension)

5. Walk through example 3 in the text. Draw the table on the board and make sure the students know where you are getting the probabilities to plug into the formula.

6. Go onto example 4. This is dealing with making tree diagrams for the probabilities. Make sure the students know where the probabilities go on the tree diagram. They should know how to check to make sure they used the right probabilities and put them in the right spot. You do this by making sure each level on the branch adds up to be 1.00. Tell them to Make a hypothesis of what a problem would be that would result in 3 or 4 branches somewhere in the tree. (Bloom's Synthesis)

7. Assign the homework problems1-9, 12, and 13. Be sure to point out that 1-4 correlate with example 1 in the text, 5-8 with example 2, 9 with example 3, and 12 with example 4. Let the students work in groups if they remain quiet. (Gardener's Interpersonal)

Closure: Tell them that this is used in many different fields of study. The article mentioned using it in the medical field. The article also had another part that I left out that talked about probability in a court room. In the next chapter we will be learning how to analyze the data we have been finding.

Adaptations/Enrichment:Student with ADD:

When the examples are done on the board, have the student draw the tables on the board for you to refer to. This will get them out of their seat momentarily and will give them something hands on to do so you do not loose them.

Pick one of the students that is friends with the student with ADD that when you give them time to work on homework in class he is to work with the student with ADD. Tell them to make sure he stays on task and does not get distracted with something else.

Give the student a highlighter. Tell them to highlight all the percentages in the hand out as we get to them while we read. This will help him follow along and give him something to do rather than just sit there and follow along.

Give the student a peace of paper and tell them that when we get to the part of the lecture about tree diagrams, he should use the concept to draw a tree. The concept being how one line breaks off into 2 or 3 lines. This will make them remember that it is called a tree diagram and it will give them something to do while they can still listen.

Reflection:I will be able to use the homework to see how well the students understood the materials.

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I can see how well they followed along by seeing if they found the definition in the paper. I will also be able to see if they thought the handout was interesting or useful in understanding the concept by how they are acting during and after I read it. If I can not tell I can always ask how they liked it and what they thought when I am done before I start my lesson.

Work Cited:Strogatz, S. (2010, April 25). Chances are. The New York Times, Retrieved from

http://opinionator.blogs.nytimes.com/2010/04/25/chances-are/

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Chances AreBy STEVEN STROGATZ, April 25, 2010, 5:00 pm

Have you ever had that anxiety dream where you suddenly realize you have to take the final exam in some course you’ve never attended? For professors, it works the other way around — you dream you’re giving a lecture for a class you know nothing about.

That’s what it’s like for me whenever I teach probability theory. It was never part of my own education, so having to lecture about it now is scary and fun, in an amusement park, thrill-house sort of way.

Perhaps the most pulse-quickening topic of all is “conditional probability” — the probability that some event A happens, given (or “conditional” upon) the occurrence of some other event B. It’s a slippery concept, easily conflated with the probability of B given A. They’re not the same, but you have to concentrate to see why. For example, consider the following word problem.

Before going on vacation for a week, you ask your spacey friend to water your ailing plant. Without water, the plant has a 90 percent chance of dying. Even with proper watering, it has a 20 percent chance of dying. And the probability that your friend will forget to water it is 30 percent. (a) What’s the chance that your plant will survive the week? (b) If it’s dead when you return, what’s the chance that your friend forgot to water it? (c) If your friend forgot to water it, what’s the chance it’ll be dead when you return?

Although they sound alike, (b) and (c) are not the same. In fact, the problem tells us that the answer to (c) is 90 percent. But how do you combine all the probabilities to get the answer to (b)? Or (a)?

Naturally, the first few semesters I taught this topic, I stuck to the book, inching along, playing it safe. But gradually I began to notice something. A few of my students would avoid using “Bayes’s theorem,” the labyrinthine formula I was teaching them. Instead they would solve the problems by a much easier method.

What these resourceful students kept discovering, year after year, was a better way to think about conditional probability. Their way comports with human intuition instead of confounding it. The trick is to think in terms of “natural frequencies” — simple counts of events — rather than the more abstract notions of percentages, odds, or probabilities. As soon as you make this mental shift, the fog lifts.

This is the central lesson of “Calculated Risks,” a fascinating book by Gerd Gigerenzer, a cognitive psychologist at the Max Planck Institute for Human Development in Berlin. In a series of studies about medical and legal issues ranging from AIDS counseling to the interpretation of DNA fingerprinting, Gigerenzer explores how people miscalculate risk and uncertainty. But rather than scold or bemoan human frailty, he tells us how to do better — how to avoid “clouded thinking” by recasting conditional probability problems in terms of natural frequencies, much as my students did.

In one study, Gigerenzer and his colleagues asked doctors in Germany and the United States to estimate the probability that a woman with a positive mammogram actually has breast cancer, even though she’s in a low-risk group: 40 to 50 years old, with no symptoms or family history of

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breast cancer. To make the question specific, the doctors were told to assume the following statistics — couched in terms of percentages and probabilities — about the prevalence of breast cancer among women in this cohort, and also about the mammogram’s sensitivity and rate of false positives:

The probability that one of these women has breast cancer is 0.8 percent. If a woman has breast cancer, the probability is 90 percent that she will have a positive mammogram. If a woman does not have breast cancer, the probability is 7 percent that she will still have a positive mammogram. Imagine a woman who has a positive mammogram. What is the probability that she actually has breast cancer?Gigerenzer describes the reaction of the first doctor he tested, a department chief at a university teaching hospital with more than 30 years of professional experience:

“[He] was visibly nervous while trying to figure out what he would tell the woman. After mulling the numbers over, he finally estimated the woman’s probability of having breast cancer, given that she has a positive mammogram, to be 90 percent. Nervously, he added, ‘Oh, what nonsense. I can’t do this. You should test my daughter; she is studying medicine.’ He knew that his estimate was wrong, but he did not know how to reason better. Despite the fact that he had spent 10 minutes wringing his mind for an answer, he could not figure out how to draw a sound inference from the probabilities.”

When Gigerenzer asked 24 other German doctors the same question, their estimates whipsawed from 1 percent to 90 percent. Eight of them thought the chances were 10 percent or less, 8 more said 90 percent, and the remaining 8 guessed somewhere between 50 and 80 percent. Imagine how upsetting it would be as a patient to hear such divergent opinions.

As for the American doctors, 95 out of 100 estimated the woman’s probability of having breast cancer to be somewhere around 75 percent.

The right answer is 9 percent.

How can it be so low? Gigerenzer’s point is that the analysis becomes almost transparent if we translate the original information from percentages and probabilities into natural frequencies:

Eight out of every 1,000 women have breast cancer. Of these 8 women with breast cancer, 7 will have a positive mammogram. Of the remaining 992 women who don’t have breast cancer, some 70 will still have a positive mammogram. Imagine a sample of women who have positive mammograms in screening. How many of these women actually have breast cancer?Since a total of 7 + 70 = 77 women have positive mammograms, and only 7 of them truly have breast cancer, the probability of having breast cancer given a positive mammogram is 7 out of 77, which is 1 in 11, or about 9 percent.

Notice two simplifications in the calculation above. First, we rounded off decimals to whole numbers. That happened in a few places, like when we said, “Of these 8 women with breast cancer, 7 will have a positive mammogram.” Really we should have said 90 percent of 8 women, or 7.2 women, will have a positive mammogram. So we sacrificed a little precision for a lot of clarity.

Second, we assumed that everything happens exactly as frequently as its probability suggests. For instance, since the probability of breast cancer is 0.8 percent, exactly 8 women out of 1,000 in our hypothetical sample were assumed to have it. In reality, this wouldn’t necessarily be true.

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Things don’t have to follow their probabilities; a coin flipped 1,000 times doesn’t always come up heads 500 times. But pretending that it does gives the right answer in problems like this.

