# Cluster 3: Investigate chance processes and develop, use, and evaluate probability models. (Supporting Cluster)Archived Export Print

Clusters should not be sorted from Major to Supporting and then taught in that order. To do so would strip the coherence of the mathematical ideas and miss the opportunity to enhance the major work of the grade with the supporting clusters.

General Information
Number: MAFS.7.SP.3
Title: Investigate chance processes and develop, use, and evaluate probability models. (Supporting Cluster)
Type: Cluster
Subject: Mathematics - Archived
Domain-Subdomain: Statistics & Probability

## Related Standards

This cluster includes the following benchmarks.

## Related Access Points

This cluster includes the following access points.

## Access Points

MAFS.7.SP.3.AP.5a
Define the probability of related events given a situation of chance.
MAFS.7.SP.3.AP.6a
Make a prediction regarding the probability of an event occurring; conduct simple probability experiments and compare results to predictions.
MAFS.7.SP.3.AP.7a
Compare actual results of a simple experiment when numbers of instances are increased.
MAFS.7.SP.3.AP.8a
Determine the theoretical probability of compound events (e.g., two coins or two dice).
MAFS.7.SP.3.AP.8b
Use tree diagrams, frequency tables, organized lists, and/or simulations to collect data from a two-step simulation of compound events (using two coins and/or two dice).

## Related Resources

Vetted resources educators can use to teach the concepts and skills in this topic.

## Formative Assessments

Work Clothing:

Students are asked to make a tree diagram to determine all possible outcomes of a compound event.

Type: Formative Assessment

Number List:

Students are asked to make an organized list that displays all possible outcomes of a compound event.

Type: Formative Assessment

Coat Count:

Students are asked to design a simulation to generate frequencies for complex events.

Type: Formative Assessment

Automotive Probabilities:

Students are asked to find the probability of a compound event using a tree diagram and explain how the tree diagram was used to find the probability.

Type: Formative Assessment

Probabilities Cubed:

Students are asked to estimate the frequency of an event given its probability and explain why an expected frequency might differ from an observed frequency.

Type: Formative Assessment

Hen Eggs:

Students are asked to estimate the probability of a chance event based on observed frequencies.

Type: Formative Assessment

Game of Chance:

Students are asked to estimate the frequency of an event given its probability and explain why an expected frequency might differ from an observed frequency.

Type: Formative Assessment

Marble Probability:

Students are asked to determine probabilities based on observed outcomes and to determine if the outcomes appear to be equally likely.

Type: Formative Assessment

Number Cube:

Students are asked to determine probabilities based on observed outcomes and to determine if the outcomes appear to be equally likely.

Type: Formative Assessment

Probability or Not?:

Students are asked to determine whether or not a given number could represent the probability of an event.

Type: Formative Assessment

Technical Difficulties:

Students are given a scenario and asked to determine the probability of two different events.

Type: Formative Assessment

Likely or Unlikely?:

Students are asked to determine the likelihood of an event given a probability.

Type: Formative Assessment

Likelihood of an Event:

Students are asked to determine the likelihood of an event given a probability.

Type: Formative Assessment

Errand Runner:

Students are asked to determine the probability of a chance event and explain possible causes for the difference between the probability and observed frequencies.

Type: Formative Assessment

This lesson unit addresses common misconceptions relating to probability of simple and compound events. The lesson will help you assess how well students understand concepts of:

• Equally likely events
• Randomness
• Sample sizes

Type: Formative Assessment

## Image/Photograph

Clipart ETC: Probability:

Clipart images that relate to probability.

Type: Image/Photograph

## Lesson Plans

Genetics and Proportions Design Challenge:

Students will explore principles of heredity through an activity where they design a themed Potato Head toy set.

Type: Lesson Plan

Water, Water Everywhere - Natural Disaster Water Filtration:

Students will be tasked with an engineering challenge to design an effective and efficient portable water filtration system. The designs will take dirty water and make it clear so it can be boiled for safe drinking. This lesson aligns to both math and science content standards.

Type: Lesson Plan

3D Printing: Designing Robots Using Heredity and Probability:

This lesson explores the importance of Punnett squares in determining genetic characteristics. It uses a 3D printer to demonstrate these characteristics.

