# Teacher Resources

## Prize Winner, Part 2

On this page you will find lesson objectives, standards, instructional notes and UDL opportunities.

**Estimated Time: **45 minutes

### Lesson Objectives:

- The students will determine the conditional probabilities given a two-way table.
- The students will determine if two conditional events are independent or dependent.

**Key Common Core State Standards:**

S.CP.3: Understand the conditional probability of *A* given *B* as *P*(*A* and *B*)/*P*(*B*), and interpret independence of *A* and *B* as saying that the conditional probability of *A* given *B* is the same as the probability of *A*, and the conditional probability of *B* given *A* is the same as the probability of *B*.

S.CP.4: Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities. *For example, collect data from a random sample of students in your school on their favorite subject among math, science, and English. Estimate the probability that a randomly selected student from your school will favor science given that the student is in tenth grade. Do the same for other subjects and compare the results.*

S.CP.5: Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations. *For example, compare the chance of having lung cancer if you are a smoker with the chance of being a smoker if you have lung cancer.*

S.CP.6: Find the conditional probability of *A* given *B* as the fraction of *B*’s outcomes that also belong to *A*, and interpret the answer in terms of the model.

**Supporting Common Core State Standards:**

7.SP.C.8: Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation.

7.SP.C.8a: Understand that, just as with simple events, the probability of a compound event is the fraction of outcomes in the sample space for which the compound event occurs.

7.SP.C.8b: Represent sample spaces for compound events using methods such as organized lists, tables and tree diagrams. For an event described in everyday language (e.g., “rolling double sixes”), identify the outcomes in the sample space which compose the event.

8.SP.4: Understand that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a two-way table. Construct and interpret a two-way table summarizing data on two categorical variables collected from the same subjects. Use relative frequencies calculated for rows or columns to describe possible association between the two variables. *For example, collect data from students in your class on whether or not they have a curfew on school nights and whether or not they have assigned chores at home. Is there evidence that those who have a curfew also tend to have chores?*

**Standards for Mathematical Practice Emphasized:**

1. Make sense of problems and persevere in solving them.

2. Reason abstractly and quantitatively.

3. Construct viable arguments and critique the reasoning of others.

4. Model with mathematics.

5. Use appropriate tools strategically.

6. Attend to precision.

7. Look for and make use of structure.

8. Look for and express regularity in repeated reasoning.

## Instructional Notes:

This lesson applies what the students know about probability and conditional probability, using two-way tables. While this lesson is independent of Lesson 1 in this Module 6, the context and real-world applications are related. If the students are in need of additional practice and activities, consult Illustrative Mathematics . Under the high school standards for statistics and probability, the Titanic 1, 2 and 3 illustrative tasks have the students solve conditional probabilities with two-way tables You can create more real-world two-way tables with actual data using WEAT: Web Enabled Analysis Tool . This site automatically generates two-way tables using categories that can be selected by the user.

Take time to read the Progressions for the Common Core State Standards in Mathematics , pp 14 – 17. These documents provide an overview of conditional probability, providing real-world applications as well as simulations and two-way table examples. The examples provided in these documents can be used as an assessment for learning as students work through the lesson.

It is important for the students to logically think about what probability they are looking for in terms of the given situation. It is far more critical for a student to have a contextual understanding of probability, rather than to simply memorize formulas. The games used in this lesson are simple to construct. Consider allowing the students the time to “act out” the games so they can get a feeling for what the probabilities are asking while developing an understanding of experimental probability.> Go to Prize Winner, Part 2 lesson

## Sample Responses

For sample responses to the Algebra II Journal questions, visit the Algebra II Journal in the Teacher Resources.

## UDL Opportunities:

**Principle 1: Provide Multiple Means of Representation**

Checkpoint 1.1 Offer ways of customizing the display of information.

Checkpoint 1.3 Offer alternatives for visual information.

Checkpoint 2.1 Clarify vocabulary and symbols.

Checkpoint 2.2 Clarify syntax and structure.

Checkpoint 2.3 Support decoding text, mathematical notation, and symbols.

Checkpoint 3.1 Activate or supply background knowledge.

Checkpoint 3.2 Highlight patterns, critical features, big ideas, and relationships.

Checkpoint 3.4 Maximize transfer and generalization.

**Principle 2: Provide Multiple Means of Action and Expression**

Checkpoint 4.2 Optimize access to tools and assistive technologies.

Checkpoint 5.3 Build fluencies with graduated levels of support for practice and performance.

Checkpoint 6.2 Support planning and strategy development.

Checkpoint 6.3 Facilitate managing information and resources.

Checkpoint 6.4 Enhance capacity for monitoring progress.

**Principle 3: Provide Multiple Means of Engagement**

Checkpoint 7.2 Optimize relevance, value, and authenticity.

Checkpoint 7.3 Minimize threats and distractions.

Checkpoint 8.4 Increase mastery-oriented feedback.

Checkpoint 9.1 Promote expectations and beliefs that optimize motivation.

Checkpoint 9.3 Develop self-assessment and reflection.