# Predicting Results Using Ratio & Proportion

## Overview

Students will compare expected results to actual results by first calculating the probability of an event, then conducting an experiment to generate data. They will use an interactive to simulate a familiar event—rolling a number cube. Students will also be introduced to terminology.

# Key Concepts

- This lesson takes an informal look at the Law of Large Numbers through comparing experimental results to expected results.
- There is variability in actual results.

Probability terminology is introduced:

*theoretical probability*: the ratio of favorable outcomes to the total number of possible equally-likely outcomes, often simply called*probability**expected results*: the results based on theoretical probability*experimental probability*: the ratio of favorable outcomes to the total number of trials in an experiment*actual results*: the results based on experimental probability*outcome*: a single possible result*sample space*: the set of all possible outcomes*experiment*: a controlled, repeated process, such as repeatedly tossing a coin*trial*: each repetition in an experiment, such as one coin toss*event*: a set of outcomes to which a probability is assigned

# Goals and Learning Objectives

- Predict results using ratio and proportion.
- Compare expected results to actual results.
- Understand that the actual results get closer to the expected results as the number of trials increase.

# Number Cube

# Lesson Guide

Have students take a few minutes to discuss the Opening questions with a partner.

# Mathematics

Students will briefly discuss with a neighbor whether they think one event is more likely than the other. Since each side of a cube is equally likely to land face up, each number must have equal probability.

SWD: Consider the prerequisite skills for this lesson. Students with disabilities may need direct instruction or guided practice with the skills needed to complete this lesson. Make sure that students understand vocabulary used in the activities in this lesson.

- theoretical probability
- expected results
- experimental probability
- actual results
- outcome
- sample space
- experiment
- trial
- event

Pre-teach these terms to students with disabilities.

## Opening

# Number Cube

If you roll a number cube containing the numbers 1 through 6, are you more likely to get a 1 or a 5? What is the probability of each event?

Think about the situation, and then discuss your ideas with a partner.

# Math Mission

# Lesson Guide

Discuss the Math Mission. Students will determine the theoretical probability of events and experiments to compare actual results with expected results.

## Opening

Determine the theoretical probability of events and experiments to compare actual results with expected results.

# Rolling a Number Cube

# Lesson Guide

Students will work individually as they calculate the probabilities for 30 rolls and use the Number Cube interactive to get experimental results for 30 rolls.

Create a line plot for the results for 30 rolls. Conduct an informal poll to collect the class results, showing the frequency for each of the six numbers.

SWD: Students with disabilities may demonstrate difficulty initiating tasks because they are introduced to new vocabulary or because they are asked to predict outcomes from the interactive tools. Make sure that students have a clear understanding of the task expectations and directions. Provide students the option of listening to the content in this task (via Text-to-Speech, or via read aloud by teacher or peer) to support comprehension.

# Mathematics

Students should see that each number is equally likely to be rolled, although in actuality it is not likely that each number will be rolled the same number of times. This is the difference between the expected results and the actual results.

For 30 rolls, students should predict that there will be five of each roll (knowing that this probably won't actually happen). Assuming there are 30 students in the class, students will probably think of multiplying the probability by 900 (or figure out what number times 6 is 900) to predict the results for 900 rolls, since 6 ⋅ 150 = 900. Point out that this is because of the ratios involved, $\frac{1}{6}=\frac{150}{900}$, and that proportions can be used.

Each of the outcomes for rolling a number cube are equally likely and do not depend on one another. So, although we could say that the probability of rolling 1, 2, 3, or 4 is $\frac{4}{6}=\frac{2}{3}$because there are four favorable outcomes out of six total outcomes, we could also say that the probability is $\frac{1}{6}+\frac{1}{6}+\frac{1}{6}+\frac{1}{6}=\frac{4}{6}=\frac{2}{3}$. If events are mutually exclusive, the probabilities can be added.