Type: Lesson Plan

Genetics Has Gone to the Dogs!:

This lesson uses pooches to teach about pedigrees and the impact of artificial selection on individuals and populations as well as to drive home math concepts already discussed in lessons on Punnet squares.

Type: Lesson Plan

Hair or No Hair- Please tell me Punnett Square:

This lesson is designed to teach students how to read and interpret Punnett square with the final goal of them creating their own squares. The students will be able to determine possible genotypes and phenotypes of offspring based parent alleles.

Type: Lesson Plan

Let's Flip Out:

This lesson is designed to follow a lesson that teaches theoretical probability. The students should have a strong foundation in theoretical probability, understanding how to find theoretical probability for a given situation. In this lesson students will roll a number cube and flip a coin to find the relative frequency for a given simulation.

Type: Lesson Plan

Pick and Roll:

This lesson is designed to teach students about independent and dependent compound probability and give students opportunities to experiment with probabilities through the use of manipulatives, games, and a simulation project. The lesson can take as long as three hours (classes), but can be modified to fit within two hours (classes).

Type: Lesson Plan

Independent Compound Probability:

During this lesson, students will use Punnett Squares to determine the probability of an offspring's characteristics.

Type: Lesson Plan

Garbage Can Hoops:

This lesson guides students through an experiment to learn about relative frequencies and probability of events occurring. Students discover what happens when repeatedly tossing a paper ball (balled up paper) from:

• a relatively short distance (5 ft.) from a garbage can;
• a relatively medium distance (10 ft.) from the garbage can;
• a relatively long distance (15 ft.) from the garbage can.

Students participate in the experiment and compare their predictions to the experimental outcomes of others. They propose and refine conjectures about relative frequency probability.

Type: Lesson Plan

Understanding Probability of Compound Events:

This lesson uses guided teaching, small group activities, and student creations all-in-one! Students will be able to solve and create compound event word problems. They will also be able to identify what type of event is being used in a variety of word problems.

Type: Lesson Plan

Permutations and Combinations:

This is a seventh grade lesson that should follow a lesson on simple probability. This is a great introduction to compound probability and a fun, hands-on activity that allows students to explore the differences between permutations and combinations.  This activity leads into students identifying situations involving combinations and permutations in a real-world context.

Type: Lesson Plan

How to Hit it Big in the Lottery - Probability of Compound Events:

Students will explore a wide variety of interesting situations involving probability of compound events. Students will learn about independent and dependent events and their related probabilities.

Lesson includes:

• Bell-work that reviews prerequisite knowledge
• Directions for a great In-Your-Seat Game that serves as an interest builder/introduction
• Vocabulary
• Built-in Kagan Engagement ideas
• An actual lottery activity for real-life application

Type: Lesson Plan

Basic Probability Concepts:

This lesson is meant to build on students knowledge of probability. It is designed to teach and reinforce the concept that all probabilities are a number somewhere between zero and one.

Type: Lesson Plan

Tree Diagrams and Probability:

This lesson is designed to develop students' ability to create tree diagrams and figure probabilities of events based on those diagrams. This lesson provides links to discussions and activities related to tree diagrams as well as suggested ways to work them into the lesson. Finally, the lesson provides links to follow-up lessons designed for use in succession with the current one.

Type: Lesson Plan

This lesson unit addresses common misconceptions relating to probability of simple and compound events. The lesson will help you assess how well students understand concepts of equally likely events, randomness and sample sizes.

Type: Lesson Plan

This lesson is designed to introduce students to random numbers and fairness as a precursor to learning about probability. The lesson provides links to discussions and activities related to probability and fairness as well as suggested ways to integrate them into the lesson. Finally, the lesson provides links to follow-up lessons designed for use in succession with this one.

Type: Lesson Plan

Probability:

This lesson is designed to develop students' understanding of probability in real life situations. Students will also be introduced to running experiments, experimental probability, and theoretical probability. This lesson provides links to discussions and activities related to probability as well as suggested ways to integrate them into the lesson. Finally, the lesson provides links to follow-up lessons designed for use in succession with the current one.

Type: Lesson Plan

Chancy Candy:

In this lesson students will use candy to find the probability of independent compound events, determining the sample space from a tree diagram. They will then do an experiment to test the theoretical probability. Once the experiment is complete, the students will compare the theoretical and experimental probability.

