Unit Project: Probability
Unit Project: Probability
Maggie Sharp
Lauren Nowak
Chanel Keyvan
Table of Contents
|Rationale |3 |
|Unit Calendar |4 - 6 |
|Mini and Full Lessons: |7 - 20 |
|Mini lesson day 1 |7 |
|Mini lesson day 2 |8 |
|Mini lesson day 3 |9 |
|Mini lesson day 4 |10 |
|Mini lesson day 5 |11 |
|Mini lesson day 6 |12 |
|Full lesson (Lauren) day 7 |13 - 25 |
|Mini lesson day 8 |26 |
|Mini lesson day 9 |27 |
|Mini lesson day 10 |28 |
|Full lesson (Chanel) day 11 |29 - 38 |
|Full lesson (Maggie) day 12 |39 - 48 |
|Mini lesson day 13 |49 |
|Mini lesson day 14 |50 |
|Assessment | |
Rationale
For this unit, we wanted to accomplish both mathematical and social justice goals. The mathematical goals we targeted were concerning basic probability topics. We wanted students to leave this unit with a concept of what probability is in general, how to calculate probabilities in certain scenarios, and how to interpret the results. We included topics such as experimental vs. theoretical probability, law of large numbers, area models for probability, expected value, combinations, permutations, etc. We tried to integrate lessons that allowed students to recognize real world contexts of probability models. To do this, we incorporated problem-based lessons, heavy student involvement in simulations, and applications to real world contexts. Though we attempted to incorporate many problem-based lessons, we felt that it was best to teach the combinations and permutations lessons in a more traditional manner. This is because these topics are more involved and complex, and this is integral for our social justice project. We wanted to set students up to successfully complete the final project and to provide enough support for the students. We believe that these methods will help emphasize and encourage students to view mathematics as a tool to help understand the world around them.
Alongside our mathematical goals, we thought that the probability unit would lend itself well to a social justice project. The main purpose of this project is to expose students to using mathematics as a way to interpret and understand the world to develop a mathematical power. In particular, we want students to realize that since this much of this information is public, they have the opportunity to conduct their own research. We also want students to take advantage of their education and realize that they can make valuable contributions to society. Finally, we want students to have the freedom to discuss their own opinions and recognize that they have a voice within the classroom, and so structured our project so the last day is a discussion with a few guiding questions but lots of room to explore topics that students found engaging. We chose the lottery because it’s a topic that is rich in mathematical content, but also is a surprising source of a social justice issue because it is not inherently obvious or widely publicized as anything other than a game. While it is good to investigate obvious injustices such as food shortages or racist hiring practices, we felt that students might be more surprised and more inclined to critically evaluate all aspects of the world if they investigated an issue that seemed to be relatively neutral.
We are assuming that this is a unit we would teach towards the end of the year. Students would have an understanding of the classroom norms such as group work and problem-based learning. We assume that students have been introduced to probability in middle school, and are comfortable with basic arithmetic.
We have a classroom with 25 students, 50% are considered under-achievers, students show potential but need motivation, and 2 of the students struggle with reading and are allowed extended testing time. To accommodate for these different factors, we tried to incorporate activities that were engaging and welcoming. The activities we specifically chose incorporate students physically moving around, topics that are relevant or interesting for the students, and accessible yet challenging activities. We also designed our worksheets and assessments to be easy to read and not overwhelmed with unnecessary jargon. Our assessments primarily take the form of projects, homework assignments, and journal reflections; with these sorts of assessments, time pressure is less likely to affect performance and students can work at their own pace. We introduce an article for students to read on the second to last day of the unit. To accommodate for this, we use the Directed Reading Thinking Activity to help students understand what they are reading and teach them tools to develop their own reading skills.
Unit Calendar
|Date |Brief Description of Content and Lesson |Technology, Special Activities, Manipulatives, |
| | |Problem-Based, Instructional Strategies |
| |Introduction To Probability: |Unit Hook, Computers, Dice, Spinners |
| | | |
|Day 1 |Students are playing probability games that will eventually | |
| |be analyzed later in the unit. The games include a | |
| |predicting the sum of two dice, interactive computer games, | |
| |a spinner to determine how much money you will receive, and | |
| |a Word Game. | |
| |Theoretical vs. Experimental, Law of Large Numbers: |Technology lesson, Calculators, Problem-Based |
| | |Lesson |
|Day 2 |Students use calculators to analyze a dice game to compare | |
| |experimental probabilities, theoretical probabilities, and | |
| |law of large numbers. They will use simulations to help | |
| |further their understanding of the relationship between | |
| |theoretical and experimental probabilities. | |
| |Multiplying Probabilities: |Cup, Blue and Green Marbles, Problem-Based |
| | |Lesson |
|Day 3 |Students use a geometric area model to begin to examine | |
| |why/when probabilities are multiplied. The main problem will| |
| |be to arrange marbles in cups to provide the greatest | |
| |probability to draw a green marble. | |
| |Tree Diagrams: |Student Acting |
| | | |
|Day 4 |Students further explore how multiplying probabilities works| |
| |and then begin to explore how tree diagrams are used to help| |
| |calculate theoretical probabilities for different events. We| |
| |will use a story/play for students to understand tree | |
| |diagrams. | |
| |Expected Value Intro: |Spinner, Problem-Based Lesson |
| | | |
|Day 5 |Students analyze the spinner game in the first day of the | |
| |class and calculate the expected value of the game. They | |
| |will simulate a few rounds and then explain expected value | |
| |in the context of the problem. | |
| |Expected Value Application: |Manipulatives for simulation activity, |
| | |Problem-Based Lesson |
|Day 6 |Students explore the one-and-one situation in basketball to | |
| |further understand expected value compared to most likely | |
| |outcome. They will be using the area model and simulations | |
| |to determine the differences between expected value and most| |
| |likely outcome. | |
| |And/Or, Venn Diagrams (Full Lesson Plan: Lauren) |Incorporates Physical Movement |
| | | |
|Day 7 |Students use Venn Diagrams to break down categories into | |
| |mutually exclusive events. They use an interactive activity | |
| |at the beginning to gather class data to use in the lesson. | |
| |Permutations: | |
| | | |
|Day 8 |Students analyze the “Word Game” and other carefully chosen | |
| |problems as an introduction to permutations. Students will | |
| |understand that permutations are order specific. | |
| |Combinations: |Problem-Based Lesson (At beginning) |
| | | |
|Day 9 |Students replicate and evaluate the “Handshake” problem. | |
| |They begin a worksheet to further their understanding of | |
| |combinations that shows how combinations originate from | |
| |permutations. | |
| |Integration of topics/ Synthesis |Problem- Based Lesson |
| | | |
|Day 10 |Students work in groups to solve the problem of the day | |
| |which incorporates an integration of both combinations and | |
| |permutations. They will be determining all the different | |
| |ways to create a seating chart. | |
| |Part I – Can playing the lottery be profitable? (Full Lesson|Social Justice Lesson |
| |Plan: Chanel) | |
|Day 11 | | |
| |Students analyze how profitable Fantasy 5 is through | |
| |combinations and expected value. They will determine if the | |
| |lottery is profitable based on their results of the expected| |
| |value. | |
| |Part II – Who benefits/harmed by the lottery? (Full Lesson |Social Justice Lesson |
| |Plan: Maggie) | |
|Day 12 | | |
| |Students determine who benefits/harmed by Fantasy 5 through | |
| |an analysis of income and types of people who play the | |
| |lottery. Based on the percentage of income spent, students | |
| |analyze whether or not the lottery should be considered a | |
| |regressive tax. | |
| |Part III – How are the proceeds used? |Social Justice Lesson |
| | | |
|Day 13 |Students evaluate how funding for the lottery is related to | |
| |the state education budget. They analyze the proportion of | |
| |funding different income brackets contribute to the budget, | |
| |and read an article critically analyzing the lottery’s role | |
| |in district funding in Chicago. | |
| |Project Discussion |Social Justice Lesson |
| | | |
|Day 14 |Students spend the class sharing their findings from the | |
| |past week. They discuss what their answers were to the | |
| |previous day’s prompts regarding the lottery and fairness. | |
Mini and Full Lessons
|Day #: 2 |Lesson Title: Experimental v. Theoretical , Law of Large numbers |
|Goal: |Compare experimental and theoretical probability |
| | |
|Objectives: |Students will construct accurate theoretical models for summing dice |
| |Students will generate data using their calculators and construct histograms on the calculator from this data |
| |Students will define the differences and similarities between theoretical and experimental probability |
| |Students will explain how number of trials affects the outcome of the experimental probability |
|Lesson Summary (one|When students come to class, calculators are provided for them. We explain that students will be using their calculators to model|
