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Instructional Plan Template

Mathematics Governor’s Institute 2004

(Download as Microsoft word document: Instructional Plan Temp1.doc)

Names of group members: Tina Mitchell, Millie Lavelle, Richard Blessing, Margaret Watson

Topic/Theme: Experimental and Theoretical Probability

Level: Grades 6-8

Time Element: 2 – 50 minute class periods

NCTM Standards Addressed:

Data Analysis and Probability

In grades 3-5 all students should

· Collect data using observations, surveys, and experiments.

· Represent data using tales and graphs such as line plots, bar graphs, and line graphs.

· Propose and justify conclusions and predictions that are based on data and design studies to further investigate the conclusions or predictions.

· Describe events as likely or unlikely and discuss the degree of likelihood using such words as certain, equally likely, and impossible.

In grades 6 – 8 all students should

· Use proportionality and a basic understanding of probability to make and test conjectures about the results of experiments and simulations.

· Compute probabilities for simple compound events, using such methods as organized lists, tree diagrams, and area models.

PA Math Standards Addressed:

2.7.5.A

Perform simulations with concrete devices (e.g. dice, spinner) to predict the chance of an event occurring.

2.7.5.C

Express probabilities as fractions and decimals.

2.7.5.D

Compare predictions based on theoretical probability and experimental results.

2.7.5.E

Calculate the probability of a simple event.

2.7.8.C

Analyze predictions (e.g., election polls).

2.7.8.D

Compare and contrast results from observations and mathematical models.

2.7.8.E

Make valid inferences, predictions and arguments based on probability.

Math Assessment Anchors Addressed:

ME.1

Formulate or answer questions that can be addressed with data and/or organize, display, interpret or analyze data.

ME.3

Understand and/or apply basic concepts of probability or outcomes.

ME.4

Develop and/or calculate inferences and predictions or draw conclusions based on data or data displays.

Reading Assessment Anchors Addressed:

R7.A.2

Demonstrate the ability to understand and interpret nonfiction text, including informational, e.g., textbooks, print media (magazines, brochures, etc.), autobiographies, biographies, editorials and speeches appropriate to grade level.

Objectives:

Describe the difference between theoretical and experimental probabilities.
Make predictions, conduct experiments, and compare and contrast data to confirm or reject conjectures.
Determine that as the number of trials of an experiment increases the experimental probability approaches the theoretical probability.

Instructional Strategies and Plan (include strategies used to help different types of learners, i.e. auditory, visual, etc):

Anticipatory Set

The students will be given Activity 1 page 1 to complete.
The activity will allow the students to calculate theoretical probabilities of flipping a coin and landing on either heads or tails.
The activity will allow the students to make predictions of how many times a coin will land on heads or tails if it is flipped 25, 50, 100 and 1000 times.
The activity will allow the students to convert between fractions, decimals and percents.

Lesson

· As the previous activity is reviewed, definitions of theoretical and experimental probability will be discussed.

· The main activity will be introduced by modeling.

· Students will be in groups of 3 or 4.

· Several activity stations will be located around the classroom.

· Each group will rotate to each activity station and compile data for that particular activity.

· Once students have visited each activity station, they will compile and record their GROUP results on the appropriate transparency as well as discuss any findings (as described in their activity packet).

Activity Station 1: Penny Power (model this activity by flipping the coin 10 times)

Each group member is to flip a penny 25 times.
The number of heads and tails will be recorded in a table.
The students will then compare their experimental probabilities to their answer in the anticipatory set activity.
The group members will compile their data (how many heads they flipped as a group and the total number of flips they completed as a group).
The group will then calculate the experimental probability of the group data.
The students will then discuss how their group data compares to the predicted probability and the theoretical probability.
The group will then put their group results on the transparency labeled “Penny Power.”

Activity Station 2: Wheel of Fortune (Model this activity with 10 spins.)

Each group member is to spin the spinner 25 times.
The number of each value the spinner lands on will be recorded in a table.
The students will compare their experimental probabilities to their predicted probability.
The group members will compile their data (the number of each value they got as a group and the total number of spins they completed as a group).
The group will then calculate the experimental probability of the group data.
The students will then discuss how their group data compares to the predicted probability and the theoretical probability.
The group will then put their group results on the transparency labeled “Wheel of Fortune.”

Activity Station 3: Red Head (Model this activity with 10 ‘flips’ using the calculator.)

The students will simulate flipping a coin and getting heads or tails on their graphing calculators (directions attached).
1 = Heads 2 = Tails
The student will then spin a spinner (red, yellow, blue or green or what ever colors available!)

Make sure students flip and then spin, flip and then spin, and so on.

The objective is to get a Red Head.
It will be reviewed with the students how many possible outcomes there are. Make a diagram with the students of the different colors they would get with either Heads or Tails.

1R 1B 1Y 2R 2B 2Y
Ask what the theoretical probability is of getting a Blue Head (1/6).

