| I. MOTIVATION & RATIONALE | 3 |
| A. Teaching philosophy | 3 |
| B. Why Statistics and Probability? | 4 |
| C. Summary of the unit | 5 |
| II. GOALS OF THE UNIT | 9 |
| III. OVERVIEW OF DAILY LESSONS | 11 |
| PART I: THEORETICAL VS. EXPERIMENTAL PROBABILITIES AND THEIR USES | 11 |
| LESSON I: (2-3 days) Introduction To Probability & Backgammon | 11 |
| LESSON 2: (1-2 days) Backgammon Scenario | 12 |
| LESSON 3: ( 1 day) Theoretical Probability of Backgammon Scenario | 13 |
| LESSON 4: (1-2 days) Experimental Probability | 15 |
| LESSON 5: (1 day) Reflect and Summarize | 16 |
| PART II: SURVEYS SEEN AS PROBABILITIES | 17 |
| LESSON 6: ( approx. 7 days) Conducting Mini Surveys | 17 |
| LESSON 7: ( 1 day) Article Research and Commercial Analysis | 19 |
| PART III: CORRELATIONS | 19 |
| LESSON 8: (2 days) Introduction & Collection of Class Measurements | 20 |
| LESSON 9: (1 day) The Scatterplot | 21 |
| LESSON 10: (1 day) Analyze Data and Reflect on Correlations | 23 |
| LESSON 11: (1 day) The Best Fit Line & The Equation of a Line | 24 |
| LESSON 12: (1 day) Slope of a Line | 25 |
| LESSON 13: (1 day) TheY-intercept | 26 |
| LESSON 14: (3-4days) Piecing it all Together | 26 |
| UNIT REVIEW & FINAL ASSESSMENT | 27 |
| POSSIBLE EXTENSIONS TO THIS UNIT | 28 |
| IV. DAILY NARRATIVE ACCOUNTS AND DOCUMENTATION OF UNIT IMPLEMENTATION |
29 |
| PART I: THEORETICAL VS. EXPERIMENTAL PROBABILITIES AND THEIR USES | 29 |
| LESSON I: Introduction To Probability & Backgammon (DAYS 1&2) | 29 |
| LESSON 2: Backgammon Scenario (DAYS 3&4) | 35 |
| LESSON 3: Theoretical Probability of Backgammon Scenario (DAY 5) | 44 |
| LESSON 4: Experimental Probability (DAYS 6&7) | 47 |
| LESSON 5: Reflect and Summarize (DAY 8) | 54 |
| PART II: SURVEYS SEEN AS PROBABILITIES | 58 |
| LESSON 6: Conducting Mini Surveys (DAYS 9-14) | 58 |
| LESSON 7: Article Research and Commercial Analysis (DAY 15) | 65 |
| PART III: CORRELATIONS | 67 |
| LESSON 8: Introduction & Collection of Class Measurements (DAYS 16&17) | 67 |
| LESSON 9: The Scatterplot (DAY 18) | 71 |
| LESSON 10: Analyze Data and Reflect on Correlations (DAY 19) | 75 |
| LESSON 11: The Best Fit Line & The Equation of a Line (DAY 20) | 75 |
| LESSON 12: Slope of a Line (DAYS 21&22) | 79 |
| LESSON 13: The Y-intercept (DAY 23) | 82 |
| LESSON 14: Piecing it all Together (DAYS 24-26) | 83 |
| UNIT REVIEW & FINAL ASSESSMENT (DAYS 27&28 + take-home project) | 85 |
| V. EVALUATION OF WHAT STUDENTS LEARNED | 86 |
| VI. REFLECTION AND ANALYSIS | 90 |
| VII. APPENDICES | 93 |
| Appendix A. Backgammon board and set up | 93 |
| Appendix B. Backgammon Rules | 94 |
| Appendix C. Backgammon Scenario Board | 95 |
| Appendix D. Backgammon Reflection Sheet | 96 |
| Appendix E. Backgammon Scenario Worksheet | 97 |
| Appendix F. Backgammon Worksheet #2 | 98 |
| Appendix G. Assessing Your Understanding about theoretical and experimental probability | 99 |
| Appendix H. Thumbtack/ Theoretical vs. Experimental Assignment | 100 |
| Appendix I. Data Collection Worksheet | 101 |
| Appendix J. Survey Information Sheet | 102 |
| Appendix K. Requirements for Survey Presentation | 103 |
| Appendix L. Article Packet (selection only) | 104 |
| Appendix M. Relationship/Correlation Packet | 106 |
| Appendix N. Scatter plot graphs | 107 |
| Appendix O. Compiled list of class measurements | 108 |
| Appendix P. Reach Height vs. Jump Height Packet | 109 |
| Appendix Q. Slope Worksheet | 112 |
| Appendix R. Slope & y-intercept worksheet | 113 |
| Appendix S. Assessing Your Understanding | 114 |
| Appendix T. Selection of correlation graphs | 115 |
| Appendix U. Unit test | 119 |
| Appendix V. Project description | 123 |
A. Teaching Philosophy
The National Council of Teachers of Mathematics has been a driving force in the mission to improve the quality of school mathematics. The instruction of mathematics today encompasses an environment where students adapt an active role in their own learning. Thus, the learning of mathematics transcends from a transmissive model towards a reconstructionists framework. A transmisive model is one that views knowledge as objective, where the teacher is an authority who uses directive pedagogy (E. Swartz, 1992). A reconstructionists model views knowledge as socially constructed through interaction. The teacher presents knowledge as a problem-posing pedagogy, not a banking approach (E. Swartz, 1992). In this model, mathematics is presented as a subject that is not necessarily pre-determined. Rather, it is a subject where one can assume the role of an investigator or explore, forming hypothesis and testing outcomes.
My teaching philosophy is one that exemplifies the teaching of mathematics through constructionist view point. This involves viewing students as active learners who are capable of developing their own sense of understanding of mathematical content. Students are encouraged to utilize what they know or perceive to further investigate, explore and modify their understanding of mathematical phenomenon. According to Magdalene Lampert, "Knowing mathematics in school therefore comes to mean having a set of unexamined beliefs, whereas Lakatos and Polya suggest that the knower of mathematics needs to be able to stand back from his or her own knowledge, evaluate its antecedent assumptions, argue about he foundations of its legitimacy, and be willing to have others do the same" (Lampert, 1992). The teachers role is to organize collaborative and critical activities that allow students to communicate and express their thoughts. The discourse in a mathematics classroom should be directed by both the teacher and the students. Moreover, the students should be given opportunity to provide examination and cross examination of one anothers thoughts and solutions. Value is placed on EVERY students' questions, arguments, beliefs and suggestions.
Incorporated in this unit is a pedagogical style that is reflective of the constructionist model of teaching mathematics. Students are active participants and explorers in their learning. This exploration involves mathematical discovery, "hands-on" activities, and real world application. Moreover, social interaction and effective communication plays a dominating role in how the students construct their thinking and understanding of the mathematical content. The following practices were utilized in this unit in order to promote this teaching philosophy:
Explain to me why you think....?
How is it that..........?
What lead you to your conclusion?
You said....meaning what?
How do you see this pertaining to my question?
Show me what you mean by...?
Do you think that....? Why?
B. Why Statistics and Probability?
This unit has been designed to bridge a connection between statistics and probability. There is perhaps no other branch in the mathematical sciences that is as important for ALL students , college bound or not, as the study of statistics and probability. A knowledge of both these areas are necessary in order for students to become intelligent consumers who can make critical and informed decisions. Even a cursory glance at newspapers shows the extent to which the language of statistics and probability have become a part of everyday life. This particular unit examines the question of why statistics and probability were created. Both subject areas can be viewed as tools that are necessary to predict a multitude of events (from game theory, to weather forecasts, stock market, medicine etc.). Moreover, statistics and probability offer students an opporuntunity to examine how it is that mathematics can portray a curriculum that is open to alternative solutions, personal opinions and arguments as well as defined to have a purpose for examination. Students should be given the opportunity to explore how statistics and probability can be used and misused and how misconceptions can lead to erroneous decision making. Examining a multitude of probabilities and statistical scenarios and drawing conclusions is a concept that perhaps a large number of adults have not mastered, nor have been granted the opportunity to explore. According to Huff: "So it is with much that you read and hear. Averages and relationships and trends and graphs are not always what they seem. There maybe more in them than meets the eye, and there may be a good deal less" (p. 8, 1982). Because of society's expanding use of data for prediction and decision making, it is important that students develop an understanding of the concepts and processes used in analyzing data (NCTM 1989, p. 105).
The study of statistics and probability arises from a need to make informed judgments about uncertain events and data around us (Moore, 1989). For example, to establish an appropriate inventory, the manager of a shoe store must decide which sizes and brands to restock. The weather bureau must issue reports that indicate the likelihood of rain, sunshine, hurricanes, tornadoes, and other conditions. Consumers are bombarded with often conflicting claims about product quality from which they use to make informed purchasing decisions. The medical arena conducts thousands upon thousands of experiments in order to assess the effectiveness of a drug or the likelihood of contracting certain diseases. According to Moore, "interest in teaching statistics is certainly due in part to recognition of the place that working with data plays in everyday life and in many occupations. Its is increasingly common to teach mathematical topics that are of direct use, rather than to select topics simply because they lead to later topics in mathematics. Statistics is such a topic (1989).
C. Summary of the unit
The following NCTM standards (p.105, p. 171, 1989) will be reflected in the statistical exploration of this unit. Students will explore statistics and probability in real world situations so that they can:
This unit is comprised of 3 mathematical sections designed to predict the likelihood of events from both a probabilistic and statistical point of view. The following is a summary and motivation for each section in the unit.
SECTION I
Lesson 1, Introduction to Probability & Backgammon: The introduction to this unit involves allowing students to share their experience with luck, chance and probability. Moreover, students will be able to surface their misconceptions in the field of probability. They will be asked a set of open ended questions that focuses on students perception of how probability plays a role in variety of different scenarios (i.e. carnival games, the lottery, board games, sports, etc.). As students are engaged in sharing their experiences, the introduction of backgammon can surface. Students should be aware that the game will be used to explore if and how chance may play a role in winning the game. The game of Backgammon will serve as a real world context for developing a true understanding of probability. By the time students approach the 8th grade, they will have had experience playing with spinners and dice. The use of Backgammon will serve as a newer manipulative that students will not be so familiar with.
Lesson 2, Backgammon Scenario:
Students will be asked to take a closer look into a specific backgammon scenario. This investigation involves having students examine "what the best move is" at that particular point in the game. They are encouraged to provide the reasoning behind the solutions that they are providing. This lesson allows students to provide a hypothesis on what they think would be the safer move. The set up for this lesson involves students working first in small groups and than moving to presenting their hypothesis.
Lesson 3, Theoretical Probability of the Backgammon Scenario:
This lesson allows students to explore how they would check to see what the safest move really is, thus leading into theoretical probabilities. The development of the sample space and the formula for theoretical values will be arrived through the students' own exploration and former knowledge. Students will be expected to find the theoretical probabilities for each move in the scenario.
Lessons 4-5, Experimental Probability:
The next step is to examine how much truth is involved with the theoretical probabilities that have been determined. For example, what does 18/36 really mean and what would it take to believe in the truth of the value? Students will begin to test the theoretical probabilities and examine the law of large numbers. In addition, they will be able to contrive the formula for experimental probabilities as well as understand the role that such probabilities play in terms of testing the validity of theoretical values. Moreover, they will explore experimental probabilities in other contexts such as research, weather forecasts, and other unequally likely events. The goal is to have students understand that experimental probability can test as well as create theoretical probabilities. Students will also be able to identify where and to what extent are experimental probabilities apparent in our society.
SECTION II
Lesson 6, Conducting Mini Surveys:
Section II of this unit consists of allowing students to conduct their own survey of choice. Their results will be seen as "experimental" probabilities. Thus, probability will transcend from games of chance to its role in statistics and research. Through their own surveys, students will develop an understanding of causation, biases, and sampling techniques and how each can effect the outcome. They will realize, just as they did using the law of large numbers in backgammon, that the closer they arrive to a representative sample as well as a population count, the stronger their experimental results will become. The middle school years are probably the best age group to investigate surveys. The students are very interested in themselves and become even more interested when they are empowered to conduct a survey of their own choice. At an early age, students will begin to learn how to interpret results and critique or analyze conclusion drawn from not only their own surveys, but from research articles, opinion polls, consumer reports and commercials. Students will have developed a sense of how experimental results can indeed act as a tool to predict future results or expectations.
Lesson 7, Article Research and Commercial Analysis:
Students will be given real life articles to reflect and apply what they have learned thus far. The articles present interesting topics to analyze and critique.
SECTION III:
Lessons 8-10. Correlation:
This last section, will take students a step further in examining how statistics can act as a tool for prediction through the exploration of correlations. Students will be given a situation where a detective deduces that the suspect in a crime is tall due to a large footprint that was left in the mud. How close do you think this detectives' assumption is? Students are asked to back up their reasoning when trying to decide the degree of the relationship ( i.e. very strong, moderate, very weak). Students will collect and draw scatter plots of their classmates body measurements. They will learn to interpret scatter plots and to look for linear patterns.
Lessons 11-14, Predicting from the Best Fit Line & the Equation of a Line:
Students will use the classes' reach height/jump height data to examine the purpose of the best fit line in order to predict, describe, check, and compare relationships between two variables. Students will develop a procedural as well as a semantic meaning of the equation of the line (to include, slope and y-intercept). This last section will allow students the opportunity to learn these algebraic concepts through the data they have collected. Thus, meaning is given to each variable in the equation of the line in the context of their own graphs.
This particular unit was implemented to approximately 100, eighth grade students at a suburban school in western New York. The average class size consists of 25 students. The majority of the students will be tracked at the NYS regents level curriculum. However, just as any 8th grade classroom, the students exhibit a variety of mathematical capabilities.
This unit examines how probability and statistics act as tools for prediction of events in our every day surroundings. Students will investigate the concept of prediction first from a probabilistic point followed by a statistical framework. The lessons can be made adaptable for 7th-12 grade students. The level of difficulty can be modified, depending on the students' mathematical background and skills. The following is a description of the goals for the three sections in this unit as it is applied to 8th grade students.
PART I: Theoretical vs. Experimental Probabilities and Their Uses
PART II: Surveys Seen as Probabilities
PART III: Correlations
PART I: Theoretical vs. Experimental Probabilities and Their Uses
LESSON I: (2-3 days) Introduction To Probability & Backgammon
| Materials: |
| Backgammon Setup and blank boards (Appendix A) |
| Rules to Backgammon (Appendix B) |
| Pennies |
| dice |
Facilitate a discussion to capture the students' first beliefs about chance and probability. This preliminary discourse will allow student to share their experiences and perceptions of the subject. Broad questions will allow the students to guide the remainder of the discussion. Students will most likely mention carnival games, the lottery, gambling and perhaps even sports. As a beginning manipulative have a few students try to throw a penny into a cup. Have the class try to predict whether or not their classmates will make the shot. At the 8th grade level, many students will associate luck and probability together. Moreover, they will claim many events that are dependent upon the outocme of another. The goal of this introduction is to get a feel for what students' beliefs are about probability. More importantly, as the unit progresses, students' will have a chance to modify their thoughts on the many roles that probability can play, to include revisiting their misconceptions. The following questions can help facilitate this discussion:
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With the mention of board games (which in most cases the students' will first mention) bring out the introduction of Backgammon. Explain to students that they will be exploring how chance or probability or luck may come into play through a board game such as Backgammon. Pass out instructions to the game, the game board and the chips. A great substitution for paper chips would be to have each student bring in 15 pennies. One player can be heads while the opponent can be tails. Allow approximately 1.5 - 2 days for students to play so that they become familiar and comfortable with the rules of the game.
NOTE: During the introduction of this game it is recommended to make over heads of the game board as well as an overhead of the board set-up.
HOMEWORK: Practice playing Backgammon.
LESSON 2: (1-2 days) Backgammon Scenario
| Materials: |
| Backgammon Scenario Board (Appendix C) |
| poster paper |
| Backgammon Reflection Sheet (Appendix D) |
Once students are comfortable with playing the game present them with the scenario in App. C. Using an overhead explain to the students that they are to assume the position of the black chip which is on its way to the inner table. The roll is a (4,3). How would you move? According to the rules of backgammon they can move the chip either a total of 7 spaces, or 4 spaces, or 3 spaces, or not choose to move at all. Most students will want to head towards the inner table with out thinking about the chance of getting "hit" by the white chips. Because the underlying theme for this lesson is using statistics and probability to predict events, this particular scenario will allow students their first opportunity to use their insight in hypothesizing what the best move would be according to the rules of Backgammon. Have them work on this scenario in pairs and present to the class their decisions and rationale. Some students will focus their attention on getting closer to the inner table. Other students will catch-on to realizing that the "safest" move is not to move because the probability of the white chip hitting the black chip would mean that he/she must roll a sum of 11-a less likely roll. Many students will provide some good arguments that regardless of what the safest move may be, the idea of taking a chance in a game can work for ones' best interest. Thus, analyzing every move is not necessary to become a winner.
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HOMEWORK: Backgammon Reflection (App. C)
LESSON 3: ( 1 day) Theoretical Probability of Backgammon Scenario
| Materials: |
| Backgammon Scenario Worksheet (Appendix E) |
After students have finished presenting their strategies, the following questions can facilitate a discussion leading into trying to find what the safest move really is. However the next set of questions should be prompted by what rationale the students' provided. In other words, the teacher should only use the students' reasoning and explanation to lead the discourse:
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Once students have brought out the possibility of trying to figure out how many ways or times a particular number may come out, the Backgammon Scenario Worksheet (App. E), can be handed out. This worksheet can act as a guide to finding the theoretical probabilities for getting "hit" on each of the moves. Some points to keep in mind when developing the idea of theoretical probability:
The equation for theoretical probabilities is: # of times the favorable event can occur/total number possible outcomes Note: Students are familiar with basic probabilities such as a die or a penny. The ratio should be developed from their own understanding. Thus try to avoid "giving them" the rationale/formula. Rather let them arrive at an understanding of where the ratio is coming from. If students have trouble take them back to concepts they are familiar with to lead into the rationale. For example: What is the probability of getting a 6 on one die? How did you know that? |
Have students finish page 2 of the worksheet which involves finding the theoretical probability of each move in the scenario. Once they have done so proceed to discuss and form conclusions on what may be the best move:
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HOMEWORK: Have students write out their thoughts on how much they trust the theoretical probabilities that they have found. Discuss how one would test the theoretical values.
LESSON 4: (1-2 days) Experimental Probability
| Materials |
| Backgammon Worksheet #2 (Appendix F) |
| dice |
| Assessing Your Understanding (Appendix G) |
| poster paper (optional) |
The next step is to explore how believable theoretical probabilities really are as well as bring meaning to experimental probabilities. In backgammon as well as any other kind of game, we only get one chance to roll or take a turn. Thus, really how much weight do theoretical probabilities hold? Moreover what do these values really tell us or help us do?
