Place the vocabulary words from this lesson on the board and ask students to define them in their vocabulary journal as the lesson progresses. Keep a supply of vocabulary journal pages (M-7-2-1_Vocabulary Journal Page.doc) on hand for students to use when they need them. Write: “Words you should know by the end of the lesson” above them. Have a large spinner, as shown below, drawn on the board or projected on the overhead screen when students come in to class.
“You are going to play with a partner to see who gets to (do any kind of incentive or reward that would be exciting for them). The person sitting on the right gets a point when the spinner lands on the shaded section, while the person on the left gets a point when the spinner lands on the unshaded section.” (Instead of left and right you could use boys and girls, earliest birthday, etc.) “When you get to 10 points, you win.”
Students should immediately react with, “That’s not fair!”
“We can all see that that would not be a fair game. The shaded part of the spinner is much smaller than the unshaded part.” This is a good time to talk about chance (experimental) versus expected (theoretical) outcomes with students. Expected (theoretical) value is a good extension to challenge those students at or going beyond the standard. Bring in expected value when possible. An extension activity for finding expected value is available at the end of the lesson.
“Some probability situations are not as obvious. Today we will review tree diagrams, organized lists, and charts. We will learn how to use area models to determine outcomes and theoretical probabilities. These methods are used so that theoretical probability can be studied and predictions can be made for all kinds of real-world applications like games, sports, weather, attendance patterns, and many more.”
Begin the lesson with a story: “Calvin and his brother both like to ride in the front seat of the car. In order to avoid arguing about it every day, Calvin made up a game to decide who sits in the front seat. The boys each flip a coin. If the coins match, then Calvin wins and gets to sit in the front. If the coins don’t match, Calvin’s brother wins. Do you think Calvin is being fair or trying to trick his brother?” Students will probably have two main ideas about this:
- The game is unfair because there are 3 outcomes: H-H, T-T when Calvin wins, or one H and one T where his brother wins. Calvin has more chances.
- The game is fair because there are 4 outcomes: H-H, T-T, T-H, H-T. So each boy has the same chance to win.
This is a good opportunity to test the idea of whether or not T-H is the same as H-T. Let students play with partners for 20 games. They can keep track of results on notebook paper. Let them record the “one of each” situation however they like at this point. It will become apparent later that they are indeed unique outcomes. As you walk around the room, make note of students who are trying to determine if they flipped H-T or T-H. Since the coins are identical, you may see some groups trying to keep the coins “separate” when they toss, or tossing them one at a time and recording results as first and second because they realize the order probably makes a difference. Some students may mark the coins in some fashion for the same purpose.
“How many people think the game is fair?” Ask individual students why they think so. “How many thought it was unfair before they started but now think otherwise?” Again, ask for their reasoning. “How many think the game is unfair?” Someone will invariably think so if one person scored a significantly larger number of points.
Have students share their outcomes by recording the number of games won by Calvin and his brother on the board or overhead. Be sure to point out that when the data from the whole class is added together, it appears much closer to the theoretical probability. If possible, add the data from previous classes to observe the change in experimental probability. This is a good time to talk about chance (experimental) versus expected (theoretical) outcome. Even though you would expect to get heads half the time with a single coin, it is possible to flip a coin many times in a row and get heads every time. This is a good extension area for students who are at or going beyond the standard, and is a great review of the previous exploration on the law of large numbers.
“This situation, like many, can be analyzed by making an organized list. The problem is that we have not agreed on whether ‘getting one of each’ is one outcome or two different outcomes. Let’s think of them as coin 1 and coin 2.” Write the outcomes H-H, T-H, H-T, T-T on the board. “This is called the sample space or a list of all the possible outcomes. It is a simple probability situation, so the list is easy to make. Who would like to come up and make a tree diagram for this situation?”
Have a student create the tree diagram.
Discuss the situations when tree diagrams are easy and not easy to make. Ask for examples. If it doesn’t come up, suggest finding the probability of getting a specific sum when rolling two number cubes and start making the tree diagram. It quickly gets messy, so don’t carry it all the way through. “Some of you have seen this situation organized this way before.” Begin making the classic 6 x 6 chart for the outcomes. Write the numbers 1 to 6 down the left side and again across the top. Ask for a volunteer to fill in the table (with the sums).