The number of times each student gets a Red Head will be recorded in a table.
The students will compare their experimental probabilities to the theoretical probability.
The group members will compile their data (the number of Red Heads they got as a group and the total number of spins/flips they completed as a group).
The group will then calculate the experimental probability of the group data.
The students will then discuss how their group data compares to the predicted probability and the theoretical probability.
The group will then put their group results on the transparency labeled “Red Heads.”

Closure (class discussion)

For the first activity have a group give you their data (ask for the decimal representation as well) and ask the class, “How does this group’s data compare to the theoretical probability?”
Add another group’s data to the first and ask the same question – ask students to calculate the decimal and percentage representations.
Add the third group’s data to the first and second and ask the same question – have students calculate the decimal and percentage representations.
Keep doing this until there is a class total.
Ask students to write down anything they notice about the data that has been collected and the corresponding probabilities.

They will notice that as the number of trials increase the experimental probability gets closer to the theoretical probability.

If students have not figured this out, do the same process for the second activity. Otherwise verify their conjecture with the second activity.
If necessary do the same for the third activity.
Tell students that I am their classmate and I was absent today and that I need them to write me a paragraph describing what they learned about theoretical and experimental probabilities and how they relate to each other.

Materials/Resources:

TI-83 graphing calculators & Overhead View Screen
Spinners
Pennies
Number cubes
Transparencies

Interdisciplinary Connections:

· Reading

· Following directions

· Making predictions

· Making inferences

· Technology

· TI-83 Graphing Calculator

· Manipulatives (spinners, pennies, number cubes)

· Other

Assessment Strategies:

· Formative Evaluation (checking student understanding during the lesson):

o Teacher observation

o Class discussion

o Individual question/answer

· Summative Evaluation (How will it be determined that the objectives were achieved?):

o Activity Packet (3 separate activity sheets)

o Written conjectures of students (Closing Activity)

o Vocabulary and concept review

o Homework Assignment (Follow-up Activity)

Correctives/Remediation:

· Fraction, Decimal, Percent Conversions

· Comparing Fraction and Decimals

Extensions/Enrichment:

· Enrichment Activity, Stars That Rock

Special Accommodations (special needs students)

· Description of the Special Needs student selected:

This student is functioning three years below grade level. She is receiving learning support services for language arts and math and has difficulty following direction in all instructional areas. She has difficulties in language comprehension, vocabulary, direction following, event-sequencing and working memory. She can answer literal comprehension questions in all content areas and usually answers 1 out of 5 inferential questions correctly. She tends to jump into reading tasks without previewing material but has success when instructions are broken down and accompanied by modeling. She has difficulty with basic math facts for multiplication and division as well as problems involving multiple steps.

· Accommodations to use with this student:

o Each activity will be modeled by the classroom teacher.

o Manipulatives will be used to complete assigned activities.

o More space between sentences will be allotted on activity sheets.

o Vocabulary will be pre-taught.

o Calculators will be available for use

Calculator Instructions

To use the random number generator on your calculator, follow the instructions below.

1. Turn calculator ON

2. MATH PRB 5 (randInt)

3. randInt( should appear on your screen

4. randInt(1, 2) ENTER

(1,2) represents the range of values you have

Continue pressing enter until you complete the exercise.

Name ____________________________ Class _____

THEORETICAL VS. EXPERIMENTAL PROBABILITY

Part A

Activity #1: Penny Power

1. Calculate the following probabilities:

When flipping a coin, the probability of getting a head is:

FRACTION = ______ DECIMAL = ______ PERCENT = _______

When flipping a coin, the probability of getting a tail is:

FRACTION = ______ DECIMAL = ______ PERCENT = _______

2. Imagine you flipped a coin 25 times.

How many times would you expect to get heads? __________________

How many times would you expect to get tails? __________________

3. Imagine you flipped a coin 50 times.

How many times would you expect to get heads? __________________

How many times would you expect to get tails? __________________

4. Imagine you flipped a coin 100 times.

How many times would you expect to get heads? __________________

How many times would you expect to get tails? __________________

5. Imagine you flipped a coin 1000 times.

How many times would you expect to get heads? __________________

How many times would you expect to get tails? __________________

PENNY POWER

Part B

PUT IT TO THE TEST!

Flip a coin twenty-five times.
Create a frequency table of your results below.

6. In twenty-five trials you got ________ heads and ________ tails.

7. Based on these results calculate the experimental probability of getting a head.

FRACTION = _________ DECIMAL = ___________ PERCENT = ________

8. How many heads did you get as a group, and how many total flips were made?

Heads ________ Total Flips ________

Discuss the following questions with your group.

9. How does your individual data compare to the prediction you made in the introductory activity?

10. How does your individual data compare to your group’s data?

Name ____________________________ Class _____

THEORETICAL VS. EXPERIMENTAL PROBABILITY

Activity #2: Wheel of Fortune

1. Predict the chances you have of getting $ 1,000 on your first spin. ______

2. Predict the chances you have of getting $ 2,000 on your first spin. ______

3. Predict the chances you have of getting $ 3,000 on your first spin. ______

4. Predict the chances you have of getting $ 10,000 on your first spin. ______

5. Why did you make these predictions?

Take a chance on winning it all!

