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Learning Probability

With Technology

And Manipulatives

A Five-Day Lesson Plan for 7th Grade Mathematics By: Anthony J. Nazzarett

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Standards that will be addresses in this lesson plan. New York State Learning Standards: 7.S.1 Identify and collect data using a variety of methods 7.S.3 Convert raw data into double bar graphs and double line graphs 7.S.6 Read and interpret data represented graphically 7.S.8 Interpret data to provide the basis for predictions and to establish experimental probabilities 7.S.9 Determine the validity of sampling methods to predict outcomes 7.S.10 Predict the outcome of an experiment 7.S.11 Design and conduct an experiment to test predictions 7.S.12 Compare actual results to predicted results NCTM Standards:

• Number and Operations • Data Analysis and Probability • Problem Solving • Communication • Representation

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Materials:

1. Classroom set of Dice 2. Classroom set of Standard Playing Cards 3. Classroom set of Spinners 4. Classroom set of Coins 5. Classroom set of Marbles 6. Classroom set of Snap Cubes 7. Classroom set of Texas Instruments TI-84 Silver

Edition and TI View Screen 8. Access to Internet

Objectives for Students:

1. Students will be able to collect, organize, display and analyze data.

2. Students will be able to make predictions that are based upon data analysis.

3. Students will be able to understand and apply the concepts of probability.

4. Students will be able to compute the probability of an event.

5. Students will be able to communicate their mathematical thinking coherently and clearly to peers, teachers and others.

6. Students will be able to analyze and evaluate the mathematical thinking and strategies of others.

7. Students will be able to use the language of mathematics to express mathematical ideas precisely.

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Daily Objective Overview:

Day 1: 1. Introduction to Probability 2. Introduction to Vocabulary of Probability

Day 2:

1. Introduction to Calculating Single Event Probability 2. Introduction to Manipulatives

Day 3:

1. Introduction to Calculating Multi Event Probability Day 4:

1. Introduction to the Concept of Replace and Not Replace 2. Introduction to M&M Experiment

Day 5: 1. Introduction to Technology for Probability

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Day 1 of Lesson Plan: Daily Objectives: Today’s class will give students the elementary building blocks and vocabulary necessary to understand and compute probability problems. Today’s Topics: This class will deal with the topics of experiments, sample space, fair games, outcomes and tree diagrams. Students will be given a worksheet that will be worked on in class with instruction from the teacher and input from students. Outline of Today’s Lesson: 1. Opening discussion 2. Introduction to manipulatives and vocabulary. 3. Worksheet to be finished in class by students with teacher and peer

instruction. 4. Closing discussion on class and preview of next class. 5. Homework assignment explanation. Opening Discussion: Class will start with the teacher doing a few experiments to give the students an introduction to probability. Experiments will include a coin toss, dice roll and spinning a spinner. Students will then be asked a series of question regarding the experiments and their beliefs on what probability measures.

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Instruction: Experiment -- an activity where results can be observed and record

Teacher example: Teacher will flip a coin and roll a die. Fair -- an object, game, or experiment where all of the outcomes are equally probable Teacher example: Teacher will flip a coin with heads and tails to show

fair game. Teacher will flip a two-headed coin to show students an example of an unfair game, showing that tails is not possible.

Sample space -- the set of all possible outcomes of an experiment

Teacher example: Teacher will roll a die to show the students all the possible outcomes: 1,2,3,4,5 and 6.

Outcome -- one of the possible results of an experiment

Teacher example: Teacher will pull a marble out of a bag that contains red, green, blue and yellow marbles. Teacher will show that picking a red marble is one possible outcome.

Tree Diagram – a visual representation that illustrates a sample space Teacher example: Teacher will construct a tree diagram to represent rolling a die.

Roll Die

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Name:_____________________ Date: ________________

Worksheet 1

Vocabulary: Experiment -- an activity where results can be observed and record Fair -- an object, game, or experiment where all of the outcomes are equally probable Sample space -- the set of all possible outcomes of an experiment Outcome -- one of the possible results of an experiment Tree Diagram – a visual representation that illustrates a sample space Applying Definitions:

1. List three activities that would be considered EXPERIMENTS: A. B. C.

2. List two experiments that would be considered FAIR games. List two

experiments that would not be considered fair games. Fair: Not Fair: A. A. B. B.

3. From the definitions above list a possible OUTCOME for the following experiments. A. Coin toss B. Standard Dice Rolling C. Picking a red card out of a standard Deck of Cards. D. Pulling a marble out of a bag that contains a red, blue, green and

yellow marble.