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At this point Worksheet #2 (App. F) can be used to begin experimenting and testing whether the theoretical probabilities hold true. Thus, the first use of experimental probability is to use it as a tool to check a theoretical probability. Have students perform the experiment using the table on the worksheet to record the results. Once all the values have been tested, have each student record their experimental probability on the board. I would advise taking only one event to analyze thus to avoid having a cumbersome chalk board ( students should perform experiments on all 4 of the outcomes). Once one is analyzed as a class, students will automatically become interested in finding the experimental probabilities for the other three and comparing their results.
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By the end of the discussion students should form an understanding that experimental probabilities work in the context of testing a theoretical probability as well as developing a theoretical probability.
HOMEWORK: Assessing Your Understanding (App. G)
LESSON 5: (1 day) Reflect and Summarize
| Materials: |
| poster paper |
| Thumbtack/ Theoretical vs. Experimental Assignment (Appendix H) |
One class period should be devoted to summarizing what was learned thus far. This can be done as part of reflection for class notes or poster paper. The purpose of this reflection is to bring a form of closure to the big ideas that were brought out through the many activities that were performed thus far. At this point discussion of theoretical and experimental probabilities can be brought to events beyond the game of Backgammon. Students should be able to provide examples of where they see both kinds of probabilities in the real world. The following are some questions that students should be capable of answering as part of the notebook or poster paper reflection:
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HOMEWORK: Thumbtack and Experimental vs. Theoretical Worksheet (App. H)
PART II: Surveys seen as probabilities
LESSON 6: ( approx. 7 days) Conducting Mini Surveys
| Materials: |
| Data Collection Worksheet (Appendix I) |
| Survey Information Sheet (Appendix J) |
| Requirements for Survey Presentation (Appendix K) |
| poster paper |
This begins the transition from seeing probability and experimentation through games of chance towards how experiments are applied to statistics and research. Students will begin to see that not only can experiments act to prove theatrical probabilities, but they can also become the guide to developing both theoretical probabilities and making predictions.
Rather than formally teaching students how to conduct surveys, the purpose of this section is to have students explore questions and problems that are common to conducting surveys, polls and other forms of research. At the 8th grade level, students show some degree of knowledge regarding surveys. Therefore the introduction to this section should involve discussing what the students' already know. They should be empowered to conduct their own surveys with the intent for them to recognize and discuss some problems, biases or generalizations from their own explorations.
The first Worksheet to guide this investigation is found in App. I called "Data Collection". Some leading questions into this area may include:
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Biases play a major role in surveys, opinion polls and research. At the 8th grade level students may be familiar with the word "bias". However, they have not yet developed a true understanding of what is meant by this terminology nor have they had working experience with this concept. Therefore, this provides a good opportunity to strengthen their understanding of the term:
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When assigning the survey assignment, students should first start with a small sample such as their math class to begin. This will act as a sort of pilot, to test their question and observe what the outcomes are for a small sample.
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Allow time in class for students to work on this assignment. App. J consists of a worksheet for students to complete during class time. This sheet is not only a model that students can use, but it also is valuable for the teacher since it will be difficult to monitor how every group is doing with such short class periods.
App. K is the requirements for the survey presentation. Each student should be given a copy of this in preparation for their presentation. When students begin presenting, encourage the class to participate in asking questions. Students will quickly catch on to the kinds of questions to ask their peers as well.
LESSON 7: ( 1 day) Article Research and Commercial Analysis
| Materials: |
| Article Packet (Appendix L -- selection only) |
At this point the students will have confronted important concepts that effect data collection and prediction through their own survey investigation. This includes identifying where biases are present and the effect they can cause, choosing an appropriate sample size as well a representative sample, recognizing the impact age and sex may have on results, and designing a survey question in order to eliminate any assumptions or questions that survey takers may confront. A good transition at this point is to relate what the students have learned through their own mini surveys to articles of research and surveys in society today. The article packet in App. L offers some good articles or advertisements that are valuable for a classroom discussion. Students will realize that though much of survey results, research and opinion polls are based on in depth experimentation, there may be areas that become difficult to predict or detect a relationship amongst. Thus, the leading question into the last section of this unit is to explore how and when can one make mathematical predictions when working with extended data collection. In statistics the topic of correlation offers a broad range of opportunity to provide "hands-on" experience with predicting from data collection. Moreover, correlation will assist students in understanding the degree of strength that relationships have and the effects that biases or extraneous factors have on predicting events. During the discussion of these articles, it may be a good idea to document students thought on poster paper.
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HOMEWORK: The last page of the packet in App. L.
PART III: Correlations
LESSON 8: (2 days) Introduction & Collection of Class Measurements
| Materials: |
| Relationship/Correlation Packet (Appendix M) |
| Measuring tape |
| rulers |
This section will take a closer look at research based on two variables. This section will take students a step further in utilizing statistical analysis and graph extrapolation in order to predict events. Specifically they will be examining the study of correlations and how they are used as tools for methods of prediction. This particular introduction will allow students an opportunity to discuss relationships that they may have heard about previously. The question becomes how do we know such relationship are true? If they are true, to what extent?
Begin by distributing the Relationship/Correlation Packet (App. M). Have students read the first page. The following questions can help drive the discussion:
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The following chart should also be at the front of the room:
| Strong Moderate Week No Relationship |
| foot/height |
| foot/forearm |
| smoking/lung cancer |
| height/running speed |
| studying/GPA |
| jump height/reach height |
| age of car/worth of car |
| income/IQ |
| hair loss/age |
| items on sale/# of buyers |
This chart will act as the introduction for this section. Have students
discuss the kinds of relationships, if any, the above categories may have.
At first, most students will bank on "the exceptions to the rule".
It is important to clarify that the study of relationships from a statistical
standpoint does not imply "all cases hold". Rather, the goal is
to examine the strength and the nature of the relationship. For example,
if studying and GPA has a moderate relationship, it is true that there will
be a degree of variance amongst students. However, there is recognition
that a general relationship does exist-regardless whether or not all cases
hold. During this discussion the first goal is for students is to describe
relationship verbally.
What would it take to convince you how strong or weak certain relationships are? |
Reflect on the above chart and ask students how one would go about finding what sort of relationship exists between forearm/foot, height/foot size, and jump height/reach height. By this point students will know that they need to collect data.
DAY 2: Collection of Class Measurements:
Referring to App. M, have students work in pairs to take the required body measurements. Have them place their measurement on the miniature table (see App. O from the next lesson). These tables should be collected by the teacher in order to compile the results.
HOMEWORK: None
LESSON 9: (1 day) The Scatterplot
| Materials: |
| Scatter plot graphs (Appendix N) |
| Compiled list of class Measurements (Appendix O) |
To begin this lesson, pass out the class measurement (App. O) and also have the completed chart in Lesson 8 available for the students to view. Up to this point students only have had experience describing relationships with words such as "strong", "moderate" and "weak".
| Looking at your data tables, how would one attempt to determine what kinds of relationships are apparent? Does the chart tell you anything regarding the strength of the relationship? |
| What can we do with our data to see if we can detect a relationship? |
The goal here to have students realize that perhaps they need some kind of picture or graph that allows them to label relationships that are weak, moderate or strong. Some students may be quick to assume that a bar graph or circle graph can display the data and show the relationship amongst any two variables. If this is surfaced, try to simulate the idea to show that in fact a bar graph will not give the information that they are looking for.
If a student does suggest to graph the data, ask the class if that may work. If the class seems to agree with this idea have them beginning plotting the points on the graphs in the Relationship/Correlation Packet ( App. M).
At the 8th grade level, scatter plots are not very familiar to the students.
Thus, this will be regarded as new material. Up to this point, the students
are accustomed to connecting the dots and forming line graphs. This is an
important topic of discussion. Displaying appendix N will fuel leading discussion
into this topic. These 8 graphs can be shown to the student before or after
they plot their class measurements. The main purpose of this is to have
students explore how scatter plots are able to tell us the degree of strength
in a relationship.
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If students have not done so thus far, have them graph their class' data and label what kind of relationships the graphs show.
HOMEWORK: Last page in App. M.
LESSON 10: (1 day) Analyze Data and Reflect on Correlations
Have students take a closer look at their graphs as a class. Ask students
to compare what they previously predicted to the actual results of the class.
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Give students time to extrapolate themselves what sort of evidence the graphs are producing. For example, the relationship between forearm and foot size and wingspan and height may not necessarily show a strong degree of correlation, though they actually should. Students should be able to provide reasons why the results are not satisfying. Some variables such as palm height, can be used to facilitate a discussion on the importance of defining exactly what is being measured. For example, questions such as these can be presented to the class:
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Before beginning to use the correlation graphs to predict future outcomes, this may be an opportune time to reflect on the concepts that have been learned thus far. Such summarization can be performed via the notebook or poster paper. At this point, students should have developed concrete understandings of the following:
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HOMEWORK: Ask students to think about how statisticians may use scatter plots to predict the outcome of values that are not shown on a graph.
LESSON 11: (1 day) The Best Fit Line & The Equation of a Line
| Materials: |
| Reach Height vs. Jump Height Packet (Appendix P) |
Choose from one of the graphs in the packet to use as a model for extrapolating and predicting data. The Reach Height vs. Jump Height graph will be used as the model for this lesson plan (App. O). The Reach Height vs. Jump Height packet (App. P) provides step by step instruction and explanation on the procedures for understanding how to predict from such graphs. A motivating question for the students would be to ask: How can we use our graphs to predict the jump height of the tallest women in the world or your favorite basketball player? Students should follow along in their packets as the teacher models and provides explanation for each step.
Begin with a discussion reflecting on what students said was so special about a line. Also discuss how the stronger the relationship a graph displays, the closer it becomes to forming a straight line.
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Students will begin to quickly pick up on how the best fit line can be used to predict anybody's'' jump height. Ask students to use the line to predict their own jump height. Have them discuss how far off they are from their predictions?
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Introduce the students to the equation of a line-a more precise way of predicting using the best fit line. Some points that will help in starting this material:
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LESSON 12: (1 day) Slope of a Line
| Materials: |
| Slope Worksheet (Appendix Q) |
In order to use the equation of a line to predict outcomes, the first goal is to break down each variable in the equation and explore its meaning as well as its purpose as it related to the graph. Thus, beginning with slope pose the following questions:
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At the 8th grade level, a good way to begin the instruction of slope is to view it as the ratio between the rise and the run or the change in vertical distances versus the change in the horizontal distance. According to the data specifically students should view slope as a rate of change between the jump height and the reach height.
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HOMEWORK: Slope worksheet that will help students become familiar with finding the ratio (App. Q).
LESSON 13: (1 day) TheY-intercept
| Materials |
| slope & y-intercept worksheet (Appendix R) |
The y-intercept is a fairly basic concept for students to grasp. However they should be able to recognize that (x) will always be 0. They should also form an understanding that the slope and the y-intercept are the exact and only ingredients needed for someone to be able to draw your exact line! The y-intercept would be the location of where one would begin to draw the line. Students will realize that it seems simple to just read off where the line crosses the y-axis. However in their reach height/ jump height packet, one will notice that graph is broken off. Therefore, the y-intercept is not visible.
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At this point 8th grade level students may feel a bit confused or uncomfortable with using the equation of the line to find the y-intercept. They do not have the mathematical background to fully appreciate the usefulness of this equation. Give ample time for students to practice this process. Assign student helpers who show a concrete understanding of the material thus far to offer help to those students who are seeking further assistance.
HOMEWORK: Slope and y-intercept worksheet (App. R)
LESSON 14: (3-4days) Piecing it all Together
| Materials: |
| Assessing Your Understanding (Appendix S) |
At this point the students will have all the pieces of the equation and thus ready to predict their own jump heights.
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HOMEWORK: Assessing Your Understanding #3 (App. S)
NOTE: The next few days should be devoted to solidifying the process of using the equation of the best fit line to predict from graphs. Appendix T provides a selection of correlation graphs that can be worked on during class time as well as homework. At the 8th grade level, it is advised to provide a variety of different context for students to investigate in order to pull the strings together in understanding why this method works.
ASSESSMENT MATERIALS IN THIS UNIT & FINAL PROJECT
UNIT REVIEW & FINAL ASSESSMENT:
Appendix U is the unit test, which was designed to assess all the concepts that students' have learned starting from the probability section and ending with the correlation unit. The assessment materials were designed to test both, the students cognitive understanding of the subject area as well as the processes of solving a variety of mathematical problems. Some questions check students' understanding of fundamental concepts, while others are problem-solving questions that probe more deeply to assess how students are reasoning about what they know.
UNIT PROJECT:
Appendix V offers an authentic assessment which specifically allows the students to develop their own correlation study. This project can be implemented when students feel comfortable with the correlation material. With this particular school, the project was implemented after the unit test. Expectations include:
POSSIBLE EXTENSIONS TO THIS UNIT
For the middle school level this unit can be extended to include a brief lesson on measures of tendency such as mode, median, mean, range or variability. Typical 8th grade level students will have already had experience investigating these measures. However, these central measures can be analyzed at higher cognitive level. For example:
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Another extension for the middle school level students is to use this unit to fuel a closure in depth look at solving equation with 2 variables.
For the high school level, students should be exposed to examining how close their estimates are when predicting from the equation of a line. Standard deviation or the root mean square error and variance will allow students to formulate an understanding that such measures represent the average distance from a point of reference. Through a real life context students will formulate an in-depth understanding of what the average error really means. The correlation coefficient is another means of calculating the average error that students can be exposed to as well.
This particular unit only focuses on linear graphs. However it can also provide a smooth transition into exploring non-linear graphs and functions.
The majority of the following narrative is generated mainly from the 5th set, 8th grade mathematics class. I felt it necessary to capture other interesting narratives that happened outside of the 5th set class as well.
PART I: THEORETICAL VS. EXPERIMENTAL PROBABILITIES AND THEIR USES
LESSON 1: Introduction To Probability & Backgammon
DAY 1 (2/10)
This begins the first day of the implementation of this unit. My cooperating teacher explained that his students will be interested and are ready for a nice change. Sure enough students were quick to become engaged in conversation about what comes to their minds when they hear words such as "chance", "luck", or "probability". The intention behind this introduction is to surface students' mathematical thoughts and abilities on probability. I felt it necessary to have students guide the discussion as often as possible.
Teacher: What comes to mind when I mention words such as "chance", "luck", or "probability"? Brian: What are the chances of winning something? Like when you play games and stuff like that. Christian: Everything is luck with that stuff. Teacher: How so? Christian: No one ever wins anything. Teacher: Well, why don't we talk about that. It seems like all of you have played carnival games before. Why don't people ever seem to win. Brian (and other voices): Because it's rigged. How the heck are you suppose to throw something this big into a container this small. I hate those stupid games. Teacher: Nice example. I agree with you guys, some of these games are unfair-or as you have stated rigged. What makes something unfair? Mike: When you have a slim chance of wining. The odds are against you. Tim: You have a greater chance of loosing than winning. If that didn't happen than the booth would lose allot of money. Teacher: Good point. You guys don't think skill is ever a factor? Dave: It depends though on the game. Like some games are based on skill, others just luck. Teacher: Ok. Then what is luck based on? Dave: Luck, it's hard to explain-luck you know, winning or loosing. Teacher: Anybody want to help Dave? Tim: Luck, is like something beyond your control. Teacher: I agree, but how do you know when something is based on luck versus skill? |
At this point, the class had a tough time trying to explain what "luck" really is and whether it coexists with probability. This is a rather big issue for young students to inquire about. However, I'm glad to have captured the opportunity for them to attempt answering such "hard to explain" questions. In their exploration of Backgammon, I will ask them to take a closure look at the game and try to compare where they see luck as a factor versus where they see probability as a factor. In other words, when can chance be recognized and used to win the game in making decisions.
Dave: Whenever someone mentions that word "probability", I think of those colored balls in a bag and trying to find the probability of choosing a specific color. Teacher: Funny you should say that, I sort of have that vision permanently ingrained in my brain. Don't you worry though, I promise you'll have other things to think about "probability" by the time I'm through with you. Ame: My parents play the lottery all the time. I think it's such waste of time. Teacher: Why? Ame: Because your chances of winning are one in a trillion! Teacher: Wow, that some chance!! Kristen: I heard that you have a greater chance of getting hit by lightning than winning the lottery. Teacher: Where do you think that statement came from? How could someone arrive at that conclusion? Brian: Well, first of all I'm sure that some sort of scientist group have calculated someone's' chance of getting hit by lightning and than you just compare it to winning the lottery. Teacher: Well I guess so, there probably is some statistics on the number of people that get hit by lightning per year? I unfortunately do not know how many people win the lottery in a year. Does anyone know? Christian: Maybe like 5-7 people or so? Maybe less? Teacher: I'd probably guess about 3-4 win the whole pot. Teacher: Where else do you guys hear about things that are tied with "chance"? Jessica: I play basketball and my foul shot average is something like 1 out of 5. Teacher: What does that mean? Jessica: It means that I have a 20% chance of making a foul shot-I'm not very good. Teacher: Well, your young, I'm sure with more practice you'll get better. Speaking of making shots. I'm going to give Jessica this dime and ask her to try to get it in this cup. But first, Do you guys think she'll hit or miss? Some students yell hit while the majority yell miss. Teacher: Oh please, how do you guys know? Matt: Well, she's not very good at basketball, but at the same time that's not really hard to do. Teacher: Go ahead Jessica make the shot. Jessica misses the first time. The class responded "told you so". Teacher: Ok. Let's have her do it again, what do you think will happen. More students said she'll miss. However, she ended up making the shot. Teacher: Wow, Ok so what happened? Many voices: "luck"! Teacher: Luck huh, okay let's have her try again. What do you think will happen. Interesting thing happens here-more students said she'll make it! Fortunately she did. Teacher: What if we had Jessica do this another time. What would happen. Brian: She'll miss, the other tries were just luck. |
This small little activity really got the kids riveted. More importantly, it gave me insight to how the students tended to utilize the outcome of previous shots to make a prediction of what would occur next. Students again referred to making the shot as luck. I think this is because they haven't seen Jessica perform enough trials in order for them to resort to skill.
Brian: Ms. Maine, why is it that when you toss a coin it can only land hands or tails? How come we don't include landing on it's side. Teacher: Good question Brian-any reactions? Tim: That's a dumb question-the chance of it landing on it's side is nearly impossible-so that's why we can't count it. Teacher: Why can't it land on it's side? Tim: It won't balance, it will just fall. Teacher: You're right-the mechanics of coin make it very difficult for it to land on it's side. How many times do you think you would need to toss it until you get it to land on it's side? Brian: About a million. Which doesn't matter anyway, because by that point you'll have 50% heads and 50% tails. Teacher: Excellent point Brian!! Do you guys see that? How many tails to you think you'll get with a million tosses? Many voices: "half", "500,000" |
I was very impressed with this particular conversation. However, I realized that only a fraction of the class really were in sink with Brian. Yet, I could see the "wheels" start to spin in their heads as they continued to think about the probability of a coin landing on it's side and why we usually do not consider it an option. Moreover, as seen by the dialogue students seem very comfortable using percentages and ratios to describe the chance of winning or getting something.