“This is called an organizational chart or a sample-space organizer. Usually, the specific outcomes are what are inside the grid. When we are trying to calculate probabilities, we circle all the favorable outcomes that make up the event. Remember the event is the outcome or set of outcomes that is favorable. If I asked for the probability of getting a 5, you would circle all the 5s.”
Write “P (getting a 4) =” on the board and ask what the probability is. When a student says “3 out of 36” ask, “How did you get that?” Students should recognize that there are 36 sections inside the grid and 3 of them contain a sum of 4. This is also a good time to review that the probability could be written as a fraction in simplest form, a decimal, or a percent. “You are really using this sample-space organizer as an area model, which is what we are going to be learning about today.”
“Let’s go back to Calvin and his brother’s situation. I am going to make a 10 x 10 grid. How many sections are in the grid? Who would like to come up and divide the grid so that one side is the probability of getting heads and one side tails?” The fact that they only need the grid to be two by two will probably be obvious and easy for students to see. “Now I will write Heads and Tails next to the table (on the left or the top) by each of the sections so we can keep them straight. I will also write Coin #1 on this side so we know the horizontal (or vertical) line that divides the section shows the two possible results for the first coin.” Now write “Coin #2” on the other side of the square (either top or left, whichever was not labeled the first time). Label the two remaining columns (or rows) as Heads and Tails and you should end up with something like the following, except it will be blank inside.
“Who can come up and label the outcomes inside the grid where they belong?” If no student volunteers at first, fill in one of the sections with H-H. “Who wins in this situation?” Add the words “Calvin wins.” Students will quickly understand the rest. When the grid is filled ask, “How many possible outcomes are there in total? Is the game fair or not fair? What are the probabilities in each section?” Write them on the grid.
“This type of area model is used quite often to model probability situations where there is more than one dependent or independent event. The two-coin toss activity is independent because neither coin is affected by the outcome of the other. However, if I set the game up so that after getting heads on the first coin flip I have to quit, yet I get to flip a second time if I got tails on the first flip, these would be dependent events. We are only going to examine independent events today, but area models can be used for both.
“You may have noticed that I used a 10 x 10 grid to make the area model. It is easy to ‘count’ the probability of each section and express it as a percentage because the grid is divided into 100 equal sections. Since the outcomes of flipping a coin are or and , it is easy to find the dividing lines. You do not need to use a grid to make an area model, however. The round spinner from the beginning of the lesson was an area model also. Since it was clearly divided into and , the probability was easy to determine.”
Demonstrate making another area model using the spinner and the flip of a coin. Do not use a grid. Simply start with a rough square (see below). Explain that a square or rectangle is useful for a compound event since it has sides. A circle or other shape can generally only be used for simple or one-stage events.
“Let’s say we are designing a new game and we want to know if it is fair. In our game, we have a spinner with three equal sections (A, B, C) and a bag with one clear marble and two blue marbles. The game involves spinning the spinner and then drawing one marble out of the bag.” Hand out copies or display a transparency of the Matching Game (M-7-2-2_Matching Game.doc).
“How many people think that getting a ‘match’ (for example the letter B and a blue marble [Bb] or getting the letter C and a clear marble [Cc]) is likely?” Many students may think that it is likely because more than half the letters on the spinner are Cs or Bs. “How many think it is equally likely to get a match as to not get a match?
“You are going to work in groups to create area models of the new game. After you have finished, decide if the game is fair and be ready to support your decision. If you think the game is unfair, create a new set of outcomes that you think would be fair (using the same spinner and marbles). If you think it is already fair, see if you can devise another game that is almost fair but seems close enough to trick someone into taking the less likely outcome.” If groups are using the handout, the instructions are repeated on the sheet. If groups are using regular grid paper, write the instructions on the board or overhead for students who have difficulty processing multiple verbal instructions. As groups begin to work, walk around the room to make sure they are setting up the area models correctly and that everyone is participating.
