Theoretical vs Experimental Probabilities

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Objective

SWBAT predict outcomes by using proportions and theoretical or experimental probability.

Big Idea

students complete classwork in pairs and reflect on the difference between theoretical and experimental probabilities, including why they do not always equal each other when an experiment is carried out

Do Now + HW Check

15 minutes

Students enter silently according to the Daily Entrance Routine. They are handed their Do Now assignment at the door. A timer is displayed at the front of the room to show that they have 6 minutes to complete the two questions on their paper.

The first question aims to review fraction subtraction, a skill whose mastery I wanted to check since many struggled earlier in the year with these types of problems. The second question aimed to review the concept of mean and absolute deviation and consistency. Without calculating it, we can visibly see the data is clumped together more for class B than class A. I let students know that it is not necessary to calculate the MAD, but instead talk about how it most likely compares between the two classes, justifying their answers by explaining what the MAD is and what it describes. Many will get stuck, thus we will spend the last 4 minutes of this section looking for the complete answer in the notes from two previous lessons. [link for day 128 here]

For students who do not have access to those notes, I will have a print out of the complete Notes to hand out.

Sample answer:

Class B seems to have more consistent scores because most are clumped around the same value. If we were to calculate the MAD for each class, Class A would most likely have a larger MAD value because those scores are more spread out, making them less consistent.

After reviewing the answers to the Do Now, we review the answers to the homework. Again, it is important to point out that when asked for the probability of an even we must give answers in the form of a fraction. It is also important to review the probability continuum in reference to each answer. For example:

  • In question one, you are asked to name an outcome with a probability of 0. Where does this probability fall in the continuum? Where is it on the number line? Is it likely, unlikely, something else? An outcome with a probability of 0 is impossible

 


Check TS + Class Notes

15 minutes

5 minutes: Students take 5 minutes to discuss or finish the Thinking Skills assignment.  Thinking Skills is a period of the day each morning, from 7:25 – 8 am, where students work independently to complete a worksheet or reading provided by a different 7th grade teacher. On Monday mornings, students work on Math Thinking Skills. I include material that has already been taught and needs to be spiraled; so for this assignment I choose a sheet out of the Marcy Mathworks binder which includes questions about measures of variability. Based on the last quiz, students showed they needed to continue practicing finding median and other quartiles. The “thinking Skills” asks students to identify quartiles and calculate interquartile range. Answers are correlated to letters which answer the riddle. Those who complete the riddle correctly, and can show evidence of having completed the worksheet will receive extra credit points on the next quiz. As I walk around while students are working to finish, I am identifying those who are still struggling to master these skills; ideally we would have time during class to review on the spot, which is best. However, since I am short on time this year, I will take a remediation block to review these topics with students.

 

After collecting final Thinking Skills work, I ask students to take out a sheet of paper to take our class notes. The first direction is to complete a heading and copy the aim from the powerpoint. Today we will be answering question involving theoretical and experimental probability. Students must first copy the definitions of these terms. After copying the terms students are asked to read them to each other, in pairs. They must box terms they are not able to understand together. I answer questions or make sure to check for understanding on those key terms in the definitions: likeliness, outcomes, ratio, event, trials. The ultimate essential question:

 

what is the difference between experimental and theoretical probability?

 

Next we move on to some examples. I display these examples (i.e. “we flip a coin. What is the probability it will land on tails?”) and then ask, “is this an example of theoretical or experimental probability?”.

 

Next, we complete a quick experiment. I take out a coin, I ask students to remind me about the theoretical probability of flipping a coin on heads (1/2). I ask them to make a prediction, if I flip a coin 10 times, how many times can I expect to land on heads? 5, half as many as the number of trials. This is a great opportunity for reviewing that same vocabulary from the key definitions.

 

After carrying out the experiment, the likely result will be that I will NOT land on heads exactly 5 times. This is when I introduce the questions students must be discussing and thinking about as they complete today’s classwork:

 

We don’t live in a “perfect” world. Explain how this statement relates to the question: why aren’t the theoretical and experimental probabilities the same? Why didn’t the coin land on heads exactly 5 times?

 

This essential question is meant to help students understand and explore the difference between the two terms introduced today. If we DO get heads 5 times, I will either retry the experiment (given time) or ask, “do you think we’re always going to get 5 times if we repeat the experiment?” and connect the essential question then.

 

I end this section by explaining that both theoretical and experimental probabilities (along with proportions) can be used to make predictions about data. This will be further explored in the class work.

Class Work

20 minutes

After receiving Classwork, students will be asked to work in pairs to complete the sheet. As I walk around, I need to keep an eye out for students who still do not understand the difference between the two terms. Some will use the theoretical probability to answer a question that includes experimental results. These flagged students will immediately be pulled into a small group of 6 – 8. A student leader ought to also be selected to continue helping the students in the group in case I need to visit and help other students. In order to identify these student leaders I will need to be able to think of questions that quickly assess their mastery and understanding of the concepts:

  • What does your proportion look like?
  • What is the initial ratio?
  • Is this an example of theoretical or experimental probability?
  • How do you know?

It is also important that the student leaders selected be able to motivate other to continue persevering to understand the problem (MP1). Student leaders will be given index cards with the same questions as above to help question and push their teammates to understand and explain the answers. Often I have noted that students are better cheerleaders than myself. I have also noted that the “highest achieving” student isn’t always the best cheerleader or leader. This may be an opportunity for “middle achievers” or even “low achievers” to stand out as leaders. I have one student for example, she constantly struggles this year to meet mastery, but she is the hardest working student I have. She never gives up and is always asking questions. She was a great leader during this activity, and I paired her with a “high achieving” student, but I made HER the leader of the group. 

Closing

10 minutes

After completing the classwork, students will be asked to take out their journal to answer the essential question posed during class notes as well as one additional question:

  • In our whole class experiment when flipping the coin, why didn’t our experimental probability match our theoretical probability of (1/2)?
  • What are two things you learned today? OR what are two questions you still have about theoretical and experimental probabilities? Feel free to write about a question in the classwork that really confused you; make sure you explain why.