Admittedly the logic is a little shaky — that’s why the textbooks look down their noses at this approach, compared to the more rigorous but hard-to-use Bayes’s theorem — but the gains in clarity are justification enough. When Gigerenzer tested another set of 24 doctors, this time using natural frequencies, nearly all of them got the correct answer, or close to it.

Although reformulating the data in terms of natural frequencies is a huge help, conditional probability problems can still be perplexing for other reasons. It’s easy to ask the wrong question, or to calculate a probability that’s correct but misleading.

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Lesson Plan #4

Lesson: Analyzing Data (12-3) Length: 45 min Age or Grade Level Intended: Algebra II Honors

Academic Standard(s): A2.9.2: Use the basic counting principles, combinations, and permutations to compute probabilities.

Performance Objective(s): Given 12 problems on counting and probability, students will answer them with 80% accuracy.

Assessment: 12 problems from the text will be given as homework and will be handed in tomorrow.

Advance Preparation by Teacher: No preparation needed.

Procedure: Introduction/Motivation: Yesterday we went over how to do conditional

probability. Today we will be looking more on how to analyze data that we have been finding probabilities for. Have the students draw the face of a cat and a boxing ring on a piece of paper. Ask for 3 or 4 volunteers to draw both of their pictures on the board. If you can walk around and pick some that are a lot different that would work best. Ask the other students what all the cat faces have in common and all of the boxing rings have in common. The cats should all have whiskers, the boxing rings should all be rectangles or squares. (Gardner’s Bodily/Kinesthetics)

Step-by-Step Plan: 1. Ask the students to hypothesize why they think I would be talking about whiskers and

boxes in math class. If no one answers this correctly ask them if they have ever heard of a box-and-whisker plot. (Bloom’s Synthesis)

2. If any of them have questions ask them to draw one on the board. If no one has, ask if anyone thinks they know what one would look like and wants to draw it on the board. If they draw it wrong just correct them unless another student wants to try. If no one wants to, then just draw it up yourself.

3. Explain how you make a box-and-whisker plot when you want a way to model a group of numbers. Label the median, the upper and lower quartile, and the medians of the quartiles.

4. Tell them you are going to rewind and tell them how to find the median.5. The median goes in a group of 2 other vocabulary words, mean and mode. Ask the

students if they know what any of these mean. If they don’t tell them. (ANS: mean= the average, median= the middle number in the group, mode= the number occurring most

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often)6. Do example one on the board:

Mean: 98+95+99+97+89+92+97+62+90= 819, 819/9=91,Mode: write in order; 62, 89, 90, 92, 95, 97, 97, 98, 99; mode= 97 (occurs twice)Median: take the numbers in order, find the middle number. Median= 95

7. Ask the students what you would do if you had an even number of numbers, so when you tried to find the median you had 2 numbers in the middle. (ANS: find the average of them 2 numbers, so add them up and divide by 2) (Bloom’s Analysis)

8. Take them back to the box-and-whisker plot and tell them that for this one, the middle of the box would go over 95 on the number line. Explain that the ends of the box are the median of that half of the numbers. So for the last example you would take 62, 89, 90, and 92 and find the median to fine the lower quartile (the median of the lower half of the numbers). This would be 89.5.

9. Ask a student to find the upper quartile. (do this the same way with the highest 4 numbers, ANS: 97.5) (Bloom’s Application)

10. Now you point out that the whiskers are made by drawing a line from the box to the min and the max numbers in your original set. Ask one of the students what the min and max is for our set. (ANS: min=62, max=99)

11. This will complete the box-and-whisker that you should now have on the board for them to see. (Gardner’s Visual/Spatial)

12. Tell them that the only time the min or the max isn’t the lowest and the highest number in the set is if there is an outlier. This means a number that is way higher or lower than the rest of the numbers. See ex 6. 98 is obviously much larger than the numbers in the set.

13. Let them know that example 3 is another example of a box-and-whisker plot for them to use on their homework, but we aren’t going over it now.

14. Next we are going over exercise #5. In this we are finding what numbers are in what percentile in a set of numbers. They list 20 numbers and then put them in order and ask to find the numbers in the 20 percentile and the numbers in the 65 percentile. To find the 20 percentile you take the amount of numbers you have (20) multiplied by the percent (20%). So you have 20x.20=4. So the lowest 4 numbers are in the 20th percentile. So 21, 24, 31, and 45 are in this. so the number that is at the 20th percentile is 47.

15. Ask the students why it’s not 45 (ANS: since 45 is in the lower 20 percentile it can’t be AT the 20th percentile, so it is the next higher one). (Bloom’s Evaluate)(Gardner’s Logical/Mathematical)

16. Ask the students to find the 65th percentile on their own. Walk around the room and see how they are doing. (Gardner’s Intrapersonal)

17. When they are done have someone that got it right do it on the board. (ANS: 87 is at the 65th percentile)

18. Assign problems 1-6, 8-11, and 14-15 and have them work by themselves. Also hand them the second half of the vocab words at this time. Tell them to fill in the vocab as they get them. This is not going to be for a grade, but it will be useful to study for the test.

Closure: In the next lesson we will learn more things to do with sets of data. We will learn how to find the standard deviation and the interquartile range. Standard deviation can be used in a number of different ways. We can find how much the height of the students deviate, as well as test scores, shoe sizes, etc.

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Adaptations/Enrichment:Student with learning disabilities:

I will copy the notes for the lesson and give it to him/her before class starts. By doing this they can fully focus on what I am lecture about and on my examples rather than trying to write down what I am saying.

While I am walking around the class on step 16 I will ask them if they understand what we have gone over so far. If they then I will tell them I will help them this is the last example and I will help them right after this while the rest of the students start their homework.

I will tell the student ahead of time that I am going to call on them for an answer. This way they know it is coming and they have time to think about it and don’t have the stress of getting called on unexpectedly and having to think of an answer quick.

When they work on their homework, I will let this student work with someone else. I will match them with one of the higher level students after previously talking to the other student about it making sure it is ok. If it sometime easier to learn when one of your peers tells you how to do it

Reflection: I will be able to make sure the students know what they are doing by calling on them and asking if any of them have questions. The hardest thing in this lesson is finding what number is at a certain percentile. This is why I am giving them one to do on their own while I walk around and see how they are doing.

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Vocab Second Half12-4Measure of Variation=

Range of a Set of Data=

Interquartile Range=

Standard Deviation=

Z-Score=

12-5Random Sample=

Sample=

Sample Proportion=

Margin of Error=

12-6Binomial Experiment=

12-7Normal Distribution=

Standard Normal Curve=

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Lesson Plan #5

Lesson: Standard Deviation (12-4) Length: 45 min Age or Grade Level Intended: Algebra II Honors

Academic Standard(s): A2.10.1: Use a variety of problem-solving strategies, such as drawing a diagram, guess-and-check, solving a simpler problem, writing an equation, and working backwards.

Performance Objective(s): Given 10 problems where diagrams are needed and writing equations is used, students will answer them with 80% accuracy.

Assessment: 10 problems from the textbook will be given as homework and will be handed in tomorrow.

Advance Preparation by Teacher: Nothing is needed for this lesson, just the textbook and this lesson plan.

Procedure: Introduction/Motivation: Talk about how yesterday we learned how to analyze

data. Today we will be looking at how to find the standard deviation of the data. Ask the students, “When you get a test back and you ask your friends what they got on it, do you ever wonder what everyone else got in the class? Maybe you got a B- but everyone else got C's and D's. Do you ever wonder what the average score was in the class. How far are you away from the class average? Where the grades all over the place, or were they clustered together where most of the class got about the same grade? These are all things you will find with the standard deviation.”

