**Additional Resources**

**Materials Required**

**Estimated Time Needed**

*(Times are approximate and will depend on the needs of the students.)*

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.***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?*

Then ask:

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?*

*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.]**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?*

You may also want to ask students:

In trials, students have found card H challenging.