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4. For the following experiments list all possible outcomes or SAMPLE SPACE for each. A. Coin toss B. Standard Dice Rolling C. Picking a red card out of a standard Deck of Cards. D. Pulling a marble out of a bag that contains a red, blue, green and yellow marble.

5. Draw a TREE DIAGRAM that represents the following sample

spaces. A. Coin toss B. Standard Dice Rolling C. Picking a red card out of a standard Deck of Cards. D. Pulling a marble out of a bag that contains a red, blue, green and yellow marble.

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Student Name: ___________________ Date: ________________

Homework Assignment 1

1. Toss a coin five times and record each outcome. 1. 2. 3. 4. 5. 2. Roll a die five times and record each outcome. 1. 2. 3. 4. 5. 3. Pull a card out of a deck five time and record color. Replace card before each experiment. 1. 2. 3. 4. 5. 4. Draw a tree diagram to represent a person first flipping a fair

coin and then rolling a die.

5. Show the sample space if a person flips a coin three times.

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Day Two of Lesson Plan: Daily Objective: Today students will begin working on the outcomes of single events. Students will start working with manipulatives to help them better understand probability. Today’s Topics: This class will deal with experiments and outcomes. Students will be given manipulative and a work sheet to work on in class. Outline of Today’s Lesson:

1. Opening activity will involve determining the probability of a four colored spinner

2. Middle activity will involve determining the probability of a six-sided die.

3. The closing activity will involve determining the probability of a picking a colored marble randomly out of a bag of marbles.

Materials: Colored Spinner, Dice, Jar of Multi Colored Marbles. Opening activity: The first activity will involve a Four Colored Spinner.

Teacher will show the students that the spinner has four different colors and that each color has the same chance of the pointer landing on it. Teacher will then have students determine the sample space for this activity, which are red, blue, green and yellow. Students will then be instructed that the probability of an event is the number of way an event can occur divided by the total number of possible outcomes.

The Number Of Ways Event A Can Occur P(A) = The Total Number Of Possible Outcomes

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P(Color) = Color/Total number of colors P(Green) = ¼ P(Yellow) = ¼ P(Red) = ¼ P(Blue) = ¼ Middle Activity: This activity will involve rolling a die. The teacher will give each student a die for this activity. Students will first determine the sample space for this activity. Once the sample space has been determined the teacher will have students determine the probability of getting each number on the die by enter the numbers in the formula.

The Number Of Ways Event A Can Occur P (A) = The Total Number Of Possible Outcomes

P(Number) = Number/ Total numbers on die or 6 P(1) = 1/6 P(2) = 1/6 P(3) = 1/6 P(4) = 1/6 P(5) = 1/6 P(6) = 1/6

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Closing Activity: Jar of Marbles. For this activity we will use a jar containing five red, green, blue and yellow marbles for a total of 20 marbles. For this activity students will first determine the sample space and then determine he probability of getting each color.

The Number Of Ways Event A Can Occur P(A) = The Total Number Of Possible Outcomes

P(Colored Marble) = Color/ Total number of Marbles P(Red) = 5/20 P(Green) = 5/20 P(Blue) = 5/20 P(Yellow) = 5/20 Daily Assessment: Students will break up into groups of four. Students will be given the following worksheet that will contain some questions that will require manipulatives and some questions that require the students to draw an example to answer. Worksheet will be handed in at the end of class to be graded.

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Worksheet 2

Question 1. A jar of marbles contains 40 marbles: 15 green, 10 red, 12 yellow and 3 blue? List the probability for each color?

Now have one student pull a marble out of the bag and determine the probability of picking that marble.

Question 2. A deck of card contains 26 red and 26 black cards.

What is the probability of picking a red card? Now have one student pick a card and list the probability of picking that card.

Question 3. A bag contains 11 snap cubes of which 7 are green and 4 are red.

Now have one student pick a snap cube out of the bag and list the probability of picking that snap cube.

Question 4. A fair coin is tossed. What is the probability of the coin landing on heads? Now have one student toss a coin and list the probability. Question 5. A bag contains 10 snap cubes of which 2 are red, 4 are black and 4 are white. What is the probability of picking a red snap cube?

Now have one student pick a snap cube and list the probability of picking that snap cube.

Question 6. A person goes into a supermarket that has 12 rows, but only 1 row has potato chips.

What is the probability of picking the row with potato chips the first time?

Question 7. A person rolls 2 dice.

What is the probability that the sum of the 2 dice will be an even number?

Question 8. A bag contains each letter of the alphabet. What is the probability of picking a vowel?

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Question 9. A cooler has 12 cans of soda, 4 cans each of Pepsi, Coke and Root Beer? What is the probability of picking a Pepsi? Question 10. There are 12 players including you on your baseball team. What is the probability of the coach picking you to bat first?