Step-by-Step Plan:1. Use the values given in example #2 and say they are test scores. To find the standard

deviation you want to first find the mean. For these values, add them up and divide by 5. Once you know the mean you can set up a table as shown. The first column is the value, and the second column (denoted by x bar) is the mean we just found. (Gardner's Visual/Spatial)

2. Next we subtract the value from the mean and put those answers in another column. Tell them that it does not really matter if you take the value minus the mean or the mean minus the value. The only difference is one answer will be negative the other one, and this will give us the same outcome once the number is squared in the next step.

3. After this we square all the differences and make that the last column.4. Lastly we find the average of the last column. We do this by adding them all up and

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dividing by 5 since there are 5 values. For this example it is 31.1/5. 5. Once we find that number (6.22) we take the square root of it. This gives us 2.5, therefore

our standard deviation of the values is 2.5.6. At this point make sure they can see how our table helps us plug things into the equation

shown under it on the same page. We basically used our table to find (x- xbar)2. Then when we add them up, we are doing the summation part of the equation. Since we had 5 values our n=5, so that's where we subbed that into the equation. All we have to do now is take the square root. (Gardner's Mathematical/Logical)(Bloom's Knowledge)

7. If It asks to determine the number on standard deviations include all the numbers, you simply find [mean-st.d.<=x<=mean+st.d.] This represents 1 standard deviation. To find the 2nd standard deviation you would use [mean-2st.d.<=x<=mean+st.d.]. So if the value x makes the first statement true, then that value lies within the first standard deviation. If it does not, you move onto the second, you keep doing this until all the values are met.

8. How many standard deviations do you think encompass the data values we were using earlier? (Bloom's Evaluation)

9. Do this with our data. 50-2.5<=x<=50+2.5; 47.5<=x<=52.5. Do all the values lie within these constraints? (ANS: no, 53.2 and 46.6 do not fall into that)

10. So you do 2 standard deviations. 50- 2(2.5)<=x<=50+2(2.5); 50-5<=x<=50+5; 45<=x<=55. Do all the values lie in this? (ANS: yes)

11. Therefore, all the values lie within 2 standard deviations.12. Take time here to answer any questions the students might have.13. Once all questions are answered do example #1, except use the values we have been

working with.(The values are 53.2, 52.3, 49.9, 48.0, 46.6)14. Ask one of the students to find the median. (ANS: 49.9) (Bloom's comprehension)15. Ask another student to find the quartiles of these values. (ANS: [48.0+46.6]/2= 47.3 =Q1,

[53.2+52.3]/2= 52.75=Q3.)16. Explain to them that once you have this information you can find the interquartile range

by subtracting Q3 and Q1. After saying this ask what the interquartile range is equal to. (ANS: 52.75- 47.3= 5.45)

17. After this move on to example #3. This is how to find the z-score. To find this we need a value, the mean, and the standard deviation. The example tells you the mean= 85, standard deviation=6, and you are finding it for value=76. You simply plug this into the equation shown and solve.

18. Find the z-score of 48.0 given our last set we used from example 2. We already have found the mean =50.0. We found the standard deviation= 2.5. Our value is given to be 48.0. Let the students solve this and then ask them for their answers. (ANS: [48.0-50.0]/2.5= -2.0/2.5=0.8. Therefore the z-score=0.8) (Bloom's Application)

19. Assign problems 1, 2, 5, 6, 9, 10, 11, 12, 15, and 22. Allow the students to work in groups if they want to. (Gardner's interpersonal)

20. 5 minutes before the bell rings tell the students to write down a couple sentences on how they feel about the material covered. Did they understand it well, not well, did they think they understood it then struggled on the homework.... (Gardner's Intrapersonal)

21. This last step is a good way for me to see how comfortable the students are with the material I taught today. I is a good change from the normal and the students will be able to think about how well they actually do understand the material and not just leave class and forget about it.

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Closure: Problem 22 talks about something we did not really go over in class but I think you guys can do it. Tomorrow we will be working with different samples and maybe finding the standard deviation and z-scores of different samples.

Adaptations/Enrichment: Gifted and talented student:

Let this student do the problem in step 7 and explain to the class how they got the answer. Before class ask the student before hand if this is ok. Be ready to help him/her if they struggle at all.

If one of the students asks a question during the lesson, ask the gifted student to go over it on the board. Sometimes after college and higher level math classes, its hard for teachers to translate it into words students can understand. This is why it is very helpful sometimes for a student to help other students.

Have the student write a simple scenario where standard deviation might be used in real life situations. Have him/her hand this in with his homework. This will not be graded on grammar or anything other than the idea.

For homework do problems 1, 2, 6, 9, 10, 15, 17, 18, 19, and 22. These are a bit more difficult than the other problems given to the rest of the class.

Reflection: After the homework is handed in I will be able to see how well I covered this topic. If the objective is not met the I will go over the problems tomorrow before class. Also frequent questioning will go on during my lesson to make sure the students are understanding me.

How can my unit be enriched for gifted students:Math is a somewhat hard subject to help students that are gifted. You do not want to just

give them more work. In other subject you can give them different things to read or change their requirements on papers to make it more challenging. I have thought of a few ways that I think would be a good way to challenge gifted students.

First I think giving them a slightly different assignment that is a bit more challenging. In this lesson I omitted some questions and added some other challenging ones. If they still are not being challenged enough you can let them jump ahead and do chapters on their own. This might not work for every student though. Some students that are good at math might not be strong readers. I know when I try and read the chapter before we go over it in class I will not know what to do, but when someone explains it to me, I am good to go. If this is needed I might be able to teach them their mini lesson while the others do their homework.

Another thing I plan on doing is using this to everyone’s advantage. Having a more advanced student can help me be in 2 places at once. They can help the other students in the room and help you explain things that the students do not seem to be understanding. Like I explained before, sometimes if is hard to translate what you know to something students can

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understand. This is where an advanced student can help. They are able to understand what you say, and then explain it in a way the other student or students can understand.

I hope these ideas will work for my gifted students. If it does not I am sure there will be other teachers that are more than willing to help me. There will probably be other math teachers that had the student the year before I did. I plan on using my resources to my benefit as much as I need to.

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Lesson Plan #6

Lesson: Working With Samples (12-5) Length: 45 min Age or Grade Level Intended: Algebra II Honors

Academic Standard(s): A2.10.2: Decide whether a solution is reasonable in the context of the original situation.

Performance Objective(s): Given 10 problems over sample spaces, students will have to evaluate their answers to make sure they are reasonable and will answer them with 80% accuracy.

Assessment: 10 problems will be given out of the textbook as homework and will be handed in tomorrow.

Advance Preparation by Teacher: Nothing more than the textbook and notes.

Procedure: Introduction/Motivation: So far, all the problems we have done the textbook

give you the data. This lesson will tell you how to get a sample yourself and how to make sure your sample will give you accurate predictions. Tell the students that they are a sample of the whole high school. They are grouped in the class according to the math classes they have already passed and that is it. Tell them to stand up if they are in sports. Count everyone that raises their hand and write the number on the board. Then count the total number of students that are in the class. (Gardner's Bodily/Kinesthetic)

Step-by-Step Plan: 1. Once you have both of those numbers written on the board label the number of students

that play sports as x, and the total number in the class as n. Ask the students how they would find what percentage of the students in the class play sports. (ANS: x/n) (Bloom's Comprehension)

2. Ask the students if it would be accurate to assume that (x/n)% of students in the school play sports. (ANS: Since the sample is not biased towards athletes this could be considered somewhat accurate.) (Bloom's Evaluation)

3. Ask them how they could make this percentage more accurate. (ANS: Ask more people.) (Bloom's Synthesis)

4. Ask the students to stand up if they have passed Algebra 2 class. Write the number on the board of the students that stood up (this should be 0).

5. Again ask them to make this a percentage. (ANS: 0%) Again ask if it is accurate to assume that 0% of the students in the high school have passed algebra 2. (ANS: No)

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(Bloom's knowledge)6. Ask them to explain why this is not accurate but the last one was. (ANS: This is bias

because this sample is chosen from a sample space based on the math classes they have had)

7. Go over example #1. In this example their x is the number of people that have not ever make a snow sculpture and n is the number of teenagers they asked. So to find the percentage they used the equation we used, x/n, to find the percentage, which they show to be 84%.