Allow groups to present their reports. The script below should summarize the use of the area model and the difference from an organizational grid (as shown in the drawings.)
Some groups may have set up the area model more like a sample-space organizer than a true area model. Display only the finished chart shown above on the left. “Many groups used a chart like this one to answer the question. This model divides the spinner/marble problem into nine equal parts because there are nine distinct outcomes that are equally likely. That makes this more of a sample-space organizer than an area model.” Encourage any debate about the fact that the separate, similar outcomes from the chart would still equal the same probabilities from the area model.
Begin a new square on the board. “The choosing-a-marble situation has only two outcomes: clear or blue. But they are not equally likely. What are the probabilities in fractions for the two outcomes?” (P(c) = and P(b) = )
“How can you divide the grid into and ?” Continue completing the chart with student participation. The final product from the class discussion should appear as the grid on the right above.
Discuss with students that the sample is a list of all of the possible outcomes for the problem they are working on. For this problem (spinner with A,B, C and marbles clear, blue, and blue), students could list the outcomes by using the capital letters for the spinner sections, and the first letter of the marble color to show all the possible ways one of each could be combined. For example: Ac means you got A on the spinner and drew a clear marble.
Have students assist in creating the sample space below. Discuss the importance of having an organized way of creating it so outcomes do not get missed.
Ac, Ab, Ab, Bc, Bb, Bb, Cc, Cb, Cb
Also discuss how the sample space can help predict an outcome. Describe how in real-life experience, since chance still plays a role, predictions will not always be correct.
Before building the last area model for the whole class, check group work to see that the sum of the probabilities of individual outcomes is 100%. Correct any misunderstandings during the reports. On the day following the end of the lesson, give students a partner warm-up activity to assess their understanding of the differences among organized lists, charts, tree diagrams, and area models.
One question might be “Give an example of a compound probability situation and make an organized list, tree diagram, organizational chart, and area model for the situation.”
Another option is to ask, “When would it be convenient to use a tree diagram? When would it be more convenient to use an area model? Give examples of both.”
A third activity could be a vocabulary presentation in which each group is given one word and a graphic organizer (M-7-2-2_Four-Square Vocabulary Organizer.doc) to fill out and present to the class. Students could use a paper copy or transparency of the graphic or draw an enlarged version on chart/poster paper to use in their presentation. During presentations students in the class could complete their own definitions in their vocabulary journals or on handouts of a journal page (M-7-2-1_Vocabulary Journal Page.doc or M-7-2-2_Four-Square Vocabulary Organizer.doc) so they could be collected.
- Routine: Emphasize the proper use of probability vocabulary in both student work and classroom discussions involving probability situations. Throughout the lesson the following words should have been added to the vocabulary journals: area model, independent event, organized list, chart of outcomes, sample space, and simple and compound events. Review these terms and be sure students have accurately recorded each in their vocabulary journals. Another option is for students to share and compare their vocabulary entries for 15 minutes in small groups at the end of each lesson. Have extra copies of the journal pages available for students who need them.
- If students are having difficulty with the idea of area models or constructing them, send home the supplemental area models worksheet (M-7-2-2_Lucky Winner and KEY.docx).
As a warm-up activity after the last day of the lesson, have these students work with partners who understand probability to go over the supplemental worksheet. They could also create an organized list or tree diagram to go along with the area models. If students are having difficulty with the idea of multistage probability situations, provide additional easy compound situations for them (spinners, choosing blocks, flipping coins, etc.) to practice in groups.
- Expansion: Ideas for extension work for students meeting or going beyond the standard include:
- Find the expected value for each player on the Who Will Be the Lucky Winner? worksheet (M-7-2-2_Lucky Winner and KEY.docx).
- Create an organized list, chart, tree diagram, and area model for a compound probability with three events (for example: spin a spinner, roll a number cube, and flip a coin). Discuss the advantages and disadvantages of each method for finding outcomes and probabilities.
- Create a conditional probability situation and ask students how to adapt tree diagrams or area models to show conditional probabilities.
- Given an area model, have students create situations that could be represented by the model.