Teacher: What if we tossed a marshmallow? What are the chances of it landing on it bigger side? Tim: What kind of question is that? Kristen: 50% Tim: Either it lands on it's side or it's top- 50%. Teacher: Do you guys agree with that. Mike: Yup. But it might not be that exact. Brian: Frankly, I don't think anything is ever exact in probability? Teacher: Why is this so? Brian: Well, if we are talking about chance, chance it's not definite-it could or could not happen. Teacher: Nice intuition Brian. What do you guys think. Kristen: Probability and chance are not definite. Like suppose in the Coca Cola game they say you are suppose to get a free coke in one out of every 8 cans. So like if I buy 8 cans-I should get a free coke-really, I mean how often it that true? Tim: Yeh, I mean if you bought a 24pack there's no way you'll get 3 free cans. Teacher: Nice example, your right. I think the packaging may not necessarily allow you to literally get a free coke with every 8 cans. I guess what we would need to explore is how random the process is. Christian: Yeh I mean how are they getting that answer 1 out of 8? How did they make sure that every 8th can will have a cap that says you win a free coke. Ame: And than what cans go together-like is it done fairly? |
Now this particular conversation was very impressive as well. They were really thinking and applying my line of questioning to real life examples. The students showed that they have some basic understanding of investigating really what games like the Coca-Cola Game really mean. One very important concept behind their thinking is that they recognized that probability is not a definite outcome. In other words, they know that it acts as way of determining how likely something will/will not happen.
However, one big misconception I know they have is assuming that everything has a 50% chance-either you win or you loose. In other words, students tend to ignore that a percent should be placed on the likelihood of the event literally occurring. This is only associated with the idea of winning and loosing.
Although it may seem like these conversations caused "chaos" in the classroom, it provided me with some great insight into what the students already know and what will need to be worked on throughout the investigation. More importantly, the students were able to display some analytical skills and probabilistic concepts that I can always refer to when we begin to formally learn about theoretical and experimental probabilities.
Since I felt pretty good with what has begun to surface, I decided to begin the introduction to the game of Backgammon. I had a goal of not spending more than 2 days in learning this game. I explained to the students what we will be doing over the next couple of weeks.
| Teacher: You guys, brought up some great points that I will refer to as we proceed in this unit on probability and statistics. We will be doing some interesting activities that investigate the purpose for certain concepts in these areas. One major theme over the next number of weeks will be thinking about topics in probability and statistics that help us predict future outcomes or events. I want you to begin viewing mathematics as a tool or key to exploring many topics in our lives. To start off we will be playing a game called Backgammon. Over the next couple of days I want you to think about any forms of strategy that is involved with this game. Before I begin to teach you the rules of this game, I want you to think about how many of you have said that luck is something you can't control. Well, suppose that there is strategy to winning this game. I want you to think about whether such strategies will change your chance of winning and to what extent? What role does chance play in this game? |
I distributed backgammon boards to every students. Some students brought in their own which was a big help. I had made out paper chips for students to use. I introduced the game using the overhead and the chalkboard. The class learned how to set up the game and I discussed the rules of the game. As students began playing, I went around the room individually teaching the game to groups who were confused . I had students who were familiar with the game go around helping others as well. The paper chips were difficult for students to use as they would easily fly away if someone sneezed let alone moved. Thus, I told students to bring in pennies and use the head/tail sides to differentiate amongst the two players.
DAY 2 (2/11)
This day was devoted to playing backgammon. I found it difficult to make sure that every student was on the right track. My cooperating teacher provided additional help as well. Most students were able to finish one game of backgammon in this period. However, I felt a little bothered by the fact that about 10 minutes before the period ended many students got up and walked around while others tried to finish their game. On this particular day, not too many students showed enthusiasm about the game. However, most of the students did get the general rules and objective of the game. At this point, I gave them their first Reflection Worksheet (Appendix D) for homework. I also decided that a great way to get these kids enthused would be to bring the point of why they are being asked to learn this game into light. I decided to jump into the scenario with only 1.5 days of learning the game. I thought that the scenario would really get these students to think a little deeper into the game and reach a stronger understanding of the rules. It is interesting to note here that while I was planning these sets of lessons, I originally thought that we would spend 1 week learning this game! I'm so glad that I was able to reflect on really what is important and what I trying to get at with this game.
LESSON 2: Backgammon Scenario
LESSON 2: Backgammon Scenario
DAY 3 (2/12)
This marks the middle of the week before break. The students were starting to get that "I can't wait till break" excitement- a little on the hyper active side. I collected their Reflection Sheet (App. D) that they had for homework and placed the Backgammon Scenario up on the overhead (Appendix C):
Teacher: Okay guys, I know some of you have a better handle on this game than others. Today, I'd like to present you with a specific scenario. Now, believe or not, while I watched most of you play, a large number of you had this kind of situation happen. Here's the board as it stands. You are the black chip. You are on your way to clearing the table. You have these white chips coming your way. You rolled a (4,3). What move are you going to make. What are your choices? Christian: You can move the black chip 7 Teacher: Okay that's one option. I'll put it up on the board, the rest of you tell me what other options there are. Ben: You can move it 4 and than move some other black chip 3 or the reverse. Teacher: I agree, so now we've got 3 choices. Anything else we can do? Brian: You know what, Ms. Maine, you don't have to move it at all. You can move other black chips as well. Erin (sitting next to Brian): I think we have a less chance of getting hit-I think Brian is right. Christian: No, what are you talking about, you are close to winning the game-go for the chance and move that one chip 7. If those white chips don't get you, you can get them. Kacey: I think 7 is too much to move that one chip. I think we should move it 3 or 4 to stay on the "safer" side. Teacher: (as the class begins to get very involved), okay guys, I love all the thinking that's going on here. I'm not going to make a comment on anything that was just said because, you all have excellent points. I think this is a good time to get in groups, take some poster paper and tell me what move you would do and why. When your done, you can present it to the class. Use pictures, tables or computation to help you explain your position. Go to it! |
This was exactly how I wanted the class to start off with. The students were all thinking from a probabilistic point of view. There were other students in addition to Brian and Erin who were able to deduce that the "safer" move would be not to move. If the chip is not moved, the white chips would need to get an 11 to hit the black chip (a probability of 2/36). While I was walking around the room to observe students. I noticed that very few were attempting to figure out what the probabilities really were. Yet, they would use reasoning such as "less chance of getting hit", "take the risk at get home", "even numbers are going to come up more often". I was very impressed by there intuitive thinking skills. I also realized that the students really enjoyed working in groups to come to a consensus of what the best move would really be. Moreover, much of the discourse that was occurring amongst students really forced a crucial goal in the learning of mathematics- "backing up your thinking". Some groups even began rolling two dice to see what kinds of numbers they would get! What is even more exciting is that the students weren't going to be happy until they could convince themselves that their arguments made sense. Here are a few examples of what students offered as explanations:
Kacey & Emily: Don't move, the white chips will come up on you-but they won't get you and than you can get them! Teacher: How do you know you won't get hit? Emily: Because you have less of a chance. These white chips have to get an 11 to hit you. That's a big number. Usually the smaller numbers are what I keep getting. Teacher: You tested it? Kacey: No, but when we play the game, I don't remember getting an 11 allot. Teacher: Do me a favor, think about a way of finding out the chances of getting an 11. Tell me when you think you've got something. Morgan &Matt: It's a 50% chance of getting a 4, so we said go for it! Move 7. Teacher: Where did 50% come from? Matt: We rolled 20 times, and according to the rules of backgammon we got a 4, 10 times. But who cares-take the risk-you've got a 50/50 chance! Teacher: Do you think that if another group rolled 20x they would get a 4 10 times as well? Morgan: (a little hesitant), well, I mean yeh-why not? They would at least get close to it. Teacher: What if you rolled it 30 times? Morgan: We'd still get 1/2. Teacher: Okay-you very well could. But how do we know for sure what your chances are? Matt: You can't know for sure, it's the luck of the dice. Teacher: You mean you can't control what the dice say-I agree, but what about after you've rolled-is there a way of looking at the numbers and getting an idea of the probability of getting hit? Morgan: Well, I guess we have to list all the pairs and see what seems to come up the most. Teacher: Nice thinking, Morgan, hold on to that thought-I think we very well may use it! Ame: We decided that if you move 4, than the white chips needs a 7 to hit you. Since seven is odd, it won't come up as often and you'll be a little closer to home and probably on the safer side. Teacher: So you think it's good to get your guy moving across the board. But, tell me more about your view on the odd number 7. Ame: Well, look how many ways can you get a 7, doubles won't get you it, only (2,5) and (4,3), (1,6). That's it- like 3/12. Teacher: Mmm. Interesting. Your right, those are the ways to get a 7. How did you get 3 out of 12? Ame: there are two dice with 6 choices on each so 6x2 = 12. Teacher: Anyone want to comment on that? Brian: I thought it would be 30. You gotta list all the pairs! Teacher: Any comment on what Brian is saying? |
Ame was one of the last presentations. At this point, many students begin playing around with trying to develop a list of possible outcomes. However, some were more focused on rolling the die an "x" number of times to see what happens. At any rate, this particular lesson was quite enjoyable for both I and the students. Every student had something valid and substantial to share with the class. I'm not so sure that every student really understood how impressive their arguments were. Since their work was so valuable, I decided to hang the students'' posters on the back wall. Some of the rationale the students had could easily be referred to when we begin to move towards closer exploration of theoretical and experimental probabilities. Thus, the posters would act as a frame of reference for the students to remind them that they indeed, had the right idea! Since class was ending soon, I decided to have the students think about ways we could find out if our predictions were true. In other words I asked them to think about how one could find the exact chance of getting hit on the move they chose. This would begin our next class.
DAY 4 (2/13)
I regret, not assigning the homework as written work! When I asked students with what they came up with, they had nothing to say! It was evident that they didn't take time to think about it. This was my fault in the sense that with this age group, unless an assignment isn't literally assigned on a piece of paper or book, the students' simply won't consider it as a "real" assignment. At any rate, the students' had surfaced more than enough to begin a discussion of examining how we could try to find out the chances of the moves in the scenario. My goal for today was to get through finding the theoretical probabilities of each of the moves and than comparing it to what students' predicted as the "best" move.
Teacher: Okay, yesterday we started examining the question: How could you find out the chances of getting hit. Well, today I want to proceed with this and see if we can develop a process that indeed will allow us to check our predictions. Now I know some of you began mentioning fractions and percents when doing your poster presentation. Others, used comparison, in what numbers would come up more often. For example an odd number may not have as many choices as an even number (at least according to the rules of backgammon). Why are probabilities talked about in terms of fractions or ratios? Tim: Because you are comparing the chance of getting something to not getting it. Dan: Yeh, it's like the percent something will happen out of 100. Teacher: I agree. But what about the fractions that some people came up with. Why are fractions used? Erin: Can't you think of it like say the probability of getting a six on a die is 1/6 because you only have one way of getting a six and there are 6 choices on a die so 1/6. Teacher: Erin, that's exactly what I'm looking for-let's use something that you guys learned in 6th grade, a simple die. What are the chance of rolling any number? Class: "1/6" Teacher: okay, can you use words to describe that fraction? What is 1/6-Erin said it-can anyone repeat it. Brian: It's how many times you can get any number. Teacher: okay, you guys agreed that the probability of getting a 6 in one roll of a die is 1/6. What does the 6 represent in the fraction? Brian: it's 1,2,3,4, 5, 6. Teacher: what are those numbers representative of? Brian: all the numbers on the die. Teacher: exactly-it's all the choices we have! Now, can someone explain to me what 1/6 means in terms of rolling the die. Matt: It means that if I roll it once, I'll have a 1/6 chance of getting the number, say like the number 2. Kacey: You can get a 2 at least once in 6 rolls. Teacher: Matt and Kacey have got it. Are you all with us. (most heads are nodding at this point). Ben, since you guys know that the chance of rolling a 2 on one dice is 1/6- and you didn't have to test it, is it possible to figure out the moves in our backgammon game the same way? Ben: yeh. Teacher: okay, I'm going to pass out this worksheet (Appendix E) that will help us find out what the safest move really is. Now follow along and let's see if we can get some actual answers-I don't know about you but the suspense is killing me! (class sighs....) Now, let's look at our scenario. Say we decide to move 7. The white has to get a 4 to hit us. What are the chances of getting a four? What do I need to do? Brian: well, since we are talking about 2 dice and not one we got to figure out all the pairs that come up like double ones, double twos, etc.. Teacher: Okay, so what am I trying to do? Christian: you need to find out all the ways of getting a four and compare it to all the possible pairs. Teacher: Perfect. Now, I want you guys to take 2 minutes and see if you can develop your own system for finding all the choices we have with 2 dice. Anybody, want to take a guess at how many it will be. Class: 12, 30, 24 Teacher: well, let's go ahead and see what we come up with. |
At this point, I walked around the room to find that most of the students were not using the counting tree. Rather they were listing any pairs that came to their heads. Some were doing all the ones first and pairing it up with all the numbers and than doing the same for the remaining numbers. I also know that I had a couple of problems that would need to be addressed. The first was to explain that something like (1,3) and (3,1) are not the same. The second is to convince them that 12 is not the total number of possible outcomes. These two situations are problems that many students have. They think: "since one die has 6 choices, two dice will have 12 choices". The (1,3), (3,1) scenario was going to be a tough pill for these students to swallow as well.
| Teacher: okay, 2 minutes is up-go ahead guys start shouting out pairs and I'll right them on the board. |
I intentionally decided not to immediately teach them the counting tree or the punnette square just yet. Rather I wanted students to realize that listing all the possible outcomes with out any framework or structure may result in an incomplete list. I want the students to fall into asking themselves: "how do we know if we covered all the possibilities" -especially when their is no organization. The students began listing numbers.
Teacher: hey hold up, I can't right this fast, I wrote down (1,2), but what about (2,1)? Gaelyn: It's the same. Teacher: Does everyone agree??? Class: yeh. Teacher: No one wants to challenge that? Christian: How can you-it's the same thing. Teacher: Okay, everyone stop for a minute-this is really important. What is 1 + 2 =? Class: 3 Teacher: What about 2 + 1 =? Class: 3 Teacher: So how many ways do we have of getting a 3 right now? Class: 2 Teacher: Now tell me if you are convinced that (1,2) and (2,1) count as 2 separate choices? Brian: Well, I guess that makes sense. Emily: But it's still the same-you get a sum of 3. Erin: Why do we have to count it as separate? |
If someone were to take a snap shot of the class at this point, one would see that probably half the class accepted this passively, while the other half really were not convinced. I decided to use a white die and a brown die to give them another way of looking at it.
Teacher: Okay these are two dice that are different colors. Suppose I get a 2 on the white dice and a 1 on the brown dice. That's the (2,1) outcome. Now can't I get a 1 on the white dice and a 2 on the brown one? Matt: yeh but our dice is not colored. Teacher: I know that but suppose I gave you colored dice, or suppose I took spray paint and painted one of your dice-would that change things? Matt: I guess not since it's the same dice. Teacher: Right, I'm only changing the color. Who's with me. |
The students needed time to think about this for a moment. A few more people began to understand. But, for some reason I wasn't convinced just yet! Off the top of my head I decided to give them another way of thinking about it.
Teacher: okay, guys, I'm not convinced that you guys are with me. Let's try this. Brian you are on the track team right? Brian: yeh Teacher: Suppose you are running against one person. So someone will get first place, and the other will get 2nd. Right. Class: yeh Teacher: Folks, how many ways can Brian place? Brian: I can get first place and he get second. Teacher: OR Brian: okay or I can get 2nd and he get 1st-but that will never happen. Teacher: Erin does that make sense. Erin: yeh, I see what your saying-so that means we have to do the same for all those pairs of numbers. Teacher: you got it kid! So let's continue. Give me more pairs. |
The class continues to increase the list until no one has anything left to offer.
Teacher: How do you know if we have everything-it's hard to look at this list and see what's missing. Did anyone organize things differently. If so come up to the board and show us.
Two students came to the board and actually had counting trees. When the students saw the trees, many of them quickly remembered with a "oh yeh". The counting tree was relatively easy to explain. In addition, I taught them the punnette square method as well. It reminded them of either their science class or the multiplication table. Students took to this method pretty well also. At this point, students saw that in fact they had 36 possible outcomes.
Teacher: Well look at the diagrams on the board. How many pairs do we have? Class: 36 Teacher: Now, once again what does 36 stand for Matt? Matt: That's all the choices with 2 dice. Teacher: You got it, it's the total possible outcomes. Now, guys look at this list. What are the chances of getting a 4-according to our outcomes? |
The students begin to mention the pairs as I would circle them. In backgammon, any pair of double is worth 4 times. This needed to be reminded to the students. They arrived at 18/36.
Morgan: Holy Cow-we were right Ms. Maine. Yesterday we told you it was 50%! Teacher: You better believe it! What's even more fascinating is you did it without doing what we just did. What does this fraction mean in your own words: Morgan: 18/36 is 1/2 so you have a 50% of getting a 4 in the game. Teacher: Matt, what is another way of thinking about this fraction? Matt: It's like out of 36 possible a 4 comes up 18 times-just like your grid says. Teacher: very good. Ben-what else is 18/36? Ben: I don't know. Teacher: Well, if someone tells you that you have a 1/2 chance of winning something-what does that mean in terms of your chances specifically. Ben: 50% Teacher: yeh-but use the fraction and put it in words-can someone help him. Christian: I think I got, it means that 1 out of every 2 rolls you'll get a 4. Teacher: Thank you-Perfect. Did everyone hear that! |
One of my main focuses, have been to get the students' to really understand and interpret mathematics in their own words. If this is not accomplished than a fraction just becomes a fraction with no meaning. In other words, what have they learned?
Teacher: Okay, now we have this fraction 18/36. Kacey what does the 18 represent? Kacey: The number of times you get a 4. Teacher: Good, now in general would you guys argue if I said that the top number is the number of times an event occurs-such as 4. Class: Nods in agreement Teacher: Okay, now what does the bottom number represent-the 36. Brian: all the possible choices. Teacher: You got-it's all the possible outcomes or choices. |
The interpretation of the fraction is written on the board. I explain to the students that what they have just found is called a theoretical probability.
Teacher: When someone says in theory you have a 50% or 1/2 chance of getting a 4 what does that mean. Morgan: It means that your exact chance is 50%-but that doesn't mean it's going to happen. Christian: It's like in theory, it should happen but may not with one roll. Teacher: I like it-you guys are good-now tell me anybody, how much do you guys trust a theoretical value of 1/2? Addie: Well, you just take a chance and see. Teacher: See what? Addie: If in fact it will happen-the 4 coming up. Teacher: Okay, but where did this theoretical come from? Ben: Didn't we get it from looking at all our choices and then counting all the times a 4 came up. Teacher: Exactly-a theoretical probability is based on calculating all your possible outcomes. You are actually using mathematical analysis to figure out what should happen theoretically. Did I ask you guys to roll dice at all today? Class: No. Teacher: Exactly for a theoretical probability, we didn't have to roll the dice. We figured it out mathematically. Now, you guys are on a roll today. Go ahead and finish the rest of the worksheet in class today and for homework. You just need to find the theoretical values for the rest of the scenarios. I didn't make you read the worksheet while we were exploring the probability of getting a 4. However, tonight I want you to make sure you can piece everything together and read the sheet in it's entirety. |
LESSON 3: Theoretical Probability of Backgammon Scenario
DAY 5 (4/14)
This is the last day before winter break. The students were high strung today. However, most all of them did their homework. Because it was the last day before the break, I wanted students to reflect on the theoretical probabilities they have found and compare them to their own predictions. I thought it would be nice to give them a little "game playing time" as well. I wanted the students to have the opportunity to try playing the game once more with the idea of applying what they have gathered about theoretical probabilities of the dice.