|paragraph maximum) |the dice game they played the day before. First, we ask students to work with a partner & list all the possible ways to sum 2 |
| |dice. Walk around and help students, possibly providing a hint on the board if students are confused. When students are finished,|
| |have a group come put their results on the board. Then calculate the percentage of times they would expect to get a sum of 6. |
| |Then, students will use their calculators to create two lists of randomly generated dice rolls. They then find the sum of the |
| |rolls in a third list. We start with a low number of trials, perhaps 5 or 10. We ask students to count how many times they |
| |actually got 6 and compare their expected percentage of 6s to the actual number they found in the experiment. We then present the|
| |definitions of theoretical and experimental probability. Students discuss which situation models which probability and how the |
| |two probabilities compare. They should see that both probabilities consider total outcomes over total possibilities or trials, |
| |but that theoretical represents what we expect to happen and experimental represents what actually happened. Then we begin to |
| |increase the number of trials done in each list. We go from 20 to 50 to 100 to 1000. Each time, the students create a histogram |
| |and determine how close the experimental data is to the theoretical. Finally, we discuss how increasing the number of trials |
| |makes the probability closer to theoretical probability and introduce the term “Law of Large Numbers.” |
|HW |Write a journal entry explaining how what we did today would have changed your game strategy yesterday OR defend your strategy |
| |from yesterday based on the mathematics you learned today. |
|Day #: 4 |Lesson Title: Tree Diagrams |
|Goal: |To develop an understanding of using tree diagrams to predict the probability of different paths in a multistep event. |
| | |
|Objectives: |1. Students can visually construct a tree diagram representing all possible outcomes of a multistep event |
| |2. Students can evaluate the probability of a particular path |
| |3. Students can justify why probabilities on a path are multiplied (from prior knowledge) |
|Lesson Summary (one|At the beginning of the lesson we ask one student to come to the board and draw their area diagram for their solution. We proceed |
|paragraph maximum) |to discuss how each box represents the probability of pulling that specific marble. We ask students to assign a number to the |
| |probability of pulling each marble based on the amount of the total box that the marble represents. We then launch a discussion of|
| |how this probability can be calculated. For example: If a marble takes up a quarter of a box it means that it was in a cup with |
| |one other marble and there is a 25% chance of choosing that marble. This comes from a 50% chance of choosing the cup with that |
| |marble in it and then a 50% chance of choosing that specific marble. To get 25% we are simply multiplying the 50% x 50%. We |
| |discuss this for all or most of the boxes, depending on the divisions created. Our goal is to allow students to make connections |
| |about why we multiply probabilities. Then we ask students if any of them found a way other than the area model to visually |
| |represent their “best” outcome. If so, allow them to share. If not use this as a transition into the introduction of a tree |
| |diagram. Explain that another way to represent making “choices” in probability is to draw a diagram in the form of tree branches. |
| |We model the tree diagram of the same solution we went over with the area model before. We then explain how they will be able to |
| |multiply down a “path” to find the probability of a specific outcome. Next, we choose several student volunteers to help us act |
| |out a short, pre-prepared skit. The teacher(s) will serve as narrator and the students will act out the actions described. There |
| |will be several points in the story in which characters are asked to stop and make a choice, each one being required to choose a |
| |different path. The rest of the class will vote for their desired outcome for each character. As decisions are made, one student |
| |records the paths on the board in the form of a tree diagram. We spend the rest of the period discussing the probabilities |
| |associated with each path taken during the story. Assign homework. |
|HW |Students will be given a choice between 2 assignments: They may choose to do problems 4, 7, and 10 from pp. 673-674 OR they may |
| |create their own story problem and calculate the probabilities associated with their story. Tell students to be sure to use tree |
| |diagrams in their homework. |
Name: Lauren Nowak
Date: 5-8-2012
Grade Level: 11-12
Course: Algebra II
Time Allotted: 50 mins
Number of Students: 25
I. Goal(s):
To learn how to use Venn diagrams to break events into mutually exclusive categories, and use this knowledge to evaluate AND/OR statements.
II. Objective(s):
- The student will gather data about the different categories their classmates can be classified into
- The student will provide examples of mutually exclusive events
- The student will identify AND & OR statements
- The student will use the relationship P(A or B) = P(A) + P(B) – P(A and B) to solve for the probability of OR statements
- The student will label Venn Diagrams to show how to break an event down into mutually exclusive categories
III. Materials and Resources:
- A list of preprepared categories for the beginning game (see specifications
under “Motivation”
- Masking tape or duct tape to mark the floor
- 26 copies of activity handout
- Elmo
- (optional) transparencies & different colored markers
- 25 copies of homework
IV. Motivation (10-15 mins)
1. Begin class by moving the desks to create an open space in the middle of the room, with 24 evenly spaced x’s taped on the floor before class starts. Direct students to stand in a big circle, 24 of them each standing on an x and one in the center, with the teacher at the board acting as caller / recorder. Explain that in this game, “Whenever a category is called, if you belong to that category, then you must switch spots to a different x. You want to try not to become the person in the center. If you are in the center, then you are trying to get into one of the spots on the outside of the circle, and can move whether or not you belong to the category being called. Do not switch with a person immediately next to you. You can speed-walk, but please no running or shoving. Let’s start!”
As the teacher calls out categories (Boys switch! Girls switch! Boys wearing blue jeans, switch!), she should also record the category called and the number of each student who belonged to that category in a chart on the board. If the changing happens too fast to record the results for a particular category, the teacher should ask the students who belonged to that category to raise their hands before starting the next round.
An example list of categories is provided below. However, the categories should be created based upon the demographics and personality of the classroom, so some personalization and modification of the list below should be expected.
Creating Categories:
When playing the game, the teacher should do between 7-12 rounds, with the breakdown of category types as follows:
- 2+ “regular” categories that are mutually exclusive (girl/boy, likes/dislikes, has more than/has less than, etc.). This should be the first 2+ called out.
- 2+ additional “regular” categories
- 1+ AND combination
- 1+ OR combination
- 1+ AND/OR combination. This should be the last category called out.
Example “regular” categories to call out:
Girls*
Boys*
Students wearing blue jeans
Students with curly hair
Students who play an instrument**
Students who play a sport
Students who speak another language
Students who like hip-hop music
Students who have no siblings, one sibling, two siblings, etc.
Combinations of categories:
AND combo: Students who play an instrument AND play a sport
OR combo: Students who like hip-hop OR speak another language
AND/OR combo: Boys who are wearing blue jeans OR girls who play an
instrument
Girls who like hip-hop OR students who play sports
- These are AND/OR combos because we have:
(Boys AND blue jeans) OR (Girls AND instrument)
(Girls AND hip-hop) OR (Sports)
*Consider changing this if there are gender-queer students in the classroom; some other binary, mutually exclusive set of categories could be used (i.e. 16 and older/Younger than 16, Only child/Has siblings)
**If the class isn’t cooperative in moving for these less obvious categories (students are cheating or inconsistent), it might be best to stick with more physical, observable traits such as hair type, eye color, clothing, etc.
Transition: "Okay, now before we move the desks back together, I want to ask – raise your hand – who felt like they hardly had to move at all? Who felt like they were moving all the time? Well, what we’re going to do now is take a more precise look at what just happened in this game, and which categories had more or less students in them.”
V. Lesson Procedure (30-35 mins)
(1-2 min) 2. Have students rearrange the desks into groups of 2 or 3, and pass out the handout. As you pass it out, ask them to start recording the information from the board.
(3 min) 3. Once all the students have a handout, work through the first page on the Elmo together as a class.
Question 1 is meant to help them calculate the fractions in the third column; while students should be proficient in fractions at this point, still go through this quickly on the Elmo so they can compare their answers and make any necessary corrections. While the rest of the worksheet does not explicitly ask for these fractions, the extension would require this information, and it also serves as a reminder for students that this activity is still related to fractions and these fractions represent the probability a student belongs to that category. This connection should be explicitly stated when filling in this portion of the sheet.
