**Diagnostic Inventories for Cognition in Education (DICE)****An Assessment Tool for Probabilit**y

**Collection of Resources for Supporting Grades 6-8 Probability**

The following list of resources were developed or curated by experts in probability education on the DICE team.

Download a PDF of description of all resources here.

**Featured Resources**

The reading and lessons below are a great place to start to think about teaching some of the basic ideas in probability to middle school students.

Probability Distributions**Available from: **National Council of Teachers of Mathematics

Chapter 2 of Book Navigating through Probability Grades 6-8 (available for purchase) **Brief description: **This free sample chapter is a brief reading that provides teachers with background information that can be helpful in understanding some of the probability standards and objectives often taught in middle school.**Concepts:** sample space, outcomes, theoretical probability distributions, relative frequencies, experimental (or empirical) probability, basic probability rules.

Playing with Probability**Available from:** Shodor Foundation**Brief description: **A good introductory activity where students learn how to calculate both theoretical and experimental probability by rotating through a series of work stations.**Concepts:** sample space, experimental and theoretical probability

Gumball Machine**Available from: **STEW online lesson journal**Brief description: **This lesson gives students the opportunity to explore and discuss the variation that occurs in sampling. Students are asked to imagine that a gumball machine contains 1 red, 2 green, 3 yellow, and 4 blue gumballs that were thoroughly mixed before they were put into the machine, and then consider the probability of different events occurring. **Concepts:** sample space, theoretical probability, probability of event, experimental probability, sampling, sample size, variation from expected

**More Probability Activities and Lesson Plans**

Spinner at the School carnival (equal sections)**Available from**: STEW online lesson journal**Brief Description**: This investigation focuses on guiding students as they collect and interpret data from a spinner experiment with outcomes that are equally likely. Students will have the opportunity to organize the collected data in tables and graphs and will draw conclusions and make predictions based on the data collected. **Concepts:** sample space, theoretical probability, probability of event, experimental probability, sampling, sample size, variation from expected, bar graphs, frequency tables

Spinner at the School carnival (unequal sections)**Available** **from:** STEW online lesson journal**Brief Description:** Students are asked to predict the outcomes of an unequally divided spinner. Students collect data through actual spins and construct frequency charts. They will then have an opportunity to revise their predictions. Classroom level data are collected, graphed, and used as a basis for a discussion about experimental data and expected outcomes.**Concepts:** sample space, theoretical probability, probability of event, experimental probability, sampling, sample size, variation from expected

Comparing Probability Comparison Tasks**Available from**: Hollylynne Lee, NC State University**Brief Description:** An activity that has students grapple with similarities and differences in two tasks where they are asked to make probability comparisons. One task compares the proportion of black marbles in two bags, while another task compares the proportion of females in two samples of babies born in hospitals.**Concepts:** probability of an event, ratios and proportions, sample space, sample size, variation from expected

Families of six children**Available from:** Hollylynne Lee, NC State University**Brief Description: **A lesson that considers a family of six children and whether it is more likely to have exactly 3 boys and 3 girls in a family of six children or to have more of one gender (e.g., 4 or more boys or 4 or more girls.). Students connect this to their everyday experiences with larger families as well as use a spreadsheet to generate a large sample of families with randomly assigned gender at birth (e.g., GGBBBG, GGGGBB). **Concepts: **probability of an event, combinatorics, sample size, empirical probability.

Coin Toss Activity+ Applet**Available from:** Shodor Foundation**Brief Description: **An easy to use tool for simulating a fair coin toss and observing empirical results as ratios, a list, and a table. A student handout can be used to get them quickly engaged.**Concepts:** experimental and theoretical probability, sample size

Conditional Probability and Simultaneous Events Activity+Applet**Available from:** Shodor Foundation**Brief Description: **This lesson is based on several problems, each with a somewhat unexpected and counterintuitive answer. This very difference in expectations and actual results leads to a deeper consideration of the related mathematics and to acquiring new tools for solving problems, namely the ideas and formulas connected with conditional probability and probability of simultaneous events.**Concepts:** independent events, conditional probability

Two Dice Snail Race Activity+Applet**Available from:** Transum.org**Brief Description: **This is a visual designed to be projected for whole class play. Two dice are rolled and the sum calculated (2-12). The snail with that number takes one step forward. The first snail to the end of the race track wins. Before playing the students could be asked to predict which snail will win. **Concepts:** compound probability with two independent events, sample space, experimental probability

**Online Applets and Games to Integrate into Your Lessons**

The free applets below could be used to generate data from probability experiments with small and large sample sizes. You could integrate these applets into lessons you already use to enhance students’ experiences with generating random events and observing trends in results, and how these may vary from expected.