8. Now go over example #3, this is also something that we went over in the introduction activity. Tell them that the 2 graphs are representing 2 samples. One of them sampled 20 people and the other sampled 5. What one sampled 20? (ANS: The one on the right). Why? (ANS: They are more even and clustered together.) (Gardner's Visual/Spatial)

9. Next go over the formula for finding the margin of error, this is written in the orange box on page 665. Point out that the only thing you need to know to find this is the number of people sampled (denoted by n).

10. Go to example #5 before you do #4. In example 5 you are solving for the margin of error when you have the number of people sampled. So you sample 2580 students, so to find the margin of error you take the square root of 2580 and then take the inverse of that (take 1 divided by the square root of 2580). This is ±0.0197. This round to ±2% Once you have this number stress the plus or minus part since you will be adding it and subtracting it from a number.

11. Part B in the question is where you will be adding and subtracting. The question gives us that 9% of people are left handed. Read the question so they know how it will look. Now is were you take 2% and subtract it from 9%, then add it to 9%. This gives you the interval 7%-11% so the proportion of people that are left handed are most likely to fall between 7 and 11 percent.

12. Now go back to #4, I like to do 5 first because in #5 we solved for the margin of error given the sample space and in #4 we are given the margin of error and solving for n. This seems like a more natural order to learn in.

13. Example 4 tells us that the margin of error is ±3%. First we solve for √n. Once we do this we have √n=1/(margin of error). Now we simply sub 0.03 in for the margin of error. 1/0.03=33.33. Now we take that squared to get n=1111. So they sampled an estimated 1111 people.

14. That is the last thing you need to go over in the lesson. Tell the students to get out a piece of paper. Ask them to write on the piece of paper something they learned today, how confident they are in what they learned, and a survey they could do in the high school that would interest them. This is to be handed in before they leave class. (Gardner's Intraperaonal) [WRITING TO LEARN STRATEGY]

15. As they work write their assignment on the board: page 666-668, 1, 2, 3, 8, 9,10, 12, 13, 15, 16. Let them work in groups if they want to and remain quiet. Make sure you remind them to look over their answers to make sure they logically make sense. If you get a margin of error that is 80%, that is way too high. (Gardner's Interpersonal)

Closure: Tomorrow we will be going over binomial distributions. This is somewhat like what we did today with a few more steps. Instead of asking what percentage of people are left handed it will ask what the chances that a person is left handed and has blue eyes.

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Adaptations/Enrichment: Student With Behavior Disorder:

Allow/encourage the student to work alone on their homework. Some students that have a behavior disorder work better in small groups. They are able to focus better and stay on task.

I will seat this student in a desk that is fairly close to my desk. This will deter most bad behavior that would come if they were across the room.

For this student I will suggest that they show up after school or before school for some tutoring. If they can not make that I will suggest that they come in during my prep period. This will help them understand the content and give them confidence in my class.

Lastly if I need to I can shorten the assignments. A lot of the problems above are the same type of problem. I would assign 1, 2, 8, 9, 13, 15, 16 for this student if I would need to.

Reflection: With the writing to learn strategy, I will be able to see how confident the students are with the material I taught today. Also with the things that they write as a survey that would interest them I will learn about them a little bit and it will help me teach other lessons. If I have time to read these as they are doing their homework I will be able to let ones that are not confident.

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Lesson Plan 7

Lesson: Binomial Distribution (12-6) Length: 45 min Age or Grade Level Intended: Algebra 2 Honors

Academic Standard(s): A2.9.2: Use the basic counting principles, combinations, and permutations to compute probabilities.

Performance Objective(s): Given 10 problems where students will use combinations to compute probabilities, students will answer them with 80% accuracy.

Assessment: 10 problems from the textbook will be given as homework and will be handed in tomorrow.

Advance Preparation by Teacher: A penny for each person in the class.

Procedure: Introduction/Motivation: Give a penny to each person in the class. Tell them

to get out a piece of paper and tell them to number it down the side 1-6. This is the number of flip they are on. So the first flip will be by 1, the second by 2, etc. Tell them to flip their coin 6 times and write down how many times the get heads compared to tales. Ask them to think about what the chances are of them getting heads every time. What is the chance you get 3 heads then 3 tails? Flipping a coin, as we will soon explain, is a binomial experiment. (Bloom's Synthesis)

Step-by-Step Plan: 1. A binomial experiment has 3 different requirements. The first is that the situation

involves repeated trials. Second each trial has 2 possible outcomes (usually success and fail). Lastly, The probability of success is constant throughout the trials (aka, the trials are independent). Does our coin flipping experiment qualify as a binomial experiment? (Bloom's Application)

2. So if we had a sample where we surveyed everyone in the school like we did in our last lesson, it would it be a binomial experiment? (ANS: no) After they answer go through the 3 requirements to see which ones they fail. #1 fails because it does not have repeated trials. #2 works because you could ask them a yes or no question. #3 would work if #1 worked. If you asked all the students if they like base ball and then asked the same students again, you would get the same answers.

3. Write down the first 2 lines of example #1. Tell them not to open their books or to shut them if they are open. Now go through the questions a-c with them.

4. 1a) Describe a trial for this situations and how many trials are there? (ANS: each question is a trial because each question you are giving 5 different questions, there are 3

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questions so there are 3 trials.)5. 1b) Describe a success. (ANS: when you get an answer right) What is the probability of a

success in any single trial? (ANS: 0.2, there are 5 possibilities and each one has an equal chance of being chosen) (Bloom's Comprehension)

6. 1c) Open your books to page 672. 1c says conduct a simulation, we do not have time to do this so we are going to look at the chart of the simulation they did. Read the 3 lines before the chart out loud while they follow along. This is the paragraph that is describing how the chart is made up. There are 2 cases that the student got 2 or more right and there are 10 trials. So the probability of getting 2 or more right is 2/10, which is = 0.2 or 20%. (Gardener's Mathematical/Logical)

7. In example #2 they use a tree diagram to find the probability. Read the opening paragraph to them and then start discussing how to make the tree diagram.

8. We start with the letter P for Prize and N for No prize. The probability of success (P) is 40%. this makes the probability for N 60%. So you start the diagram with your first card. You start at a point and draw a to P and another line to N. On the line to P you put a 0.4, and on the line to N put a 0.6. This is representing the probabilities we just discussed.

9. From the P and the N we draw lines representing our second ticket. If we get a P the first ticket, on our second ticket we still have the 2 outcomes and the same probabilities for each. If we get an N on the first ticket, we again still have the 2 options on the second. So we draw 2 lines from this P and 2 lines from the N, each going toward a P and an N, each with the same probability written on the line.

10. You do this again for the third ticket, putting the P and the N with every outcome and the probabilities on the lines. This is a bit hard to picture, but the students have the picture of the diagram in their book so they should be able to follow. At this point take any questions that they might have.

11. Now you follow the line from the beginning point down to the final outcome (the one representing the third ticket or trial). Beside the final outcome write down if you follow the P or and N path. So you should have 3 P's, 3 N's or a mixture of P's and N's.

12. Also, they did not do this in the text, but I like to write down the probabilities as I go to. So for instance, for the first path that they have with PPP, I would also write (0.4)(0.4)(0.4). This will help us later in the problem.

13. So if the asks, what is the probability of 3 prizes, denoted by P(3 prizes), you would multiply the numbers we just wrote down.

14. This is where they could ask 2 different things, if they ask for P(2 prizes) it will be a different answer that if they ask for P(PNP). For the second case you just follow the path and multiply their outcomes. So P(PNP)=(0.4)(0.6)(0.4)= 0.096=9.6%.