For question 2, invite the class to supply possible answers. While there will definitely be at least one correct answer, since the first 2 categories called out were designed to be mutually exclusive, there may be other examples depending on the final list of categories used.
For question 3, work through the first pair as an example, perhaps showing how you can take any category and add “not” in front of it to create a mutually exclusive pair of categories (for example, “likes to dance” and “does not like to dance”). After this first example, tell them they have two minutes to quickly talk in their groups and come up with the other two pairs. Conduct a quick assessment after the two minutes are up by asking each group to give one of the pairs they came up with.
- Make sure to point out that mutually exclusive events do not have to divide the set completely – for example, 1 sibling and 2 siblings are mutually exclusive events/categories, but do not encompass all the possibilities within the classroom. Students may not realize this given the limited number of examples of “mutually exclusive” that they have encountered so far; for this reason, be sure to highlight if any students give an example that is a pair of mutually exclusive but not exhaustive set of categories. If no students volunteer an example like this, be sure to provide your own and ask the class to give one or two examples of this type before continuing.
(3-5 min) 4. Move on to the next page of notes. Ask students how many of them are familiar with Venn Diagrams. If anyone is not familiar with them, provide a brief explanation (each circle represents a category, and overlaps represent combinations of categories). Otherwise, ask the class to decide which category(ies) could be represented by that diagram. Have them fill in the number for an overall group next to that group’s label, and tell the students that you’ll explain why you recommend labeling in that manner in a moment. An example of how to fill in this chart is provided below. After filling in the black labels, ask, “How many girls are there, then, who don’t have curly hair?” Give wait time, and see if any student can explain why you subtract 13 – 4 using the Venn Diagram. If a student provides this answer, revoice their explanation, (otherwise provide an explanation) by stating that “We want girls without curly hair, so we take the total number of girls and subtract those who are in that group that have curly hair” Point out that that section of the diagram, “Girls without curly hair,” is mutually exclusive with the “Girls with curly hair” section. Ask them to compute how many boys have curly hair. Then, show them how, by labeling the larger circles on the outside, you can place the labels for these mutually exclusive groups on the inside of the circles and check your answers by making sure all the numbers within a circle add up to the total in parentheses outside the circle.
[pic]
(5 min) 5. Read aloud the next paragraph, then have the students work in their groups to complete question 5. Walk around and listen to student discussion as they work, answering questions and correcting any misconceptions that may arise. For the first group that finishes, ask them to write their answer on the teacher copy of the notes on the Elmo, and have them share with the class when at least half the groups have answered the problem.
Some possible misconceptions that may come up during the group work include miscounting when labeling the totals within the circles vs. the category totals, or feeling they need to gather extra data by polling the class when they have the information they need in the chart and are having trouble recognizing how to manipulate it. If the latter misconception arises, it can be addressed by allowing them to poll the class, but after doing so helping them work through it using their original data, and then comparing their answers to the class poll.
(8-10 min) 6. Repeat steps 4 and 5 with the next page of notes, this time explaining how we subtract the overlap in “OR” statements. Transparencies and colored markers indicating the different subsets of categories we are considering may be particularly useful to illustrate why this is so (how we count the overlap twice – show how the middle region gets darker using two transparencies - if we just add the two circles together, so we need to subtract the middle section). Be sure to show that the work students do subtracting the middle can be formally written as P(A or B) = P(A) + P(B) – P(A and B).
(8-10 min) 7. When moving on to the last page of notes, make sure to provide sufficient scaffolding that reflects the understanding students have demonstrated so far with using the Venn diagrams. To assist students in working with this more complex problem, have them break the last category up into its AND and OR components, and construct the intersecting circles as a class before having them work with the numbers on their own. As before, walk around and help groups that are struggling, then once the 10 minutes are up bring the class together for the closure.
The structuring of this last question will be highly dependent upon the types of questions the teacher chose beforehand; however, there are certain issues that students may struggle with regardless of the exact wording of the last category.
Students may have a difficult time dealing with multiple sets of data all at once. To help with this, suggest that they fill out the information for two-person Venn diagrams, and then have them work from there. It will also be useful to use the transparencies and colored markers again to help students visualize and pick apart different layers of categories. Finally, since students will be constructing a 3+ circle Venn diagram, they may want to come up with a system that helps them keep their values straight when marking the numbers that belong to different sub-categories (for example, keeping track on their diagram of there being 15 Girls, 10 Girls who play sports, and 3 Girls who play sports but do not like Hip Hop).
VI. Closure (5 mins)
If some groups were able to complete the final page of the notes in the time remaining, ask one of them to come up and present what they found. Otherwise, walk the class through building this more complex Venn Diagram. After each major step (ex. Constructing and labeling the circles, assigning numbers to each one, shading in different at areas), be sure to ask the class if anyone wants further clarification on why you just performed that step. As your concluding point, emphasize how breaking up the Venn diagram into different mutually exclusive sections and then adding up those sections allows you to figure out the values for each of those areas.
VII. Extension
To extend this activity, when evaluating numbers to go with the combined categories, also have the students work with the fractions associated with that category, and discuss how these fractions represent the probability of a student with those traits being chosen out of the entire class.
VII. Assessment
Assessment occurs at several points throughout the lesson. When working through the first page the worksheet together, the teacher asks students to supply answers to problems, and specifically for question 3 elicits feedback from every group of students. The next pages of notes switch between teacher-led examples and group work. During the teacher-led examples, the teacher still asks questions of the class before filling in answers, and during the group work, the teacher is around informally assessing student understanding by asking questions and checking in on each group’s progress. Furthermore, the difficulty of each assessment increases as the lesson progresses, allowing the teacher to identify the level of complexity the class is comfortable engaging with regarding mutually exclusive events.
VIII. Standards
S-CP.1. Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events (“or,” “and,” “not”).
S-CP.7. Apply the Addition Rule, P(A or B) = P(A) + P(B) – P(A and B), and interpret the answer in terms of the model.
Common Core Practice Standard: Make sense of problems and persevere in solving them
This lesson addresses the practice standard of making sense of problems and persevering in solving them, because students are asked throughout the lesson to analyze their given information, categorize it, and determine if they need more information or can work with the information they were given. They also must persevere in solving problems as they are given increasingly more difficult tasks as the lesson progresses.
Name:
Categorizing our Class!
Date: Period:
|Category |# of people |Fraction of class |
|1. | | |
|2. | | |
|3. | | |
|4. | | |
|5. | | |
|6. | | |
|7. | | |
|8. | | |
|9. | | |
|10. | | |
|11. | | |
|12. | | |
1. How many students are in our class total? __________
2. It’s impossible to fit into categories like “Adult” and “Minor” at the same time. Are there any groups of categories like that above? Which ones?
3. Categories like the ones you listed above are called mutually exclusive sets – there is no overlap between the two. Take three other categories from our list and create another category that is mutually exclusive from the one you chose.
_____________________ vs. _____________________
_____________________ vs. _____________________
_____________________ vs. _____________________
4. What categories from our list could be used to make this Venn Diagram? Please write in the labels and numbers needed to complete the diagram.
[pic]
5. This type of situation is called an AND statement, because we are looking at who belongs to one category AND belongs to another one. Formally, this probability is written as P(A ∩ B).
Take two other categories from our game and create an AND statement. How would you figure out how many students belong to that new, combined category? Do you need to gather more data?
6. What categories from our list could be used to make this Venn Diagram? Please write in the labels and numbers needed to complete the diagram.
[pic]
7. This type of situation is called an OR statement, because we are looking at who belongs to one category OR belongs to another one.
Formally, this probability is written as P(A υ B).
Once again, take two other categories from our game and this time create an OR statement. How would you figure out how many students belong to that new, combined category? Do you need to gather more data?
8. Construct a Venn Diagram for the last category called out.
Homework for Categorizing our Class! Name:
1. If you were the one in the middle, what are three categories (or combinations of categories) you would want the caller to call out so you have the greatest chance of stealing a spot? Why?
2. What are the probabilities (the fractions of the class) associated with those categories from question 1?
3. If you were a random student on the sidelines, what are three categories (or combinations of categories) you would want the caller to call out? Why?