Type: Lesson Plan

When Pigs Fly:

In this lesson students will explore probability and likelihood that an event will occur. They will place both serious and silly events on a number line, once they have assigned a value to that event. They will work with a group and then justify their classifications to their peers.

Type: Lesson Plan

Practically Probable:

In this lesson, students will differentiate between likely and unlikely event, as well as learn the difference between dependent and independent events. Finally, they learn how to compute theoretical probabilities in simple experiments.

Type: Lesson Plan

Planning the perfect wedding:

Students will decide what is the best month to celebrate an outdoor wedding. The couple is looking for the perfect wedding day. What is the definition of a perfect day? It has to be a Saturday or Sunday with a 20% or less probability of rain and sunny but not too hot. Based on the information provided , students will find the month in which the probability of having a rainy day and the probability of having a super hot day (temperature higher than 75º F) are minimal.

Type: Lesson Plan

Solving the Tomato Crisis:

We are currently being faced with many challenges to the natural resources in our environment. With this activity students can begin to find alternatives to using resources such as soil, in a healthier and "greener" way. This activity can also be used to open up discussion and debate about topics such a quantity versus quality of produce as well as cost and demand.

Type: Lesson Plan

Probabilities and Punnett Squares:

Students simulate the process of meiosis for an alien society. The students choose physical characteristics for hair, nose and eyes corresponding to genes and then generate two alien babies. Then pictures of the parents and babies are drawn, with similarities and differences noted and explained.

Type: Lesson Plan

Roll of the Dice and Some Turkey Fun!:

Students will conduct experiments on their own to see the difference between experimental and theoretical probabilities.

Type: Lesson Plan

A Roll of the Dice:

What are your chances of tossing a particular number on a number cube? Students collect data by experimenting and then converting the data in terms of probability. By the end of the lesson, students should have a basic understanding of simple events.

Type: Lesson Plan

Marble Mania:

In this lesson, "by flipping coins and pulling marbles out of a bag, students begin to develop a basic understanding of probabilities, how they are determined, and how the outcome of an experiment can be affected by the number of times it is conducted." (from Science NetLinks)

Type: Lesson Plan

M & M Candy: I Want Green:

"Students compare mathematical expectations and experimental probability; then explain any difference in the two numbers. Students use colored candy pieces (such as M & M's) for their data collection, comparisons, and explanations." from Beacon Learning Center.

Type: Lesson Plan

Who Would Have Figured? (Probability):

"Students discover what happens when a coin is tossed a few times versus when a coin is tossed many times. They discover the answer to "What is the probability of heads, and does it change as you toss the coin more times?" from Beacon Learning Center.

Type: Lesson Plan

This activity will allow students to explore the concept of simple probability using a random selection of multi-colored beads.

Type: Lesson Plan

Introduction to Probability:

This resource is designed to introduce students to the concept of probability: the probability of a rare event is represented by a positive number close to zero, the probability of a nearly certain event occurring is represented by a positive number slightly less than one. Students will indicate the approximate probability of events on a number line and determine which events are more likely than others.

Type: Lesson Plan

## Original Student Tutorials

Introduction to Probability:

Learn how to calculate the probability of simple events, that probability is the likeliness of an event occurring, and that some events may be more likely than others to occur in this interactive tutorial.

Type: Original Student Tutorial

Alice in Mathematics-Land:

Help Alice discover that compound probabilities can be determined through calculations or by drawing tree diagrams in this interactive tutorial.

Type: Original Student Tutorial

Predicting Outcomes at the Carnival:

Learn how to use probability to predict expected outcomes at the Carnival in this interactive tutorial.

Type: Original Student Tutorial

## Perspectives Video: Experts

How Math Models Help Insurance Companies After a Hurricane Hits:

Hurricanes can hit at any time! How do insurance companies use math and weather data to help to restore the community?

Type: Perspectives Video: Expert

Probabilistic Weather Modeling:

Meteorologist from Risk Management discusses the use of probability in predicting hurricane tracks.

Type: Perspectives Video: Expert

Let's Make a Math Deal:

Should I keep my choice or switch?  Learn more about the origins and probability behind the Monty Hall door picking dilemma and how Game Theory and strategy effect the probability.

Type: Perspectives Video: Expert

History of Probability and the Problem of Points:

What was the first question that started probability theory?

