November 2, 2009

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This resource guide provides links to exemplary resources and insight on how to teach mathematics and science concepts at the middle school level. Concepts supported by this particular guide include decimals, fractions, division of whole numbers, and geometry.

The guides provide information on the needed content knowledge, science and mathematical pedagogical knowledge, exemplary lessons and activities, career information, and correlations to national mathematics and science standards.

- Mathematics > General

- Grade 6
- Grade 7
- Grade 8

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**Activities for the Classroom**

These activities involve students in solving intriguing problems, each in the context of a basic probability concept. Some use virtual simulations in which students gather data from a thousand throws of a die or tosses of a coin in mere moments, allowing them to concentrate on the essentials of the problem. All are adaptable to the classroom setting, with or without a computer at hand.

**Simulations and online interactive activities**

An extremely versatile applet! In a box are shown the numbers 1-15, each a different color. Users can select which of these numbers, and how many of each, to add to the pot. As many random draws as 10000 can be made from the pot in just a few seconds. A chart shows the results of the simulated drawing with replacement from the numbers in the pot, as well as the theoretical probability of the drawing. This applet can be used to simulate flipping a coin or tossing a die or picking colored marbles from a hat.

Understanding Experimental Probability

Students can choose one of six available spinners, figure the theoretical probability of the needle landing on each color of the spinner, then spin it hundreds of times in a second to compare the experimental probability. Or they can work with two regular 6-sided number cubes or even design their own number cubes, and again experiment with the "what should happen" and the "what actually happened."

Another experiment with theoretical and experimental probability! Students create an interactive spinner with one to twelve sectors; the area of each segment is shown as a percentage in a table. As students spin, they see how many times the needle actually landed on each segment, also shown as a percentage, and compare the two probabilities.

The applet simulates the launch of a three-stage rocket; a successful launch requires that all three stages pass tests before takeoff. Students set the probability of passing at each stage on a scale from 0 to 100 percent. After each launch attempt, success or failure is reported, and the overall success rate is given as a percent and as the cumulative number of successful and failed launches. This applet is unique in that it can be used to observe a multistage event, each stage with a different probability of success.

Two [or more] players each roll a die, and the lucky player moves one step to the finish. Students can decide which rolls win, the length of the race, and how many steps the winner of the roll takes to the finish line. They can also see a second how many times each player would win out of 1000 games. Playing the game is easy; the trick is explaining why one player, over the long haul, wins so often.

Buffon's Needle involves dropping a needle on a lined sheet of paper, then counting the number of times the needle crosses a line. Remarkably, the probability of the needle landing on a line relates directly to the value of pi. Complete explanation, included for the teacher, involves trigonometry, but even without this in-depth explanation, this is an "Ah ha!" moment for your students!

**Open-ended questions and hands-on activities**

Bowl 'em over: does he have a chance?

The initial question concerns averages: Given the scores of Helix’s first five games, what does he have to bowl in his last game to win the tournament? But then the question turns to the probability of his scoring that many points in the last game. Explained here are two suggested solutions, each from a different angle.

How could I send the check and not pay the bill?

If you’re not paying attention to what you’re doing, what is the probability that you will put checks into the correct envelopes? Start with checks a, b, and c that have to go into envelopes A, B, and C. The solution is shown using three different approaches: a table, a tree diagram, and geometry.

Combination locks: I forgot the combination! How many combinations will I have to try?

In this activity, students find the number of possible combinations for a lock. The combination uses three numbers, each from 0-39. Students first consider a simpler problem, a combination lock that uses only the numbers 1 to 3. This is thoroughly explained and illustrated with a tree diagram. The solution to the easier problem is then generalized to finding the number of possible arrangements for any combination lock. . Further challenges ask students to consider the possible combinations for other types of locks and the total number of phone numbers possible for an area code.

I win!: she always wins, it's not fair!

The activity begins with playing a dice game but quickly moves to questioning whether or not the game is fair. The page offers a solution—and a challenge to students to change the game rules to make it fair!

Capture-Recapture: How Many Fish in the Pond?

How do those wildlife experts estimate how many fish are in a pond? They use a method called capture-recapture, a statistical tool that students use in this activity to estimate the total number of fish in a pond, given the numbers of fish initially tagged and released, the tagged fish recaptured, and the total number of recaptured fish. This tool allows wildlife experts to make predictions, one objective of probability theory, about population size and growth.

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