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Day Three of Lesson Plan: Lesson Topic: Calculating Probability of multiple events. Objectives: Students will be able to:

1. Understand the concept of probability 2. Determine the probability of multiple events 3. Understand the vocabulary of probability

Opening Activity: Teacher will start this activity off with flipping a fair coin and then rolling a fair die. Teacher will then construct a tree diagram to show students that there are 12 different possible outcomes.

Flip Coin

Heads

Tails

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6

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Roll Die

So with the diagram students will be able to see that there are 12 possible outcomes. Mathematically students will be shown that the probability of the coin landing on head and then rolling a 5 would be: (1/2) * (1/6) = 1/12. The 1/ 2 equaling the probability of rolling a heads or tails, and the 1/6 equaling the probability of rolling a number on a die.

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Middle Activity: Using manipulative to determine probability of multiple events. Students will get into groups of two with one students doing experiment while other student records the information. Students will switch jobs for each question. Materials: Sac of 20 Marbles with 5 green, red, yellow and blue marbles. 1 for each group Deck of Standard Playing Cards Coins- 2 per group Dice - 2 per group Four-Colored Spinner – green, red, yellow and blue

Probability Experiments

1. The first experiment will have one student flipping a coin then rolling a die. The other students will record the events.

2. The second experiment will have a student roll a die and then pull a marble out of the sac. The other student will record the events.

3. The third experiment will have a student spin the spinner and then roll a die. The other student will record the events.

4. The fourth experiment will have one student select a card from the deck and then roll a die. The other student will record the events.

5. The fifth experiment will have one student roll three dice on after another. The second student will record the event.

For each exercise 1-5 list the sample space and construct a tree diagram to represent each experiment.

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Day Four of Lesson Plan: Daily Objective: Today students will begin receiving instruction on the topics of replacing and not replacing and how it affects probability. Today’s Topics: The topic we will discuss today will involve a few experiments that deal with replacing and not replacing objects during experiments and how the outcomes of an experiment change. Materials: Deck of Cards Bag of Marbles

Opening Activity: Teacher will show a couple of example to the students: Experiment 1:

1. Teacher will place 20 marbles in a bag. Five each of red, blue, green and yellow.

2. Teacher will have a student pull a marble out of bag. 3. For example the student pulls out a red marble 4. Student will determine the probability of picking red marble

5/20 or ¼ 5. Teacher will replace marble and have another student pick a

marble. 6. For example student picks a red marble again. Probability is

still 5/20 or ¼ 7. Teacher will have another student pick a marble, this time

without replacing the first. Suppose 2nd marble picked is blue. What was the probability of picking a blue marble? ¼*5/19 = 5/76

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Experiment 2:

1. Teacher will have a student pick one card out of a standard deck of playing cards.

2. For example the student pulls out an ace of spades. 3. Students will determine the probability of picking an ace of

spades is 1/52 4. Teacher will then replace card and have another student pick a

card. 5. For example student picks a 4 of clubs. Without replacing

another student picks a card. What is the probability that second student picks a club?

6. There will be a classroom discussion about the answer. Closing Activity: The M&M experiment will end the introduction into the concept of replacing and not replacing an object during an experiment. This experiment will contain both teacher instruction and student involvement.

1. Teacher will give each student a mini bag of M&M’s 2. Each student will open their bag of M&M’s and count them 3. Each student will determine the total for each color in the bag 4. Students will determine the probability of picking each color 5. Students will then place all the M&M’s in the bag and pick one

randomly 6. Students will determine the probability of picking that color 7. Next student will determine the probability of picking the same color 8. Students will determine the probability for picking all colors to end

this exercise

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Day Five of Lesson Plan: Topic: Determining Probability with Technology Daily Objectives: Today’s class will give students the chance to see how they can use technology to better and easier understand probability. The TI-84 Plus will give us the advantage of being able to flip a coin or toss a die a thousand times in the matter of minutes. Materials: TI-84 Plus Silver Edition Activity 1: The coin toss. The probability of tossing a coin and having it land on head or tails is ½. Toss the coin ten times and record your results. Before you toss the coin write down the number of times you expect to get a heads? ______. A tails? ______. 1. T 2. T 3. T 4. T 5. H 6. H 7. H 8. T 9. H 10. T After you have recorded your results answer the following questions.

1. How many times did the coin land on heads? 2. How many times did the coin land on tails? 3. What did you expect the outcomes for heads and tails to be? 4. Do your expected values and your results equal each other? 5. If your expected values and actual results equal each other do you

believe that this is always the case? Explain. 6. If your expected values and actual results are not equal explain how

this happened? If they are the same skip to question 6.