4. What are the probabilities associated with those categories from question 3?
Homework for Categorizing our Class! Name:
5. My class of 27 students last year played this game, and had the following data. Using the data given to you, create the following Venn diagrams and fill in the missing information.
|Category |# students |Fraction of class |
|Boys |14 | |
|Girls |13 | |
|Boy OR girl | | |
|Bilingual |10 | |
|Owns a dog |10 | |
|Dog owners who are NOT bilingual |7 | |
|Bilingual dog owners | | |
|Bilingual students who do NOT own dogs | | |
|Bilingual OR owns a dog | | |
|Girls who are bilingual |5 | |
|Girls who are dog owners AND are bilingual |2 | |
|Boys who own dogs |6 | |
|Girls who are NOT bilingual and own a dog | | |
|Boys OR girls who do not own dogs | | |
Create a Venn diagram for:
(a) Bilingual & Owns a dog
(b) Bilingual & Girl
(c) Boy & Owns a dog
(d) Girl, Bilingual, and Owns a dog
|Day #: 8 |Lesson Title: Permutations |
|Goal: |Develop a conceptual and procedural understanding of permutations |
| | |
|Objectives: |1. Students identify situations which are appropriate to use permutations through discussion and worksheet problems |
| |2. Students can calculate the number of possible outcomes of a situation using the counting principle and permutations |
| |3. Students can verbally and/or in writing describe why the formula for a permutation makes sense |
|Lesson Summary (one |Students work through a notes packet that involves analyzing various situations where permutations are needed to calculate all |
|paragraph maximum) |the possible ways an event could occur. We alternate between working step-by-step through examples of how to break down the |
| |possibilities for an event, and then giving students a few minutes to work with people around them to do an exercise that |
| |applies what was just modeled. The notes packet would include an analysis of the word game, where they rearranged letters, on |
| |the first day of the unit; setting up different possibilities for a four-person relay race using their classmates; and |
| |analyzing their different locker combinations (are there enough combos for every student in the school? What if numbers can be |
| |reused? What if they can’t?). We would be sure to emphasize a conceptual understanding of why the permutation formula |
| |accurately describes a situation by referencing tree diagrams to explain the counting principle, and the idea of filling in |
| |slots with different choices to explain permutations. |
|HW |Book problems: 2(a-f: pick 4), 4, 9, extra credit problem #12 |
| | |
|Day #: 9 |Lesson Title: Combinations |
|Goal: |Develop a conceptual and procedural understanding of combinations |
| | |
|Objectives: |Students can solve the handshake problem both by making lists and by counting using combinations |
| |Students identify situations which are appropriate to use combinations through discussion and worksheet problems |
| |Students calculate possible outcomes using the combination formula in various combination problems |
|Lesson Summary (one |Students work in groups of 5 to work through the classic “Handshake” problem. They first calculate [1] how many different |
|paragraph maximum) |handshakes could occur within their group of five (which could be done with a list), then [2] how many different handshakes|
| |could occur within the entire classroom (which is more difficult to do through listing all the options), and finally [3] |
| |how many handshakes could occur within the entire school (which demands the introduction of combinations as a counting |
| |method to solve the problem). At either steps [2] or [3], when students are likely to be struggling with accurately and |
| |methodically counting the handshakes, introduce combinations as a tool they can use to answer the question and ask them to |
| |solve the problem using their new knowledge. After they finish the handshake problem, students will work through a notes |
| |sheet in a manner similar to day 8, working as a class through example combination problems and then applying what was |
| |reviewed in an exercise. |
|HW |Book problems: p. 707, 1(a)(c), 2(b)(d), 4, 11, 12 |
| | |
|Day #: 10 |Lesson Title: Integration of topics, Synthesis |
|Goal: |Students collaborate to consider an advanced problem involving permutations and combinations |
| | |
|Objectives: |Students will express consideration of how to calculate the number of possible outcomes for a complex problem |
| |Students will communicate a plan to accurately count outcomes |
| |Students will compare and contrast the differences between when to use permutations and combinations in their written explanation|
| |of their strategy |
| |Students will explain to their peers and their teacher why their plan is logical and accurate, and how they could assess the |
| |accuracy of their counting strategy. |
|Lesson Summary (one|Students spend the entire period working in groups of 3 or 4 to answer a single question: If there are 5 tables (Labeled 1-5) in |
|paragraph maximum) |their classroom, how many ways can the computer generate a seating chart where 5 students sit at each table (where the ordering |
| |within the group of 5 doesn’t matter, but the table number each group sits at does matter). This problem is neither strictly a |
| |combination problem nor a permutation problem, so students work in their groups to come up with different strategies for counting|
| |the possible outcomes. We walk around helping groups assess the accuracy of their plans, and if the class is struggling or needs|
| |more scaffolding, we bring the group together to discuss how they are analyzing the problem and what ways of framing the task |
| |might make it more approachable. We would emphasize during these discussions that the point of this exercise isn’t to quickly |
| |get a right answer, but to think strategically about how to count and rule out possibilities. One way we might provide |
| |additional scaffolding is by asking the students to explain a general process for creating a seating chart (i.e. pick 5 students |
| |for table 1, 5 more students for table 2, etc.), and only after they have a general plan for counting should they try to |
| |represent that plan mathematically. We might further scaffold then by asking students how, mathematically, they can enact each |
| |of those steps (i.e. you rule out repeat orderings by dividing by the number of different rearrangements of the same elements). |
| |Close the lesson by highlighting some of the strategies different groups used and explaining the journal entry & extra credit for|
| |the weekend. |
| | |
|HW |Journal about what your group worked on today. What was your thinking process? How did you start the problem and how did your |
| |approach change throughout the process? What did strategies did you use? |
| |Extra Credit: See if you can come up with an answer, turn in whatever extra work you did on the problem over the weekend |
Name: Chanel Keyvan
Date: 5/7/2012
Grade Level: 11th / 12th
Course: Algebra II
Time Allotted: 50 Minutes
Number of Students: 25
1. Goal
a. To determine if Fantasy 5 is a profitable lottery game.
2. Objectives
a. Students will be able to use their prior knowledge of combinations to evaluate the different ways to win different prizes in Fantasy 5.
b. Students will be able to use their prior knowledge to evaluate the probabilities of the different outcomes of Fantasy 5.
c. Students will be able to use their prior knowledge to evaluate the expected value for one play of Fantasy 5.
d. Students will communicate verbally or will write down if Fantasy 5 is a profitable game, using the previous mathematics to justify their answer.
3. Materials and Resources
a. Worksheet for each student
b. Calculators
4. Motivation
a. (5 minutes) “Today we’re going to begin our social justice project for this unit! We’ve been preparing for the past 2 weeks for this project! This is a three part project and today we’ll be starting with just part one. The goal of this project is to examine the lottery system in relation to its participants and who it may affect. So before we begin, what do you know about the lottery?” I would write students’ ideas on the board and ask further questions such as: “Do you think the lottery is a good investment? What do you think the chances of winning a particular lottery game are? How do you think some lottery game works? Etc.” The motivation is really getting students to use mathematics to thoroughly examine a social justice issue that affects each one of them.
Transition: “Before we begin our project, let’s review some of the concepts we’ve talked about earlier.”
5. Lesson Procedure
a. (5 minutes) “Take a few seconds and talk to your neighbors about the difference between combinations and permutations.” I would have one group share their thoughts and write some of the points on the board. “Okay take another couple of minutes to talk to your neighbors about what expected value is and how might you solve for it.” I would choose a different group to contribute, and I would write their contributions on the board. These will be left on the board for the remainder of the period.
b. (30 minutes) I will pass out the worksheets and have a student read, off of the worksheet, how Fantasy 5 is played and the directions for the first problem. I would double check to make sure that all of the students are clear on the game and the directions. Once this is complete, I will break the class up into 11 groups of 2 and 1 group of 3. The class would already be in pods so I will group students according to location. Since students have already been exposed to these concepts before, they will just work as a group to answer the questions on the worksheet. I will walk around and make sure that all groups are making adequate progress through the worksheet and answer any questions that the students might have.
c. The following are possible problems that student might come across during the worksheet and some ways I can address these problems.
i. Students may struggle between using combinations and permutations.
