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Evaluating Statements about Probability
During the
Lesson
  • Whole-Class Interactive Introduction (10 minutes)
      • Give each student a mini-whiteboard, a pen, and an eraser.
      • Display slide P-1 of the projector resources.
      • Ensure the students understand the problem:
        • Your task is to decide if the statement is true.
        • Once you have made a decision, you need to convince me.
      • Allow students a few minutes to think about the problem individually, then a further few minutes to discuss their initial ideas in pairs.
      • Ask students to write their explanations on their whiteboard.
      • If students are unsure, encourage them to think of a simple experiment that could simulate the statement:
        • Do you know how many red and yellow jellybeans are in each bag? Give me an example of the numbers of jellybeans in each bag. Draw a picture of the situation.
        • Can you think of a situation for which the statement is true? [For example, two red jellybeans and one yellow jellybean in bag A and one red jellybean and one yellow jellybean in bag B.]
        • Can you think of a situation for which the statement is false? [For example, two red jellybeans and three yellow jellybeans in bag A and one red jellybean and one yellow jellybean in bag B.]
      • Ask students to show you their mini-whiteboards.
      • Select two or three students with different answers to explain their reasoning on the board. Encourage the rest of the class to comment.
      • Then ask:

        • Chen, can you rewrite the statement so that it is always true?
        • Carlos, do you agree with Chen's explanation? Put Chen's explanation into your own words.
        • Does anyone have a different statement that is also always true?

      This statement highlights the misconception that students often think the results of random selection are dependent on numbers rather than ratios.

  • Collaborative Activity (20 minutes)
      • Organize the class into pairs of students.
      • Give each pair of students a copy of True, False, or Unsure?, a large piece of paper for making a poster, and a glue stick.
      • Ask students to divide their paper into two columns: one for statements they think are true and the other for statements they think are false.
      • Ask students to take each statement in turn:
        • Select a card and decide whether it is a true or false statement
        • Convince your partner of your decision.
        • It is important that you both understand the reasons for the decision. If you don't agree with your partner, explain why. You are both responsible for each other's learning.
        • If you are both happy with the decision, glue the card onto the paper. Next to the card, write reasons to support your decision.
        • Put to one side any cards you are unsure about.
      • You may want to use slide P-2 of the projector resource to display these instructions.
      • You have two tasks during small group work: to make a note of student approaches to the task and to support student problem solving.

      Make a note of student approaches to the task.

      Notice how students make a start on the task, whether they get stuck, and how they respond if they do come to a halt.

      For example, are students drawing diagrams, working out probabilities, or simply writing a description?

      As they work on the task, listen to their reasoning carefully and note misconceptions that arise for later discussion with the whole class.

      Support student problem solving.

      Try not to make suggestions that move students towards a particular approach to this task.

      Instead, ask questions to help students clarify their thinking. If several students in the class are struggling with the same issue, write a relevant question on the board and hold a brief whole-class discussion.

      • Here are some questions you may want to ask your students:

      Card A: This statement addresses the misconception that probabilities give the proportion of outcomes that will occur.

        • Is it possible to get five sixes in a row with a fair six-sided number cube?
        • Is it more difficult to roll a six than, say, a two?

      Card B: This statement addresses the misconception that "special" events are less likely than "more representative" events. Students often assume that selecting an "unusual" letter, such as W, X, Y or Z is a less likely outcome.

        • Is the letter X more difficult to select than the letter T?
        • Are the letters W and X more difficult to select than the letters D and T?

      Card C: This statement addresses the misconception that later random events "compensate" for earlier ones.

        • Does the coin have a memory?

      Card D: This statement addresses the misconception that all outcomes are equally likely, without considering that some are much more likely than others.

        • Is the probability of a local school soccer team beating the World Cup champions ⅓?

      Card E: This statement addresses the misconception that all outcomes are equally likely, without considering that some are much more likely than others. Students often simply count the different outcomes.

        • Are all three outcomes equally likely? How do you know? How can you check your answer?
        • What are all the possible outcomes when two coins are tossed? How does this help?

      Card F: This statement addresses the misconception that the two outcomes are equally likely.

        • How can you check your answer?
        • In how many ways can you score a three? In how many ways can you score a two?

      Card G: This statement addresses the misconception that probabilities give the proportion of outcomes that will occur.

        • When something is certain, what is its probability?
        • What experiment could you do to check if this answer is correct? [One student writes the ten answers e.g. false, true, true, false, true, false, false, false, true, false. Without seeing these answers the other student guesses the answers].

      Card H: This statement addresses the misconception that the sample size is irrelevant.

        • Students often assume that because the probability of one head in two coin tosses is ½, the probability of n heads in 2n coin tosses is also ½
        • Is the probability of getting one head in two coin tosses ½? How do you know?
        • Show me a possible outcome if there are four coin tosses. Show me another. How many possible outcomes are there? How many outcomes are there with two heads?
  • Sharing Work (5 minutes)
      • As students finish the task, ask them to compare work with a neighboring pair.
        • Check which answers are different.
        • A member of each group needs to explain their reasoning for these answers. If anything is unclear, ask for clarification.
        • Then together consider if you should change any of your answers.
        • It is important that everyone in both groups understands the math. You are responsible for each other's learning.
  • Whole-Class Discussion (10 minutes)
      • Organize a discussion about what has been learned.

      Focus on getting students to understand the reasoning, not just checking that everyone produced the same answers.

      • Ask students to choose one card they are certain is true and to explain why they are certain to the rest of the class.
      • Repeat this with the statements that students believe are false.
      • Finally, as a whole class, tackle the statements that students are not so sure about.
        • Ben, why did you decide this statement was true/false?
        • Does anyone agree/disagree with Ben?
        • Does anyone have a different explanation to Ben's?
      • In addition to asking for a variety of methods, pursue the theme of listening and comprehending each others' methods by asking students to rephrase each other's reasoning.
        • Danielle, can you put that into your own words?
      • You may also want to ask students:

        • Select two cards that use similar math. Why are they similar? Is there anything different about them? [Students are likely to select cards D and E.]

        In trials, students have found card H challenging.

        • What are the possible outcomes?
        • Have you listed all the outcomes?
        • Have you listed all the outcomes where there are two heads?
        • What does this show?
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