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Now lets see what happens when we increase the numbers of trials using the TI-84 Plus Silver Edition. Instruction for using Probability Simulation software o The TI-84 Plus.

1. Turn Calculator on 2. Press APPS key 3. Arrow down until the program Prob Sim is highlighted 4. Press enter and then any key to start program 5. Highlight Toss Coin and press enter 6. Press Zoom and enter the number 50 into the Trial Set: prompt and

then press the Graph key to return to main screen 7. Now press the Window key to toss the coin fifty times 8. The calculator will show a graph to represent the outcomes after fifty

tosses 9. Next we will record the outcomes after fifty. To do this we will need

to press the right arrow to see the number of tails and then press the right arrow again to record the number of heads.

10. After recording the corresponding outcome in appropriate boxes, press the WINDOW key to toss the coin another fifty times. Record the outcomes for tails and heads as explained in step 9. Continue this until you have recorded 1000 trials.

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Number of Trials Tails Heads

50 29 21 100 54 46 150 84 66 200 102 98 250 130 120 300 154 146 350 180 170 400 204 196 450 231 219 500 259 241 550 283 267 600 306 294 650 330 320 700 354 346 750 379 371 800 400 400 850 432 418 900 451 449 950 473 477

1000 496 504 Think About It. 1. What is the table telling us about probability? Answers will vary. 2. What do you think would happen if we continued on until 100,000 or 1,000,000 trials? Answers will vary.

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Name: Students Name Date: Date of Assignment

Home Work Assignment 5

Activity 1: The die roll. The probability of rolling a number 1-6 on a standard die is 1/6. Roll the die six times and record your results. Before you roll the die write down the number of times you expect to get a 1? 1 2? 1 3? 1 4? 1 5? 1 6? 1 1. 1 2. 6 3. 2 4. 4 5. 1 6. 3 After you have recorded your results answer the following questions.

7. How many times did the die land on 1? 2 2? 1 3? 1 4? 1 5? 0 6? 1

8. What did you expect the outcomes for each number to be? 9. Do your expected values and your results equal each other? 10. If your expected values and actual results equal each other do you

believe that this is always the case? Explain. Answers will vary. 11. If your expected values and actual results are not equal explain how

this happened? Answers will vary.

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Now lets see what happens when we increase the numbers of trials using the TI-84 Plus Silver Edition. Instruction for using Probability Simulation software o The TI-84 Plus.

1. Turn Calculator on 2. Press APPS key 3. Arrow down until the program Prob Sim is highlighted 4. Press enter and then any key to start program 5. Highlight Roll Dice and press enter 6. Press Zoom and enter the number 50 into the Trial Set: prompt and

then press the Graph key to return to main screen 7. Now press the Window key to roll the die fifty times. 8. The calculator will show a graph to represent the outcomes after fifty

rolls. 9. Next we will record the outcomes after fifty. To do this we will need

to press the right arrow to see the number of 1’s and then press the right arrow again to record the number of 2’s and continue on until all numbers are recorded.

10. After recording the corresponding outcome in appropriate boxes, press the WINDOW key to roll the die another fifty times. Record the outcomes for each number as explained in step 9. Continue this until you have recorded 1000 trials.

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Number of

Rolls 1 2 3 4 5 6 50 13 6 9 5 5 12 100 27 10 18 11 18 16 150 36 15 28 16 26 29 200 48 19 42 24 35 32 250 57 28 50 33 43 39 300 64 37 58 40 50 51 350 68 50 70 47 56 59 400 77 57 78 53 64 71 450 86 69 87 59 69 80 500 98 78 94 62 82 86 550 107 88 97 69 93 96 600 113 96 110 76 105 100 650 120 107 117 86 110 110 700 129 117 121 98 115 120 750 134 127 133 109 122 125 800 139 135 143 117 131 135 850 144 142 153 127 138 146 900 158 146 161 135 148 152 950 165 154 167 142 157 165

1000 172 160 177 152 165 174

Assignment: Explain your results and draw a bar graph that represents the outcomes for each number.

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Resources

1. Epp, Susanna S. (1995). Discrete Mathematics with

Applications. Belmont, California: Wadsworth Group/ Thomson Learning.

2. Lappan, Glenda. (1993). Connected Mathematics: Data Analysis and Probability. Glenview, Illinios: Prentice Hall

3. Dressler, Isidore. (1990). Integrated Mathematics. New York, New York: Amsco School Publications.

4. http://education.ti.com 5. http://nysed.gov 6. http://nctm.gov

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