Class started with comparing the answers the students had and discussing their thoughts:
Teacher: okay, so you all should have figured out a percent and a fraction for the 3 remaining scenarios. Any thoughts on the numbers you got? Actually, what is the "safest" move according to the theoretical probabilities you found? Emily: The safest move is the move where the chance of getting hit is 1/18. Teacher: Which move is that? Emily: Just stay where you are. Teacher: Does anyone disagree with that? What does the fraction or percent stand for? anyone? Ben? Ben: It means the chance of getting an 11 is 1 out of every 18 rolls or 2 out of 36 rolls. Its like a 6% chance of getting an 11. Teacher: So your chances are pretty slim. Nice interpretation Ben. How many of you said not moving is the better idea when we did our presentations? Less than one half of the class rose their hands. Teacher: Those of you who chose other moves? What are you thinking right now? Kacy: The probability of moving a 3 or a 4 really doesn't matter. Teacher: Explain to me what you mean Kacy. Kacy: I mean that if you move 3 you have a 16% chance of getting hit and if you move 4 you have a 19% chance-really there's not much difference. So it doesn't matter too much if you pick one or the other. Teacher: Your right those percent are pretty close. Christian: You know it doesn't even matter because those percents are really low to begin with so I would just go for the move-you only have a 16% chance of getting hit -BIG DEAL. Teacher: Hey, Christian is getting a little feisty! Let's think about what he said. Who disagrees with him? Tim? Brian? Erin? Brian: I think he's right-but we are still much safer if we stay. Let the white chip take the next roll and than nail him!! Teacher: That's one way of looking at it. But, let's think about this for a minute. How much do you guys trust these values? Like do you really think that the probability of getting an 11 means that it would take 18 rolls to get an 11? Christian: probably not-I mean it could happen more often. Teacher: Could it happen less often as well? Christian: I guess so. Teacher: So than what the heck is the purpose of this stuff? Ame? Ame: It sort of helps us with what may happen. Kristen: yeh-it's like we get an average of what may happen. Matt: It's not suppose to be definite. Teacher: What do mean by that. Matt: Well, we said that a probability tells you your chance of getting something-it does not necessarily guarantee it. Teacher: So how does what we did help you? Do you see any value in what we have done? Matt: yeh, because we got a chance to check if our guesses were correct. Teacher: Absolutely. What else? Anyone? Morgan? Morgan: When you play the game you can think about what the move is and see if the other guy will have a high chance of "hitting" you. Teacher: Excellent point. What do you guys think. Can you use this process to better your game strategy? Class shows general agreement. Brian: Ms. Maine, I don't think its necessary to do this with every move!!! I mean only in certain parts it may be good-like if its a close game. Teacher: I agree Brian, you don't need to use it with every move. However, I wanted you guys to see that analyzing the "luck of the dice" may work in your favor some times. So again tell someone what the benefits are of these theoretical values. Jessica: It helps you determine if something will happen or the chance of it happening. Teacher: Nice. Every body see that? |
At this point the class seems pretty much in sync with what theoretical probabilities were. However, I wanted them to begin talking a little bit more on how much does one trust these values. Students were pretty quick to conclude that the theoretical probabilities are what is expected to happen. However, most students I don't think are completely convinced that it will actually happen. I almost felt that the students believed that more often than not a theoretical probability may be off. I think this thought process will really fuel an interesting investigation of experimental probabilities. I thought this would be a good time to stop and allow for some play time.
| Teacher: Okay, you guys had some great insight into the rationale for theoretical probabilities and I think we all agree how such concepts can help us with our strategy when playing the game. So having said that, I'm going to allow you the rest of the period to play the game and really think about some of these things we talked about. Some of you might even play the game a little differently. Over the break try to play a couple games with your brothers, sisters, parents or friends. When you approach a move which seems difficult to make, try analyzing your choices and report to us next Monday on your experiences with the game. |
The class began playing the game. Interestingly enough, they were far more engaged in this game than when I first taught it! Students were definitely placing more thought into the game before they made there moves. Thus, they were in fact applying what they had learned-one of my more important goals in teaching mathematics.
2/17-2/21 FEBRUARY BREAK!!!!!!!!!!!!!!!!!!!!!!!!!!!!
LESSON 4: Experimental Probability
DAY 6 (2/24)
At this time, I feel it necessary to discuss the event that took place in my first class of the day. It was their first day back from break. The students were pretty tired and quite. I on the other hand am rearing to go!!! I passed out the next worksheet on experimental probability (Appendix F). My goal was to begin with this worksheet WITHOUT THE INTENT OF ANY REVIEW!!!!!!!! What is even more distressing, is that the thought of "reviewing" what we did 2 weeks ago never crossed my mind! About 3/4 for the class did not remember what had happened before break. Yet, here I had this goal of having them get through the worksheet and realize the difference between a theoretical and an experimental probability! I tried to remind them of the probabilities that we discovered such as 18/36 for getting a 4. Students had no clue where the 36 came from! One girl though it represented the triangles on the backgammon board! Moreover, I had the students work in groups on the worksheet. Thus there really wasn't much classroom discourse going on. At the end we were able to discuss #1 and #2 on the worksheet. Yet, I found myself offering the answers. I felt completely miserable. The students had no idea what was going on! To make my start even worse Judy happened to be sitting in on that day.
At the end of class, I looked at Judy and asked for some serious help. My cooperating teacher explained to me that this was a typical day back from a break and that nothing really surprised him. Judy explained to me that there was nothing in the room that would help trigger the students' memories. I realized that I should of started class with perhaps a brainstorming session on poster paper and perhaps use the worksheet as a homework! Moreover, I should of at least used the back wall of the room to display the work they had done 2 weeks ago. At any rate, I had 3 more chances to rescue myself. The following is the continued documentation of the Set 5-a much better experience.
Teacher: Okay guys so how many of you played backgammon over the break? About 1/3 of the class rose their hands. Teacher: So did you guys find yourself playing differently or becoming better. Tim: I actually have been thinking about the moves and what someone would have to get to hit me. Teacher: Good. Why don't we start off refreshing our memories on what we did two weeks ago. I gave you a scenario. What did I ask you to do with it? Emily: Make a prediction or a guess at what the best move would be. Erin: Oh yea, we had to get into groups and present what our move would be. I remember. Teacher: Thank G-d-Who does not remember this? How did you guys decide what move would be the best? Ben? Ben: Well, it was like common sense, just move the chip so that you are less likely to get hit. Teacher: For some reason I don't remember it being that simple! But you are right, most of you tried to make a judgment call on trying to get home versus trying to avoid getting hit. Okay what next? Brian: We presented our posters? Teacher: Yes you did, and there up in the back of the room. (At this point I am in front of the posters). So we predicted and than what did I ask you to think about. Brian: We had to figure out our chances. Teacher: Chances of what? Ame: Chances of getting hit on the moves. Teacher: Yes, now what was the first thing we needed to do to figure this out. Does anyone remember? Try to reflect on simpler things that you've done already like the chances of getting 6 on a die. Christian: we had to do that list of pairs. Kristen: yeh and that table graph thing that you did. Teacher: You got it. What was the purpose of doing the lists, charts or trees? Kristen: to get everything. Teacher: Give me more. Kristen: It gave us all the ways of how numbers would come up. Teacher: Exactly it's all your possibilities or choices. So with the two dice, how many pairs did we come up with. Pull out your sheet from last week to help you out. Morgan: It was 36 pairs. Teacher: Yes, and what does 36 stand for? Class: all your possibilities. Brian: I remember the 36 was in the bottom part of a fraction and the top part was like a event that we were looking at. Emily: We used the probability of moving 7 which means getting hit by a 4-that's what is on my sheet. Teacher: Thanks Emily, Let's just use that example to take us through this. What is the probability of getting a 4 in backgammon? Class looked at their sheets and replied : "1/2" Teacher: Right, 18/36 which is 1/2. What does that fraction mean in your own words. Ame: It's a 50% chance of getting a 4. Teacher: okay Ame, that's right but let's stick with the fraction-spell it out to me using words. Ame: Well, out of 2 rolls you should get a 4 once which is half the time. |
It was very important to me to make sure that students understood the fractional interpretation. Student have known 1/2 is 50% since the 3rd grade. However, I wanted more from them as it relates to the situation at hand. This articulation of thought is crucial to making sure that students are truly understanding the purpose of fractions in the area of probability.
Teacher: Okay, is everyone with me thus far? (heads nod) Now, why do we call something like this a theoretical probability? Morgan: Because it what should happen but may not. Matt: It is there to give us an idea of what might happen, determine our chances. Teacher: very nice! So the probability of getting a 4 is 18/36. How much do you guys trust this? Is it true that for every 36 roles you'll get a 4, 18x? General Responses: "sometimes", "it depends", "in most cases". Teacher: How do we know this is true if I did not ask you to test it? How do we know if it will or will not happen? Brian: we have to test it. Teacher: Guys listen up I think Brian is on to something (at this point students begin to discuss amongst themselves whether they believe it is true or not). How would we test it Brian? Brian: By rolling it 18 times. Teacher: Why 18x? Brian: No wait I mean 36 times-since that's our possibilities. Teacher: Wait what if it doesn't happen in 36 times? More students begin to mention rolling it 100x Teacher: What is so special about 100? Morgan: You can get a definite proof that it is 1/2 b/c you do it allot of times. Teacher: So you guys are saying that a way to test a theoretical probability is by literally rolling the dice either 36 or 100x? Christian: Well, you could roll it any number of times and just keep watching what happens. I mean even in the first 36 rolls, I could close to 50%. The class confirm what Christian has just said. Teacher: Okay, but it seems like in general you guys are saying that all we need to do is roll the dice at least 36x-first to see what happens. What if we rolled it less? Brian: Well, we should get it 1 out of every 2 rolls since its 50% chance-but that's a theoretical and one roll is not going to prove a theoretical. Teacher: Let's talk about that for a moment. How do you guys think mathematicians came up with the chance of getting H or T when tossing a coin is 50%. Class Responds: "they tossed it like a billion times", "100x", "they did lots of tosses". Teacher: Wow, why do you think they did lots of tosses? Erin: To get some results. Like you have to keep testing to get at what you need. Teacher: Bravo Erin! I like that word "testing" or "experimenting". Okay, we are almost out of time. The sheet that is on your desk is for homework. You guys should not have a tough time with it. You did well today. |
What has just taken place is not only a review of what they have learned thus far, but the students' have managed to follow a discussion that not only treated an experimental probability as a way of checking the theoretical but also a way of developing theoretical. This was not stated in so many words. The students definitely had some great insight into the ability of proving events. I'm not even sure if the students are really aware of the transition they have made between experimental probability and theoretical probability and vice versa. However, this discourse was much more than what I had hoped for. Thus, I was quite pleased. What I also realized is that we sort of conducted a discourse that followed the worksheet (app. F). Yet, we really didn't refer to it. I assigned this worksheet for homework.
DAY 7 (2/25)
Today, we began class by discussing the homework. For #1 and #2 the students were mentioning allot of the points that have been coming up during our class discussion. Question #3 lead us right into conducting actually experiments in order to see how close we would come to our theoretical probabilities. At this point in the discussion of the homework, the students were convinced that a good way to start was just to begin rolling 36 times-since that was the way they interpreted the fraction "out of 36 rolls". I agreed with them that this would be a good way to start and asked them to turn to the chart on their worksheet where they can place all their data. I also decided to discuss the # of roles someone may choose to do later on in our investigation. Students got into pairs and began rolling the dice. I decided to focus on only testing the probability of getting a four according to the rules of backgammon. This would act as a model for the 3 other moves which they will be assigned to do on their own. As students arrived at their fractions they began giving me the results to place on the board:
| 18/36, 18/36, 16/36, 14/36, 19/36, 16/36, 17/36, 20/36, 18/36, 19/36, 17/36 |
As one can see, the class did pretty well. Moreover, I think this hands on experimentation really gave them a grasp on what the fractions mean. I began the discussion as follows:
Teacher: Okay, now just to make sure you are on the right track? Why am I making you do this? Various voices: "to test what should happen". Teacher: Right, we are testing our theoretical value of 18/36. Did we conduct an experiment to get this theoretical value? Class almost in unison: "NO" Teacher: So, now look up at the board, are you guys impressed with what your experiments resulted in. We got some people who actually got 18/36. Brian: Ms. Maine how come some of the values are not exact. Teacher: Yet, what is interesting Brian, is that they are not that far off! I'm glad you asked that question. If some people did not get 18/36 does that make the theoretical wrong? Brian: I guess not b/c it is close. Teacher: Suppose I were to make you do this 90 times. Then What? Christian: It will be closer to 1/2. Teacher: Why? Christian: Because it is more trials. Emily: Ms. Maine I just figured out the class average. It's 17.4. Teacher: Explain to me what you did. Emily: Well I just took the average of what everyone rolled-just the top numbers and divided by 11 and got 17.4 out of 36-not bad. Teacher: You are right. I never thought of looking it at that way. But indeed a class average is a good way of analyzing your results. Guys, I'm going to save you from rolling the dice anymore. Let's just pool all our results together. |
On the chalkboard, I begin to explain how many roles that actually occurred in the class as a whole. They followed along with me and punched in the amounts in their calculators to arrive at 48%.
Teacher: Mmm. You know for 396 rolls, why are we not at 50%? Morgan: First off, I think Tim and Dan who got 20/36 were rolling their dice across the floor. Brian: yeh so their dice rolled allot more each time-the rest of us just rolled it on the desk. Teacher: Wow, that is in fact a very good point. The way you roll the die can influence your results. Why else? Emily: I didn't count right-I forgot that doubles will also allow you to move four-so my fraction is incorrect-14/36 that's wrong. Teacher: Thanks Emily, you are helping us clarify what happened. So now what guys, Do you have faith still in theoretical probabilities. Jaessica: Yes, b/c it gives us a pretty good idea of what will happen in general. Teacher: Does the number of experiments you use matter when comparing it to your theoretical values? Kyle: Well, the more you do the closure you get to what is expected. Teacher: What if you guys only rolled 6 times. Do you think you will get a 4, 1/2 the time? Kristen: Well, yeh you might. Teacher: Suppose you do. Would you be absolutely convinced that if you roll another 6 times you get the same results? Various Voices were quick to disagree. They in fact began talking amongst themselves that you would have to get a large number of trials to get a consistent result. The more you do the closure you get to your theoretical. Before the class was going to end, I wanted to see if student could come up with an interpretation for the experimental probability fraction: Teacher: Remember when you guys were able to look at the theoretical value of 18/36 and tell me what the top and bottom number signify resulting in this kind of equation (begin to write it out on the board): Theoretical= # of times event occurs Total possible outcomes Well, let's look at what you guys got for the experimental probability. Let's use 17/36-what does the 17 stand for, where did it come from? Various Voices; the number of time we got a 4 Teacher: okay so it's the number of favorable or successful events-that's right (writes on the board). What about the 36? Various voices- "the number of rolls we did", "The number of times we rolled". Teacher: Exactly, it is the number of trial you performed. Thus, the experimental probability can be though of as: Experimental = # of favorable outcomes # of trials conducted |
The students were fairly quick at this point to contrive what the fraction really means. I think this is partly do the fact that they literally performed the trials, and counted the outcomes thus relating it directly to the fractions they came up with. I decided to stop, here to let the students soak in what they had done thus far. Interestingly enough, students began testing the other theoretical values without my direction. I allowed them the remaining of class time to compare the experimental values with the theoretical values. In general the class seem pretty convinced that these theoretical values do indeed make sense. My hope is that it will be brought out on the reflection sheet they will have for homework.
| Teacher: Tonight I want you to do this sheet for homework (Appendix G). It is another sheet that asks you to write out your thoughts so please use anything to help you articulate your thinking. |
LESSON 5: Reflect and Summarize (DAY 8)
DAY 8 (2/26)
Thus far, I felt that although the students in general were following along with the activities and formulating some strong and important concepts there was still something missing. I felt that things needed to be more grounded-almost like a mini closure to experimental and theoretical probabilities. When I described this feeling to my cooperating teacher, he responded that the best way to "ground" the concepts in was to go straight to the notebook. He showed me his math notebook and said that he wants these students to have something that they can refer to whenever they need it. Moreover, he explained: "if you put it in the notebook, these students will consider it more official". Since I had not had much experience with giving notes, I decided to do exactly what my cooperating taecher suggested. He simply makes up the notes the night before and gives it to the students to write down in their notebooks. However, before I began with the notebook I decided to start summarizing some points that the students might have to offer on poster paper. The following boxes show the questions that were asked and the comments that the students made:
total possible outcomes
# of trials |
This was a nice start for the class. However, since I knew I had notes to give them I decided to stop and begin with the notes. There were a couple of items that I wanted students to discuss, however, I though I would just do it in the notebook. For example, I wanted students to reflect on what they know so far and decide whether or not everything has a theoretical value. I wanted them to also why some things have theoretical values while others cannot-this would bring us into equally likely and unequally likely events. I also decided to use the notebook to place other examples outside the dice. I felt that is important in order for students to get a true understanding of these concepts.
