Try, Try Again
Seventh Grade Poster Problem
Statistics and Probability
In this poster problem, students get experience with both theoretical and empirical probability for compound events, that is, situations where something happens more than once and they try to find the probability of a particular combination of results.
Common Core State Standards for Mathematics:
A probability is a number between zero and one (inclusive), so you can represent it as a fraction, a decimal, or a percent. Which should students use? At this stage, it really doesn’t matter.
But consider the following...
For theoretical probabilities using dice, fractions are exact while the others are approximations. If a student always says that the probability of rolling a three is “point one six seven,” they might be missing the elegance of “one sixth.”
If students use fractions, they may have developed the reflex that they have to express every answer in lowest form. Not true! When they do two-dice sums, for example, the probability of rolling a 5 is 4/36. This is better than 1/9 because it contains useful information: there are four ways to get a five among the 36 possibilities.
For empirical probabilities—which are approximations by nature—decimals and percents make it easier to compare.
The Lesson Plan:
Engage students in a discussion about probability. You do not need to explore all of the following questions. Mainly, we want to establish that a probability is a number between 0 and 1, and get some intuitive idea about what the values mean.
You could then ask about something more sophisticated, but qualitative:
Having taken time with generalities in a context, now choose questions like the following to get specific and quantitative.
2. POSE A PROBLEM
Student responses in the “Launch” will give you an idea what the students can do and what they’re clueless about. Although students will work in groups—and can therefore work together to help figure things out—you might need to give them some help with the following, either beforehand, or (ideally) at the moment they need it.
The task is to explain why Aloysius is wrong—using two different approaches. The first is theoretical: students should make a diagram or systematic list. The second is empirical: they should simulate Steve going to the free-throw line many times.
Here are some questions you can ask to help students as they work. As a general question,
Then, for the “theoretical” part, you might ask the following (with the most general questions first, more “leading” later).
For the “empirical” part, you might ask the following.
Upon completion of handout,
Show Slide #1.
Have students compare theoretical approaches to what the they did.
Use Slide #2 to compare an empirical approach to what they did.
Some groups will probably get empirical results close to Aloysius’s wrong answer from the handout, ⅓. Students need to learn that simulation naturally gives variable results. If you combine the class data, however—putting all their simulated shots together—you get a larger sample, and it will (probably) agree better with theory.
Students need to learn that variability due to randomness is often much more “random” than we expect. See the Teacher Tuneup on variability and what randomness looks like.
Show Slide #3 that contrasts the “Two Approaches.” Explain that this slide is for the Steve problem that the students just did. You can always try to use both approaches, but sometimes only the second one is possible.
Now you’ll try to apply what you did with Steve to a new problem in a new context.
Distribute Handout #2, “Penelope’s Group.”
Here is the relevant text:
Penelope works in a group with Quentin, Rochelle, and Sid.
The teacher always chooses a group member at random to present the group’s solution to the class. On this day, Penelope’s group has to present twice.
What’s the probability that the same student gets chosen both times?
Ask students if they have questions. Here’s an issue that will likely come up: even if you’ve been chosen before, the chance that you’ll get chosen again is the same. You don’t get any immunity!
If you’re worried that students might not understand the task, ask questions like these before they get started:
Note: Some successful groups will get the answer 1/16. Chances are excellent that they are answering the question, “What is the probability that Penelope gets chosen both times?” After probing gently to verify that this is the case, simply ask, “Are you sure you’re answering the question that was asked?”
4. POST, SHARE, COMMENT
Have students post their work around the classroom.
Encourage students to view the posters other groups created. Encourage students to pose questions to other groups by attaching small adhesive notes.
During this time, review the posters and consider which ones to highlight during the next phase.
A: Simulation by drawing cards. n = 20. Here, students conclude that p = 7/20 or 35%. No theoretical result.
B: Systematic list or table, 16 entries (PP PQ PR PS QP QQ QR QS …) with four matches marked. Result, p = 1/4 or 25%. No empirical, simulated result.
