Pre-Algebra: Week of March 30, 2020 - Home - Great Hearts ......Mar 07, 2020 · Pre-Algebra: Chapter 11 Probability March 30 – April 3 1 Packet Overview Date Objective(s) Page

Pre-Algebra: Week of March 30, 2020 Time Allotment: 40 minutes per day

Student Name: ________________________________ Teacher Name: ________________________________

Pre-Algebra: Chapter 11 Probability March 30 – April 3

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Packet Overview Date Objective(s) Page Number

Monday, March 30 Calculate the number of permutations of n things taken r at a time.

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Tuesday, March 31 Calculate the number of combinations of n things taken r at a time.

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Wednesday, April 1 Identify key words to determine when to apply formulas for permutations or combinations.

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Thursday, April 2 Calculate the probability of an event given equally likely outcomes. Calculate the probability of the complement of an event.

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Friday, April 3 Calculate the probability of the complement of an event.

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Additional Notes: Thank you students for all of your hard work and commitment to Pre-Algebra!

Email: [email protected] or [email protected]

Each lesson will end with a set of math problems pertaining to that particular lesson of the day; some will have problems in the lessons themselves you should do as well. Please create an “Exercise Packet” which is to include all your work and completion of these daily exercises. Each day is to have a title with the date followed by the name of the lesson. Please include a title page and staple all the completed exercises. At a later point, we will ask you to turn your exercise packet. Do not worry right now about whether that will be online or in person, simply do the problem set as I instruct with the proper titles and labels.

For extra support and instruction, see the module at Kahn Academy on Intro to Theoretical Probability: https://www.khanacademy.org/math/ap-statistics/probability-ap/randomnessprobability-simulation/v/basic-probability [The last section on “Statistical Significance” is not important for this lesson.]

We appreciate all of you. Have a great day!

***How to Learn from a Math Book***

A big thank you to Mrs. Boyd who shared with us a great resource – “How to Learn from a Math Book.” Many have realized this past week that reading a math book is far different that reading a typical literature book. The entire document Mrs. Boyd provided is posted on the GHNO website where you can download the weekly packets. However, we wanted to emphasize a couple of items as well; those are listed on the next page. Hopefully this will help students with this next week’s packet. As always, ask questions!!


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HOW TO LEARN FROM A MATH BOOK

Reading a math book is different from reading other types of books and reading a math textbook is different from the traditional way students are taught to read textbooks in general. Here are some tips on how to learn math from your math textbook.

1. Every word counts. Math books are usually not repetitive, so there is little chance of

picking up missed information from reading on. Writers of math texts believe that extra words and repeats get in the way of clarity. Never start in the middle of the book, the chapter, or the page. Each page assumes you have mastered the previous pages.

2. Words and symbols of math have very specific meanings. If you are at all uncertain about the meaning of a term look it up, or ask someone to explain it. Keep a list of these new vocabulary words and new symbols for easy review.

3. Create a resource for yourself by recording key points on a separate piece of paper or into a notebook. Make use of index cards, concept sheets, math diaries, etc. to record formulas, algorithms, theorems, important derivations, mathematical terms and symbols, and relevant examples.

4. What should you do if while reading the text you get to a point where you cannot understand the material?

a. Locate and review any diagrams, examples, or rules that explain the misunderstood material.

b. Refer to another math textbook, website, or DVD that expands the explanation of the misunderstood material.

c. Prepare questions for your instructor on the confusing information and contact your math tutor or math instructor for help in understanding the material. Prepare yourself to ask those questions at the next class meeting

Academic Honesty

I certify that I completed this assignment independently in accordance with the GHNO

Academy Honor Code.

Student signature:

___________________________

I certify that my student completed this assignment independently in accordance with

the GHNO Academy Honor Code.

Parent signature:

___________________________


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Introduction to Unit We are starting a whole new unit today called Probability. In this unit, we will be looking at ways to analyze random events. For example, what it the likelihood of you picking a card from a deck of standard playing cards and drawing a heart. In order to begin an analysis of any random event, the starting block is determining all the possible outcomes first. The easiest way to do this is counting them, such as counting the number of cards in a deck of standard playing cards. However, simply counting all possible outcomes can be very difficult and time-consuming, thus this unit starts with mathematical tools that helps us determine the number of possible outcomes in more complex situations. Monday, March 30 Pre-Algebra: Chapter 11 Probability Lesson 1: Permutations Objective: Calculate the number of permutations of n things taken r at a time. Intro to Lesson: Permutations are arrangements of things where the order of the elements of the group matter. The first question that naturally arises in such a situation is the number of possible permutations of a set of objects given a certain initial group of things out of which the set is chosen. Permutations considered in this why are an example of a basic counting principle. These basic counting principles can be challenging, but they form the foundation of discrete mathematics and find wonderful applications in things like the Binomial Theorem later on in mathematics. NOTE: I will be providing some examples specific to this lesson. Please highlight the key points as you read through this entire lesson or take notes in the margin. At the end of the lesson, the list of questions in the exercise are to be completed to ensure mastery. Preparing for this section: Before getting started, let’s learn some definitions. In mathematics an arrangement of a group of things in a particular order is called a permutation.