I A theoretical probability is what we can expect to happen if we perform an experiment many times. -Theoretical probabilities are found by using math to analyze all the possible outcomes of an event. Theoretical Probability: # of times an event can occur/ Total # of possible outcomes A first step in finding theoretical probabilities is to list all your possibilities in a given situation: Ex: Suppose you have two spinners A and B: What are the chances of spinning a black on one spinner and a white on another spinner? 1. Find all the possible outcomes of this event: Write down all your possibilities This can be found by making a list, a chart or a tree diagram: B W Total outcomes to choose from is 4 B | BB BW W |WB WW 2. Analyze the ratio. How many times can we get a Black and White scenario # of times event occurs/ total possible outcomes 2/4=1/2. Ex 2: List 4 tapes or CD's you have. If you had to pick one without looking what is probability you'll pick your favorite? Do all situations have theoretical probabilities? Tossing Marshmallow? Coke Can? ________________________________________________________________________ Experimental probabilities: A probability based on collecting data from conducting an experiment. If the experiment is conducted a large number of times you are likely to get closure to a theoretical value. To compute an experimental probability a ratio is formed: Experimental probability: # of the desired outcome occurs/total # of trials or turns Many uses of probability in daily life, such as weather forecasts and sports predictions, are based on experimental probabilities. Name examples of other things to experiment: tossing a can, your average. time for running a mile, foul shot average etc. |
The students had some nice examples of where they saw experimental probabilities. One student even proposed that the number of times a radio station plays a certain song in 2 hours is an experimental probability. However, when I asked students about whether or not everything has a theoretical probability they were a little thrown off:
Teacher: Do you guys think that everything has a theoretical probability? Various voice-mixed yes and no Teacher: Take the weather man-do we get mad at him/her when they are wrong? Various voices: "they are always wrong" "yes". Teacher: Why-why are they not exactly right all the time? Brian: Because it's like a 30% chance of rain-so they are really giving us a number from all their equipment that they use-but it doesn't tell us for sure what can happen. Teacher: You bet Brian. That is exactly right. So we can say that it's almost like they are using equipment to get at a experimental probability or chance of rain. So what other things do not have a theoretical probability? Class is sort of stuck at this point. Therefore I decide to proceed with another example to think about. Teacher: What about tossing a marshmallow? What are the chances it will land on its flat sides? Class was taken by surprise with this example. Christian: What kind of question is that. Teacher: Why can't you answer it? Christian: Well, I mean either it lands on its side or its flat side so 50/50. Teacher: Wait a minute-does everyone agree with that the chances for a marshmallow to land on its flat side is 50%-like a coin!!!!! A marshmallow has a 50% chance of landing on its flat side? Ame: No, it will probably land on its flat side more. Kristen: It depends if you are talking about big or small marshmallows. Teacher: I agree the mechanics of a marshmallow make it very difficult to determine the chances of it landing on its flat side theoretically. Christian: okay so we'd have to just keep tossing it and record the results. Teacher: okay guys suppose we do what Christian is saying. Let's say we get a result. Then what do we do with that? Do we have anything to compare it to? Like suppose we found out that the marshmallow landed on its side 30% of the time. Now what? Brian: well, than according to our test that's what the answer is. Teacher: Is that a theoretical answer-how do we know if it's a good value. Like we know that flipping a coin as a 50/50 chance of heads or tails. What about the marshmallow. Kristen: I see what your saying-it doesn't have a value to compare it to. Like we can only go by the number we get based on our own experiment. Teacher: Exactly, and why is this so class? Ben: Because nothing is proven with a marshmallow coin they have proved already. Teacher: Is everyone with me? See how a marshmallow will not have a theoretical value. The weight and shape of the marshmallow makes it too difficult to determine a theoretical value. What about a coke can. Many Voices: "it will land on it's side allot more times". Teacher: How did you guys know that? Matt: Because its hard to throw a can and have it land straight up. Teacher: I agree, it almost seems impossible-do you know the percent of that occurring? Class in general says "no". Teacher: So what does that tell you about the coke can and theoretical probability? Many Voices: "There isn't going to be one", "you would need to test it to see what the chances are". Teacher: If you test it, will that be your experimental or theoretical? Class: "experimental" Brian: Ms. Maine if we spent years testing it-we could perhaps get close to a theoretical value. Teacher: you are absolutely right-and this just confirms our notion that we can use experiments to get theoretical values. But we do have to do a great many experiments and analysis-just like the scientists do. |
I already had the foresight that students would view objects that have unequally likely events as 50/50-thus referring to either it can happen or not. In other words, they avoid having to offer what the theoretical value is. This is because they simple do not know it-yet they fail to realize that "not knowing" it, is actually more than legitimate with obscure objects such as soda cans, marshmallows and thumbtacks. I chose a homework sheet that I think will be very valuable to their understanding of when things exhibit a theoretical probability vs. when they exhibit a experimental probability (Appendix H).
Reflecting back on this particular class, the students were pretty active in the discussion. However, I realized that perhaps I really did not need to have students use "my" notes as a means of summarizing and compiling what we have learned thus far. This will be explained further in Part IV of this section. Moreover, there are some changes I would ad before going into Part II of this unit. These changes will be discussed in Part IV as well.
PART II: SURVEYS SEEN AS PROBABILITIES
LESSON 6: Conducting Mini Surveys (Days 9-14)
DAYS 9-11 (2/27-3/3) Conducting mini-surveys
We started class by going over the homework (Appendix H). Most of the students were able to identify situations that were experimental or theoretical. The thumbtack worksheet presented a bit more problems. First, students didn't realize that they should be using the table to predict what would happen in 50 and 100 trials. Thus many students concluded that the tack has a 50/50 chance of landing on its side vs. top. Of course, on the last question of the homework students argued that there is a possibility of finding the theoretical value for the thumbtack. However, they did mention that they may need an expert in thumbtack dynamics to be able to calculate the chances of it occurring without testing it. This reiterated to them the importance of realizing that a theoretical value has already been proven, in some cases without being tested. Rather, by analyzing all the possible outcomes. The thumbtack discussion lead us into taking a closure look at experimental probabilities.
Teacher: I know that you guys have mentioned many times before that an experimental probability, with further testing, can get us into a theoretical probability. We know this is indeed true from the thousands of research project that are carried out every day in hopes of reaching a theory. Even with this thumbtack discussion, you guys believe that nothing is exactly impossible-and that there could be a way of finding the theoretical value. Moreover, we know that experiments can help us make predictions. So like if you experimented with this thumbtack, you may be able to predict the next outcome. Have you guys ever conducted an experiment before? Or collected data? Most of the students have collected data in their sciences class. Teacher: Okay, do you guys think data is valuable in conducting an experiment? Class of course confirms this question. Teacher: Well, than what is the data used for? Justin: To find if what ever you are testing is true. Matt: Or like you have to gather data to explore something-like when we write papers. Teacher: Nice example Matt. Have you guys ever done surveys or been in one? Brian: Sometimes in the mall they stop you and ask you questions. Ame: Have you guys ever been stop to watch a preview for a new movie-my parents once did. Teacher: I did that once too. Then they make you fill out the survey! Why do you think surveys are performed? Morgan: To find out like what's your favorite movie. Teacher: They definitely have a goal. Some surveys measure the populations' opinion on certain topics-for example politics. Let me ask you a question. Do you think that these surveys and opinion polls arrive at a theoretical or experimental probability? Class: "experimental" Teacher: How come? Kacy: Well, its like they are measuring peoples opinion and stuff like that-so there is not one definite outcome. Teacher: I agree. Christian: Okay, its like we are conducting an experiment or a survey so obviously that will only get us an experimental value. Teacher: Oh, what was I thinking to ask that question (class laughs)! Of course you guys know this already. Great than, if we know that surveys, opinion polls and even scientific research is based on carrying out experiments or tests and collecting data-what will it help us do in the long run? Class: "we will know what will happen", "we will know what the general trend is". Teacher: So what if it will help us know this? Brian: Than you will know what happens in general or even what will happen in the future. Like before the election, the opinion polls already told us who is going to win. Teacher: Nice job Brian-these surveys and research will allow us to predict what may happen. Like suppose I wanted to know what CD's 8th graders listened to. What would I need to do to conduct this survey? (At this point I write the topic on poster paper, and begin writing what the students offer). Morgan: You'll need to break it up into music groups-cause there's like a million CD's 8th graders listen. Teacher: I agree-okay so let's offer some music groups-now what? Kacy: Go out and ask 8th graders. Teacher: okay-how many? Brian: well, probably the whole class-cause that's 100 people and than you could speak for the entire 8th grade. Christian: No, If you asked 50 that's pretty good. Teacher: okay both of you have some good points-will your results vary depending on the # of people you ask? Class in general agreed that the number is important. Teacher: Okay so suppose I asked 50 people and a got all the results, now what? Class: "get the percent of each music group", "ratio". Teacher: Right, so we need to compile our results into fractions or ratios so that we can form a conclusion of what percent likes what. How would you present your data? Class: "Bar Graphs", "graphs", "charts", "list". Teacher: Okay, wow, you guys are like almost experts with this stuff. I just have one more question for you? Has anyone heard of the term bias before? Most of the students have heard the word used. However, they had a tough time trying to explain what is meant. Many of the students used the phrase "the results were swayed or unfair". Lina; Well, let's say I had someone who decided to ask people whether they preferred to bungee jump or water-ski. Do you see a problem with that question? Class thinks for a moment but doesn't come up with much. Teacher: How many people here have bungee jumped? (0 students), How many water ski (4 students). Mmm. Well, guys what's the problem with this question. Matt: Oh, not everyone has tried both. Teacher: Good, so how does that effect the question? Brian: You are like asking them to make a choice-but probably most people will lie and say they have done both. Teacher: True, or you could be forcing the question. People will have to say one or the other even if they haven't tried either of them. Do you think you'll get trust worthy data? Class: "no" Teacher: So in a way the question was biased or swayed to force people to picking one or the other. How could we change this question to eliminate the bias? Morgan: You can ask two questions. Like how many people water ski before and how many bungee jumped. Matt: yeh, but than you don't know which one they prefer. Teacher: Good point Matt, it's like you changed the question Morgan-but you could do that if you wanted to know how many 8th graders have done those things. Anybody else? Brian: You can put another option that says-I can't answer the question. Teacher: That's one way. Anybody else? Christian: You can put a choice that says "neither" because maybe there are people who did do both and hate it. Teacher: Very good, I didn't think of that one. Thanks. Ame: Actually you could put an option that says that you haven't tried either of them. Teacher: Another good suggestion. You guys got the hang of it. Suppose I took this question and went to survey the elderly community. What do you think of that? Are my results going to be telling the truth? Class: "no way", "old people haven't done that stuff", "actually they may only have water ski-but I don't know if they would remember that". Teacher: So than you're saying that the elderly population wouldn't be telling me what most people prefer. Good. See who you survey is also very important. If you only surveyed 8th graders-your results would change as well. So we need to be thinking about how we can make our question fair as well as how many and who do I survey. Now having said this, I think you guys are ready to do your own survey. I want you guys to have an opportunity to collect your own data and arrive at your very own experimental probability! You can't ask for a better assignment. |
I passed out the Data Collection Worksheet (Appendix I) to start the students off with what the assignment would be for the next few days. I explained to them that they could get into groups of 2-3 and come up with a question they wanted to explore. I wanted them to feel free to do whatever they were interested in. Moreover, I wasn't giving them guidelines as to how many people they should survey nor how to phrase their question. I felt that through their own investigation they will arrive at their own methods and conclusions. The discussion above, gave me more than enough proof that the students did not need their hands to be held. Moreover, this discussion acted like a model of things that they should think about when they proceed with this assignment. The whole point of this section is for students to be able to apply what they know.
The first part of this exploration, I wanted the students to only survey their math classmates. This would allow them a chance to test out their question as well as compare their predictions.
Appendix J was given to students on the second day of this assignment, 2/28. Since, I was going to give them time in class to work on their survey and compile their results, I wanted them to stay on task. App. J was to be turned in at the end of class.
Appendix K is the survey presentation requirements. This was given to them on the second day as well. It outlines the presentation expectations as well as the grading system. Students used this sheet as a check list. They needed to be prepared to answer each of the topics listed on the sheet.
The students really enjoyed doing this assignment. Topics that students chose include things like: sports, colors, actors/actresses, movies, soda, fast food restaurants and of course, music. Many students chose a number of athletes to use in their question for sports. For example who is the better player: Rodman, Jordan, or Grant? Many students asked if they could survey teachers. Because this was a substantial question I left it up to them-but I also reminded them to prepare to talk about it. What was most interesting to watch is that after practicing their survey question on their math class, many students revisited their question to rephrase or even take away or ad more choices. In other words they were critically thinking about what they were asking. By the third day, students began to realize that only surveying their friends wasn't going to give them a representative sample. Students would comment: "your friends are like you, so we have to get more people".
During the class presentations, the students had some amazing posters and well thought out explanations. In fact, they are still hanging up in the classroom. The students wanted them to take home, so my cooperating teacher and I decided that it would be best to leave them at the school. As you can see the students were proud of their work. Each presentation was interactive amongst the presenters, the class, and myself. Everyone was expected to participate and most everyone did. By the 2nd day of presentations, the students were assuming the role of the teacher in asking questions to the presenters.
DAYS 12-14 (3/4 - 3/7) Survey presentations
The following are some excerpts of what took place during the presentation:
Which brand of shoe do you like the best? Nike, Reebok or Adidas?
This particular group, knew that their question was forced in the sense that they did question how many people really have tried all the brands.
Charlie: We decided to limit our question to these 3 top brands b/c if we didn't we would have too much data to deal with and most likely people would fall into one of these 3 categories anyway. Teacher: Good point Charlie. Let's try to talk about these brands from a marketing view point. Why do you think people choose these shoes? Larry: Well, when we asked people what brand they prefer, I don't think they were necessarily talking about a comfort level. Charlie: yeh, like these brands are influenced by famous people. Miken: Not everyone has tried each shoe to compare. Teacher: I agree, I chose Reeboks only because I own a pair. |
What is your favorite subject in school?
Matt: I had students choose from Math, Social Studies, English, Science, Gym and Foreign Language. Classmate: Why didn't you put lunch? Matt: Because everyone would pick it and that's not really an academic subject. Teacher: Did students tell you why they picked the subjects they chose. Or better yet, what do you mean by favorite subject. Matt: My interpretation is that it is a subject that you do well in and enjoy. I think that's why lots of people chose Gym. But I also think that your favorite teacher will make you choose your favorite subject. I could see where the word "favorite" is maybe too broad-yet I look at the results and it seems to confirm what I thought most students would do. |
Which artist do you like best in the rap group Wu-Tang?
Teacher: Do you think there are any biases in your question? Rich: Yeh, we know that some people don't know who the group is. Tom: Also, when we surveyed people, they were just picking one of the choices-and we knew they really didn't care or have a preference. But if we did this survey in like New York City with a bigger population -more people will know the band-we'd get better results. Teacher: What would you guys do different next time? Rich: I think we'd have people listen to the music, so that they can become familiar with it. Plus, we would first ask people if they knew the group or not and if they didn't than we'd move to the next person. |
What's your favorite sport?
Dave: We think that one of the big problems with this question is that it varies within the seasons. Rob: Ms. Maine you know how you asked us to do our math class first-well that's only 18 people. I put the math class in a pie chart and than I took the 50 people we did and put them in a pie chart-and look at the difference. I can see how the more people you have the better numbers you get. Teacher: Very nice analysis Robbie. By any chance did you find that taller people were choosing basketball? Dave: Well, it's tough to say, I mean what is tall to me, may not be tall to Robbie. Kyle: Wait look at some of the NBA players. The point guards are short. Teacher: Wait a minute Kyle, if we are talking NBA-I think the majority of the players are "tall". Dave: yeh she's right. But still in our survey I don't think I could see that relationship-maybe if I went to the high school it might happen-those guys are taller. |
Needless to say the students really came up with some intuitive thoughts and ideas. One of my students drew 4 pie charts on what is your favorite sport. The charts comprised of what 25 adults said, what his math class said, what 60 8th graders said, and the last one put all the population together. His charts gave us a great look at what happens to results when the sample is varied. Another group ended up distributing their survey question at the mall to get a random sample! I think one of the most impressive points in this whole survey section is that the students really took ownership of their own learning. They only asked questions to verify their thoughts. I'd say that my assistance ended up to be more support than anything else. The students had displayed some critical thinking with their questions and how it fits into the bigger picture of trying to predict what the general population will say.
DAY 15 (3/7) Article analysis
By the last survey presentation, the students had a solid understanding of what they have been exploring. Thus, rather than beating a dead horse and taking away the excitement of what they had mastered, I decided to close this section up by taking a look at some articles and consumer advertisements. Appendix L was passed out and my intention was to pick and choose some articles or excerpts to analyze as a group. I had students read the articles aloud in class. At some points I'd ask students to stop reading, so that we can interpret some lines that may be a little more difficult to understand.
The following are a few articles that were discussed:
Teacher: How could it be that Ann Landers comes out with a poll that says 70% will not have children, while the national poll comes out with 91% will have children? Mike: First off all the national survey has to have more people in it. Teacher: yeh, but my gosh, how many more people to get the complete opposite result? Brian: Well, how many people can respond to this anyway. It depends on how many people got the article in the first place. Teacher: now you are on to something Brian. How did Ann get people to respond to this article? Jessica: Well, she just put it in her column and whoever wants to respond will do it. Teacher: Jessica is about ready to nail it!-Can you give me a little more. What kind of people responded? Jessica: People who hate their kids. Ben: Oh, I get it its like a question that normal people don't have time to deal with. So all the people with kid problems responded-it's like a huge bias. Teacher: Do you guys see a problem with what the health magazine is trying to do? Matt: Not everyone is a reader of this magazine-how many people own health magazine anyway-probably like all the doctors and nurses and medical people that take care of themselves. Teacher: Okay, so perhaps the readers are from a select population. What else? Ame: They are trying to lie because they told people who take lots of vitamins to respond. So the 93% is from people who take lots of vitamins as oppose to what the regular people do. Teacher: I agree-they are swaying the results. Christian: How could someone say that it's the vitamins that improved your health. I mean I know they are good for you-but some people could be naturally healthy. This is a rather difficult article for 8th graders to read smoothy. However, the contents of the article is in their level of understanding at this point. I had students read the first paragraph and than conducted a discussion with them: Teacher: What would it take for you to conclude that airbags save lives? Ben: First you'd have to find out how many people died in car crashes. Christian: All cars have seat belts but not all have airbags so obviously more accidents are going to happen in cars with only the seatbelt-so its like hard to track it. Kristin: You know those airbags don't save kids because they are too little to reach the airbag. Justin: How the heck are you suppose to know if people were wearing their seat belt in the first place? Tim: yeh, like what if they were wearing the seatbelt and have an airbag but died anyway. Teacher: All of you have some good points and it is exactly these points that you guys have mentioned that are in this article. I knew you guys didn't even need to read it! Does everyone see why the title of this article makes sense. |
The students like talking about these kinds of controversial topics-such as the airbag. I know that in a math class it doesn't happen very often. I'm glad these students caught on to allot of the biases that such everyday articles portray. I assigned the last page of the packet (Appendix L) for homework.
PART III: CORRELATIONS
LESSON 8: Introduction & Collection of Class Measurements
DAY 16 (3/10)
This begins section III of this unit. The purpose of this section is to have students take a closure look into how statistics is used to predict the likelihoods of events or outcomes. Students thus, far have explored how probability is used as a tool for prediction. They have also conducted surveys and explored how data collection can effect conclusions. The next, step is to move further into statistics, specifically exploring correlations and their interpretations. My goal for the first day was to discuss with students their thoughts and ideas of certain relationships that they may have heard of previously, but really never explored.
