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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:

What do you think of when I mention words like luck, chance or probability? Do these words all mean the same thing? What do they mean to you? What are your experiences with carnival games? Sports? The lottery? How many people do you know always win at carnival games? Why do you think they win? Do you consider board games to be the same as carnival games? Are your chances of winning any different? When the weather forecasts states there will be a 30% chance of rain, what does that mean? What does it mean that the chance of getting a head when flipping a coin is 1/2? Suppose I asked one of you to try to throw a penny into this cup. What are the chances that someone will make it in? How did you arrive at that answer? (Have students begin to try. If a particular student makes the shot, ask the class if he/she will make the next shot).

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.

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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.

As students discuss with one another which move they think they would make, the teacher should visit with groups to make sure they are providing the motive or reason behind their thinking. Their maybe a few students who will try to figure out the probabilities of getting "hit" on each of the moves. Ask them to explain how they came up with such values. Ask students to not only back-up their chosen move, but also to explain why they are avoiding other moves.

HOMEWORK: Backgammon Reflection (App. C)

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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:

Some of you have decided that the safest move was to try to get as close to the inner table (as well as the opponent white chips) as possible. Others chose not to move because there may be less of a chance of getting it. Some preferred moving the chip only for one value of the dice. Is there a way of finding out which move is the safest? In most cases some students may begin to give fractions or percents. Ask them where they arrived at such values. This will help lead into the discussion of listing the sample space. Some students may say that simply playing the move and finding out where the opponent goes will verify how "safe" the move was. In game playing, this is a good point. However ask the students whether they think everyone will have the same outcome. Moreover, will that outcome occur all the time or half the time or less than half? How would you decide?

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:

It is not necessary for the teacher to quickly give the students a way of organizing the sample space of the two dice. Give them an opportunity to list all the possible outcomes themselves. Have them compare with other classmates. Only after students have attempted to write out the sample space, may you offer other methods of making sure all the outcomes are accounted for (counting tree or punette square). At the 8th grade level, students will not necessarily recognize that (1,3) and (3,1) are to be counted as two separate events. Be prepared to discuss why this is so. You can refer to colored dice or the addition of 1 and 3 and 3 and 1-two ways of getting a sum of 4. Here is a perfect opportunity to bring some kind of formal understanding of what probability really is. Students at this level will know that probability appears in the form of fractions and percents. Have them recognize that probability values are ratios which also mean that they are numbers between 0 and 1. Now that you have a complete list of all your possible outcomes, how can that help our quest to determine the safest move? If students are quick to give the fractional answer for the probability of getting an 11, ask them if they can provide a formula or method of how they arrived at the answer. 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:

Which move has the greatest chance of getting hit? the least chance? Does this sway your opinion in any way? What would you do? Would you take the chance or would try to remain "safe". How much do you trust these theoretical probabilities? Do you think everything has a theoretical probability? Can everything be assigned a ratio? Those that are considered "good" players of Backgammon, do you think its' luck? (Hopefully, at this point many students will begin to revisit their thinking on games and what is meant by the "luck of the die". Have students begin practicing interpreting what the ratios mean according to the concept. For example if the ratio of getting a 4 is 18/36, what does that mean? Encourage them to use their own words: "out of 36 rolls, I'll get a 4, 18 times". Why was theoretical probabilities ever invented? (The idea here is to begin thinking about predictions.

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.

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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?

How would you run a test to find out whether these theoretical probabilities are accurate? If you roll your dice 36 times, do you think you will get the exact theoretical value for any one of these outcomes? What if the outcome of your test does not match your theoretical value? What if we rolled less than 36 times? more than? What do you think would happen?

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.

Are you surprised by your results? Compare those ratios that came out exact as to those that were off. Can you offer any explanations to why this is so? If you did not get the exact theoretical value, does this make the theoretical probability invalid? What would happen if I added the class data together and determined the percent of getting a 4? If students respond that it may get closer to the theoretical value, ask them why they think this is so? Have students read the quote by Jakob Bernoulli and ask them what it may mean? It may also be a good time to go beyond the Backgammon game and ask students to think about how theoretical probabilities were developed and what is the purpose(s) for experimental probabilities. Revisit the question asked earlier: Does everything have a theoretical probability? If needed provided topics to talk about such as weather forecasts, sports predictions, medical research, the lottery, throwing a soda bottle, weighted dice, etc.

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)

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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:

What are theoretical probabilities based on? Experimental Probabilities? Explain some events that may not necessarily have a theoretical probability but rather only an experimental. What are the characteristics of the event that allows it to only have an experimental probability? Organize a list of possible outcomes for scenarios other than dice. Understand how probability can be seen as tool to help predict outcome of events. Understand the difference between chance and luck.

HOMEWORK: Thumbtack and Experimental vs. Theoretical Worksheet (App. H)

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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:

Do you have any ideas on how national polls and surveys are conducted? Suppose I wanted to survey the 8th grade on what their favorite CD is. How should I go about doing this? Who should I ask? How many people? Does it matter? What ways are you familiar with when representing data? (i.e. circle graphs, bar graphs, etc.). Should I give choices for people to respond to? What if I do not -would it matter?

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:

Suppose I surveyed people on their preference between bungee jumping or water skiing? Do you see any problems with this question? Suppose I went to an elderly home and asked this question. What do you think the results will turn out to be? Will my results be representative of the general population? Suppose I surveyed your age group? What would happen to the results? Is the question fair? Can you think of ways to rephrase it? Would you consider this survey a experimental or theoretical probability?

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.

Though you at first will only be surveying your class, what do you think will happen to your results when you go outside of the classroom? Try to predict what you think the outcomes will be.

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.

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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.

