The Language and Logic of Probability
Lesson 9 of 12
Objective: SWBAT analyze problems using precise language aligned to probability concepts.
Although the students have worked with calculating probabilities, this lesson will challenge them to evaluate claims and test decisions associated with probability problems. At the start of class, circulate the Entry Activity Worksheet. This worksheet has three simple claims on it that the students are to analyze. Be sure to explain to the students to read through the statement carefully, answer the question, and thoroughly explain their reasoning.
It is important that the students work entirely alone for this entry event, and that you provide absolute minimal assistance. Typically, the students can respond to the three questions in 7-8 minutes. It is important that the students work alone because this activity will bring several misconceptions in language to the forefront for discussion. Allowing the students to make mistakes will help drive home the points later in the lesson.
Once all students have quietly finished their responses, I provide the following secondary instructions:
Based on the nature of your response, create a question that you could ask a person with an opposing viewpoint (an answer opposite yours) that would “stump” them into reversing their answer.
This may require a little further explanation, however, it is a great way to get the students to reason abstractly (MP2) and construct viable arguments and critique the reasoning of others (MP3). For example, looking at question #1, a student may have correctly responded “NO – it does not imply a 50% chance of rain” (they should also provide further explanation for their reasoning). To create a question that “stumps” someone who thinks YES into changing their answer, the student could create the question(s): What factors affect whether it will rain tomorrow or not? What is the chance of rain for tomorrow? Next week? Next year?
Putting the students through this mental exercise is a great way for them to practice logic and abstract thinking. These are important mathematical skills to develop, that especially manifest themselves in proofs applications. Although you do not have to expose the students directly to proofs, it is beneficial for them to engage in this line of thinking and questioning.
Other Sample Questions:
In what way does the fact that there are already four boys in the family affect the sex of the next child?
What is the probability that the baby will be a girl?
Is it more difficult to throw a six than a two?
Is it more difficult to throw a six, then another six OR a two and then a three?
Students who answer the question incorrectly will have a difficult time coming up with these types of questions. They will quickly realize that they need to rethink their response. This, in and of itself, is a great and worthwhile exercise!
Whole Class Introduction
Provide each student with a mini whiteboard (my students use Educreations on their iPad). Explain the following scenario to the students and draw the corresponding bags on the board:
I have two bags. Both contain red and yellow jellybeans.
There are more red jellybeans in bag A than in bag B.
If I choose one jellybean from each bag am I more likely to choose a red one from bag A than from bag B?
I allow the students to think about the problem individually and then discuss their ideas with their peers. Following these discussions, I ask the students to write their explanations on their whiteboard and place it on their desks. When the students are finishing up, I rotate around the room and see how they have done.
If the students are struggling, encourage them to question similar to the Entry Activity…
Do you know how many red and yellow jelly beans you have? Give an example of the numbers of jelly beans in each bag. Can you draw a picture of the situation?
Can you think of a situation for which the statement is true?
Can you think of a situation for which the statement is false?
Once all students have had a chance to complete the task, I ask a few of them to share their findings and encourage the rest of the class to comment and engage in the discussion.
Finally, ask the students to use their whiteboard to rewrite the statement so that it is ALWAYS true… Have the students share out loud different versions of their true statements - - another great chance to emphasize mathematical thinking and communication! (MP2, MP3)
Attached you will find the True, False, or Unsure ANSWERS.
I have tried a few different methods to begin our class discussion over the student results, but the most effective method that I tried was to ask the students first which ones they were absolutely sure they had the correct responses and reasons to. Rather than going right down the line in numerical order, asking the students to identify which problems they were SURE they had right helped to allow for more discussion and engagement on the others. Rarely will the students get any questions wrong from want they are sure is correct, and most of the time the students agree during this time. In my past experience, the students will agree on #2,3,5,6 and 8 - and get all of these problems correct. It is not that these problems are easy for the students, but the collaborative nature of the previous activity helps filter out faulty reasoning and poor logic.
#1, 4, and 7 usually generate great class discussion, and I work hard not to reveal the answer to the students until they have reached a uniform conclusion as a class. With each student claim I question them and probe for specific examples that help to drive home their point. My goal is to make the students think for themselves, and back up their reasoning. Eventually, the class usually works their way to the correct answer!
This activity may take more than 5 minutes, so be prepared to spend a little more time in discussion in the next class period.