1. “Does the order of the winning numbers matter?”
ii. Students may struggle with finding the number of ways to match 5, 4, 3, and 2
1. “Think of the winning cards as there are 5 ‘good’ numbers and 34 ‘bad’ numbers. If you want to get all 5, you would want to choose all 5 ‘good numbers and zero ‘bad’ numbers.” Same reasoning would follow for 4, 3, and 2
iii. Students may struggle with finding the number of ways to match 0 or 1.
1. “Ask yourself if these events are mutually exclusive or not”
2. Since they are mutually exclusive, students should remember to just add the combinations together
iv. Students should remember that to find probability you take the total number of ways for what you want divided by the total number of ways to choose 5 from 39 cards without repeating
v. Students may not know how to calculate the average winnings for each section
1. “First think about what happens if you win with one play. How much do you win and how much did you have to pay?”
2. “Then think about if you were to play more than once and win every time, how could you find out how much you earned on average?” I would want students to come to realize that you take how much you would earn if you won multiplied by the probability of winning such an event.
vi. Students may not know what the expected value means or how to interpret it
1. “When we calculate for expected value, what are we essentially doing?” I would want students to realize that we are adding the average winnings for each event together.
2. “Why would we want to add all of the average winnings together? What do all of them added together describe?” I would want students to realize that adding all of the average winnings together describes how much we would win if we play Fantasy 5 once.
3. “If you were to play 100 times, how much would you expect to earn or lose?”
d. If students are working individually and not as a group, I would remind students that part of their analysis at the end of the unit (and their grade) is reflecting on their work and commenting on what they would change or how they might have thought differently about different problems. As a result, it’s worthwhile to work together because you will then have a basis to formulate your own thoughts and create a richer analysis of this project.
e. I would also want groups to help each other out as well. Thus, if group A has a question in which I’ve noticed group B has a good understanding of, I would prompt group B to help group A.
f. To check that students are working together, if one student has a question, I would ask another student in the group what the question was. I would also work to address all individuals in the group instead of the student who initially asked the question.
Transition: “If you haven’t finished the worksheet, don’t worry. You’ll have time to complete it
at home. For now, let’s regroup and talk about some of the data that we analyzed today.” I am aware that some groups may not get through the entire worksheet based on the nature of the students. However, I would anticipate that students would get through filling out the majority of the chart and at least begin to work on the expected value problem.
6. Closure
a. (10 minutes) I would regroup the class and have the groups share their conclusions about Fantasy 5. We would first write the answers on the board that everyone has finished. Then, we would look at the expected value problem. In this lesson, expected value is determined by the probability of an event multiplied by the average payout that we would receive if we win. Then we would add up all of these events to determine the average amount we would win/lose if we were to play this game multiple times. It is the average winnings over a long period of repeated plays. Part of this would include students sharing their justifications with each other to practice communicating mathematics. I want students to explicitly take away that the lottery is not profitable because you would be expected to lose $0.26 every time you played a card. “Tomorrow, we’ll continue our project and look at who benefits or is harmed by playing the lottery. Before you leave, I’d like you to write down something that you’ve learned today as your exit ticket.”
7. Extension
a. For the extension, I would pose the question “How could we change Fantasy 5 so you would have a net profit from the playing? What aspects of the game could we change?” I would have students discuss this in their groups and include this as part of their exit ticket. Some aspects of the game that they could change could include the price of the ticket, including a prize for getting 0 and 1 point, and changing the number of points that they could make for each selection.
8. Assessment
a. Walk around and see if students are able to successfully fill out the worksheet at the current step
i. This assessment strategy is used to check for all of the objectives as they come up during the group’s progress on the worksheet
b. Having students explain their solutions to each other and myself throughout the group work
i. This will assess all of the objectives as they come up during the group’s progress on the worksheet.
c. Having students write down what they’ve learned
i. This is assessing all of the objectives as they come up during the group’s progress on the worksheet
9. Common Core Standards Addressed in this lesson
a. S-CP.7. Use the rules of probability to compute probabilities of compound events in a uniform probability model
i. Apply the Addition Rule, P(A or B) = P(A) + P(B) – P(A and B), and interpret the answer in terms of the model.
b. S-MD.2. Calculate expected values and use them to solve problems
i. Calculate the expected value of a random variable; interpret it as the mean of the probability distribution.
c. S-MD.5. Use probability to evaluate outcomes of decisions
i. Weight the possible outcomes of a decision by assigning probabilities to payoff values and finding expected values
ii. Find the expected payoff for a game of chance.
iii. Evaluate and compare strategies on the basis of expected values.
d. The Common Core standard that was perfect for this lesson was Model with Mathematics. This real world problem really calls for students to think about how the mathematics is providing an analysis of data that students may not have considered before, and how their new knowledge may be applied.
Name:__________________________________
Part I: Can Playing the Lottery be Profitable?
How the Game Works:
How to Win:
Exercises:
1. For the Fantasy 5 lotto game, find:
a. The number of ways to draw the 5 random winning cards (without repeating)
b. The number of ways to match all 5 cards
c. The number of ways to match 4 cards
d. The number of ways to match 3 cards
e. The number of ways to match 2 cards
f. The number of ways to match 1 or 0 cards
2. Fill in the following table by solving for the following
a. Calculate the probabilities to match 0 or 1 (treat as one event since both indicate that you’ve lost), 2, 3, 4, and 5.
b. Use Table I to find the average payout for each of the winning events (matching 3, 4, and 5 of the numbers).
c. Determine how much you win or lose on average in each of the five possibilities (matching 0 or 1, 2, 3, 4, or 5)
|Matched |Probabilities |Average Payout |Average Winnings |
|0 or 1 | | | |
|2 | | | |
|3 | | | |
|4 | | | |
|5 | | | |
3. Find the expected value for one play of Fantasy 5 lottery.
4. How much on average will you win or lose each time you play Fantasy 5? Use your calculations to justify your answer and explain whether or not the lottery is a profitable investment.
[pic]
Solutions to the worksheet
Exercises:
1. For the Fantasy 5 lotto game, find:
a. The number of ways to draw the 5 random winning cards
39C5 = 575,757
b. The number of ways to match all 5 cards
(5C5)( 34C0) = 1
c. The number of ways to match 4 cards
(5C4)( 34C1) = 170
d. The number of ways to match 3 cards
(5C3)( 34C2) = 5610
e. The number of ways to match 2 cards
(5C2)( 34C3) = 59840
f. The number of ways to match 1 or 0 cards
(5C0)( 34C5) + (5C1)( 34C4) = 510,136
2. Fill in the following table by solving for the following
a. Calculate the probabilities to match 0 or 1 (treat as one event since both indicate that you’ve lost), 2, 3, 4, and 5.
b. Use Table I to find the average payout for each of the winning events (matching 3, 4, and 5 of the numbers).
c. Determine how much you win or lose on average in each of the five possibilities (matching 0 or 1, 2, 3, 4, or 5)
|Matched |Probabilities |Average Payout |Average Winnings |
|0 or 1 |.886 |-$1 |-$0.886 |
|2 |.104 |$0 |$0 |
|3 |.00974 |+$16.55 |+$0.161 |
|4 |.000295 |+$378.76 |+$0.112 |
|5 |1.73 x 10-6 |+204,562.50 |+$0.354 |
3. Find the expected value for one play of Fantasy 5 lottery.
add up average winnings = -$0.259
4. How much on average will you win or lose each time you play Fantasy 5? Use your calculations to justify your answer and explain whether or not the lottery is a profitable investment.
Because of the negative expected value for a given play, you can expect, on average, a return of -$0.26 every time you spend $1 on the lottery. Thus, it is not profitable because you are losing money in the long run. If you played 100 times, you’d pay $100 upfront and then expect to lose $26.
Name: Maggie Sharp
Date: 3/8/12
Grade Level: 11th/12th
Course: Algebra II
Time: 50 min
Number of Students: 25
I. Goals:
1. Analyze who benefits from the lottery through probability applications
II. Objectives:
1. Students will apply the addition rule of probability to solve the first question on the worksheet part one problems one and three.