15. For the first case every one that will have 2 prizes will have 2 (0.4)'s and 1 (0.6). So we just count the number of outcomes have 2 prizes and multiply that by (0.4)(0.4)(0.6). As you see in the book there are 3 outcomes with 2 prizes and they plugged the 3 into the equation we just talked about.

16. After knowing this, what going in the grey boxes for P(1 prize) and P(0 prizes)? (ANS: P(1 prize)=3(0.4)1(0.6)2, P(0 prizes)=1(0.6)3.

17. Now bring up the orange box on page 673 that says “Definition”. This has the definition of Binomial Probability. Read this definition out loud as they follow along in their books.

18. Go over this using the last example for P(1 prize). n=3 and x=1 p=0.4 and q=0.6. So we get the equation 3C1(0.4)1(0.6)3-1= 3C1(0.4)1(0.6)2.

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19. Tell them to close their books and so we can go over example 4 without having the answers in front of us. Read them the paragraph. (if you can put the directions on the projecter so they can refer back to them. We are using the expansion (p+q)n. N is the number of trials. So what does n equal in this problem? (ANS: 3) What does p and q equal? (ANS: p=0.4, q=0.6)

20. Now write this equation on the board. (p+q)3=1pq+3pq+3pq+1pq. Ask them how the book got the 1, 3, 3, and 1? (ANS: From the nCx we learned earlier where x=3, 2, 1, then 0) (Gardener's Visual/Spatial)

21. Ask a student to go up to the board and fill in the first 2 boxes. (ANS: 3 and 0) (Gardener's Knowledge)

22. Ask a student to go up to the board and fill in the second 2 boxes. (ANS: 2 and 1)23. Ask if they can see the pattern. 24. Tell them to turn back to example 4 if they do not see the pattern. If they do not get it yet

explain that they all add up to 3 and as the first number goes down one, the second goes up. They should see it when they look at their books.

25. When we piece together what we know with this equation we see that it is the same as P(3 success)+P(2 success)+P(1 success)+P(0 success). So if we are looking P(at least 1 success)=P(1 success)+P(2 success)+P(3 success). Now that we know this you can easily follow the rest of the example. They simply added in the probabilities and added them together.

26. Tell them to compare example 4 with the problem we did as an example of binomial probability. How are they the same? How are they different? (Bloom's Analysis)

27. Assign 1, 5-19 odds, and 14. Let them work in groups if they stay quiet. (Gardener's Interpersonal)

Closure: We can use this information in almost any case that you can make the outcome either a success or a fail. This can be tossing coins like we did earlier to winning and losing games in sports. Tomorrow will will be learning about another distribution. This new distribution is called normal distribution and its used to show data that vary randomly from the average.

Adaptations/Enrichment: Student with Autism:

I will place this student in the front of the room so they do not get distracted by any other students as well as the posters that I have in my room.

I will give this student some squeeze balls or stress balls to squeeze during the lesson to reduce rocking and fidgeting.

I will give this student a copy of my notes so he/she is not overwhelmed by taking notes and listening to the lecture at the same time.

I will tell him/her some of the questions I will ask during my lesson so they can have time to prepare and think about it. I will circle the parts in the examples that I ask questions or write them in my notes that I give him/her.

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Reflection: I will be able to see if the students are keeping up with the lesson by the questions I ask throughout the lesson. This lesson is a bit longer, but a lot of the material is trivial and can be covered somewhat quickly. This is why I have to ask questions during the lesson so I know I am not going to fast. We will grade the homework tomorrow at the beginning of class so the students can ask questions and I can go over and problem spots.

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Lesson Plan 8

Lesson: Normal Distribution (12-7) Length: 45 min Age or Grade Level Intended: Algebra II Honors

Academic Standard(s): A2.10.1: Use a variety of problem-solving strategies, such as drawing a diagram, guess-and-check, solving a simpler problem, writing an equation, and working backwards.

Performance Objective(s): Given 11 problems, the students will draw diagrams to help them answer the problem with 80% accuracy.

Assessment: 11 problems out of the textbook will be given as homework and will be handed in tomorrow.

Advance Preparation by Teacher: No special preparation is needed for this lesson.

Procedure: Introduction/Motivation: If I were to ask you if you wanted me to grade the

next test on a normal curve what would you guys say? At first you might want to say you would, but after this lesson you will learn that that is the very wrong answer. (Bloom's Synthesis)

Step-by-Step Plan: 1. You may ask what the difference is between binomial distribution (what we learned

yesterday) and normal distribution. The difference is binomial distribution is a discrete random variable that is used for finding the probability of x success in n independent trials. Normal distribution is a continuous distribution.

2. Ask the students if they know what a normal curve looks like. If any of them say yes, let them draw it on the board. If they do not draw one up there for them.

3. Ask them why this curve might be bad for grading tests. (ANS: A lot of C's are given out, less B's, and only a few A's) (Gardner's Logical/Mathematical) (Bloom's Comprehension)

4. Tell them all to get out a piece of paper and draw a normal curve on it. Tell them to use that curve to draw a picture of something, anything at all involving the curve. (Gardner's Visual/Spatial)

5. Ask some of them what pictures they came up with. Hopefully some of them will be bells. If not draw yours up on the board, make it a bell. Tell them that a normal curve is also known as a bell curve.

6. Turn to page 681 and go over Exercise #6. Tell them to refer to example #2 for a reference. Let them do this on their own for a few minutes as you walk around the room.

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Ask one of the students you know got it right to get up and put their answer on the board. Ask another student to explain how the first student got that answer. (Gardner's Intrapersonal) (Bloom's Application)

7. Now direct their attention to the standard normal curve on page 679. Read the 2nd

paragraph as they follow along. 8. When you are done with this turn to example #4. a) 2 standard deviations above the mean

is 13.5% of the population sampled. Ask them how they would find the answer to (a). (ANS: Since 174 students were in the class you multiply 174 by 0.135. 174(0.135)=23.49. so between 23 and 24 students.)

9. In (b) we are finding the number of students that scored a 61 or below. Ask the students how they think we would approach this problem. (ANS: On the graph we see that below 61 includes about 13.5% and 2.5% of the population. Add 13.5%+2.5%=16%. now we do the same thing we did in (a). 174(0.16)=27.84. So about 28 students got a 61 or below.)

10. Once you are done with this go back to example #3 where you find the z-score of a standard normal curve. We are given the mean and the standard deviation. In (a) we are given the z-score and we are solving for the value that has that z-score. This is done for us in the example, the students can just follow along.

11. In (b) we see that the mean is 13h and the standard deviation is 3h. This means 10 and 16 are both 1 standard deviation away from the mean. From the paragraph you read out loud we know that 68% of the sample is within 1 standard deviation of the mean.

12. Before they you give them their assignment ask them to take out a piece of paper and write a couple sentences on what they learned today and what they are confused about from today. (Bloom's Evaluation)[WRITING TO LEARN STRATEGY]

13. Assign from the textbook 1-13, 17, 21-25 all odds. 1 and 3 are very simple so we did not do an example in class, but if they struggle, example #1 is the same kind of problem. Let the students work in groups if they would like to. (Gardner's Interpersonal)

Closure: This is the last section in chapter 12. We will be doing a review tomorrow so look over old home work problems and come with questions. I will be giving a homework assignment that will include some problems you have done before and some new problems. If you have already done the problem, try to do it without looking at your old homework. This will be for your benefit because you will need to know how to do these things for the test.

Adaptations/Enrichment: Student with Mild Mental Retardation:

I would copy my notes on the lesson and give them to this student. This way he does not have to multitask and try and understand what I am says as he is taking notes.

I would reduce this students homework assignment down a bit. A lot of the numbers I assigned are the same kind of problem, I just assigned numerous ones for more practice. For this student I would assign 1, 5, 9, 11, 13,17, 21, and 25. This reduces it down but they still have every kind of problem covered.