Type: Perspectives Video: Expert

The Titanic 1:

This task asks students to calculate probabilities using information presented in a two-way frequency table.

Tossing Cylinders:

The purpose of this task is to provide students with the opportunity to determine experimental probabilities by collecting data. The cylindrical objects used in this task typically have three different resting positions but not all of these may be equally likely and some may be extremely unlikely or impossible when the object is tossed. Furthermore, obtaining the probabilities of the outcomes is perhaps only possible through the use of long-run relative frequencies. This is because these cylinders do not have the same types of symmetries as objects that are often used as dice, such as cubes or tetrahedrons, where each outcome is equally likely.

How Many Buttons?:

This resource involves a simple data-gathering activity which furnishes data that students organize into a table. They are then asked to refer to the data and determine the probability of various outcomes.

Waiting Times:

As the standards in statistics and probability unfold, students will not yet know the rules of probability for compound events. Thus, simulation is used to find an approximate answer to these questions. In fact, part b would be a challenge to students who do know the rules of probability, further illustrating the power of simulation to provide relatively easy approximate answers to wide-ranging problems.

Rolling Dice:

This task is intended as a classroom activity. Students pool the results of many repetitions of the random phenomenon (rolling dice) and compare their results to the theoretical expectation they develop by considering all possible outcomes of rolling two dice. This gives them a concrete example of what we mean by long term relative frequency.

Rolling Twice:

The purpose of this task is for students to compute the theoretical probability of a compound event. Teachers may wish to emphasize the distinction between theoretical and experimental probabilities for this problem. For students learning to distinguish between theoretical and experimental probability, it would be good to find an experimental probability either before or after students have calculated the theoretical probability.

Sitting Across From Each Other:

The purpose of this task is for students to compute the theoretical probability of a seating configuration. There are 24 possible configurations of the four friends at the table in this problem. Students could draw all 24 configurations to solve the problem but this is time consuming and so they should be encouraged to look for a more systematic method.

## Text Resource

Shuffling Shenanigans:

This informational text resource is intended to support reading in the content area. A student in love with magic card tricks asks and answers his own math questions after pursuing a career as a mathematician in order to solve them. How many times must a deck be shuffled to achieve a truly random mix of cards? The answer lies within.

Type: Text Resource

## Tutorials

Finding Probability:

This video demonstrates several examples of finding probability of random events.

Type: Tutorial

The Limits of Probability:

This video discusses the limits of probability as between 0 and 1.

Type: Tutorial

Comparing Theoretical to Experimental Probabilites:

This video compares theoretical and experimantal probabilities and sources of possible discrepancy.

Type: Tutorial

Finding Probability of a Simple Event:

This video demonstrates how to find the probability of a simple event.

Type: Tutorial

Making Predictions with Probability:

Watch the video as it predicts the number of times a spinner will land on a given outcome.

Type: Tutorial

Constructing Probability Model from Observations:

This video demonstrates development and use of a probability model.

Type: Tutorial

Compound Sample Spaces:

This video explores how to create sample spaces as tree diagrams, lists and tables.

Type: Tutorial

Probability of Compound Events:

This video shows how to use a sample space diagram to find probability.

Type: Tutorial

Die Rolling Probability:

The video will show how to use a table to find the probability of a compound event.

Type: Tutorial

Count Outcomes Using a Tree Diagram:

This video shows an example of using a tree diagram to find the probability of a compound event.

Type: Tutorial

## Video/Audio/Animations

Compound Probability of Independent Events:

This 6-minute video provides an example of how to work with compound probability of independent events through the example of flipping a coin. If you flip a coin and it lands on heads, is the next flip more likely to be tails? Or are those events independent?

Type: Video/Audio/Animation

Probability Explained:

This 8-minute video provides an introduction to the concept of probability through the example of flipping a coin and rolling a die.

Type: Video/Audio/Animation

## Virtual Manipulatives

Fire!!:

In this activity, students burn a simulated forest and adjust the probability that the fire spreads from one tree to the other. This activity allows students to explore the idea of chaos in a simulation of a realistic scenario. This activity includes supplemental materials, including background information about the topics covered, a description of how to use the application, and exploration questions for use with the java applet. This activity would work well in mixed ability groups of two for about 30-35 minutes if you use the provided exploration questions and 10-15 minutes otherwise.