2. Students will compare percentages to solve part one problem two.
3. Students will use averages to solve part two problem one.
4. Students will provide a mathematical analysis (in writing) to justify the answer to the overall question of who the lottery benefits
III. Materials and Resources:
1. Each student will need a worksheet (provided at the beginning of class)
2. Students should also use the worksheet completed in the last lesson as a reference
IV. Motivation: (5-10 min)
“Before we begin today, I would like to ask the class what you took away from yesterday’s lesson. First, can someone please summarize what you did yesterday? (Allow students to raise hands and call on one or two students to summarize the lesson covered yesterday)
“Does anyone have anything to add or any questions to ask?” (Again, wait to see if hands are raised. This is a good opportunity to discuss any misconceptions or misunderstandings from yesterday’s lesson. If students do raise questions, open them up for class discussion. Be sure to check in with that student after the discussion to be sure he or she has a better understanding. If not, it may be helpful for me to summarize myself in order to move the discussion along.)
“What probability concepts did you apply yesterday to get the results we have been discussing?” (Allow multiple students to provide answers. Be sure not to move on until they have identified combinations and expected value. When students discuss these topics ask them to identify when they used them.)
“Was anyone surprised by any of the results you saw yesterday?” (Give students time to discuss their reactions to the data. Be sure to prompt students to talk as a class but to one another and not just to me. If students are having trouble getting a productive discussion started, have these questions prepared: Did you predict the game would be profitable or not profitable? Did the expected values seem higher or lower than you thought they would be?)
“Thank you for sharing your ideas. Keep your work from yesterday in mind as we begin today’s lesson. Feel free to refer back to your worksheet from yesterday to look at your work or reference the tables. If anyone needs another copy of the worksheet please let me know! Right now I am passing out a copy of the work we are going to do today. We are going to attempt to answer the question of who the lottery actually benefits.” (As I am talking, I will be passing out papers so the class stays engaged.)
Transition: “Can someone please read the direction on the top of the worksheet to the class?” (Call on one of the students with his or her hand raised to start reading.)
V. Lesson Procedure: (40 min)
1. As soon as the student was done reading the introduction to the worksheet, I would ask another student to summarize what we will be doing in this activity. I would cold call on a student because they would be used to the norm of summarizing what another student had read or said by this point in the year. At this point I would ask students if they had any points which needed clarification.
2. After answering any questions that come up from the class, I would ask students to make predictions before we begin the assignment. I would want them to predict who they think the lottery most benefits and why. As students share their ideas, I would summarize them on the board, so everyone could see them. I anticipate that students may assume that the lottery benefits low-income families most because they need the money most. If we are lacking predictions, I may ask them to consider who they believe plays the lottery the most and where the money spent on the lottery goes. Making predictions will give the students a chance to consider the question before they begin the assignment. It is also a way to engage less motivated students because they may become more interested in the problem if they can compare it to the predictions they have made. At this point, I would ask if students had any questions before we began the assignment.
3. Students will be given most of the rest of the period to work on the worksheet with the other students they sit with (Class is arranged in clusters of desks to promote group work, so students will already have a group to work with). The calculations on the worksheet are not too difficult given their knowledge of probabilities and averages, so they should not require too much support. However, as students work on the problems, I will walk around the class to listen to groups and answer questions if they run into problems. As I begin walking around, I will be sure to remind students that as they do group work, they should always bring up questions to their group members first. If they have discussed the question as a group and are still puzzled, I will be happy to help and scaffold them to see where they should go. (Each time a group asked a question, my first response will always be, “Did you ask your group members?” and then, “What do you think right now?” I will ask these questions to promote group communication and to assess the thought process of the group so I can understand where the misconception lies).
4. I anticipate that the following issues or questions may come up:
a. The first problem on the worksheet is probably the hardest. This problem asks students to consider how many times they need to play the game in order to ensure a 30% chance of winning either first or second prize (matching 5 or 4 numbers).
i. First I would ask them to consider the probability of winning first OR second prize. I would ask questions like: “What does the ‘or’ mean when we calculate probability?” “How did we evaluate the ‘or’ statements in previous lessons?” This should help students recognize that they need to add the probability of winning first prize plus the probability of winning second prize.
ii. Students may still struggle because they may not be sure how to then calculate probabilities based on number of times played. First I would ask, “What is the probability that you will win if you play twice?” Students may initially want to multiply the probability by itself. In this case, I would say, “If I multiply the probability, it means I am looking at how likely it is that I will win both times. Is this what we want?” Students should then recognize that this is not correct, but may not understand what they should do. I would then say, “I am concerned with the probability that the player will win after two tries, which means they can win in either the first try OR the second. How would you calculate this kind of probability? What kind of statement is this?” At this point students should see that they would need to add the probabilities together.
iii. Finally, students may struggle with putting together how calculating the probability of two tries connects to finding when we will end up with a 30% chance of winning. If students were asking this question after having gone through the last two steps of scaffolding, I would ask them to consider the probability after three times and then after four. Then I would ask, “When we keep adding the same number to itself multiple times, what are we actually doing?” Allow students time to consider this question for a while, they should see that this is simply multiplication of the probability by the number of times we play. Then I would say, “So now can someone please read the question back to me? What is it asking us to do?” For some students this will probably be enough for them to see that they should multiply the probability by some unknown quantity (say n) and set it equal to the desired probability, which is 0.30. Then they would solve for n. If students still don’t see it, I would say, “What is your desired probability?” They should easily answer 30%. Then I would say, “What is the unknown quantity we want?” They should answer that we want number of plays. I would ask, “Can you set up an equation with a variable for the unknown to get the desired probability?” This should be plenty of scaffolding for this part.
b. The next question simply asks students to consider the percentage of average salary two different families would use in order to play the lottery enough times to ensure the chances they calculated above.
i. As I was walking around, I would be sure to check that the students were using the appropriate values from the tables because their results are dependent on that. If I notice that a group was using the wrong value for one or both of the families, I would ask them to explain to me where they got their value and why. I would use their explanation to inform how I would help them find the correct values. Most likely I would simply have to ask them in which table they should look for each Family and I would remind them that both are families of 4.
ii. In order for students to calculate percentages, they would need to understand that they would spend $1 each time they play, so the amount spent would be equal to the number of times they play (calculated above). I do not anticipate many students having a problem with this, but if they do, I would remind them that every play cost $1. Students would have enough prior knowledge to calculate percent of total income each family uses based on the amount they spend to play. This is a simple calculation which students have seen before.
iii. Finally, students need to consider the comparison between the two percentages. If they did not know what this meant I would say, “I am looking for you to tell me how much more percent of income Family A is spending compared to Family B.” This should make it clear that they need to find the difference between the 2 percentages.
c. The next question asks them to calculate how many times you would have to play to get a 30% chance of winning just first place. Then they should consider 5%.
i. If students are confused, I would ask them to refer to their work from the first question but remind them that they are only considering first prize this time, so they can eliminate the consideration of second prize. They should follow the same procedure from question 1, so this would provide enough guidance.
d. The next question asks students to calculate an average, a concept they are very familiar with. They should not need any further scaffolding than what is provided by the hint on the worksheet.
e. The next question asks students to compare the average median income of lottery winners compared to the average median income of people in the state of Georgia.
i. They should be able to calculate the difference (which is close to $20,000) with very little difficulty. This section needs no scaffolding because we will discuss the implications of this value in our closing discussion.
5. Some strategies I will use to promote group work and discursive moves would be:
a. When scaffolding strategies with the groups, I anticipate that certain group members may come to a realization before others. When this happens, and a solution is discussed by one or two group members, I will ask another member who was not talking to explain why the solution makes sense. (Prompt for Restatement)
b. If groups do not seem to be working collaboratively go up to the each group and ask certain members to explain the strategies their partners are using. Remind groups that collaboration is an important part of mathematics and that they can gain a richer and more in depth understanding of the problem by working together.
c. When working with groups and listening to explanations from group members, I will be sure to orient varying explanations when possible. I will ask students to consider if one or both of the explanations make sense and why. This forces group collaboration and forces students to communicate mathematical reasoning.