Coin Tossing Applet**Available from:** Statistics Applets**Brief Description: **In this applet, you can set the true probability of heads for your virtual coin to be any value between 0-1, then toss the coin any number of times. Multiple representations are used to show the results so that students can attend to the sequence of results as well as the proportion of tosses that produce heads. There is an option to show the theoretical probability for comparison.

Adjustable Spinner**Available from:** Shodor Foundation**Brief Description: **A free applet that allows you to design a spinner with 1-12 sections and to also adjust the size of each section directly on the spinner. The theoretical probability of landing on each section will be displayed and users can spin the spinner to collect small or large number of trials.

CODAP and the Sampler plugin**Available from:** The Concord Consortium**Brief Description:** Within the free online data analysis and visualization tool CODAP, there is a plugin called a Sampler. *When in a New Document, click on the icon Plugin on the tool shelf and choose Sampler*. With the Sampler, you can design many different models and simulate data collection from that model. Watch an introductory video here that describes how to model a simple carnival game in the Sampler and generate simulate data.

**Learn More: Readings About Teaching and Learning Probability **

Below are several articles that provide explanations and activities for use with middle school students learning probability.

Probability Experiments with Shared Spreadsheets**Available from** National Council of Teachers of Mathematics

Mathematics Teaching in the Middle School (Author: Darin Beigie)**Brief Description: **An article that illustrates how to collect student data from an experiment with two dice to consider a larger sample of data from the class to compare experimental and theoretical probability for the sum of two dice.

Target geometry and probability using a dartboard**Available from**: National Council of Teachers of Mathematics

Mathematics Teaching in the Middle School (Authors: Nicole WIlliams and Carolyn Bruels)**Brief Description: **An article that describes an activity that connects area and probability through collecting data of random “throws” at 3 different dartboards.

What a Pip! Probability and Efron’s Dice**Available from**: National Council of Teachers of Mathematics

Mathematics Teaching in the Middle School (Authors: Ben C. Sloop and S. Megan Che)**Brief Description: **An article that describes a classroom activity and the probability concepts that students can learn by playing a game with two non-traditional “loaded” die.

Student’s understanding of variation in a probability environment**Available from** J. M. Shaughnessy and M. Ciancetta**Brief Description: **A brief reading that presents results from a research study of students in grades 6-12 considering the sample space of a compound event (two spinners) and determining the probability of a specific event. Typical ways that students reason and instructional strategies for helping develop better concepts are discussed.

Designing and using probability simulations**Available from:** Chapter 5 of Preparing to Teach Mathematics with Technology: An integrated approach to data analysis and probability (Authors: Hollylynne Lee, Karen Hollebrands, P. Holt Wilson)**Brief Description: **A reading and activity that helps teachers attend to how an empirical approach to probability that can be used to foster understanding of the effect of sample size and variability when comparing empirical data to an expected outcome based on a theoretical estimate of probability. In addition, teachers are exposed to the usefulness of using intervals as estimates in probabilistic situations rather than single value estimates.

Using data and simulations to motivate sampling distributions**Available from**: Chapter 6 of Preparing to Teach Mathematics with Technology: An integrated approach to data analysis and probability (Authors Hollylynne Lee, Karen Hollebrands, P. Holt Wilson)**Brief Description: **A reading and activity that allows teachers to consider pedagogical issues concerning use of real data as a springboard for probability lessons, the use of repeated samples in simulations to conceptualize effects of sample size, and ways in which simulations can support understanding sampling distributions and computation of theoretical probabilities using the binomial formula.