### Michaél Dunnivant

#### Description

As an introduction to probability, students use tree diagrams to predict the possible outcomes of coin tosses. The data they collect and graph also help them predict the likelihood of getting heads or tails when tossing coins the next time.

#### Objectives

The student solves problems by generating, collecting, organizing, displaying, and analyzing data using histograms, bar graphs, circle graphs, line graphs, pictographs, and charts.

The student uses models, such as tree diagrams, to display possible outcomes and to predict events.

#### Materials

-Pennies (one per pair of students)
-Heads or Tails Recording Sheet/Graph (one per pair of students and one for overhead as found in the Associated File)
-Class Graph on which to compile data
-Crayons
-Pint baskets (one per pair of students) in which to toss their coin
-Chart paper

#### Preparations

The teacher needs to:
1. Read up on probability and appropriate learning experiences that encourage investigation into probability.
2. Collect pennies and baskets for each pair of students.
3. Copy one -Heads or Tails Recording Sheet/Graph- for each pair of students. This can be downloaded from the Associated File.
5. Prepare chart for tree diagram. The tree diagram is a graphic organizer that shows each of the possible outcomes when tossing a coin, heads and tails, one on each branch.
6. Prepare chart for class graph.

#### Procedures

1. Review the problem-solving steps:
Step 1. Understand the Problem
Step 2. Decide on a Plan
Step 3. Carry out the Plan
Step 4. Look back and Review

2. Brainstorm with the class about tossing coins and experiences they have had regarding coin tosses.

3. Chart their ideas as the class discusses their experiences.

4. Tell students that today, -We are going to use the problem-solving steps to figure out this problem, 'What will happen when I flip this coin?'- State that this also helps us begin to learn about probability and how it works. Introduce probability as the chance that something will happen.

5. After students respond, clarify what it means to get a -heads- or -tails- when you flip a coin. Ask, -Do we understand the problem? That's step one of the problem-solving steps.-

6. Draw a tree diagram as a picture to introduce -heads- and -tails- as possible outcomes when you toss a coin. Explain that a tree diagram is a model that shows all of the possibilities or outcomes you may get when you conduct an experiment, such as a coin toss.

7. Review by showing the backs (tail side) of a variety of coins, not just pennies. Ask, -What is this, heads or tails?- Refer to the tree diagram each time heads or tails is discussed for each coin and reinforce that they are the two possible outcomes when tossing a coin.

8. Ask students to predict what they think will happen if they tossed a coin 20 times. Record their predictions for the number of heads and tails on a chart in random order accepting all responses. You might want to ask -why- in some instances, to clarify their responses, but it is important to take all predictions.

9. Ask, -Which of the predictions seems likely?- and -Why?- Ask students for their plan to solve the problem.

10. Model the plan for solving the problem as you -think out loud- to make your prediction. Record your prediction on the overhead -Heads or Tails Recording Sheet.- Explain that predictions are neither right nor wrong, but as the students go along they may want to change their prediction based upon what their -Heads or Tails Recording Sheet/Graph- is showing.

11. Continue modeling as you toss the coin a total of twenty times and record the results each time on the -Heads or Tails Recording Sheet/Graph.- State that this is Step 3 of the problem-solving steps, -Carrying out the plan.- Take the data from the results of the coin toss experiment and begin to create the class graph.

12. During the experiment, stop and ask students to look at the data and make -true statements- about the data. Model a -true statement- for the students (i.e., I've rolled the die five times and tossed heads three times and tails two times.) Ask others to respond to the data by making true statements. Give non examples of -true statements,- like -Tails always wins.- State that this is Step 4 the part of the problem-solving steps where we -Look back and Review.-

13. Make predictions about what would happen if we conducted the experiment again. Ask the students to justify their reasoning and whether we have solved the problem.

14. Now students are ready to investigate the problem in pairs repeating the same process. Give each pair a -Heads or Tails Recording Sheet,- one penny, one basket to toss the coin in, and crayons, pencils, etc. Students predict the outcome, conduct the experiment, and record results on the Heads or Tails Recording Sheet/Graph in pairs, as modeled.

15. Pairs then make true statements about the data and record the true statements on the -Heads or Tails Recording Sheet,- or in a journal.

16. Record results on the class graph.

17. As a class, analyze the results of the class graph. Discuss what the results show as compared to the tree diagram.

#### Assessments

Formative assessment criteria:
Explains in writing:
- how to use a tree diagram to show possible outcomes
- how to use a tree diagram to predict events

Graph:
- accurately displays data collected in a bar graph after tossing a coin 20 times

True Statements:
-analyze data that is organized on the bar graph

Possible prompts are:
-How did we use the tree diagram?
-Did the tree diagram help us solve the problem? Why or Why not?
-What does a tree diagram tell you about what might happen the next time you toss a coin?

This formative assessment should guide your next teaching steps. As students write about experiences to explain the concept of probability, graphing, and analyzing data, it gives you a glimpse into their understanding of it. Your next teaching steps are thus, based upon where your students are in their understanding. The assessment is not designed for assigning a score or grade. It is more important to give specific feedback on performance at this point.