I handed out the Relationship/Correlation Packet (Appendix M). I asked them to read the first page. This immediately struck up a conversation:
Teacher: Do you think the detective is right to claim this relationship between height and foot? Jessica: not all tall people have big feet-so no its not true. Teacher: So b/c not everyone may fall to this description, you are claiming theres no relationship what so ever? Jessica: No there may be one but it's not always true. Teacher: I can agree with that. I know you guys know people who are tall with small feet and vice versa. So how do you think we could find out what kind of relationship exists between height and foot? Ben: We'd probably have to take allot of measurements and do something like find the average. Teacher: An average, so what compare the average height with the average foot size. I don't know really how much that will give us-it looks like it will just tell us the average height and foot of allot of people. Well, we are going to try to sort of tackle this topic. We are going to figure out if there's a way of determining how strong a relationship maybe between two variables like height and foot. Teacher: Do you guys think that this relationship is strong? Class: "maybe" "not very strong" "no". Teacher: well, lets see what you guys have to say about these relationships: |
The following poster paper was placed in the front of the classroom:
| Strong Moderate Week No Relationship |
| foot/height |
| foot/forearm |
| smoking/lung cancer |
| height/running speed |
| studying/GPA |
| jump height/reach height |
| age of car/worth of car |
| income/IQ |
| hair loss/age |
| items on sale/#of buyers |
Teacher: Now, thus far we have been taking a closure look into data collection and utilizing it to provide us with information. However, we haven't had an opportunity to really see how statisticians interpret data. Moreover, how do they see statistics as a tool for prediction? What we will begin to do for the next couple of weeks is talk about variables that are sometimes said to hold a relationship amongst one another. Take a look at the poster up here. I want you guys to tell me to what degree you think these relationships hold. Let's first take a look at the amount of studying someone does and their GPA. Jackie: I don't think that's fair to say people who study are the only ones who do well. I mean I'm an A student and I never study. Brian: yeh, I know people who don't open a book and get the grades. The class in general seems to be agreeing Teacher: okay, but guys, think for a minute when we talk about relationships you need to go above and beyond using yourself and your friends as the norm or the population. We learned this when we did our surveys. I want you to think of "in general". Relationships that exists only do so in a general sense-they never represent every single person. Christian: Well, than I think what we are saying is that it has a moderate or week relationship. Jackie: I say its a week one. Teacher: Jackie what do you think college students need to do to get the grades. Jackie: oh, they study day and night-but that stuff is tougher than the 8th grade. Teacher: yes, but don't you think that college students are in this picture as well? Brian: Okay lets say its moderate. Teacher: okay, I'll agree with that. |
The 8th graders were asked to complete the rest of the chart as part of a class discussion. More often than not, they kept referring to their own personal experiences and the exception to the rule. At certain points I asked students how it was that they had arrived at their decisions for how strong or week something was:
Teacher: How are you guys determining the strength of these categories? Ame: We think about if its always the case to if its not always. Like its all different. Teacher: I agree, notice that even if you say something has a week relationship, you are still saying that there is a relationship but it is not very strong-there's allot of variance. Like let's take a look at money and your IQ. Larry: yeh, we said that it's a week relationship. Teacher: okay, why did you say it was week. Brian: Because there are some people that do have high IQ's and make allot of money. But we can't say that everyone who makes allot of money has a high IQ. Teacher: Exactly so even though we know the case can't hold for all-we are still saying that there is a slight relationship. Thus, this is why relationship are thought of for "general cases" not all cases. Therefore you guys don't need to be thinking about your exceptions all the time. Teacher: I noticed here that you said the height and speed is a moderate relationship. How come not strong? Kristen: well, because there may be other things like the weight of a person. Tall people can also be fat and thus run slow. Ben: yeh and what about how athletic someone is. Brian: also if someone is light they will also run faster than the average person. Teacher: See you guys already are thinking in terms of questions that you need to ask when considering strengths of relationship. Really what causes people to run faster? You guys made some good points. |
The class ended up with a chart like this:
| Strong Moderate Week No Relationship |
| foot/height x |
| foot/forearm x |
| smoking/lung cancer x |
| height/running speed x |
| studying/GPA x |
| jump height/reach height x |
| age of car/worth of car x |
| income/IQ x |
| hair loss/age x |
| items on sale/# of buyers x |
After we had finished the above graphs there was very little time left. Since this was a lesson to warm up the students for what coming up next, I decided not to give any homework.
DAY 17 (3/11)
Today I wanted to have them perform their measurements. However, I wanted them to provide some insight on how would one go about finding the relationship that we had talked about yesterday.
Teacher: So picking up from yesterday-you guys said that there is a moderate relationship between foot and height, and a strong between height and forearm-how would we test to see if they are true? Ben: We need to get the measurements of our forearm, height and foot. Teacher: Sound like a plan. Why don't you guys pair up and get the requested measurement that are in your Relationship/Correlation packet. Hand them into me before the end of class and I will compile them for you on a master chart. |
The remaining time was spent getting the students' measurements.
LESSON 9: The Scatterplot
DAY 18 (3/12)
To begin the class, I passed out the compile list of class measurements (Appendix O). Up to this points, the students have only been able to define relationships using words such as "week", "moderate", or "strong". The idea today, is to try to get the students familiar with scatter plots which can viewed as a way of drawing the adjectives they have been using thus far.
Teacher: You guys are looking at the class data. Tell me, does this chart tell you what kind of relationship exist between say reach height and jump height? Class in general try to follow the chart and see if they can find a relationship amongst the numbers. Soon they realize that it is simply to difficult to do. Teacher: Well, lets think about this, how can we, get our data to make more sense to us we need to organize in a way that will allow us to see whether a relationship exists or not. Any ideas? Class begins to mention things like "bar graph" , "histogram", and line graph. Teacher: Let's take that bar graph idea first. Do you guys think a bar graph will tell us the kind of relationship two variables have? I started to make a bar graph. Showing the students that it really didn't tell us how strong the relationship was. Teacher: Pretend this is the bar graph for reach height and jump height-what is this bar graphs telling you. Morgan: only how high people jump in the range of a certain reach height. Teacher: Right. Does it tell you that this is a strong or week relationship? The class in general sees that it doesn't make sense. Teacher: Someone suggested to make a line graph? Well lets try it and see where it takes us. Now if we need to make a graph first what do we need to do? Class: Plot the points. Teacher: okay take a look at these graphs I have up here on the poster paper (Appendix N). Suppose these were the graphs of some of the relationships we are trying to compare. Just by looking at these graphs, which one would you say has a strong relationship. Class: Graph A Teacher: Why? Erin: Because its like the points are closer together. Teacher: I agree, what are these points trying to form or close to forming. Christian: Oh we did this in science, it's trying to be a straight line. Teacher: very good so you guys are familiar with this type of relationship. Where have you seen it before? Ame: It's like speed and acceleration graph-direct relationship. Teacher: Exactly. Why do you guys call graphs that have this sort of straight line direct? Ben: Because its like as speed increases the acceleration increases in proportion-we did this like a month ago. Teacher: Well, tell me what you think is so "idea" about a straight line-why does it make something a "strong" relationship? Brian: It's like exact or perfect-you can read it easily. Teacher: Very good, now tell me something, if these graphs are to represent our data points-how come I just don't go ahead and connect the dots. I mean you have been connecting the dots all your life. How come we don't connect the dots on these graphs up here (App. N) The class really doesn't know why connecting the dots is not an option. Teacher: well, Dave why don't you come up here and connect the dots to this graph. Dave began to start connecting the points, within in 1 minute he realized that there were too many dots and that he couldn't tell which way to go. Teacher: Dave, why did you stop? Dave: Theres too many points-I can't connect them-it will be just a bunch of zig zags. Teacher: So what does this tell you about connecting the dots with this stuff? Kristen: It won't get us anywhere. Teacher: Exactly, the zig zags are only going to tell where it increases and where it drops. Its not going to tell you the strength of the relationship. So let's go back to these 8 graphs, tell me how you would order the strongest graph to the weakest graph. Matt: It would be A is the strongest and than D, C, B. Teacher: How did you know that? Matt: The farther the points are the weaker the relationship. Teacher: Does everyone agree with that. How far do the points have to be to still consider a relationship. Brian: Well, the points can't go everywhere b/c than your data is everywhere and there is no relationship. Teacher: Good. Would you guys agree that the closer the points are to a line the stronger the relationship. Class agrees. Teacher: Now tell me what is different between graphs a-d and e-h? Morgan: The direction of the points are different. Teacher: In what way? Morgan: One goes up the other goes down. |
Teacher: Does anyone know how to interpret what Morgan is saying using other words-try to remember your science class-when she told you about speed and acceleration what were both those variable doing at the same time. Class quickly understand what I'm referring to and they respond "increasing". Teacher: Great, now which graphs show both variable are increasing? Class: the first set a-d. Teacher: Then what are the other graphs doing? Class: "decreasing". Teacher: Wait a minute both variables are decreasing?? Let's pretend I put age on the x-axis and hair on the y-axis. What do you guys think happens to hair when you get older? Class: You have less. Teacher: So as age increases, what happens to hair? Class: It's decreasing. Teacher: So tell, me what graphs e-h are doing? Ben: As one increases the other decreases-I see it now. |
Since it seemed that most of the students understood the difference between the two sets of graphs, I proceeded to give the graphs formal words-negative and positive relationship. The remaining of class time, allowed student to begin graphing their data and the starter graphs that were provided in the packet (Appendix M). For homework they were to complete the last page of the packet. This was to get them familiar with drawing scatter plots and detecting which graphs show strong relationships and those that show weaker relationships.
The class seemed to go pretty smoothly. I think what they had learned in their science class prepared them well for this class. I noticed that there were some students who were not so ready to give up the line graph! Clearly this material was new to the students. Even though they have seen the direct positive relationship in the science class, they are only aware of it in that particular context. Moreover, they have never seen scatter plots before. Yet, they were able to pick up this introduction rather quickly. Again, allot of important information surfaced today. However, nothing has been written down. Thus, I decided tomorrow would be a good day to go to the notebooks to try to formalize these concepts.
LESSON 10: Analyze Data and Reflect on Correlations
DAY 19 (3/13)
Today was dedicated to writing notes from what we had learned yesterday. Again, I had developed the notes and asked the students to interact with the questions that I pose. The following notes were compiled:
There are two kinds of correlations: 1. Positive Correlation: When both variables will increase. The cloud of points on the scatter plot will slant from lower left to upper right corner. Examples of the graphs are drawn here (similar to those in App. N). Example are discussed: travel time and distance, studying and GPA. 2. Negative Correlation: When one variable decreases while the other increases. The cloud of points slant from upper left to lower right. Example of the graphs are drawn here (Appendix N). Examples are discussed: product on sale and number of buyers, age and worth of the car, sports training per week and pulse rate. |
(3/14) SNOW DAY-NO SCHOOL
LESSON 11: The Best Fit Line & The Equation of a Line
DAY 20 (3/17)
I started class by passing out the Reach Height/Jump Height Packet (Appendix P). The students took out their graphs (Appendix O). I explained that the new packet would act as their guide to what we will be doing the next couple of days. Before introducing the concept of the best fit line the following took place:
Teacher: If someone told you that scatter plots like the one you are looking at can be used to predict the jump height of your favorite basketball player, what would you say? Class: "no way", "may be-but how they are all taller than us". Teacher: Well, guess what guys I challenge you and will prove to you that indeed these graphs can help us predict, data that is not even there. Teacher: Okay, you guys are looking at your graphs. Tell me where you think someone with a reach height of 87 would jump? The students began looking at their graphs only to notice that no one had a height of 87 inches. Thus, they tried to estimate where the point would fall. Class: "100, 103, 98, 102". Teacher: How did you guys determine that? Class: "We just tried to go along with the rest of the points", "the data is going up, so we need to follow it". Teacher: Exactly, Remember when we had talked about the importance of a straight line and how you guys said its easier to read. Well, I'm wondering do you guys see a place where we can put a line through or data to try to capture as many points as possible? Christian: You mean like how you said that a correlation is strongest when the points are close together and they form a line? Teacher: yes. However our graph probably looks more like a moderate relationship-but still can you imagine a line through your points? Class in general sees that there is an upward trend and that a line can be drawn in the middle of the points. Teacher: Now, I'm going to draw a line through the middle of my data, you guys go ahead and do the same. Now use your line to predict the jump height of someone at 87 inches. Tell me what you got. Students were quick to get the hang of using their line. Class: 102,103,101,104, 98 Teacher: Compare that two the values you got without drawing that line. Ben: It's like the same. |
Teacher: Exactly see, what you guys did is predict a value by imagining a line was there-of course you may not have known you did this, but you did. Then I asked you to literally draw a line. This line is know as the best fit line. What is its purpose? Morgan: To predict data that may no be there. Brian: It suppose to go through the middle of your data. Teacher: Exactly, it is a line that tries to summarize your data points. To gain more practice with using the line. I asked students to try to predict their jump heights using the line. Teacher: How far off are you from your actual? Nate: I'm like 8 inches off. Teacher: How many people are more than 4 inches off? About 10 students are more than 4 inches off. Teacher: Why is this so? Morgan: First off, Nate is like really tall, of course he will jump allot higher than the rest of us. Ame: Yeh, besides we all have different lines anyway. Teacher: I agree, is it okay to be off from the line? Brian: Yeh, b/c its impossible for everyone to be on the line, we all jump differently. Teacher: and what is the purpose of that line again. Kristen: to give you the average. Teacher: yeh-that's one way of seeing it. The line is trying to hit as many points as it can. You usually draw your line where the average number of point are. Nice. Matt: I'm really short so. of course I won't be on the line! Teacher: Guess what Matt-the same goes for me! |
As students began to use the line, they all saw it as an average. This was an extremely important insight on their behalf and will be explained in section V of this unit. The students were also able to see that it was okay for the line not to hit all the points-which is almost impossible anyway. They provided some good thoughts as to why certain people were off the line.
Teacher: Suppose I wanted to determine how high Michael Jordan jumps. Our graphs don't go that high-what should I do? Brian: Can't you just extend the line and read off like we've been doing. Class in general agree with his suggestion. Teacher: That is definitely one way, but look I don't have enough graph for the this huge guy. Now, I deliberately used Jordan as an example to introduce to you another way of using your scatter plot to predict. Moreover, this way will be more precise than just reading it from your line. Take a look up here. What does this equation mean to you? Class: nothing. Teacher: Why nothing? Brian: Because we don't know what the heck it means. Teacher: Oh so you don't know what those terms and letters mean. That makes sense. What do we need to do in order for you to understand what this equation is? Class: Tell us what it means. Teacher: Okay, that sounds like a deal. But wait, before we move on- What's an equation? Matt: It's like a bunch of letters. Ben: It's like you put numbers in the letters and than get an answer. Teacher: What are equations used for? Tim: To get an answer or solve for something. Teacher: True. I sort of think an equation as a way of writing a picture. Like right now you have a graph in front of you with a line through it. This equation is like the blue print of your line. What 's a blue print? Matt: It's like what architects use to build a house? Teacher: Yes, what do they show you? Matt: what your house will look like. Teacher: Exactly, so let's think of this equation as a way of writing your best fit line. What we will do beginning tomorrow is break the equation piece by piece and than show you how to use it to predict anyone's jump height. |
I was not surprised by the fact that students did not know what the purpose of an equation is. However, I think that viewing the equation of a line as a blue print allowed them to meaning to something they are familiar with. The last thing that I wanted to do was build fear in the students for what was coming up ahead. I think the students felt comfortable in knowing that we would break the equation apart and kind of work through this nice and slow.
LESSON 12: Slope of a Line
DAY 21 (3/18)
The students had their graphs and their packet ready to go. I had the equation of a line written on the board. I began class with the following:
Teacher: So yesterday we left off with the ideas that this equation is going to give us a way of writing our own best fit lines. The first word in that equation is "slope". Well you tell me, what comes to your mind when you think of slope. Jessica: Like slant. Many students agree Justin: Is like a degree of slant. Brian: yeh, like the angle defines the steepness-like when your skiing. Teacher: okay, let's take that skiing example. Because a ski slope is a incline, do you guys no what kind of distance you are covering when you go up or down the hill. Ben: It's just the height of the hill. Teacher: Well, not exactly, its more like this. |
I began drawing the vertical and horizontal distance that is covered on a line. I explained to them that slope is a ratio of the distances and modeled how to find it using my own graph. An easy way for them to remember it was "rise over run". I had students figure out the slope of their own line. This took about 15 minutes. I had assigned student helpers to go around the room when ever necessary just as long as they had displayed an understanding of how to find the slope. Once everyone had a slope, the following took place:
Teacher: Now, guys, pay attention this is how I want you to interpret slope on your graphs. First off as I was going around the room, allot told me slope is like the rate that the line slants upwards. Well, that's a nice way of seeing it. In a way, it is like a rate of change of your line. Now tell me what is on our x-axis. Class: Reach height. Teacher: the y-axis? Class: Jump height. Teacher: Matt, what did you get for your slope? Matt: 2.4 inches. Teacher: okay, here it comes, the slope that you found means that suppose that Nate over hear is one inch taller than you. For every one inch that he is taller, he will jump 2.4 inches higher At this point the class was really engaged and interested in the comparison that was happening. Brian: So wait, Nate is like 5 inches taller than Matt. So that means he'll jump like 10 inches higher! Teacher: Yup, its amazing isn't it? So Lindsey-tell me what your slope means according to your data. Lindsay: I got a slope of 1.9 so it means that for every inch taller someone is they will jump almost 2 inches higher. Teacher: Good! okay, look at your graph is that a positive or a negative correlation? Class: Positive. Teacher: Good, is your slope a positive or negative number? Class: Positive. Teacher: okay, tell me suppose I drew this graph of physical fitness, and pulse rate. Is this a negative or positive correlation? Class: Negative. Teacher: what is increasing what is decreasing. Tim: As someone is more fit they have a lower pulse rate. Teacher: okay, now tell me do you think the slope will be positive or negative? Class: Some say positive while others say negative-they were just guessing. |
Well :Suppose I got a slope of 3. What does that mean in the date. Ame: For every 3 levels of fitness you are you will get one pulse rate lower. Teacher: Well, your close but you are saying the opposite. Remember what every variable is on the y axis is what will be changing per every increment on the x-axis. Brian; For every one level increase in you fitness, your pulse will be 3 higher. Teacher: Wait, but you said as your fitness gets better your pulse gets lower? Brian: oh yeh, I mean it will be 3 lower. Teacher: okay, so guys listen up when every you have a negative correlation, you are going to have a negative slope-in this case -3. This means your pulse rate will be 3 times lower. |
At this point I went around the room asking people to give me their interpretation of slope. It came quicker for some than others which is naturally expected. However, I knew that I would give them plenty of opportunity to understand this concept. I should mention here that when I modeled the method of finding slope I did mention that they are allowed to pick any 2 points on the line. However, I did not tell them why, nor did they ask. I think this is because it is still early in the investigation. I also think that students knew that they all had different lines anyway so it didn't matter which points were chosen. I knew that this would be something I would have to show the students. At this time it did not occur to me to let this concept become a discovery. Section V, will discuss my view on this after the fact. The homework assigned was a slope worksheet (Appendix Q).
DAY 22 (3/19)
Today, the students wanted to go over the homework, graph by graph. Since this was new to them. I decided to go ahead and grant their wish. Some of the graphs had a negative slope which needed to be reviewed as well. By the end of class almost all the students were able to figure out slope on their own. More over, some students wondered how is it that most of the class is getting the same answer on the homework when they chose different points. Rather, than realizing that the students were on to something I gave it away to quickly. I explained to them that slope never changes, as long as the line never changes. Therefore they can pick any point on the line. I also used the ski slope to help them understand the concept as well. They understood that on a hill, the steepness won't change unless the hill drops or straightens.
LESSON 13: TheY-intercept
DAY 23 (3/20)
I decided to start off the class by getting the slope into their notebooks. My cooperating teacher wanted to make sure we put this in the notebook as well.
Slope of a line is a measure of steepness. It is a ratio between the vertical an the horizontal move along a line. It can also be seen as a rate of change. Formula: rise/run + slope goes uphill - slope goes down hill |
Next, we began to discuss the y-intercept. I first explained to students by drawing a number of graphs that the y-intercept is where the line will cross the y-axis. I first explained to students what the y-intercept was using a variety of graphs. This was relatively easy. It simply involved reading where the line crossed the y-axis. Students also realized that by looking at the few exercises we did together that (x) will always be 0. However, they had not yet figured out the slope on their own graphs:
Teacher: okay, it seems like you got the change of this. Now take a look at your graphs. Tell me what the y-intercept is. The class takes a look at quickly respond: "0". Teacher: What how do you see that? Brian: Well look my line is like going in that direction. Teacher: Guys, see those zig zags at the corner of your graph. That means that your graph did not start at 0. Its a broken graph. I did that so that I can fit all your data. A whole portion of your graph is missing. That's why you can't tell where the y-intercept is. Here I'll show you. |
At this point I superimposed the graph to show them how much of the graph was missing. This worked out great because the students had a visual to help them understand why they couldn't simply read off their graphs. Moreover, they caught on to the fact that we needed another way to try to figure out how to calculate the y-intercept.