Fill in the Blank Answer: order matters! It is important for us to implement the Fundamental Counting Principle in order to understand how to find the number of permutations that exist for certain situations.


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An example applying the Fundamental Counting Principle: The school cafeteria offers a choice of two main courses - grilled cheese sandwiches or the soup of the day and five desserts – jello, pudding, fruit cups, sundaes, or granola bars. How many different lunches could you have?

Main Courses: G, S Desserts: j, p, f, s, b

Answer: There are 2 branches because there are two choices for the main course.

There are 5 branches because there are 5 possible choices for the dessert. If the first action can be performed in 2 ways, and the second action can be performed in 5 ways then it will be 2 x 5=10 possible lunches.

We can demonstrate this example below by using a “Tree Diagram” which shows all the possible Main Courses matched up all the possible desserts.

The tree diagram is a very nice way to organize the information, but we do not want to draw a tree diagram for each scenario so we will use multiplication.

In our example above since there 2 Main Course and 5 Desserts, we determined that 2 x 5 = 10, which means there are 10 possible ways to arrange both a Main Course and Dessert.


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How many ways can 5 students line up for lunch? Notice how the number of permutations of 5 things is 5 X 4 X 3 X 2 X 1. So, we can write 5! to represent the expression. Since 5! = 5 X 4 X 3 X 2 X 1 this means that 5! = 120 since 5 X 4 X 3 X 2 X 1 = 120. Exercise: Cover the right side below and determine the Correct Answers for the Following Factorials on the left. Then check your answers!

1! = ____ 4! = ____ 1! = 1 4! = 24

2! = ____ 5! = ____ 2! = 2 5! = 120

3! = ____ 6! = ____ 3! = 6 6! = 720

Some of you might be asking what is 0! Think about it and then look at the answer on the next page.


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Proof of 0!:

Permutation Formula: In the previous examples or arranging the three cards A B and C, the group size was 3 and we were all 3 cards. Thus, was used 3! However, we can also write this another way. (Note: the “P” stands for Permutation!)

means n number choices taken n at a time. P(n,n) is also how it may be written.

Example = !𝑃! = 3 X 2 X 1 = 6. This is the same as 3! However, there are situations where you start with the total group size, but you are not going to use the entire group. Counting the number of permutations depends on how many things can be in the group and how many elements the group has -- and is written as -- where n is the number of things than can be in the permutation (i.e., how big is the group), and r is the number of elements the permutation has (i.e., how many items are you selecting from the group).

means n number choices taken r at a time. P(n,r) is also how it may be written.

=""𝑃# = 11 X 10 X 9 X 8 X 7 = 55,440 Example:

n rP


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Example to try on your own: The answer is on the Answer Key. After completion, check your answer before moving on.

Directions for completing the exercises:

1. Create a packet for completed exercises. I want you staple together several papers with a title page “Exercises for Pre-Algebra”

2. At the top of each exercise completion page, title as such: Exercises for Monday, March 30 Permutation. 3. After you finish the entire exercise, please check your answers with the answer key at the end of the

packet and attempt correction. Exercises for Monday March 30, 2020: Please do # 1- 10 all

Any questions right now? Please email: [email protected] or [email protected]


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Tuesday, March 31st Pre-Algebra: Chapter 11 Probability Lesson 2: Combinations Objective: Calculate the number of combinations of n things taken r at a time. Intro to Lesson: In the previous lesson, we learned about Permutations which are arraignments of things where the order of the elements of the group matter. We also learned how to determine the number of permutations of various sets of objects. In this lesson, we extend that knowledge to combinations, i.e., groups in which the ordering of the elements does not matter. Definition: A combination is a grouping of distinct objects without regard to the order in which the grouping is made.

Fill in the Blank Answer: order DOES NOT matter!


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This last formula may look complicated but can be quickly derived from the math problem.