Type: Virtual Manipulative

Spinner:

In this activity, students adjust how many sections there are on a fair spinner then run simulated trials on that spinner as a way to develop concepts of probability. A table next to the spinner displays the theoretical probability for each color section of the spinner and records the experimental probability from the spinning trials. This activity allows students to explore the topics of experimental and theoretical probability by seeing them displayed side by side for the spinner they have created. This activity includes supplemental materials, including background information about the topics covered, a description of how to use the application, and exploration questions for use with the java applet.

Type: Virtual Manipulative

Interactive Marbles:

This online manipulative allows the student to simulate placing marbles into a bag and finding the probability of pulling out certain combinations of marbles. This allows exploration of probabilities of multiple events as well as probability with and without replacement. The tabs above the applet provide access to supplemental materials, including background information about the topics covered, a description of how to use the application, and exploration questions for use with the Java applet.

Type: Virtual Manipulative

The students will play a classic game from a popular show. Through this they can explore the probability that the ball will land on each of the numbers and discover that more accurate results coming from repeated testing. The simulation can be adjusted to influence fairness and randomness of the results.

Type: Virtual Manipulative

Random Drawing Tool - Individual Trials (Probability Simulation):

This virtual manipulative allows one to make a random drawing box, putting up to 21 tickets with the numbers 0-11 on them. After selecting which tickets to put in the box, the applet will choose tickets at random. There is also an option which will show the theoretical probability for each ticket.

Type: Virtual Manipulative

## Student Resources

Vetted resources students can use to learn the concepts and skills in this topic.

## Original Student Tutorials

Introduction to Probability:

Learn how to calculate the probability of simple events, that probability is the likeliness of an event occurring, and that some events may be more likely than others to occur in this interactive tutorial.

Type: Original Student Tutorial

Alice in Mathematics-Land:

Help Alice discover that compound probabilities can be determined through calculations or by drawing tree diagrams in this interactive tutorial.

Type: Original Student Tutorial

Predicting Outcomes at the Carnival:

Learn how to use probability to predict expected outcomes at the Carnival in this interactive tutorial.

Type: Original Student Tutorial

The Titanic 1:

This task asks students to calculate probabilities using information presented in a two-way frequency table.

Tossing Cylinders:

The purpose of this task is to provide students with the opportunity to determine experimental probabilities by collecting data. The cylindrical objects used in this task typically have three different resting positions but not all of these may be equally likely and some may be extremely unlikely or impossible when the object is tossed. Furthermore, obtaining the probabilities of the outcomes is perhaps only possible through the use of long-run relative frequencies. This is because these cylinders do not have the same types of symmetries as objects that are often used as dice, such as cubes or tetrahedrons, where each outcome is equally likely.

How Many Buttons?:

This resource involves a simple data-gathering activity which furnishes data that students organize into a table. They are then asked to refer to the data and determine the probability of various outcomes.

Waiting Times:

As the standards in statistics and probability unfold, students will not yet know the rules of probability for compound events. Thus, simulation is used to find an approximate answer to these questions. In fact, part b would be a challenge to students who do know the rules of probability, further illustrating the power of simulation to provide relatively easy approximate answers to wide-ranging problems.

Rolling Dice:

This task is intended as a classroom activity. Students pool the results of many repetitions of the random phenomenon (rolling dice) and compare their results to the theoretical expectation they develop by considering all possible outcomes of rolling two dice. This gives them a concrete example of what we mean by long term relative frequency.

Rolling Twice:

The purpose of this task is for students to compute the theoretical probability of a compound event. Teachers may wish to emphasize the distinction between theoretical and experimental probabilities for this problem. For students learning to distinguish between theoretical and experimental probability, it would be good to find an experimental probability either before or after students have calculated the theoretical probability.

Sitting Across From Each Other:

The purpose of this task is for students to compute the theoretical probability of a seating configuration. There are 24 possible configurations of the four friends at the table in this problem. Students could draw all 24 configurations to solve the problem but this is time consuming and so they should be encouraged to look for a more systematic method.

## Tutorials

Finding Probability:

This video demonstrates several examples of finding probability of random events.

Type: Tutorial

The Limits of Probability:

This video discusses the limits of probability as between 0 and 1.