Teacher: Let's look at our nifty equation. We know slope and look we are trying to figure out the y-intercept. Now, if I told you that this equation will help you predict a jump height as well as reach height, what must (x) and (y) be in that equation? Matt: What? Teacher: Well, I told you that you can predict using this equation. But if you can predict someone's reach height or jump height-they have to be part of the equation somewhere. Kacey: So the reach height as to be a letter in the equation. Teacher: Right. What is on your x-axis? Class: Reach height. Teacher: So what is (x) in our equation? Class: Reach height! Teacher: There you go, you got it. Kacey: So (y) is the jump height. Teacher: Perfect. Now we know the slope, and we know our (x) and (y), can't we use all this to find the y-intercept? |
At this point I began to model the process. The students were given the rest of the time to find their own y-intercept. They also needed to brush up on solving for equations. The one thing, I knew they understood was why we needed to use the equation to find the y-intercept. For homework I handed out Appendix R which consists of graphs that required them to find the slope and y-intercept.
LESSON 14: Piecing it all Together
DAY 24 (3/21)
Many student came in today, very confused on the homework. This is understandable since they were getting quite deep into algebraic and graphical manipulation. Thus, I decided to spend class time going over the homework. This was absolutely necessary because if students did not have a concrete understanding thus far, they would never be able to follow along in the remainder of the unit. I found it interesting that since the students didn't have concrete variables to use in their homework (they were just plain graphs), they had a tough time plugging in an (x) and a (y). Sometimes they even got the numbers they used for slope confused with the point to use for the y-intercept equation. At this point, the students arrived at mechanical errors. I was confident that these problems would straighten out with more practice. Once everyone's' questions were answered I decided to piece everything together:
Teacher: Well guys, it looks like we've got everything in our equation that we need. Now tell me how this equation can be used to find your jump height. Brian: Don't we just need to get our reach height and put it in for x? Teacher: Class what do you think? Kristen: yeh, I guess he's right. So we just get our reach height and put it in for x. Teacher: okay guys let's try it. Brian give me your reach height so I can predict your jump height. |
I began to model the process using Brian's reach height. The students were really engaged in finding out their own jump heights. It was nice to see them so interested and not overwhelmed with all the stuff they have learned thus far.
Teacher: Brian, I got a jump height of 100, what is your actual. Brian: Wow, Ms. Maine your only one off. I got mine exactly! Ame: Oh my g-d, so did I. Katie: me too! Teacher: Any body else want to make a comment? Nate: I was about 5 inches off-but still that's not too bad! Again I'm the guy that is way out there anyway. Teacher: So what do you guys think of this method? Do you trust it? Or would you rather just guess how high someone can jump? Jackie: Ms. Maine I wasn't hear when we did our measurements, so I predicted Kacy's jump height and I was 2 inches off! Teacher: Impressive huh! Ben: This stuff really works-I just don't know if I can remember everything to do it again! Teacher: I agree guys, this was allot of stuff. We are going to be doing this stuff using different variables for the next couple of days. Moreover, I want to give you a unit test before Easter. Plan on next Thursday as your unit test. All next week we will review. Don't worry about this particular stuff-right here-that will be tested in a project. However, I want you to be able to figure out slopes and y-intercepts for the unit test. |
The students seemed pretty impressed with what they had just explored. However, there was still much confusion in the air. The best thing for these students right now was to look at something different and start from step 1 to the end! For homework they were assigned a reflection sheet (Appendix S).
DAYS 25 & 26 (3/24 -3/25)
Students worked on various exercises in Appendix T.
These few days provided the opportunity to work out the steps without any pauses or breaks. Thus, the students were able to see how each piece fits into the big picture. I think this marks the first time that the students were able to really understanding how to go from one step to the next.
UNIT REVIEW & FINAL ASSESSMENT
DAY 27 (3/26) Students work on review sheet for unit test
DAY 28 (3/27) Unit test (Appendix U)
3/31 - May cooperating teacher takes over his class again. He decided to discuss mean, median, mode as well as the proper format in creating a histogram.
4/2 - PROJECT ASSIGNED!!! (Appendix V)
In light of the magnitude of this unit, the 8th grades students certainly went above and beyond the usually "8th-grade level" material. Some of the math concepts are normally taught at the 9th-10th grade level. From a cognitive view point, students became experienced in thinking about data around us at an intuitive young adult level.
In the probability section students were able to view probability not only in situations that involve chance (games, lottery, etc.), but also through a much broader picture (sport predictions, medicine, stock market, consumer ads etc.,). The students displayed some interesting thoughts in understanding the transition between experimental and theoretical probabilities. For example, approximately 70% of the students were able to conclude that although theoretical probabilities stem from a sample space, they are also created by a continuous number of trials. Some students had explained that if someone had been playing darts for quite some time and are aware of how "good" they are. They will tend to use their scores has an average of their ability. This average which is only based on practice, is almost viewed as a theoretical, even though their performance will fluctuate. I thought this to be very insightful in the sense that the whole point of experimental probabilities is that even though they may not be proven or hold much mathematical weight, they indeed often times are thought of as "theoretical". In many circumstances we do not have anything to rely on other than experimental results.
The notion of chance and luck in the first part of this unit was a bit of a challenge. In the introduction the students really tended to related the two as one. Most students had some kind of understanding that probability tells you the "chance" of getting something. However, they never saw chance as a mathematical arena. In other words although you could find out your chances of winning, actually winning, is a matter of luck beyond your control. The backgammon scenario seemed to have brought the concept of probability into light. Approximately 75% of the students began to realize another way to view "the luck of the dice". They began to believe that really analyzing your chances in any game could actually act as a strategy. In other words, once the die is rolled, the values can be used in your favor. Moreover, students also realized that although one can determine the likelihood of a certain event occurring, sometimes taking a chance exactly means what it is intended to mean-go for it!
The law of large numbers brought out some interesting insight as well. Students saw that in board games a player will only get one chance or one shot at something. Thus, even though the probability of an event happening could be something like 1/18, the likelihood of that particular event happening in one shot, really is not worth the worry. Therefore as they began exploring the law of large numbers, approximately 65% of the students had a handle on the fact that it would take a certain number of trials to have an outcome occur. Some students (about 30%) were able to take the experimental probabilities that their classmates got and figure out the average experimental probability-which in fact came fairly close to the theoretical. About half of the students caught on to the fact that certain experimental probabilities would be slightly off due to the fact that every student rolls the die differently. Thus, after performing the experiments, approximately 80% of the students did find faith in the theoretical values. The students not only learned that the more trials one performs the closure they arrive at the theoretical value but more importantly they were able to think about what this means in a given context, such as the game of backgammon. Through the experimentation they were able to conclude that if they could calculate a theoretical probability, it could be used to see what would happen over the long run, without having to rely on experimentation alone.
Approximately 85% of the students were able to deduce that though some events may not have theoretical probabilities, if experiments were carried out consistently, in general they would know what would happen. For example, in the coke can situation, the students knew more often than not it would land on its side, regardless of the fact that there is no theoretical claim behind it. By the thumbtack discussion, they finally realized that to say a thumbtack has a 50/50 chance of landing on its side vs. top actually does not make sense since they do not know how often each would actually happen. However, they knew that due to the mechanics of a thumbtack, it would probably land on one side allot more often than the other. Moreover, they quickly were able to come up with many situations that only hold experimental values. This realization, I do not think happens in the adult population. In other words experimental probabilities are every where.
The survey section of this unit was a remarkable transition from experimental probabilities. Students had the practical experience of understanding exactly why their results were viewed as "experimental". Every survey presenter was more than prepared to discuss how their results would change if geography, population, sample size were varied. They also realized that many of their questions could not represent the general norm because their is so much room for variance. In other words, asking people in New York what their favorite sport is could not hold the same for people in Canada. Students were extremely proud of their work. So proud, that they did not want me to take their work with me! I think what students really felt was that this was something they did on "their own". More over, students understood how their surveys played a role in the bigger picture. For example a question that asks what your favorite kind of music is, could very well target the radio stations, music agents as well as the music industry on what kinds of records are going to sell. They were able to take their survey a step further to discuss some things they would need to do in order to make their results more significant. When I began student teaching, I told the student that you only know something if you can teach it. Well, during the survey presentations the students were able to almost assume the role of the teacher and ask some interesting questions about the data collection and the conclusions that were discussed. Almost every student saw the importance of interpreting the results in the context of who was surveyed and how the survey was performed. This came to light in the discussion of the articles in Appendix L. Interestingly enough, during the articles discussion, one class began talking about the Trident commercial: 4 out of 5 dentist recommend Trident chewing gum to their patients who chew gum. At that age level, students know enough not to believe commercials. However, they really never took an in depth look at how certain commercial can be faulty. Students were able to deduce that 4/5 means that 80% of dentist will recommend Trident. However, they knew to ask, really where 4/5 came from and exactly how many patients discuss with their dentist the kind of gum they chew. Thus, they applied their learning to everyday events. Yet, how many really take the time to understand how such statistics can be faulty.
The correlation unit was a little more mathematically demanding for the students. I knew with such young students, the only way they could get a handle on this section would be to focus on the meaning of the material. Though the students may not have mastered the material as expected at a 9th or 10th grade level, at least 80% of them have at least developed some important concepts about the best fit line, the equation of the line, and graph prediction. First an foremost at this age level, students do not have a large degree of experience with equations. Thus, when the investigation of predicting from the equation of a line began, I asked students to think about such an equation as a way of writing a picture. Students viewed it as a blue print for what their line looks like. Having begun this exploration with this view, the students knew that breaking y = m(x) + b piece by piece would allow them to understand what each variable really meant.
Defining slope came from what students saw it as: "steepness". As I began drawing the triangles to show what slope meant, about 40% interpreted it as how fast something was moving up or down a line. Now, even though this is not the mathematical description of slope per say, the students had the right idea of what slope means to a line. It is in fact a rate of change. I also required them to define slope according to the data. Though they were unable to come up with this on their own (since it is new material), they were very interested at what this value meant when looking at a graph. Most students will learn slope as another formula to use when graphing a line. However, these students learned it not only as a process but as a defined purpose. As student gained more experience with finding slope approximately 60% of them realized that one could choose any two points on the line to find the slope. In fact, I thought I would literally have to teach them that concept but they were quicker than me.
The y-intercept was a very interesting experience for the students as well. At first finding the place where the line crossed the y-axis seemed easy. However, what happens when the line does not reach the y-axis? Many students (about 75%) ended up automatically extended the line. When I asked them why they did this they explained that it seemed like the right thing to do. Upon further discussion, I realized that students saw a graph as a continuous grid. In other words they explained to me that although we only took reach heights and jump heights of our class, we can still assume that taking heights of another younger or older group can also be included in the graph. In other words, they saw that the data can be taken of anyone and plotted on the very graph they were looking at-regardless of where the best fit line was drawn. Upon discovering this they also understood that it is a good idea to extend the line whenever possible to predict further data. Thus, sometimes you do not need to use the equation of the line in order to predict.
When I introduced the best fit line, I asked students to think of it as a way of capturing your data. Throughout the next couple of lessons, I would often quiz the students and ask them to tell me the purpose of the best fit line. What is so interesting is that the students did not use my words per say. Rather, approximately 75% of the students saw the best fit line as the average of the points. Now, I allowed students to take on this view. However, I and my cooperating teacher sat down and thought about what the students were gathering about the best fit line. We had realized that the best fit line is actually capturing where the average amount of points are on the scatter plot. Moreover, at a higher math level, the best fit line actually shows the average distance points are away from a line! This was an amazing analysis on the students behalf.
From a graphical interpretation, approximately 80% of the students were able to understand their own point on the scatter plot as it relates to the best fit line. They knew that if their own point was above or below the line, it was because they were taller or shorter than the average classmate. They also saw how some short people had a high jump height, and where their points were located from the majority of the points (outliers). This was an important insight for the students because they finally understood how "the exception to the rule" fits in with correlations. In other words, the students are able to explain that although you have some short people who can jump high, there is a general trend that seems to say the taller you are the higher you can jump.
The more practice they were given piecing everything together the more they could understand how to check to see if their answers made sense. This is a very valuable and important point to mention. Thus far, and especially in their future, students will use the idea of checking their answers by performing substitution. However, in correlations, checking the answer means asking oneself does it make sense? Approximately 65% of the students were able to arrive at an answer and than reflect on their graph to see if indeed the answer they got made sense with their original data.
Due to the mathematical content, many students were having problems with the basic kinds of skills that are necessary in mastering this material (i.e. solving for an equation, understanding that a/b means that (a) will go in the box, while (b) is your divisor, and remembering which variable go where in the equation). These kind of skills seemed to come together with practice. Aside from this, these 8th graders gained some amazing insight into material that they will get into once again next year. Their caliber of understanding is exactly what will make it easier for them next year.
This unit took up the majority of my student teaching experience. As I reflect on it, indeed it is quite massive. Moreover, the correlation aspect of this unit was my second time around. At the 8th grade level, I was able to provide better means for helping these younger students understand higher level concepts. For example, treating the equation y = m(x) + b as a blue print, certainly made a difference in helping students understand the role that it would play later on. I was proud of these students. They really did some impressive things. This section will be devoted to looking at some of the sections of this unit that may have given me a difficult time. This section will also discuss some changes that may make a difference in the next time it is implemented.
The backgammon section was implemented during the first week of my student teaching. During this time, I had no idea of where, or how far I would take the game. Moreover, I had not yet thought of this unit. Only at about the middle of the section did I realize that what students were really exploring is probability and statistical tools that are used for predicting. At that point the bigger picture was coming to light. The backgammon game is an excellent way of exploring chance and theoretical versus experimental probabilities. It provides a new context for them to explore. Moreover, the idea of chance versus luck unfolds in a nice fashion. However, backgammon offers a limited context of what can be explored in probability. First off, the majority of what we had explored was actually through dice. I wanted to give them an opportunity to look at things that were unequally likely. The purpose for this is to provide more context for them to rethink their misconception of "everything has a 50/50 chance". Also I think it would be beneficial for them to practice listing sample spaces for different manipulatives as well. I have a strong belief that if students are to be taught a concept, they should be able to explore the concept across a variety of levels.
Another interesting episode that caught me a little off guard was the (1,2), (2,1) dilemma. One of the major drawbacks was that I did not have colored dice for students to play with. In other words, they were unable to "see for themselves" that both these events are not the same according to the rules of probability.
Earlier in this unit, I had mentioned that it is not necessary to necessarily have students take down the "teachers" notes when it came time for reflection. This is one of the biggest realizations I had. Unfortunately, this realization did not come to me until I had moved to the 7th grade! At any rate, for the kinds of activities that were performed in this unit, having students compile the notebook would be an ideal form of assessment. Not only would students see their own language and interpretation on the board, but I as the teacher, would be able to assess what the students comprehended.
The slope and the y-intercept went relatively well with the students. However, they needed a large amount of practice in understanding the concepts. Some were confused on the purpose for these concepts. Some student would even confuse the two. Now, that I can reflect on where the misunderstanding came from I realize that there is an exercise that can enhance the understanding of these concepts. Slope and y-intercept are utilized to draw a line. Both these concepts give a location and a rate of movement on a graph. Therefore, students should be asked to draw the graph given the slope and the y-intercept. In other words, they should be able to work backwards in their thinking. I think this would have given the students not only the purpose for such concepts, but moreover a concrete understanding of what they mean according to the graph. I think this would also help to clarify confusion between how slope is found and how the y-intercept is found when using the equation of a line. As I had mentioned in section V, the students were able to pick up on the fact that slope never changes, unless the line changes. However, not every student was able to see this quickly. Therefore, when students are asked to find the slope for various graphs (Appendix Q), it is a good idea to ask the students how it is that people can choose different points on the line, but yet come up with very close values of slope? In other words, this should be a natural discovery so that the whole class can realize this together.
One of the toughest areas the students faced was understanding how the equation y = m(x) + b can be used to find the y-intercept as well as predict the missing variable. They could not understand how one equation could be used twice. This misunderstanding does not surprise me. They have not had enough mathematical experience to appreciate the fact that equations can be manipulated to solve for a variety of values. However, due to the nature of the equation, I really could not think of a way to help them straighten out this issue. Here is where the argument stands to say that it is a higher level of thinking that young students will have a tough time with. Yet, I really can not think of a method to try to conquer this problem. However, I must mention that one of the most important things I did with the y-intercept was to super impose the graph. When the y-intercept is part of a broken graph ( i.e. the graph does not start at 0), it is crucial to have students view the missing piece of the graph. This really helped them understand the hole point of why one would need to use the equation of the line to get the y-intercept.
The final project provided a great culminating activity in the sense that students would have a final opportunity to bring closure to the correlation section, from beginning to end. Unfortunately I have not been able to review the final projects that the students turned in. However, when the project was assigned I was there to assist them in understanding what was required. The students took to the project pretty well. They seemed very engaged and anxious to get started. Some students wanted to turn in an exemplary project and thus began seriously searching to create their own correlation ( in other words not using the list of suggestions). Among the choices, the more popular were miles in car and the age, number of baskets attempted vs. made in a basketball game, and height of children and their parents. One student decided to find data on the depth of a river and how fast it flows. Another chose to measure the circumference of trees and the height of their trunks. So as one can see, there were some impressive thinkers out there.
The assessment had to echo what my cooperating teacher wanted. Therefore, although I had created the project and the assessment there are some things that I would change. First and for most, I would highly encourage students to make up their own correlation that they would like to explore. This includes encouraging them to use outside resources. I also feel that a short presentation from each student would allow the class to see what other students had accomplished. I think that students should also have an opportunity to assess their own work, thus giving them an opportunity to discuss their short comings as well as their achievements.
In conclusion, I am very impressed with the caliber of this unit. Moreover,
I'm glad to know that I am teacher who does not under estimates the ability
that students have. The packets and the thought provoking questions I posed,
allowed students to quickly understand what my expectations were in their
understanding of the content. They knew, that I was not so focused on the
answers they were getting, but rather the explanations they were providing.
Moreover, I have no doubt that due to the nature of these activities, students
will remember what they have learned when it creeps upon them next year!!!
Goal: Each player must move his/her piece according to the number shown on the throw of two dice. S/he moves his/her pieces from his/her opponent's inner table along the playing board across to his/her outer table, and finally into his/her own inner table. The dark and light pieces move in opposite directions.
General guidelines:
To begin:
Movement:
Bearing off:
Doubles:
Assignment 1.
I realize that for many of you this is the first exposure to the game. Just give the below questions your best shot. Over the next couple of days, you'll be able to explore your ideas:
1. What are your perceptions thus far of how "chance" or "luck" or "probability" may influence the game of Backgammon?
2. What does it take to become a good player?
3. Do you think there may be some sort of strategy to the game? If so, or if not, please explain your answer.
Worksheet #1
This sheet will be your reference sheet from now on.