After I am done teaching the lesson I will meet with this person one on one and briefly go over the examples again and see if they have questions. The more times they hear the

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how to do the examples the more it will stick.

I will have this student sit in the front of the class to reduce the distractions going on around them. Many students with mental retardation get distracted easily. If the student is in the front they will not have the distractions of all the students around them.

Reflection: The writing to learn section will let me know what parts of the lesson they understood and what parts need to be covered again in the review. The homework will also let me know what they understand. We will check it in class tomorrow so we can review things they have questions on for the test.

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Lesson Plan #9

Lesson: Chapter 12 Review Length: 45 min Age or Grade Level Intended: Algebra 2

Academic Standard(s): A2.9.2 : Use the basic counting principles, combinations, and permutations to compute probabilities.

Performance Objective(s): Given 18 problems over probability and statistics, students will answer them with 80% accuracy.

Assessment: 18 problems from the textbook will be given as homework and will be handed in tomorrow.

Advance Preparation by Teacher: No preparation is needed.

Procedure: Introduction/Motivation: We have come to the end of the chapter. We have

gone over probability distribution, conditional probability, analyzing data, standard deviation, samples, binomial distributions, and normal distributions. In this lesson we will be going over some things we have learned this far, doing some examples, and filling out a concept map to help you study for the test we will be taking tomorrow. If you pay attention in class today and do your homework, you should do fine on the test.

Step-by-Step Plan: This is a lesson for a review, so if any students do not get the right answer either do it on

the board or call on one of the students to do it on the board1. What are some different ways we could make a frequency table with students in the

class? (ANS: height, hair color, eye color, etc) (Bloom's Knowledge)2. Pick one of these and make a frequency table quick. To make this go faster tell them to

stand up if: they have blue eyes, they have brown hair, etc. This will be a quick way to set up an example of a flow chart. (Gardner's Bodily/Kinesthetic)

3. Next we learned about probability distribution. What are two ways the book uses to show probability distribution? (ANS: A graph and a table.)

4. Represent the probability distribution for the sum of rolling 2 dice in table form and in graph form. (ANS: Look at example 3 on page 637). Have one student put their table on the board while another student puts the graph on the board. (Gardner's Logical/Mathematical)

5. What is the conditional probability formula? (ANS: P B∣A= P A∧BP A )

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6. If we asked 28 people if they liked to watch the Olympics on TV, 15 of them were males, 13 were females. If 7 males said yes and 7 females said yes, what is the probability that a person watches the Olympics on TV given they are a girl?

(ANS: P WatchesOlympics∣Female = P Female∧OlympicsP Female

= 713 ) (Bloom's

Application)7. Pick a student to define mean, one to define median, and one to define mode. (ANS:

mean= average, median= the number in the middle of the sample space (or the average of the 2 numbers in the middle), mode= the number that occurs most often.)

8. In making a box-and-whisker plot you have to find 3 quartiles. The middle one (Q2) is the median of the data. How do you find the other 2? (ANS: Take the section of the data that is smaller than Q2 and the section larger than Q2 and find the medians of those 2 parts. The lower part is Q1 and the higher is Q3.

9. Given the data 67, 73, 79, 81, 84, 88, 90, 92, 95 find the mean, median, mode, Q1, Q2, and Q3. (ANS: mean=83.22, median=84, mode=N/A, Q1= 76, Q2=median=84, Q3=91)

10. How would we make a box-and-whisker plot out of this data? (ANS: The box goes from 76 to 91, with a line at 84. the whiskers would be extending from 76 to 67 and from 91 to 95.) (Bloom's Comprehension)

11. Have a student quickly draw their bow and whisker plot on the board. (Garnder's Visual/Spatial)

12. What is the interquartile range of the data and how do you find it? (ANS: Q3-Q1 is how you find it, so its 91-76=15)

13. What is the equation for standard deviation? (ANS: σ=∑ x−x2

n)

14. What does the σ, Σ, x-bar, and n stand for? (ANS: σ =standard deviation, Σ=the summation of all the points, x-bar= the mean, n=number of data points [the numbers we have])

15. What is the standard deviation of the following data set? 1, 4, 6, 5, 3, 7, 9 (ANS: x-

bar=5, n=7. So if you plug things in you get σ= 425 =8.4≈2.9 )

16. What is the the z-score of the value 11? Round the standard deviation to 3.

(ANS: z−score= value−meanstand dev

=11−53

=63=2 )

17. If you sample 25 people over 30 years old and 10 of them are married, what sample is the sample portion for the people over 30 that are married? (ANS: x/n=10/25=0.40=40%)

18. How could we make this percentage more accurate? (ANS: asking more people)

19. What is the margin of error for this sample? (ANS: ± 1n

=± 125

=±15=0.2 )

20. So what interval is likely to contain a married person? (ANS: between 38% and 42%)21. You have berries in a bag, 6 are raspberries and 4 are blueberries. What is the probability

of choosing a raspberry, a blueberry, and then another raspberry assuming you are replacing the berry after every pick. (ANS: See page # 672, the raspberries are represented by the N and the blueberries are represented by P. so probability of NPN=0.144 or 14.4%)

22. What is the binomial probability formula? (ANS: nCxpxqn-x)

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23. If you have 4 batteries and each one has a 90% chance of lasting a year, what is the probability that all 4 will last a year? (ANS: 4C4(0.9)4(0.1)0=1(0.9)4(1)=0.6561 or 65.6%)

24. Compare binomial probability with normal probability. (Bloom's Analysis)25. At this point you are up to the review for 12-7. We did this yesterday in class and their

homework due for today was over 12-7, so they should not need a review for this. However, ask them if they have any questions on this section that they want to go over in class just in case.

26. The first part of the assignment is filling out the concept map that is below. The underlined words are the ones that need to be filled out.

27. Assign p. 639: 1, 3, 5; p. 644: 3, 7; p. 652: 5, 7; p. 660 1, 5, 11; p. 666: 1, 3, 9; p.675: 5, 7, 9; p. 681: 5, 11. Let the students work with partners and check answers with each other if they want. (Gardner's Interpersonal)

Closure: This is a good overview of the whole chapter. If you study the concept map and do your homework you should do fine on the test. This chapter is a good overview of probability and statistics. If you take any math classes in college you might look at some of these problems in more depth.

Adaptations/Enrichment: Student with hearing impairment:

I will make sure I do not talk while I am facing the board. Since they can not see my lips when I am facing the board they will not know what I am saying.

I will give this person a copy of my notes before class. This way they can read through it and they do not have to take notes during class. If they were to take notes during class they would not be looking at me, so they would not be getting everything I am saying.

Student with visual impairment:I will sit this student in the front of the room. This way if they can see anything at all, the closer they will be the better. Also since they have to use their hearing more, in the front of the room they will be far away from fans and other noisy objects.

To be able to do the homework efficiently this student will bee a braille textbook. These are much more expensive than normal text books, but we can not let this impairment hold this student back in their learning.

Reflection: As I stated at the beginning, this is a review so I will be asking constant feedback and making sure that if anyone has any problems or confusion I will be able to clear it up before the test. The assignment is the highlights of every section. They are not the hardest problems but they will be using the equations and logic that was learned in each lesson.

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Answer Key

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F. UNIT TESTSName: __________________

Chapter 12 TestDirections: SHOW ALL WORK, partial credit can be given. Be sure to put a box around all your answers so I can easily find them. If your answer is a decimal, round to 3 decimal places, and if it is a percent, round to 1 decimal place. This test is out of 50 points. The number in the parenthesis after each question indicates how much each question is worth.