Students are very familiar with the Trident commercial "4 out of 5 dentists recommend Trident chewing gum for their patients who chew gum". Ask students if they think that 80% of dentists recommend Trident. In the packet there is an article that discusses the safety of air bags and seat belts. Students will explore whether it is the air bag that is saving lives or the seat belt. This issue brings out the key question of where is the data coming from and how many cars have airbags versus how many have seat belts. The article on Ann Landers will fuel the discussion of the problems with voluntary response.

HOMEWORK: The last page of the packet in App. L.

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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:

Is the detective's assumption at all correct? Do you think in real life a detective would consider this a clue?

Do all tall people have big feet? How strong or weak do you think this relationship is? Are big feet caused by some degree of tallness?

When is something said to have a strong relation? Are relationships necessarily causal? Discuss examples of relationships that you think are causal and discuss some that may not necessarily be causal?

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.

When students begin discussing the above charts, always ask how they are arriving at their decisions. Are they only using themselves as a representation of what may happen "in general"? What determines the degree of a relationship? 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

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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.

Looking at these 8 graphs (appendix N), which graphs may look strong? weak? moderate? On what are you basing your decisions? Do you see general trends in the graph (those that are going up versus those that are going down)? Are the points in any of these graphs close to forming something What is so special about a line? Can we connect the dots on these 8 graphs? What would connecting the dots tell us? If I asked you to place more points on graph #2, where would you place them and why? (The idea here is to show students that they are actually following the pattern that the dots are displaying-in a sense they are predicting where other values may be placed according to the data. During this discussion it is a good idea to begin tossing around words such as negative and positive correlations. This should stem from students observance that the first set of graphs are moving in different direction than the second set of graphs. What are these two different kinds of graphs saying about the variables?

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.

Did certain graphs you thought had a correlation, display as such? How strong is the relationship of the graphs that seem to exhibit either a negative or positive correlation? If forearm and foot size are suppose to be equal in size, what should the "ideal" graph look like? Does this compare with our graph?

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:

How did you decide to measure palm height? How does this ambiguity effect the data as well as the interpretation of the data? Can you relate this to other statistical articles that you have read, where perhaps the variable was not clearly defined?

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:

What are correlations and define its purpose What do scatter plots show? What is the difference between a negative and a positive correlation? Understand that correlation are not intended to hold for all cases. Rather they are descriptions of relationship from a general point of view. Students should be able to list examples of both positive and negative correlations.

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.

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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.

Ask students to look at their scatter plots and see if they can predict the jump height of someone who has a reach height of 85 inches. Ask them how they did this and more importantly how accurate they think they are. The point of this is to make students realize that they are imagining a straight line going through their graph. Ask students to draw a line through their graph to capture most of their data points. Have them use this line to predict the jump height of someone who has a reach height of 85 inches. How close were you to your first prediction? How can you predict jump heights of those that have each height that scale off the graph?

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?

Provide some explanation to why your jump height is not exactly on the best fit line? Is this O.K? The idea here is have them understand that the purpose of the best fit line is to capture most of the data, realizing that a single line will not be able to get all the points

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:

What is the purpose of an equation? Suggest to them to think of it as a way of writing a picture. Does the equation y = m(x) + b mean anything to you write now? In most cases it will be completely foreign. Why do you not know what this equation is? What is missing? For the beginning stages have students view this equation as a blueprint for their best fit line.

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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:

If you were to ask someone to draw an exact duplicate of your best fit line, what would they need to know about your line? What does slant, or slope mean to you?

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.

The instruction of slope in this lesson plan should go above and beyond viewing it as another equation. Rather students should be expected to interpret what slope means in terms of the variables in question. For example if the slope of the reach height/ jump height graph came out to be 3.5 inches, students should be able to explain that for every one inch increase in reach height, the jump height will be 3.5 inches higher. Students should begin to observe that slope never changes on a line as they are allowed to choose any two places on the line. To help them grasp this, mentioning a ski slope that falls straight down will help them understand that the steepness will never change unless the level of ground changes. Ask students if it is possible to get a negative slope? What about negative correlations do you think they will have a positive or a negative slope? Why?

HOMEWORK: Slope worksheet that will help students become familiar with finding the ratio (App. Q).

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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.

What happens if we cannot see the y-intercept on our graph? Students will begin to notice that in fact a large amount of their graph is missing. Ask them to look at the blueprint of the equation of a line, is there a way of using that to find the y-intercept? It is also a good idea to superimpose the reach height jump height graph so that students can really view how much of the graph is missing and where it may look like the line will cross, although we cannot know for sure until we use the equation.

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)

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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.

In our equation y = m(x) + b, we found the slope (m) and the y-intercept (b). If we are to use this equation to predict jump heights, what must (y) and (x) resemble? Find your reach height in your tables and use the equation to predict you jump height. How far off were you? What may be some reasons of why you were off? Compare your results to your classmates. Whose line came the closest to predicting their own jump heights? Provide reasons why this is so? Predict a classmates jump height and compare results. Does what you predicted make sense with the rest of the data? In other words was your prediction reasonable? Ask students how much they trust these prediction if they were to apply it to their favorite basketball player. Students should appreciate this process in the sense that although one may not get the exact value, this form of prediction holds much more weight than a mere guess.

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.

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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:

• Data collection and organization
• Display of data using a variety of methods (scatter plot, histograms, frequency table)
• Discussion of how the data was collected as well as any known biases or other factors that may have influenced the results
• Discussion of strength and nature of the relationship
• Interpretation of slope of a line as it applies to the data
• A case where both a prediction from the line and a prediction from the equation is presented

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:

Think of instances where the mean and the median are the same? Which do you think provides a better picture when analyzing topic such as the level of income or perhaps the real estate value in particular district or county-the mean or the mode?

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.

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