C: This group used dice to simulate 100 (not shown) and got an empirical probability of 23%. They also made this two-way table, and commented that the theoretical 1/4 was close to the 23% empirical result. We like it that the group used both techniques and compared. (Note that some groups may use both approaches but not comment on the comparison.)
D: Empirical p = 29/100 (spinner) but the theory is, “It doesn’t matter who gets picked first. Whoever it is, the chance they get picked second is just 1/4.” (This theoretical reasoning shows deeper understanding—they’re not simply applying some procedure.)
E: Tree diagram. Typical of student work, the diagram is not well-planned visually. Still, it’s correct and systematic.
What Can Go Wrong
Probability can be a hard topic at this level, and the Core Standards are very demanding. Some groups will make mistakes and not get the “right” answers. Make sure you are aware of the following.
In the empirical process, students may just say, for example, “There are two possibilities: match and no match. So we’ll roll a die and use 1, 2, and 3 for a match; and 4, 5, and 6 for no match.” Ask these students to talk with another group about their procedure. If necessary, tell them to try it by simulating each question separately, so you can tell who was called each time (just as we did the two shots separately for Steve), and compare their results to their first plan.
In a systematic list, students may think that “PQ” is the same as “QP” and not count it. In that case, their list may be more like: PP PQ PR PS QQ QR QS RR RS SS, which has only 10 (not 16) members, of which 4 are doubles, giving a theoretical answer of 40%. Whether this is correct is a great topic for debate; ideally, some groups did it one way, and some the other. You can then ask how we might resolve the issue. Students might suggest an empirical approach—this is a great idea. Or they might try a different theoretical representation in order to see if it gives the same result—another great idea. For example, they might re-cast their list using a different representation such as a two-way table (as in Poster C) or a tree diagram (poster E). That organization will force them to make separate entries. Then, in the wrap-up discussion, have students explain what went wrong with the first solution.
5. STRATEGIC TEACHER-LED DISCUSSION
Facilitate a discussion of the probability answers. Select a sequence of posters to use as examples during this discussion to help all students move from their current thinking up to level 4 (or beyond!).
Level 1: Students can get the right theoretical answer (or close to it) but the list or display is disorganized, haphazard, or doesn’t clearly show their reasoning.
Level 2: Students can develop a simulation to answer the question, but their procedure is muddled or they use only a small (less than 50) number of cases.
Level 3: Students create a systematic theoretical result (with a clear display that shows their reasoning) or simulation (with a clear method and enough cases), but not both.
Level 4: Students create both a simulation and a theoretical result, but do not connect the two and comment on how well they correspond.
Level 5: Students connect their theoretical and empirical results.
Issues to raise and questions to ask across presentations
This is a tricky issue. It’s easy to think that the theoretical answer—¼—is “right” or “better.” Kids often feel that if they don’t get the theoretical result when they do something that involves probability, it means that their experience is somehow wrong. But it’s not! Random things vary. And at this stage, the empirical trials tell us how much.
To combine all the class simulation results, add the total number of “days” and the total number of “matches.” Then divide. Do not average the percentages! That only works right if every group simulated the same number of days.
You can determine probabilities either theoretically or empirically. If the probability you want is for some combination of events, you need to consider each event separately and combine them.
6. FOCUS PROBLEM: Same Content in a New Context
Project Slide #4 for the focus problem and let students work on it.
Approaches and Solutions
Students can use the same theoretical strategies they did with the problem of being picked twice. A systematic list, a chart, or a tree diagram will all work.
On the theoretical side, an area chart is particularly elegant (below).
But check out the very different table solution as well. (below.) The student explained that 1 and 6 have one neighbor, but 2 through 5 each have two. For each number, those are “out of” six possibilities.
Project funding provided by The William and Flora Hewlett Foundation and S.D. Bechtel Jr. Foundation.
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