# Representing Probabilities: Medical Testing

## Mathematical goals

This lesson unit is intended to help you assess how well students are able to:

- Make sense of a real life situation and decide what math to apply to the problem.
- Understand and calculate the conditional probability of an event A, given an event B, and interpret the answer in terms of a model.
- Represent events as a subset of a sample space using tables, tree diagrams, and Venn diagrams.
- Interpret the results and communicate their reasoning clearly.

## Introduction

- Before the lesson, students work individually on an assessment task designed to reveal their current levels of understanding and difficulties. You then review their work and create questions for students to answer to help them improve their solutions.
- At the start of the lesson, students work alone answering your questions about the same problem. They are then grouped and engage in a collaborative discussion of the same task.
- In the same small groups, students are given sample solutions to comment on and evaluate.
- In a whole-class discussion, students explain and compare the alternative solution strategies they have seen and used.
- In a follow-up lesson, students review what they have learned.

## Materials required

- Each student will need the
*Medical Testing*task, the*How Did You Work?*questionnaire*,*a mini-whiteboard, pen, eraser, - Each small group of students will need the
*Sample Responses to Discuss*, some poster paper and a felt-tipped pen. Have a few copies of the*Hint Sheet*available in case some groups of students struggle to start the task. There is a projector resource to support whole-class discussions. - Optionally, demonstrate the
*Medical Testing*applet and the interactive spreadsheet*Medical_Testing.xls*or*Medical_Testing.xlsx*using a data projector or interactive whiteboard.

## Time needed

20 minutes before the lesson, a 110-minute lesson (or two 55-minute lessons), and 20 minutes in a follow-up lesson. Timings are approximate and depend on the needs of the class.

## Lesson Type

## Mathematical Practices

This lesson involves a range of *mathematical practices* from the standards, with emphasis on:

- MP1: Make sense of problems and persevere in solving them
- MP2: Reason abstractly and quantitatively
- MP3: Construct viable arguments and critique the reasoning of others
- MP4: Model with mathematics
- MP5: Use appropriate tools strategically
- MP6: Attend to precision
- MP7: Look for and make use of structure
- MP8: Look for and express regularity in repeated reasoning

## Mathematical Content Standards

This lesson asks students to select and apply mathematical content from across the grades, including the *content standards:*

- H.S-CP: Understand independence and conditional probability and use them to interpret data
- H.S-CP: Use the rules of probability to compute probabilities of compound events in a uniform probability model