The probability of rolling a 1, 2, 3, or 4 plus the probability of rolling 5 or 6 is 100% because this covers all of the outcomes. So, if rolling a 1, 2, 3, or 4 has a probability of $\frac{2}{3}$, rolling a 5 or 6 should be $\frac{1}{3}$ (because $\frac{1}{3}+\frac{2}{3}=1$). These are complementary events.

ELL As in other lessons, when academic language is reviewed (or introduced, for that matter), show it in writing and leave it in a place that is visible so that all students can refer to it. Make sure students copy academic language into their notebook.

# Mathematical Practices

**Mathematical Practice 4: Model with mathematics**

In these problems, students are practicing their ability to take a familiar situation (rolling a number cube) and apply their knowledge of mathematics to it.

# Interventions

**Student has trouble getting started.**

- How many ways are there to roll each number?
- How many ways are there to roll a number cube?

**Student answers for the outcomes appear random, or not based on ratios.**

- How did you decide on these numbers?
- What is the probability of rolling the number?
- What fraction out of 30 is equivalent?
- What fraction out of 900 is equivalent?

# Answers

- The probability of rolling any given number is $\frac{1}{6}$: $\begin{array}{ccc}P\text{(rolling a 1)}& =& \frac{1}{6}\\ P\text{(rolling a 2)}& =& \frac{1}{6}\\ P\text{(rolling a 3)}& =& \frac{1}{6}\\ P\text{(rolling a 4)}& =& \frac{1}{6}\\ P\text{(rolling a 5)}& =& \frac{1}{6}\\ P\text{(rolling a 6)}& =& \frac{1}{6}\end{array}$
- Since the theoretical probabilities are all the same, each number would appear 30⋅$\frac{1}{6}$ = 5 times.
- Answers will vary. If there are
*x*people in the class, then each number will come up 5 ⋅*x*times. In a class of 30 people, each number would appear 150 times. - The probability of rolling a 1, 2, 3, or 4 can be calculated by adding up their individual probabilities: $\frac{1}{6}+\frac{1}{6}+\frac{1}{6}+\frac{1}{6}=\frac{4}{6}=\frac{2}{3}$. If you rolled a number cube 30 times, the number of times it came up 1, 2, 3, or 4 would be $\frac{2}{3}$ ⋅ 30 = 20 times. Similarly, the probability of rolling a 5 or 6 is $\frac{1}{6}+\frac{1}{6}=\frac{2}{6}=\frac{1}{3}$. The number of times the cube will come up 5 or 6 is $\frac{1}{3}$ ⋅ 30 = 10 times.
- Students will probably have results which are slightly different from the expected results (each number rolled exactly 5 times), they should note that this difference between the the expected results and the actual results is to be expected.

## Work Time

# Rolling a Number Cube

- What is the theoretical probability of rolling each number on the cube?
- Based on the theoretical probability of each number, if you rolled the number cube 30 times, how many times would you get each number?

1, 2, 3, 4, 5, 6 - Based on the theoretical probability, how many times would each number come up if everyone in the class each rolled a number cube 30 times?

1, 2, 3, 4, 5, 6 - Based on your answers to the previous questions, how many times would you roll a 1, 2, 3, or 4? How many times would you roll a 5 or 6?
- Now use the Number Cube interactive to roll the number cube 30 times.

INTERACTIVE: Number Cube

# Prepare a Presentation

# Preparing for Ways of Thinking

Presentations should explain the difference between the calculated probability and the actual results.

As students work, look for students who use ratios (or decimals or percentages) correctly to find the expected results for 900 rolls to share in Ways of Thinking. If any students did the Challenge Problem, have them also present during Ways of Thinking.