1) If there are 15 people in your class, 7 have brown hair, 5 have blond hair, 2 have black hair, and one has red hair, draw a frequency table for this data. (3)

2) Using the previous data, what is the probability of choosing a person with black hair if someone from the class is randomly selected?(3)

3) Below is a table of the number of students in the school and how many pets they own. What is the probability that a student at random own 3 or more pets? (4)

# of pets 0 1 2 3 4 + Total# of students

10 20 15 12 8 65

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4) You are trying to find a partner to fight crime with. You want him/her to be at least 6 foot tall and is a black belt in karate. The table below indicates the data. What is the probability that the person you are interviewing is less than 6 foot tall given they know karate?(5)

Data 6 + foot tall Less than 6 foot tallBlack belt 2 8

Not a black belt 6 24

5) Find the mean, median, and mode of the following data: 3, 6, 7, 3, 8, 9, 5, 12, 5, 10, 13, 2. (4)

6) Make a box-and-whisker plot for the previous data. (4)

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7) Find the interquartile range for the previous data. (3)

8) Find the mean and standard deviation for the following data: 5, 6, 7, 4, 5, 5, 6, 8, 8. (5)

9) If a data set has a mean of 23 and a standard deviation of 4 find the z-score for the value 7. (4)

10) Find the margin of error when your sample space has 100 samples. (3)

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11) Suppose you had a weighted coin such that you had a 60% chance of getting heads, what is the probability of flipping a heads, tails, tails in that order? (4)

12) Find the probability of 4 successes in 6 trials when p= 40% (use the choose method equation for this). (5)

13) A set of data with a mean of 25 and a standard deviation of 4 find what percent of data within the interval 21 to 29. (3)

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Name: ___(KEY)_________Chapter 12 Test (Answer Key)

Directions: SHOW ALL WORK, partial credit can be given. Be sure to put a box around all your answers so I can easily find them. If your answer is a decimal, round to 3 decimal places, and if it is a percent, round to 1 decimal place. This test is out of 50 points. The number in the parenthesis after each question indicates how much each question is worth.

1) If there are 15 people in your class, 7 have brown hair, 5 have blond hair, 2 have black hair, and one has red hair, draw a frequency table for this data. (3)

ANSWER:Brown (7 tally marks)Blonde (5 tally marks)Black (2 tally marks)Red (1 tally mark)

2) Using the previous data, what is the probability of choosing a person with black hair if someone from the class is randomly selected?(3)

ANSWER: 2/15=.133 OR 13.3%

3) Below is a table of the number of students in the school and how many pets they own. What is the probability that a student at random own 3 or more pets? (4)

# of pets 0 1 2 3 4 + Total# of students

10 20 15 12 8 65

ANSWER: P(3 pets)+P(4+ pets)= 12/65+8/65=20/65=.308 OR 30.8%

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4) You are trying to find a partner to fight crime with. You want him/her to be at least 6 foot tall and is a black belt in karate. The table below indicates the data. What is the probability that the person you are interviewing is less than 6 foot tall given they know karate?(5)

Data 6 + foot tall Less than 6 foot tallBlack belt 2 8

Not a black belt 6 24

ANSWER: P B∣A=P A∧B

P A=P Not 6 ft∣bl belt= 8/40

10/ 40= 0.2

0.25 = 0.8 or 80% or 8/10 or

4/5

5) Find the mean, median, and mode of the following data: 3, 6, 7, 3, 8, 9, 5, 12, 5, 10, 13, 2. (4)

ANSWER:2, 3, 3, 5, 6, 7, 8, 9, 10, 12, 13Mean = (2+3+3+5+6+7+8+9+10+12+13)/12= 78/12=6.5Median= 7Mode= 3

6) Make a box-and-whisker plot for the previous data. (4)

ANSWER:2, 3, 3, 5, 6, 7, 8, 9, 10, 12, 13Q2= 7Q1=3Q3=10

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7) Find the interquartile range for the previous data. (3)

ANSWER: Q3-Q1=10-3= 7

8) Find the mean and standard deviation for the following data: 5, 6, 7, 4, 5, 5, 6, 8, 8. (5)

ANSWER: mean= (4+5+5+5+6+6+7+8+8)/9=51/9=6standard deviation=

σ=∑ x−x2

n = 221212120202122222

9 = 41111449 = 16

9 =43

4/3= 1.333

9) If a data set has a mean of 23 and a standard deviation of 4 find the z-score for the value 7. (4)

ANSWER:

z−score= x−xσ

=7−234

=−164

=−4

10) Find the margin of error when your sample space has 100 samples. (3)

ANSWER:

± 1n

=± 1100

=± 110 = ± 0.1 OR ±10%

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11) Suppose you had a weighted coin such that you had a 60% chance of getting heads, what is the probability of flipping a heads, tails, tails in that order? (4)

ANSWER:P(H)= 0.6P(T)= 0.4 P(H)×P(T) × P(T)= 0.6(0.4)(0.4)=0.096 OR 9.6%

12) Find the probability of 4 successes in 6 trials when p= 40% (use the choose method equation for this). (5)

ANSWER:x=4, n=6, p=0.4, q=1-p=0.6nCxpxqn-x=6C4(0.4)4(0.6)6-4=6C4(0.4)4(0.62)=15(0.0256)(0.36)=0.138 OR 13.8%

13) A set of data with a mean of 25 and a standard deviation of 4 find what percent of data within the interval 21 to 29. (3)

ANSWER: 68%

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Name: __________________Chapter 12 Test (With Modifications)

Directions: SHOW ALL WORK, partial credit can be given. Be sure to put a box around all your answers so I can easily find them. If your answer is a decimal, round to 3 decimal places, and if it is a percent, round to 1 decimal place. This test is out of 50 points. The number in the parenthesis after each question indicates how much each question is worth.

1) If there are 15 people in your class, 7 have brown hair, 5 have blond hair, 2 have black hair, and one has red hair, draw a frequency table for this data. (3)

2) Using the previous data, what is the probability of choosing a person with black hair if someone from the class is randomly selected?(3) [Use equation x/n]

3) Below is a table of the number of students in the school and how many pets they own. What is the probability that a student at random own 3 or more pets? (4)

# of pets 0 1 2 3 4 + Total# of students

10 20 15 12 8 65

[Use this equation to start: P(3 pets)+P(4+ pets)]

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4) You are trying to find a partner to fight crime with. You want him/her to be at least 6 foot tall and is a black belt in karate. The table below indicates the data. What is the probability that the person you are interviewing is less than 6 foot tall given they know karate?(5)

Data 6 + foot tall Less than 6 foot tallBlack belt 2 8

Not a black belt 6 24

[Use this equation for this problem: P B∣A=P A∧B

P A ]

5) Find the mean, median, and mode of the following data: 3, 6, 7, 3, 8, 9, 5, 12, 5, 10, 13, 2. (4)[Be sure to rewrite these number in order first to help you. Also remember the mean is the same as the average]

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6) Make a box-and-whisker plot for the previous data. (4)[Start by looking back on the past problem where you wrote the numbers in order. Now find Q1, Q2, and Q3]

7) Find the interquartile range for the previous data. (3)

8) Find the mean and standard deviation for the following data: 5, 6, 7, 4, 5, 5, 6, 8, 8. (5)

[We found the mean in a previous problem, find it the same way in this problem. The equation

for standard deviation is σ=∑ x−x2

n]

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9) If a data set has a mean of 23 and a standard deviation of 4 find the z-score for the value 7. (4)

[The z-score equation is z−score= x−xσ ]

10) Find the margin of error when your sample space has 100 samples. (3)

[Use this equation for this problem ± 1n

]

11) Suppose you had a weighted coin such that you had a 60% chance of getting heads, what is the probability of flipping a heads, tails, tails in that order? (4)[Remember, since there is only 2 outcome, P(T) will be 1- P(H)]

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12) Find the probability of 4 successes in 6 trials when p= 40% (use the choose method equation for this). (5) [The choose method equation is nCxpxqn-x where x=#of success, n=# of trials, and q=1-p]

13) A set of data with a mean of 25 and a standard deviation of 4 find what percent of data within the interval 21 to 29. (3)

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Name: __________________Chapter 12 Test (Key for Modifications)

Directions: SHOW ALL WORK, partial credit can be given. Be sure to put a box around all your answers so I can easily find them. If your answer is a decimal, round to 3 decimal places, and if it is a percent, round to 1 decimal place. This test is out of 50 points. The number in the parenthesis after each question indicates how much each question is worth.