VI. Closure: (5 minutes)
At this point, all the groups should be finished up with worksheet. I would want to end with a discussion in which students consider the implications of this assignment. Some questions we would discuss during this time:
a. “In the last question, we saw that the average median income of lottery winners was significantly lower than the average median income in the Georgia. What does this say about who is playing the lottery?”
b. “Does this mean that lower-income people are benefitting more from the lottery?” (I would give them some time to discuss this before posing the next few questions). “Consider your findings from yesterday regarding the expected value of playing this game. Also consider how many times a low-income family would have to play the lottery to insure a high chance of winning. Recall the percentage of income families have to spend to raise their odds.” (I would throw these questions in at different times during the discussion based on how the discussion was moving along).
c. “Considering how much more low-income families need to spend on the lottery to make it profitable, is this really beneficial to these families?”
d. “Please consider this discussion and use your new mathematical knowledge to answer the journal question tonight! Remember to support your answer!”
VII. Extension:
Introduce the journal entry question: “Consider your calculations from Part I: Can the lottery be profitable?, from the exercises above about ensuring a 30% success rate, and your observations about median incomes in Georgia, comment on whether or not the lottery is a “tax on the poor.” Argue whether or not the lottery benefits or harms citizens. Use math to justify your claims!” Give students a chance to talk with their groups about their initial reactions and thoughts. Give students the opportunity to ask each other questions about their opinions and how the math helps inform these opinions. The journal entry itself will be assigned for homework, and due the next day in class.
VIII. Assessment Summary:
1. Discussion of findings from the day before: this strategy assesses whether or not there are lingering misconceptions from the previous day. We will be able to discuss and possibly eliminate these misconceptions before the lesson starts so that they do not hinder the assignment.
2. Walk around while students are engaged in group work: This strategy allows me to see if the students are making progress and it will help me catch any errors that may interfere with their understanding of the analysis of this assignment. It will also allow me to assess that the objectives are being met as I will be able to discuss with students what they are doing to uncover misconceptions and correct them.
3. Journal entry: this aligns with the last objective, which requires students to synthesize the mathematics from the problem and form an opinion based on their analysis. This also requires students to communicate through mathematics and about the mathematics. Reading these journal entries would reveal misconceptions and uncover student reasoning.
IX. Standards:
1. S-MD.5. (+) Weigh the possible outcomes of a decision by assigning probabilities to payoff values and finding expected values.
a. Find the expected payoff for a game of chance. For example, find the expected winnings from a state lottery ticket or a game at a fast-food restaurant.
b. Evaluate and compare strategies on the basis of expected values. For example, compare a high-deductible versus a low-deductible automobile insurance policy using various, but reasonable, chances of having a minor or a major accident.
2. S-MD.7. (+) Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game).
3. S-CP.7. Apply the Addition Rule, P(A or B) = P(A) + P(B) – P(A and B), and interpret the answer in terms of the model.
Standards for Mathematical Practice:
1. Construct viable arguments and critique the reasoning of others (Standard 3): Students are asked to construct an answer to the question using the data and mathematics to support their ideas.
2. Model with Mathematics (Standard 4): Students are looking at a real-world problem and using mathematics to understand and interpret it. They model the likelihood of events using their knowledge of probability.
Name:_________________
Part II: Who is harmed the most by playing the Lottery?
Introduction:
Today we are going to attempt to answer the question of whether or not the lottery can benefit the players. We are going to analyze data and answer the questions below in order to determine how much you would pay out in order to secure a good chance at winning. Then we will determine where the winners live and what implications this has. Be sure to use your worksheet from yesterday as a reference to the averages you will need. Get ready to apply your mathematical minds to see how probability can tell us how fair our lottery system is!!
Part 1: How much does it cost to play the lottery well?
1. Suppose you want to give yourself a 30% shot at winning the first or second prize. In other words, you want to ensure you have a 30% chance to match at least 4 of the numbers. How many times do you need to play Fantasy 5 to ensure a 30% chance of winning the first or second prize?
2. How much does it cost to play the lottery this many times? ____________
3. Suppose that 2 families of 4 are trying to ensure a 30% chance of winning first or second prize. Family A has a median income at the poverty line, and Family B has an income equal to the median income for families of four in Georgia. Use Tables II-A and II-B to determine the income for both families. In terms of percentage, how much more must Family A spend to play Fantasy 5 than Family B?
4. How many times must you play Fantasy 5 to ensure a 30% chance of winning first prize? What about just 5%?
Part 2: Where do the winners live?
1. Table II-C lists the cities and towns in Georgia where the most recent first prize tickets were sold. Determine the average median income for a city where winning tickets were sold. (Hint: don’t forget to include cities/towns that sold more than one ticket the appropriate number of times).
2. Next, compare the average median income for towns with winning tickets to the median income for families in Georgia. You can find data on Georgia family incomes in Table II-A. Is the average median income of towns with lottery winners more or less than the median income of Georgia families? By how much?
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|Day #: 14 |Lesson Title: Project Discussion |
|Goal: |To examine the data/results the class produced for the project |
| | |
|Objectives: |Students can interpret their results (visually, verbally, etc.) from the project and discuss with each other|
| |what implications this may have on society. |
| |Students will communicate respectfully, using mathematics to defend their arguments |
|Lesson Summary (one | This day is devoted to examining all of the results from the different projects. Students sit in a |
|paragraph maximum) |circle, including the teacher, and discuss how the data helped reveal some injustices within society. This |
| |is really a time for students to talk with each other in a safe place and define their own opinions about |
| |the subject. We make sure to emphasize that issues of income and inequity are sensitive topics, and remind |
| |students to be understanding and respectful of their peers’ thoughts. What is stressed throughout the |
| |lesson is the key role mathematics, in particular probability, played. At the end, students write down what |
| |they’ve learned from this unit/project. This will serve as their exit ticket for the day. |
| | |
|HW |If students like, they can revise their written analysis from day 13 in light of the issues examined in the |
| |class discussion |
-----------------------
|Day #: 1 |Lesson Title: Unit Hook, Probability Games |
|Goal: |Provide examples of uses for probability |
| | |
|Objectives: |Students will play games involving probability. |
| |Students will express in writing their prior knowledge involving probability. |
| | |
|Lesson Summary (one |In this lesson, we set up four stations with different games involving probability (briefly described below). The students |
|paragraph maximum) |are divided into four “teams” and travel from station to station with their teams to play each probability game. We give |
| |about 10 minutes per station. After each team has been to each station, we ask the students to answer the following |
| |questions: What do you already know about probability? How did you observe probability in one or more of the games you |
| |played today? This will serve as the exit ticket for students. They should have 5-10 minutes left of the period to complete|
| |this, so most should get it done by the end of the period, but if not they will be allowed to take it for homework. These |
| |exit tickets will give us information about what the students already know about probability and will inform our teaching |
| |of the rest of the unit. |
| | |
| |Games: |
| |Word Game: Students try to come up with as many words as possible from groups of letters |
| |Dice Game: Students pair up and take turns guessing the sum of two dice |
| |Computer games: Students play various probability minigames at laptops |
| |Spinner game: Students spin a circular spinner with different dollar amounts allocated to each section. |
|HW |Finish exit ticket if not done in class, otherwise no homework. |
| | |
|Day #: 3 |Lesson Title: Area Model of Probability |
|Goal: |Understand how to geometrically model simple probability situations |
| | |
|Objectives: |Students will use a rug diagram to calculate theoretical probability |
| |Students will represent possible outcomes of a coin flip using an area model |
| |Students will hypothesize possible “best outcomes” for marble problem |
| |Students will record predictions and analysis of probabilities during marble problem trials. |
|Lesson Summary (one |First we use an area model to describe probability. We introduce this by showing a diagram of a rug with certain parts |
|paragraph maximum) |shaded and other parts not shaded. We ask students to determine the probability that if they threw a ball onto the rug it |
| |would land on a shaded region. We remind students that they are looking at theoretical probability and ask them to work in |
| |partners to solve the problem. Then, we ask students to draw an area diagram in which they shade the probability of |
| |flipping a coin and landing on tails. We extend this by asking them to draw a diagram in which they shade the probability |