Each member spins the spinner twenty-five (25) times.
Record the amount that each group member wins by placing a tally mark in the appropriate column on the chart.

Names	$1,000	$2,000	$5,000	$10,000
1
2
3
4
5
Total

Express the number of times (total) your group landed on the following values as a fraction, decimal and percent:

Fraction Decimal Percent

$ 1,000 __________ __________ __________

$ 2,000 __________ __________ __________

$ 5,000 __________ __________ __________

$ 10,000 __________ __________ __________

Discuss the following questions in your group:

6. Based on these results what is the group’s probability of landing on $ 10,000?

7. How do these results compare with your predictions in questions 1-5?

Name ____________________________ Class _____

THEORETICAL VS. EXPERIMENTAL PROBABILITY

Activity #3: Red Head!!!

You will need the calculator instructions to complete this activity.

You will model flipping a coin by using your graphing calculator.

Heads = 1 Tails = 2

What are the outcomes when flipping a coin and spinning a spinner? Place your answers in the table below.

What is the theoretical probability of getting a Red Head? ________

Directions:

Spin the spinner
Use the random number generator in your calculator to flip a coin
Use the table below to record your data
You will complete the process 25 times

Outcomes
Tally

What is the experimental probability that you get a Red Head?

FRACTION = _________ DECIMAL = ___________ PERCENT = ________

How did your experimental probability compare to the theoretical probability?

How many Red Heads did your group get? __________

What is the total number of flips your group performed? __________

What is the experimental probability that your group got a Red Head? __________

How does the group experimental probability compare to the theoretical probability?

Name ____________________________ Class _____

THEORETICAL VS. EXPERIMENTAL PROBABILITY

Exit Problem

Your best friend was out sick today. She wants you to come over after school to go over what she missed in school today. Write a paragraph explaining what you learned in math class today. Be sure to explain what theoretical and experimental probabilities are and how they compare to each other.

Name ____________________________ Class _____

THEORETICAL VS EXPERIMENTAL PROBABILITY

Follow-up Activity

Use what you learned as you went through the group activities to complete the following problems.

1. Jeremy rolled a number cube 20 times. Five of the rolls were 3. What is the experimental probability of the rolls being a 3? How does this relate to the theoretical probability?

2. What if you roll a pair of number cubes 25 times and a pair of 5’s appear three times? Find the experimental probability of a pair of fives appearing 3 times. Then, compare this with the theoretical probability of getting a pair of fives.

3. If a spinner contains the numbers 1000, 2000, 5000, 10,000, “lose a turn”, and another 2000, what is the probability of the number 2000 being spun? Cut up 6 pieces of paper representing these six items that would be on the spinner. Label them as 1000, 2000, 5000, 10,000, “lose a turn”, and 2000. Turn them upside down. Then draw from the pile 10 times to find the experimental probability of the number of times 2000 appeared.

4. Jamie has a bag with all of the letters in it that spell PHILADELPHIA. If a letter is chosen at random, what is the probability that it will be an “L”?

5. Mrs. Blair is thinking of a 3-digit number with the digits 5, 7, and 8. What is the probability of guessing the number correctly on one try?

Name _____________________________________

THEORETICAL VS. EXPERIMENTAL PROBABILITY

Stars That Rock

The Tripl-Bubl Gum Company decides to promote its gum by including in each pack the photo of one of six rock music stars. Assuming that there are equal numbers of photos of each of the six stars and that when you buy a pack of gum your chances of getting any of the six photos are the same, about how many packs of gum would you expect to have to buy to get all six photos? (Adapted from Travers and Gray 1981, p. 327)

1. Is it possible to get all six photos with only six packs? ______ Would you expect that to happen? _____ Explain your response.

2. Is it possible NOT to get all six photos with 100 packs? _____

Would you expect that to happen? _____ Explain your response.

3. Predict how many pack of gum you would have to purchase to obtain one picture of each rock star. ________ Explain your prediction.

4. USING YOU CALCULATOR, design a random number simulation for obtaining pictures of all of the rock stars. You will need to conduct 6 trials on this simulation. A trial ends when you have obtained a photo of each rock star. Be sure to stop when all six names have been drawn, then add the tally marks to find out the number of picks.

	Trial #1	Trial #2	Trial #3	Trial # 4	Trial #5	Trial #6
Rock Star #1
Rock Star #2
Rock Star #3
Rock Star #4
Rock Star #5
Rock Star #6
Total number of picks

5. Use your calculator to find the mean and compare the mean to the two other measure s of central tendency- the median and the mode.

6. Decide as a group how many picks should be expected. ________

7. Compare the results with the original conjectures.

8. Decide whether it would be “worth it” to buy the number of packs of gum required to obtain a set of six photos of the “Stars That Rock.”

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