1) If there are 15 people in your class, 7 have brown hair, 5 have blond hair, 2 have black hair, and one has red hair, draw a frequency table for this data. (3)

ANSWER:Brown (7 tally marks)Blonde (5 tally marks)Black (2 tally marks)Red (1 tally mark)

2) Using the previous data, what is the probability of choosing a person with black hair if someone from the class is randomly selected?(3)

ANSWER: 2/15=.133 OR 13.3%

3) Below is a table of the number of students in the school and how many pets they own. What is the probability that a student at random own 3 or more pets? (4)

# of pets 0 1 2 3 4 + Total# of students

10 20 15 12 8 65

ANSWER: P(3 pets)+P(4+ pets)= 12/65+8/65=20/65=.308 OR 30.8%

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4) You are trying to find a partner to fight crime with. You want him/her to be at least 6 foot tall and is a black belt in karate. The table below indicates the data. What is the probability that the person you are interviewing is less than 6 foot tall given they know karate?(5)

Data 6 + foot tall Less than 6 foot tallBlack belt 2 8

Not a black belt 6 24

ANSWER: P B∣A=P A∧B

P A=P Not 6 ft∣bl belt= 8/40

10/ 40= 0.2

0.25 = 0.8 or 80% or 8/10 or

4/5

5) Find the mean, median, and mode of the following data: 3, 6, 7, 3, 8, 9, 5, 12, 5, 10, 13, 2. (4)

ANSWER:2, 3, 3, 5, 6, 7, 8, 9, 10, 12, 13Mean = (2+3+3+5+6+7+8+9+10+12+13)/12= 78/12=6.5Median= 7Mode= 3

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6) Make a box-and-whisker plot for the previous data. (4)

ANSWER:2, 3, 3, 5, 6, 7, 8, 9, 10, 12, 13Q2= 7Q1=3Q3=10

7) Find the interquartile range for the previous data. (3)

ANSWER: Q3-Q1=10-3= 7

8) Find the mean and standard deviation for the following data: 5, 6, 7, 4, 5, 5, 6, 8, 8. (5)

ANSWER: mean= (4+5+5+5+6+6+7+8+8)/9=51/9=6standard deviation=

σ=∑ x−x2

n = 221212120202122222

9 = 41111449 = 16

9 =43

4/3= 1.333

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9) If a data set has a mean of 23 and a standard deviation of 4 find the z-score for the value 7. (4)

ANSWER:

z−score= x−xσ

=7−234

=−164

=−4

10) Find the margin of error when your sample space has 100 samples. (3)

ANSWER:

± 1n

=± 1100

=± 110 = ± 0.1 OR ±10%

11) Suppose you had a weighted coin such that you had a 60% chance of getting heads, what is the probability of flipping a heads, tails, tails in that order? (4)

ANSWER:P(H)= 0.6P(T)= 0.4 P(H)×P(T) × P(T)= 0.6(0.4)(0.4)=0.096 OR 9.6%

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12) Find the probability of 4 successes in 6 trials when p= 40% (use the choose method equation for this). (5)

ANSWER:x=4, n=6, p=0.4, q=1-p=0.6nCxpxqn-x=6C4(0.4)4(0.6)6-4=6C4(0.4)4(0.62)=15(0.0256)(0.36)=0.138 OR 13.8%

13) A set of data with a mean of 25 and a standard deviation of 4 find what percent of data within the interval 21 to 29. (3)

ANSWER: 68%

Reason for modifications:The ways I modified my test was by giving the student most of the equations. This is very

helpful because they do not have to focus on remember the equations, they can focus on doing the actual math part. A couple of equation are hard to interpret so I put what all the variables represent. They are doing the exact same math and getting the same answers as the students with the regular test, but this way they do not have to sit and look at an empty page. This gives them a starting point on most of the problems. All the modifications I added I put in brackets so they are easy to find. I will obviously allow them more time for the tests and have someone read the questions out loud if the students needs it. The last modification I added was I put more space between the questions, allowing them more space to answer. I have found that a lot of students that need these modifications write kind of big. When I take a test and I need a lot more room than is allowed, I either have to take time erasing or sometimes I start over because I feel like I am doing it wrong since I am taking up so much space. I wanted to eliminate that stress on this test.

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G. REFLECTION PAPER

When I started this class I thought I was ready to walk into a classroom and teach without

having any troubles. I look back and think about how naïve I was. I would not have survived if I

would have been in a school setting. I did not know the first thing about how to work with

exceptional learners. In my opinion one of the most important things to learn before we start

teaching because most of us teachers never experienced being an exceptional learner, so we

sometimes forget about them. This class not only made me aware of exceptional learners, but

also taught me the best ways to handle different cases.

One of the things that never entered my mind was reading in my class room. As a math

teacher I just assumed I would be going math equations and examples and leave the reading to

the English teachers. As I thought about it, I started to understand the importance of reading in

math. Its a skill you have to learn and is totally different than reading a book. You are not just

reading for enjoyment, you have to find the things in the reading to pick out and make into an

equation. My favorite strategy that I learned was how to do read aloud. It seems like you would

just have the students read out loud to the class. I was immediately turned off by this, but then

when I read into it, I found that it actually meant the teacher reads as the class follows along. I

think this is a great idea especially in math. The teacher can emphasize things and point out

things along the way while the students follow along and soak in all the information.

The biggest thing I will take from this class is not only how to teach exceptional learners,

but also how to teach the general students as well. I think the key to teaching both of them is

trying to make things as simple as possible for the students to understand. In math some teacher

try to phrase things tricky or make student figure out how to do something on their own. I

believe these things could be used sometimes, but it will work much better if the teachers teaches

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the material in a simple manner. If you want a problems to be harder, combine two kinds of

problems they know how to do, or make the numbers bigger or fractions. Confusing the students

will not help them learn.

For exceptional learners I learned the biggest things you can do to help is make them

know they belong in your class and minimize the distractions. A very easy way to help them that

is greatly beneficial is giving them a copy of your notes before the lecture. This gives them a

chance to just watch and listen as you go through an example. When a student gets behind on

notes and tries to catch up, they will not hear a word you are saying. This is a distraction that

most teachers do not think about. This will also help the student know they belong since you are

doing extra work for him/her. If they know you believe in them and you are trying to help them

in any way you can, this will give them confidence. Confidence is a huge factor in how well a

student does in school. If they do not think anyone believes in them and they are doomed to the

resource room for their whole high school career, they will give up. So believe in them, make

sure they know you believe in them, and encourage/compliment them any chance you get.

The textbook is a great tool in math. It is probably used more in math class than any other

class. The best way I found to use it was for examples that walk through the problems step by

step. If a student asks me how to do something, I will first tell them what example it relates to

and have them think about it a little more. Every math text book has great visual aids in which I

can help the students understand the content better. Also this text book was very good at pointing

out real world problems were math can be used. I think that is the most important aspect to teach

high school kids. They have to know that they will use this after they pass this class.

The biggest tool I learned to use from this experience was realizing not everyone learns

the same way I do. I like one example for each problem, but some people need two or three.

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Some people are more hands on and some just need a visual aid. The only way to solve this is to

get to know all my students personally and figuring out their learning styles. After doing that I

will mix up my lessons to make sure they are all learning as much as they can. I will make

myself available before and after school as much as I can to answer any questions students might

have. This is very important to me because if a student is willing to put in the extra time to come

and ask me questions, I have to be willing to stop what I am doing and help. This class has not

only prepared me and gotten rid of some of my fears, but also has made me very excited to start

teaching.

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