Keep it in your Mathematics notebook at all times. It will be a great study
guide for you whenever a quiz, test or homework question is assigned.
When calculating probabilistic outcomes on equally likely
events, the following steps should be followed:
1. Ask yourself: "What are all the possible outcomes"?
What this really means is that you need to analyze what your sample space
consists of. An example of this is to look at a dime. There are only 2 possible
events when it is flipped-either a head or a tail. Thus, the total number
of outcomes with a dime is 2. First is to set up some kind of table system
to help you keep track of all the possible events.
Total Possible Outcomes = ____________________
2. Now we know how many total possible outcomes can occur. Next, lets look at finding the probability of certain events. Probability is a number between 0 and 1 that describes the likelihood that an event will occur. Because it is between 0 and 1, we know that this will be a ratio:
In essence, what you are finding is the theoretical probability
of an event or a value arrived at analyzing all possible outcomes and assigning
probabilities based on that outcome.
In our problem we need to find out the "chances"
or the probability of getting hit by the black chip. We either can move
the chip 4 times, 3 times, 7 times or none at all.
On the following page you are required to show all the
work that is being in class. Write small so that you can fit all that is
necessary.
Note: When dealing with probabilities it is sometimes helpful to deal with percents as well. Here's a quick review of the formula to convert fraction to percents:
percent % = part (is)
100 whole (of)
2a. To find the chance of
getting hit on Moving 7:
Fraction= __________ Percentage=_______________
2b. To find the chance of
getting hit on Moving 3:
Fraction= ________ Percentage=_________________
2c. To find the chance of
getting hit on Moving 4:
Fraction= __________ Percentage= _________________
2d. To find the chance of getting hit when not moving:
Fraction =__________ Percentage= __________________
Worksheet #2
Name__________________
Date___________________
Your job is to explore the following questions. Place your answers on poster paper. We will discuss what each group came up with as a class.
1. In the scenario I presented to you, we found the following theoretical probabilities as they pertain to the rules of backgammon:
P(4)= 18/36= 1/2 (i.e. reads as "the probability of
rolling a 4..")
P(7)= 6/36 =1/6
P(8)= 7/36
P(11)= 2/36= 1/18
A. Do you see any purpose in finding these probabilities? In other words, why did I ask you to find out these values?
B. How did we come up with these values?
DO NOT COMPLETE #2 UNTIL INSTRUCTED
2. In our scenario, we figured out that the chance of rolling a 4 in backgammon was 18/36 or 1/2.
A: In your own words describe what this ratio means.
B: How much do you "trust" or "believe" in these values.
DO NOT COMPLETE #3 UNTIL INSTRUCTED
3. How would you test to see if any of the above probabilities are true? Try to be as specific as you can.
DO NOT COMPLETE #4 UNTIL INSTRUCTED
4. Judging by what has happened in class, tell me how you think mathematicians came up with theoretical probabilities like: "The probability of getting Heads or Tails when tossing a coin is 1/2 or The probability of rolling a die and getting a 2 is 1/6"
5. Jakob Bernoulli-A dear friend to all mathematics teachers once said:
Further, it cannot escape anyone that for judging in this way about any event at all, it is not enough to use one or two trials, but rather a great number of trials is required. And sometimes the stupidest man--by some instinct of nature per se and by no previous instruction (this is truly amazing)--knows for sure that the more observations of this sort that are taken, the less the danger will be of straying from the mark.
Wow- that's a mouth full!! What is this mathematician getting at??
Assignment #2
NAME__________________________
DATE_________________________
Reflecting on what we did in class today, complete the following questions in the space provided.
1. From what you've gathered during class thus far: A) write a brief description of what is meant by the term "theoretical probability" and B) how does one go about finding a theoretical probability-you can use an example here if you like.
2. Do the same for experimental probabilities: A) write a brief description of what is meant by the term "experimental probability" and B) how does one go about finding an experimental probability-try to get as detailed as possible for this part.
3. We talked about probability in the context of backgammon. However, think about what probability or chance does for us in our every day application. What do people use probability for? Why was such a topic ever created? Use examples to explain your ideas.
If a tack is dropped on the floor, there are two possible outcomes: the tack lands on its side (point down), or the tack lands on its head (point up). The probability that a tack will land point up or point down can be determined by experimenting. So I tossed a tack 100 times and recorded the results in the table below:
| Outcome | Number of times event occurs |
Tack lands point up Tack lands point down |
58 42 |
A. If you dropped my tack once, what is the probability that it would land point up? What is the probability it would land point down?
B. If you dropped my tack 500 times, how many times would you expect it to land point up?
C. Is it equally likely that the tack will land point up or point down? Explain.
D. Is it possible to determine theoretical probabilities for this situation? Why or Why not?
Assignment 3
Tell whether theoretical or experimental probability is being used and Why.
1. Kelly played darts on a board made of concentric blue, red, and yellow regions. The dart landed in the red region 7 times and in the other regions a total of 13 times. Kelly stated that on her next throw, the dart has a 35% chance of landing in the red region.
2. For 10 minutes before school each day, some students from Ms. Macafee's class recorded the types of vehicles that passed by the school. They wanted to figure out whether it was more likely for a car or a truck to pass by. After a week of observing, the students used their data to predict that a car is more likely to pass by than a truck.
3. Emma is in the fun house at the amusement park. She must choose from among three exits. At one exit, visitors get squirted with water. At another exit, visitors get sprayed with whipped cream. At a third exit, visitors must walk through mud. Emma does not know which exit is which. She decides that if she selects an exit at random, she has a 1/3 chance of getting sprayed with whipped cream.
4. Dan buys a pair of weighted number cubes at a novelty store. In 30 rolls, he gets a sum of 2 eleven times. Dan figures that if he rolls the number cubes 100 times, he will get a sum of 2 about 37 times.
5. Tina keeps a pack of 20 colored pencils in her backpack. When her science teacher asks the students to design a cover for their science projects, Tina pulls out a colored pencil without looking. She figures she has about a 5% chance of picking her favorite color, orange.
Name ______________________
Date__________________
We began this unit by exploring how probability effected games of chance-such as Backgammon. You experimented with a pair of dice to find out the ratio of certain events that may occur. Experiments can help us predict what will happen in the long run. Not, only do experiments and studies help us verify outcomes that have already been established, but they can also allow us to develop new outcomes or theories. Today, there are thousands of surveys, tests, and opinion polls being conducted in every possible area of study. What's even more fascinating is that all the results and predictions come from analyzing the collected data. In the next couple of days, you will be given an opportunity to collect data from your fellow 8th graders.
A. Get into groups of 2-3.
B. Come up with a question(s) that you would like to ask each student in your class. How will you phrase your question to make sure that you get the information you are trying to explore? Will you give choices, or a yes/no question, or a number scale? Write down the question in the space below:
C. Make a prediction on how your class will answer the question(s).
D. How will you organize your data (will it be by making a chart, or by calculating a sum or average, will you need to take special things into consideration)?
E. Go ahead ask every student in this class to answer your question. Write down all the data you came up with.
Name _____________________ Date ____________________
Answer the following questions during class time. This will be due at the end of class today. During your presentations, I will be using this sheet to discuss how you conducted your survey and what the results reflect.
1. Think of when you have had to collect data or take a survey (any previous work in your science classes that involved collecting and compiling data count as an example). What things do you remember as being important when gathering data?
2. Right now, part of what you are doing involves checking to see if what you predicted comes close in actuality. Thus, you are experimenting and testing to see if your predictions were close. Explain to me how you can relate what you are doing now with what we did when we experimented with the dice in backgammon.
3. Please provide me with your survey question and the ratios or percents you calculated with each question(s).
4. As of today you were only responsible for surveying your math class. If you surveyed more people do you think they would continue to confirm the general results that you have compiled just from your own math class. EXPLAIN WHY OR WHY NOT! SOME OF YOU MAY BE UNSURE. IF THIS IS THE CASE EXPLAIN WHY YOU ARE UNSURE.
5. Do you see any biases in your question? Have you made assumptions in your line of questioning. If so state where you think the bias might be.
Name______________ Date _________________
You will be presenting on Monday. The following is what needs to be included in your presentations:
THE TOTAL NUMBER OF POINTS ON THIS IS 25. MR. SCHUECKLER AND/OR I WILL BE USING THIS SHEET TO GRADE YOU. PUT YOUR NAME ON THE TOP AND HAND IT TO US WHEN YOU PRESENT****
1. Articles included:
The introduction of air bags has saved some lives and lost some. The National Highway Traffic Safety Administration (NHTSA) has made several attempts to estimate the number of lives air bags saved since their introduction. Using computer models, the NHTSA estimated that they have saved over 17,000 lives.
A study that compared the number of drivers and front-seat passengers, who died in 2,880 cars equipped with air bags, with the number who died in crashes of 5,237 cars without air bags, concluded that there were about 10% less fatalities in cars equipped with air bags. From this they estimated that there would have been 1,136 more fatalities if air bags had not been introduced. However, the data base used for this study did not have good information on the use of seat belts and so the two groups may have differed significantly as regards to the use of seat belts. This makes it hard to determine the contribution of air bags.
It has been felt that seat belts are the chief protector and safety experts have considered air bags as supplimentary. But air bags were designed to inflate with the force to save a 169-pound man who is not wearing a seat belt. When this rule was established few people wore seat belts but now 70% do.
The NHTSA estimates that 145 people have been saved on the passenger side by air bags, all of whom are adults, while air bags have killed 36 people, all but one of whom are children. The NHTSA is proposing that the force of the bags be reduced and "smart" bags be introduced that can adjust for small children.
The changes in air bag regulations will have the effect of changing the ratio of the number of people saved to killed. The fact that these involve different groups, adults and children, makes this a difficult regulatory decision. The lack of reliable estimates for the number saved by air bags and for the use of seat belts makes this decision even more difficult.
2. Worksheet (last page of the packet)
Adapted from "How likely is it?" (Connected Mathematics Project)
Name______________________
Date_______________________
Set_________________________
Use the below data to answer the following questions. If there is not enough information to answer a question, explain what additional information you would need.
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NAME ______________________________
Now that we have graphed our data, what next? Our next goal is to examine how we can use our graphs to make predictions. Such procedures are used in every area of study, from medicine to the business world. For example, a pediatrician, needs to be able to predict the normal height and weight of an infant from its age by simply looking at a growth chart. Using our graphs can we predict how high people can jump?
This handout will be done step-by-step as a class. However, you are responsible for you own work. Each of you will come out with slightly different results-AND THAT'S OK!
PART I:
1. DRAWING THE BEST FIT LINE
Find a line that captures the general appearance of the data. The line should run approximately through the middle of your scatter.
2. PREDICTING FROM YOUR BEST FIT LINE
Your line is a model for the relationship between reach height and jump height. We can use this model to estimate how high you can jump. Find your reach height on the horizontal axis and move upward until you come to the line. Than move to the left until you come to the vertical scale. Read your predicted jump height.
WRITE DOWN THE JUMP HEIGHT THAT WAS PREDICTED __________________
How does this predicted jump height compare with your actual jump from the data you collected? What does this imply about your best fit line?
PART II: PREDICTING FROM AN EQUATION OF A LINE
So far, you've used your graph to predict your jump height and compared it to what actually happened. However, how would you go about trying to predict someone's jump height if it goes beyond your graph? The next stage will be taking graph prediction one step further by using equations. The equation of a line is y = mx + b. This means nothing to you right now, but let's break each variable down and explore how it is this very equation that will allow you to predict jump heights for anyone!
3. FINDING THE SLOPE OF YOUR LINE
Slope can be interpreted as a rate of change. According to our graph, the slope will measure the rate of change in jump height as compared to reach height. In other words, how much more will someone's jump height increase as their reach height increases? For example, let's say the slope came out to be 2.5 inches (a complete guess). This would mean that for every one inch increase in reach height, the jump height would increase 2.5 inches.
Here are a couple of ways to interpret slope as it applies to your graph:
slope = change in vertical distance/change in horizontal distance
OR
slope = rise/run
OR
slope = (y2-y1)/( x2-x1) Using points (x1, y1), (x2, y2)
CHOOSE 2 POINTS FROM YOUR LINE: They do not need to be points from your data. Make sure they are fairly far apart.
From this point you can work off your graph to calculate the distance vertically and horizontally or you may proceed algebraically:
Use the points to find the slope from this equation:
slope = (y2-y1) / (x2-x1)
Slope can be defined by the letter (m).
THE SLOPE OF YOUR LINE IS ___________________
ACCORDING TO YOUR DATA THE SLOPE MEANS___________________________________________________________________________________________________________________________
4. USING THE SLOPE TO FIND THE Y-INTERCEPT
The y-intercept is the value of (y) when x is 0. It is where your line will cross the y axis. In terms of our example, our (x) variable represents reach height. Therefore, our y-intercept would be the height of our jump at 0 reach height! Sounds obvious-huh!!! In our example this may not be 0. It could be a negative number-which still would make sense mathematically. Sometimes you will be unable to see where the line crosses the y-axis because your graph is too small. Let's see how we can find the y-intercept algebraically.
Your slope is _________________
Choose one pair of your points from #3 __________________
Use the following equation of a line:
From the above you should have a value for x, y and slope. Plug in the points and solve for the y intercept.
SHOW YOUR WORK HERE
YOUR Y INTERCEPT IS ____________________________________
5. PUT IT ALL TOGETHER
You now have all the variables you need for the equation of a line. They are:
YOUR EQUATION IS:_____________________________________________
6. PREDICTING FROM THE EQUATION
You can predict from your jump height using the equation. Your jump height will be the variable (y). Just place your height in for x, along with your slope and y-intercept and solve for (y).
SHOW YOUR WORK HERE
YOUR PREDICTED JUMP HEIGHT IS ________________________
How close is your predicted jump height to your real jump height?
_________________________________________________________________________________________________________________________________________
7. COMPARE WITH CLASS RESULTS
Let's take some time out to compare your results with those of other students. Whose line did the best job for coming closest to their actual jump height?
8. PREDICT A CLASSMATE'S JUMP HEIGHT
As you did for your own jump height in step 6, let's predict someone else jump height.
SHOW YOUR WORK
9. EXERCISE
The tallest living woman is Sandy Allen at 7ft, 7 inches. Now use the above steps to predict her jump height.
HINT: YOU NEED TO COME UP WITH A WAY TO FIND HER REACH HEIGHT.
1. What is a correlation?
2. What do scatter plots tell you?
3. What is the difference between negative correlation and positive correlation?
4. Describe the purpose of a best fit line and how it is created.
5. What is slope? DO NOT JUST GIVE THE FORMULA. PROVIDE WHAT IT MEANS IN YOUR OWN WORDS.
6. What does the y-intercept mean on a graph?
NAME____________________________
The table above gives data relating the number of oil changes per year to the cost of car repairs. Plot the data points on the grid provided.
1. Draw the best fit line. IS THIS A NEGATIVE OR POSITIVE CORRELATION?
2. Find the slope of the line. Note-pay attention to how the grid is numbered when you begin counting squares.
3. Describe what the slope represents according to the data.
4. Find the y-intercept.
5. Write the equation of the line.
6. Use the equation of the line to predict how much the engine repair would cost per year if the oil were changed 8 times. SHOW THE WORK.
NAME____________________________
The dosage chart above was prepared by a drug company for doctors who prescribe Tobramycin, a drug that combats serious bacterial infections such as those in the central nervous system, for life threatening situations.
1. Draw the best fit line. IS THIS A NEGATIVE OR POSITIVE CORRELATION?
2. Find the slope of the line. Note-pay attention to how the grid is numbered when you begin counting squares.
3. Describe what the slope represents according to the data.
4. Find the y-intercept.
5. Write the equation of the line.
6. Use the equation of the line to predict how much dosage a person who weighed 214 lbs. would need. SHOW THE WORK.
NAME____________________________
In BMX dirt bike racing, jumping high or getting air
depends on many factors: the rider's skill, the angle of the jump, and the
weight of the bike. Here are data about the maximum height for various bike
weights.
1. Draw the best fit line. IS THIS A NEGATIVE OR POSITIVE CORRELATION?
2. Find the slope of the line. Note-pay attention to how the grid is numbered when you begin counting squares.
3. Describe what the slope represents according to the data.
4. Find the y-intercept.
5. Write the equation of the line.
6. Use the equation of the line to predict how much height someone would get if their bike weighed 30 pounds. SHOW THE WORK.
NAME____________________________
The graph compares the rate that crickets chirp to temperature
in degrees Celsius.
1. Draw the best fit line. IS THIS A NEGATIVE OR POSITIVE CORRELATION?_____________
2. As the temperature increases what happens to the rate that the crickets chirp?
3. Find the slope of the line. Note-pay attention to how the grid is numbered when you begin counting squares.
4. Describe what the slope represents according to the data.
5. Find the y-intercept using the equation y= slope (x) + y intercept.
6. Write the equation of your best fit line.
7. Use the equation of the line to predict how often a cricket will chirp per a minute when the temperature is 45 degrees Celsius. SHOW THE WORK.
NAME ________________________
SET __________________________
I. On the following 4 graphs draw the best fit line (1 point each):
II. In this section, indicate whether there is a positive, a negative or no correlation. For those graphs that do show a correlation, indicate the strength: either weak, moderate, or strong. (2 points each).
3. What kind of correlation does this graph show-positive or negative?
What is the strength: moderate, strong or weak? (2points)
1. Slope: _____________
y-intercept: _________
2. Slope: ___________
y-intercept: ___________
3. Slope: ____________
y-intercept: __________
4. Slope: ______________
y-intercept:__________
1. Larry decides to listen to 98PXY for 6 hours. He keeps record of how
many times a certain song is played within that time duration. Larry found
out that 98PXY plays the same song 2-3 times per hour. Is this an experimental
or theoretical probability? (1 point). _____________________________
2. Suppose you were to spin the SPINNER below and then roll a 6-SIDED
DIE.
A. What are all the possible outcomes? (2 points)
B. What is the probability that you will get a 1 on BOTH THE DIE AND
THE SPINNER? (2 points)
C. What is the probability that you will NOT get a 1 on NEITHER THE DIE
NOR THE SPINNER? In other words, what is the probability of never getting
a 1? (2 points)
D. What is the probability of getting the SAME NUMBER on the spinner
and the die? (2 points).
E. In the above questions, did you find a theoretical or an experimental probability? (1 point)
YOUR ANSWER IS:
ASSIGNMENT:
Through observation, research or experimentation, compile a list of data that show some degree of correlation. You are expected to obtain between 20-25 values. The data will need to be represented in a frequency distribution table, a histogram and a correlation grid. In addition to these graphical requirements, you will be expected to submit a written report that includes a discussion of your findings. This should include an analysis of the mean, median and mode of your data as well as the slope and equation of your best fit line. There should also be discussion of other variables and/or biases that can influence the outcome of your results.
PROJECT TOPICS
OPTION 1: You may choose to examine any two sets of data with the following guidelines:
OPTION 2: You may choose one of the following topics to explore:
FORMAT:
ROUGH DRAFT DUE APRIL 7, 1997
EXPECTATIONS:
On April 7th you will be expected to turn in the following:
NOTE: THE ROUGH DRAFT IS REQUIRED AND IS REFLECTIVE OF YOUR FINAL PROJECT EVALUATION!!!!!!!!!!!!!! NO EXCEPTIONS!!!!!!