Type: Tutorial

Comparing Theoretical to Experimental Probabilites:

This video compares theoretical and experimantal probabilities and sources of possible discrepancy.

Type: Tutorial

Finding Probability of a Simple Event:

This video demonstrates how to find the probability of a simple event.

Type: Tutorial

Making Predictions with Probability:

Watch the video as it predicts the number of times a spinner will land on a given outcome.

Type: Tutorial

Constructing Probability Model from Observations:

This video demonstrates development and use of a probability model.

Type: Tutorial

Compound Sample Spaces:

This video explores how to create sample spaces as tree diagrams, lists and tables.

Type: Tutorial

Probability of Compound Events:

This video shows how to use a sample space diagram to find probability.

Type: Tutorial

Die Rolling Probability:

The video will show how to use a table to find the probability of a compound event.

Type: Tutorial

Count Outcomes Using a Tree Diagram:

This video shows an example of using a tree diagram to find the probability of a compound event.

Type: Tutorial

## Video/Audio/Animations

Compound Probability of Independent Events:

This 6-minute video provides an example of how to work with compound probability of independent events through the example of flipping a coin. If you flip a coin and it lands on heads, is the next flip more likely to be tails? Or are those events independent?

Type: Video/Audio/Animation

Probability Explained:

This 8-minute video provides an introduction to the concept of probability through the example of flipping a coin and rolling a die.

Type: Video/Audio/Animation

## Virtual Manipulatives

Spinner:

In this activity, students adjust how many sections there are on a fair spinner then run simulated trials on that spinner as a way to develop concepts of probability. A table next to the spinner displays the theoretical probability for each color section of the spinner and records the experimental probability from the spinning trials. This activity allows students to explore the topics of experimental and theoretical probability by seeing them displayed side by side for the spinner they have created. This activity includes supplemental materials, including background information about the topics covered, a description of how to use the application, and exploration questions for use with the java applet.

Type: Virtual Manipulative

Interactive Marbles:

This online manipulative allows the student to simulate placing marbles into a bag and finding the probability of pulling out certain combinations of marbles. This allows exploration of probabilities of multiple events as well as probability with and without replacement. The tabs above the applet provide access to supplemental materials, including background information about the topics covered, a description of how to use the application, and exploration questions for use with the Java applet.

Type: Virtual Manipulative

The students will play a classic game from a popular show. Through this they can explore the probability that the ball will land on each of the numbers and discover that more accurate results coming from repeated testing. The simulation can be adjusted to influence fairness and randomness of the results.

Type: Virtual Manipulative

Random Drawing Tool - Individual Trials (Probability Simulation):

This virtual manipulative allows one to make a random drawing box, putting up to 21 tickets with the numbers 0-11 on them. After selecting which tickets to put in the box, the applet will choose tickets at random. There is also an option which will show the theoretical probability for each ticket.

Type: Virtual Manipulative

## Parent Resources

Vetted resources caregivers can use to help students learn the concepts and skills in this topic.

The Titanic 1:

This task asks students to calculate probabilities using information presented in a two-way frequency table.

Tossing Cylinders:

The purpose of this task is to provide students with the opportunity to determine experimental probabilities by collecting data. The cylindrical objects used in this task typically have three different resting positions but not all of these may be equally likely and some may be extremely unlikely or impossible when the object is tossed. Furthermore, obtaining the probabilities of the outcomes is perhaps only possible through the use of long-run relative frequencies. This is because these cylinders do not have the same types of symmetries as objects that are often used as dice, such as cubes or tetrahedrons, where each outcome is equally likely.

How Many Buttons?:

This resource involves a simple data-gathering activity which furnishes data that students organize into a table. They are then asked to refer to the data and determine the probability of various outcomes.

Waiting Times:

As the standards in statistics and probability unfold, students will not yet know the rules of probability for compound events. Thus, simulation is used to find an approximate answer to these questions. In fact, part b would be a challenge to students who do know the rules of probability, further illustrating the power of simulation to provide relatively easy approximate answers to wide-ranging problems.

Rolling Dice:

This task is intended as a classroom activity. Students pool the results of many repetitions of the random phenomenon (rolling dice) and compare their results to the theoretical expectation they develop by considering all possible outcomes of rolling two dice. This gives them a concrete example of what we mean by long term relative frequency.