| |of getting a tail and then another tail. (They would divide a box in 2 and then divide one of the smaller regions in 2 |
| |again, ultimately shading one fourth of the diagram) Then, we pose the question: “You have three blue marbles and two green|
| |marbles. You also have two cups. Work with a partner to try and figure out the best way to arrange the marbles in the cup |
| |so that if you place the two cups in front of your partner he or she will have the highest probability of pulling a green.”|
| |We would put students in partners and give them the manipulatives to try to figure out how to maximize their success. We |
| |encourage students to use the area models to represent the possible probabilities for each arrangement, and remind them the|
| |first choice is picking a cup, which provides the first area division. The next choice is based on how many marbles are in |
| |the cup chosen, which will determine the next set of divisions. Give students time to work for the rest of the period. |
| |(Tomorrow we will discuss how they were actually using multiplication) |
|HW |Go home and provide a write-up of what you believe to be the best possible solution. Come to class tomorrow prepared to |
| |show you solution and how you got it. Be sure to include at least one area diagram in your analysis of the problem. |
|Day #: 5 |Lesson Title: Expected Value Intro (25 min) |
|Goal: |To develop an understanding of expected value |
|Objectives: |1. Students can communicate what expected value is. |
| |2. Students can calculate the expected value of a particular event. |
| |3. Students can verbally and/or write an interpretation/make sense of the expected value in the context of the problem. |
|Lesson Summary (one |Students get into groups of 5 and find the expected value of the following game through simulation. The game is a spinner, |
|paragraph maximum) |similar to the one they used in the Unit Hook, divided equally into four quadrants. The fourth quadrant is divided in half. |
| |Thus, we have three (1/4) quadrants (let’s call them Q1, Q2, & Q3) and two (1/8) quadrants (let’s call these Q4 & Q5). Q1 has |
| |$0, Q2 has $2, Q3 has $0, Q4 has $1, and Q5 has $10. It cost $1 to play the game. To play, you spin and the value the spinner |
| |lands on is the amount you receive (If you land on $0 you record a score of -$1 since you paid a dollar to play, any other |
| |number would record a value one less than what you win). In their groups, students run the trial with an actual spinner 15 |
| |times. When students are done, they calculate the average gain from each turn in their experiment. If they are struggling, |
| |remind them that average gain is the same as total gain over number of trials. Next, introduce the definition of expected |
| |value as the “average” loss or gain over time, found by multiplying the value of each event times its probability, and then |
| |summing those products. In this particular example, you would multiply the probability of landing in each space by the dollar |
| |amount won in that space and then adding all these values. This number represents the expected gain or loss of each turn. Ask |
| |students to calculate the expected value of this simulation based on the theoretical probabilities of landing in each spot. |
| |Since expected value represents the average gain or loss per turn over time, ask students to consider whether this game is |
| |profitable (If we do not get to this part, we will add it to their homework). |
|HW |Journal: Compare your results from the group experiment to the expected value calculated in class. Do your results make sense,|
| |or was your experiment a fluke? |
|Day #: 6 |Lesson Title: Expected Value Application |
|Goal: |To further develop student understanding of expected value, and differentiate between expected value and the most likely |
| |outcome |
|Objectives: |1. Students can construct an accurate area model to represent a one-and-one event in basketball |
| |2. Students can compute the probabilities associated with each section of the area model |
| |3. Students can compute the expected value of Terry’s score |
| |4. Students can verbally compare the expected value for this situation with the likelihoods they calculated |
|Lesson Summary (one |Students spend the period analyzing the following problem: Terry is a basketball player, and she successfully makes her |
|paragraph maximum) |free-throws 60% of the time. If she is in a one-and-one situation - where if she misses the free-throw, she is done, but if|
| |she makes it and scores 1 point, then she gets the chance to make a second shot to score a total of 2 points - how many |
| |points is she most likely to score? Students explore this problem by first making predictions, then running a simulation as|
| |a class, pulling blocks out of a bag to indicate the possible outcomes (0 pts, 1 pt, or 2 pts). They then work in pairs to |
| |construct an area model and compare their model to the simulation data, and finally come together as a class to discuss the |
| |difference between the probabilities for each outcome (where 0 is the most likely outcome, followed by 2, and 1 is the least|
| |likely outcome for a given trial) and the overall expected value (which is close to 1). Draw parallels between the methods |
| |used today and on Friday to calculate expected value problem. Ask students if their understanding has changed based on using|
| |points scored vs. money. Also ask students to explain how a one-and-one situation is different from just recording the value|
| |after each spin. |
|HW |Book Problems: p. 692-693, 8(a), 9, 11 |
| | |
Players can choose the numbers they wish to be entered into the drawing instead of being assigned a number. This ensures that the drawing cannot be fixed!
On the Fantasy 5 card, the player chooses 5 numbers between 0 and 39. Numbers cannot be repeated. The player pays $1 for every time he/she wants the chosen numbers to be entered in a drawing.
For example, a player chooses the numbers 8, 14, 17, 22, and 31. He wants these numbers entered in 30 drawings. Then, he fills out the lotto card with his selected numbers and pays $30 – one dollar for each drawing entered.
Five numbers are drawn at random and prizes are awarded in the following manner:
|Match |Prize Awarded |
|0 or 1 |No Prize |
|2 |1 Free Play (refund of |
| |$1) |
|3 |3rd Prize |
|4 |2nd Prize |
|5 |1st prize |
The prize amounts are determined by how many players buy tickets each day. If no player wins the first prize, then the money rolls over to the next day’s 1st prize pool. This way, if there is no winner, the jackpot continues to increase.
|Day #: 13 |Lesson Title: Part III - How are the proceeds used? |
|Goal: |To develop an understanding of where the lottery proceeds channel into and what effect this has on those participating in the lottery |
| | |
|Objectives: |Students can break up the state education budget into percentages based on income bracket |
| |Students can compare low-income families’ contributions to education funding under the current system and under a system where the |
| |lottery did not contribute to the state education budget |
| |Students will read and summarize their understanding of the article provided in class |
|Lesson Summary (one|Students work in their pods on a worksheet designed to help them analyze the following information: They use current data to determine |
|paragraph maximum) |what percentage of the state’s education budget comes from high, median, and low income families’ property taxes. They would then |
| |analyze what percentage of the state education budget comes from high, median, and low income families’ lottery spendings. They would |
| |compare this to how much each income bracket would contribute to the state’s education budget if it was solely funded by property |
| |taxes, using current graduated taxing policies. Students would look at whether low income families are contributing a disproportionate|
| |amount of money to the state budget under the current system. While groups are working, we walk around and answer questions and help |
| |students accurately calculate and evaluate the percentages. After completing this worksheet, we do a Directed Reading Thinking |
| |Activity with an article that discusses various viewpoints of how lottery funding is distributed to school districts in Chicago Public |
| |Schools and the state of Illinois. |
| | |
| |Article: |
|HW |For homework, students would write a 2-paged, double-spaced analysis of if the lottery is a fair system. They would use the article, |
| |their work from the past week, and additional research to answer the following questions: |
| | |
| |1. How would we determine if the lottery is a fair system? What information would you need and how could you use mathematics to |
| |analyze this data? Provide a specific plan for how you would research this. |
| | |
| |2. Given what we have investigated so far, do you think the lottery is fair? What data from the past week leads you to think that? |
| |What changes would you propose to how the lottery is currently run? |
|# |Learning Objective |Assessment Item |
|1 |Students will define the differences and similarities |In what ways are theoretical and experimental probability similar and in what |
| |between theoretical and experimental probability |ways can you differentiate between the two? |
| | | |
|2 |Students can compute expected value. |In Day 5, students compute the expected value of the Spinner game. |
| | | |
| | | |
|3 |Students can demonstrate how to use Venn Diagrams to |In Day 7, students are assigned problem #5 from the homework to complete. It |
| |break down categories into mutually exclusive events. |prompts students to use Venn Diagrams to fill in missing information from a |
| | |chart. |
|4 |Students can describe verbally and/or in writing the |In Day 10, students address the difficult problem by differentiating between |
| |differences between combinations and permutations. |combinations and permutations. Furthermore, students have to explain their |
| | |strategies. So although the"ISg[pic]•˜ÚÝadˆ-‹-ª¸ y may not come up with a |
| | |solution, they demonstrate a consideration of how to apply permutations and |
| | |combinations. |
|5 |Students can express how using mathematics can help |In Day 14, students discuss their interpretation of a social justice question,|
| |explain or understand the world around them. |and justify their interpretation with mathematical arguments. We will leave |
| | |time at the end to allow students to express their new perspective on how math|
| | |can inform their interpretation of the world. |
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