# Challenge Problem

## Answers

- If you roll a number cube and it does not come up 4, that means it comes up 1, 2, 3, 5, or 6.
*P*(not rolling 4) =*P*(rolling 1, 2, 3, 5, 6 ) = $\frac{1}{6}$ + $\frac{1}{6}$ + $\frac{1}{6}$ + $\frac{1}{6}$ + $\frac{1}{6}$ = $\frac{5}{6}$ - Out of 900 rolls, you would expect 900 ⋅ $\frac{5}{6}$ = 750 rolls to not come up 4.

## Work Time

# Prepare a Presentation

- Compare your actual results for 30 rolls to your expected results based on probability. Explain the differences.

# Challenge Problem

- What is the probability of not rolling a 4?
- In how many rolls out of 900 would you expect not to get a 4?

# Make Connections

# Lesson Guide

The discussion and comparison of the line plots serve as reminders of statistical tools students learned about in Unit 6.8, as well as a preview to comparing data sets in this unit.

# Mathematics

It is not necessary to discuss the results for all six of the numbers; focus on one or two for the discussion. However, do show the line plot to reinforce the concept that the results for each number are similar, but slightly different.

In discussing the results, start with a comparison of the theoretical probability for each number. Have students who present share their actual results for 30 rolls. Make sure that the following questions are addressed:

- Why do you think there is a range of results for the number?
- How many times should the number have been rolled? (Each number should have been rolled 5 times.)
- Why does each number have the same probability? (Because each outcome is equally likely.)
- How did the actual results (the students' results for 30 rolls) compare to what was expected?

Discuss mutually exclusive and complementary events:

- How did you determine the number of outcomes for these events?
- How do they compare with the actual results?

In discussing the results, students should see that the actual results get closer to expected results as the number of trials increase (e.g., comparing a student's 30 trials to the entire class's trials).

ELL: In asking prompting questions, be sure to use adequate pace and be sure that they understand meaning of the questions.

# Mathematical Practices

**Mathematical Practice 3: Construct viable arguments and critique the reasoning of others.**

When comparing predicted and actual results, students must make sense of the information and construct arguments about their observations.

## Performance Task

# Make Connections

- Take notes about how other students’ expected results compare to their actual results.

## Hint:

As your classmates present, ask questions such as:

- Can you explain how you determined the expected results for 30 rolls? 900 rolls?
- Why does each number have the same probability?
- Why do you think you got a range of results when you rolled the number cube 30 times?
- How do your actual results for 30 rolls compare to your expected results?

# Definitions

# Lesson Guide

Have students discuss the definitions with a partner before turning to a whole class discussion. Use this opportunity to correct or clarify misconceptions.

# Mathematical Practices

**Mathematical Practice 6: Attend to precision.**

Students are introduced to terms that they will use throughout the unit to describe the mathematics.

## Formative Assessment

# Summary of the Math: Definitions

Read and discuss.

** Theoretical probability**: The ratio of favorable outcomes to the total number of possible outcomes; often simply called

*probability*. The results based on theoretical probability are called the

*expected results*.

** Experimental probability**: The ratio of favorable outcomes to the total number of trials in an experiment. The results based on experimental probability are called the

*actual results*.

** Outcome**: A single possible result

** Sample space**: The set of all possible outcomes

** Experiment**: In probability, a controlled, repeated process (such as repeatedly tossing a coin)

* Trial:* Each repetition in an experiment; for example, one coin toss

** Event**: A set of outcome(s) to which a probability is assigned

## Hint:

Can you:

- Explain how expected results are determined?
- Explain how actual results, based on experimental probability, compare to expected results, based on theoretically probability?

# Reflect on Your Work

# Lesson Guide

Have each student write a brief reflection before the end of the class. Review reflections to find out what confuses students about probability.

- ELL: When writing the reflection, allow some additional time for ELLs to discuss with a partner before writing, to help them organize their thoughts. Allow ELLs who share the same language of origin to discuss in their language of origin if they wish and to use a dictionary.

## Work Time

# Reflection

Write a reflection about the ideas discussed in class today. Use the sentence starter below if you find it to be helpful.

**One thing that confuses me about probability is...**