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Note, you also write a combination as C(n, r) which means n number of choices taken r at a time. Example: C(11,5) =""𝐶# =

!!%""%"

= ""&"'&(&)&*#&+&!&,&"

= ##++'",'

= 462 Your turn to try a combination problem: C(8, 3) = C(16, 4) = Please check your answers after completion of the problems before you move on. Your turn to try a combination word problem:

Please check your answer on the answer key before you move on!! Any questions right now? Please email: [email protected] or [email protected]


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Exercises for March 31, 2020: Combination


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Wednesday, April 1st Pre-Algebra: Chapter 11 Probability Lesson 3: Review Objective: Identify key words to determine when to apply formulas for permutations or combinations. Intro to Lesson Today we spend the lesson practicing the concepts we learned the last two days. You work on the answers in your notebook, but do not need to write out the problems!


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Exercises for April 1, 2020: Study for quiz tomorrow. Any questions right now? Please email: [email protected] or [email protected]


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Thursday, April 2 Pre-Algebra: Chapter 11 Probability Lesson 4: Probability of an Event Objective: Calculate the probability of an event given equally likely outcomes.

Calculate the probability of the complement of an event. NOTE: First, please complete the quiz located at the end of the packet. Then, you may begin lesson. Intro to Lesson In this lesson, we continue our study of probability by considering the probability of an event occurring out of a range of possible outcomes. The essence of this lesson is being able to determine how many outcomes are possible, and of those how many are the favored outcome. Unless specified otherwise, all outcomes are considered equally likely and marginal cases (such as a coin landing neither on heads nor on tails but exactly on the edge) are ignored.

In general, we have the following formula:


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Exercise for Thursday, March 26, 2020 “Written Exercises” pp.407, 1-17 odd, 19, 23, 27.


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Any questions right now? Please email: [email protected] or [email protected]

Friday, April 3 Pre-Algebra: Chapter 11 Probability Lesson 5: Odds in Favor and Odds Objective: Calculate the probability of the complement of an event. Intro to Lesson In this lesson, we expand on the lessons above by considering the probability of the complement of an event. Sometimes, when we want to know the likelihood of something occurring, it is easier to figure out the likelihood of it not occurring and then using that information to find the desired probability. Note that students will struggle with the difference between probability and odds.


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Sometimes people express the likelihood of events in terms of odds rather than probabilities. The odds of an event occurring are equal to the ratio of favorable outcomes to unfavorable outcomes. In other words, the notion of “odds” is as the ration of the probability of an event occurring to the probability of an event not occurring. Odds in favor of event A= !(#)

!(%&'#) ; Odds against A = !(%&'#)

!(#)

Let’s work on this problem together. The chance of rain tomorrow is 30%. Find the odds that it will rain. Step 1: Turn the probability into a fraction in simplest form. 30% = ()

+))= (

+)

Step 2: Express the fraction in terms of favorable outcomes and total outcomes. P(rain)= ,-./0-123/456/738

'&'9:&;'?= (

+)

Step 3: Subtract the favorable outcomes from the total number of outcomes to find the unfavorable outcomes. Total outcomes – favorable outcomes = unfavorable outcomes 10 – 3 = 7 Step 4: Use the favorable outcomes to find the odds: Odds(rain) = ,-./0-123/456/738

;%@9A&B9C:>&;'?= (

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The odds that it will rain tomorrow are 3 to 7. (Note, if the question was “find the odds that it will NOT rain” it would simply be the reverse 7 to 3!) Recap: “odds of an event occurring” = !(E)

+F!(E)

and “odds of an event not occurring” = +F!(E)

!(E)

NOTE: “~” = “not”. Using that notation, write P(~E) = 1 – P(E).


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Any questions right now? Please email: [email protected] or [email protected] Exercise for April 3, 2020 “Written Exercises” pp. 411-412, 1-15 odd.


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ANSWER KEY

Try it Exercise Monday March 30th :

Exercises for Monday, March 30th:


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Try it answers for Tuesday, March 31st :

Answers for Exercises for Tuesday, March 31, 2020


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Exercises for Wednesday, April 1, 2020


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Answers for Exercises Thursday, April 2, 2020


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Answers for Exercises for Friday, April 3, 2020


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NOTE: At a later point, we will ask you to turn your exercise packet. Do not worry right now about whether that will be online or in person, simply do the problems as I instruct with the proper titles and labels.


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QUIZ

Name: ____________________________________________ Solve the following. Show all necessary work.

1. What is the difference between permutation and combination?

2. How many ways can you make a 9-player batting order from a group of 15 players?

3. 13 boys are trying out for a 10-man basketball team. How different teams could be formed?

Documents

Pre-Algebra: Week of March 30, 2020 - Home - Great Hearts ......Mar 07, 2020 · Pre-Algebra: Chapter 11 Probability March 30 – April 3 1 Packet Overview Date Objective(s) Page