Let's Take a Chance!

Designed by: Barbara Roach    School: Keenan High School

Competency Goal/Objective(s):

Find the probability of an event. (AL112A)

Apply knowledge of data and statistics to solve problems and make 
informed decisions? (AT2-C)


Grade Level: 9 - 10      Subject(s): Algebra for Technologies & Algebra One

Overview:  Probability plays an important role in our daily lives. An understanding of basic probability is essential to understanding weather reports, medical findings, political polls, employment practices, video games and lotteries. Students have many misconceptions about situations that involve chance.
Using manipulatives, games and experiments, graphic organizers, brainstorming, teacher-led discussion, Let's Take a Chance will provide an enjoyable experience for students to learn the basics of probability. 


Focus/Essential Questions:

1. What is probability?
2. What is theoretical probability?
3. How is experimental probability used to predict the outcome of an event?
4. How is sampling used to make predictions?
5. How can you use lists, tree diagrams, and the counting principle to determine outcomes?
6. What is meant by a fair game?


Time Frame: The unit requires seven to eight 90 - minute class periods. The last class session could be used to collect culminating activities and for administering the written teacher-made assessment.


Resources/Materials:

Heath Algebra One - D. C. Heath Publishing Company, 1997
Dice-Pair each student 
Centimeter graph paper
Calculators 
400 colored marbles-340 clear, 60 blue
Colored Spinners-one per pair 
Computer(s) for word processing
Paper Plates (one each student) 
Blank transparencies/overhead markers
Twelve Number Boards-one per student 
Transparencies of worksheets (optional)
Yellow, red, and blue blocks- 3 each (min) 
Poster or display boards
Assorted markers or coins 
Overhead projector 
Culminating activity with rubric 
Newspapers, magazines 
Teacher-made unit test 
Concept organizer transparency
Bag 
Clear container with top
Colored markers/pens 
Rulers
How Large Crowd transparency 
Index card for each pair

Worksheets:
How do you describe the amount of certainty?
Dated Odds
Take a Spin
Vowels 'R Us
How large is the crowd?
Class Ring
12 Number Boards


Assessment/Culminating Assessment:

1. Informal - Observation, questioning, oral responses, ticket to leave (summarizers)
2. Formal - Teacher-made unit test, worksheets, ticket to leave 
3. Culminating Activity, Throwing Dice, with rubric


Instructional Activities:


LESSON ONE

What is probability?

Launch Activity: 

Students will complete worksheet: How Do You Describe the Amount of Certainty?

After all students have completed the worksheet, the teacher will lead the discussion:

It would be easier to arrange the words if they had numbers assigned to them. Probability gives us a method of assigning numbers, which tell how likely something is to happen. If an event is impossible, 0% chance is assigned to it. If an event is as likely to happen, as it is not to happen, 50% chance is assigned to it. At the other extreme, if an event is a certainty, 100% chance is assigned to it. It is not possible to be more than a "sure thing" or 100%. You cannot have a chance (probability) of 200%. The percents used in probability range from 0% to 100%. 
Although probability can be expressed as a percent, more often it is expressed as a fraction between 0 and 1. Probability is a number between 0 and 1 that tells the likelihood that an event will occur. 

Demonstration with Discussion:

Teacher will ask: Can you describe some situations that use probability? 
Teacher will list examples on overhead as follows:

How do you think the probability is determined for these situations? 
(Accept all answers)

Mathematicians have a way of determining the probability of events. Let's
explore one way by conducting an experiment.

Materials: 
9 blocks (3 each) -red, blue, and yellow
Bag or container

Place one blue and one yellow block in a bag or container.

Ask: If a block is selected from the bag without looking, what color will he draw? Wait for responses.

Allow a student to draw. Show color to class and put back.

Ask: If a block is drawn again, what color will be selected? Repeat several times. What did you notice? Discuss.

The chance of drawing yellow or blue does not depend on previous trials. That is, blocks do not remember. Each block has the same chance of being selected.

The probability of an event is often written as a fraction in lowest terms.

Number of successes
Total number of possibilities

Probability of blue = 1/2      Probability of yellow = 1/2

Put the yellow and blue blocks back and add a blue block. Ask: What is the probability of drawing a blue block? (2/3) yellow block? (1/3)

Take out the yellow block. Ask: What is the probability of drawing a blue block? (2/2 or 1 or 100%) What is the probability of drawing a yellow block? (0/2 or 0 or 0%)

Place all nine blocks in bag. Ask: What is the probability of selecting a blue block? A red block? A yellow block?

Find the sum of the probabilities. (2/6 + 2/6 + 2/6 = 6/6 = 1) You cannot have more than 1 or 100% probability.

Take out two yellow blocks and one red block. What is the probability of selecting a yellow block? (2/6 =1/3)

Place the red block in the bag. Ask: How many blocks must be added to make the probability of selecting a yellow block 1/2? (Two yellow blocks) 

Remember that when you take away or add blocks, the total number of blocks changes and thus the probability changes.

Review important details and illustrate other examples for reinforcement.

Independent Practice:

Student will complete worksheet, Dated Odds. The teacher will circulate and offer assistance, if needed. Teacher will grade worksheet later.

Ticket to Leave: (Summarizer) 

Each student will give a written response to the following:

One of your friends says that there is a 110% chance of snow  tomorrow. Why is your friend wrong? Write your explanation.

Students will leave responses with teacher at the end of the period. Teacher
will evaluate later.


LESSON TWO

Return corrected worksheets and tickets from previous lesson and discuss. Clarify any misconceptions. 


What is theoretical probability?
How is experimental probability used to predict the outcome of an event?

Activator:

Imagine a box, which contains 26 slips of paper. A different letter of the alphabet is written on each slip. 

1. What is the probability of drawing the letter P (F)? P (K)?  Note: The probability of an event is written as P (E) where "E" represents the event being evaluated.

2. What is the probability of drawing any letter of the alphabet?

3. What is the probability of drawing a consonant?

4. If the letter W is drawn and not returned to the box, what is the probability of drawing another W from the box?

5. If the letter R is drawn and not returned, what is the P (vowel)?

To answer these questions, you used theoretical probability. You pictured a model of the situation. You assumed that each letter had an equal chance of being drawn. What should happen or what is expected to happen determines theoretical probability. 

Real-world situations are much more complicated. 
A better way to find the probability of real-world events is experimental probability. Experimental probability is based on observed outcomes, in other words, what actually happens.

Collaborative Activity: 

In pairs, students will complete the Take a Spin  Activity.

Upon completion, construct a class graph of all data on overhead using graph paper transparency. 

Ask: What conclusions, if any can you draw from the results of your graphs in comparison to the results of the class graph?
(The graphs of the individual pairs may vary greatly. The class graph should be closer to the theoretical probability) 

Independent Practice: 

Students will complete worksheet, Vowels 'R' Us. Students will need books or newspaper articles, centimeter graph paper and colored markers or pens (optional). (It is assumed that students have previously constructed bar graphs.) The teacher will circulate and offer assistance, if needed. The assignment will be graded later. 

Ticket to Leave: (Summarizer) 

Oral responses from the class: 

• Today I learned ...

• I will use the knowledge I learned ...

• I would like to know more about ...

LESSON THREE

Return corrected worksheets from previous lesson and discuss. Clarify any misconceptions. 

Present culminating activity: Assign students to groups of three, issue culminating activity and rubric and answer questions.


How is sampling used to make predictions?

Activator: 

Sometimes the size of a crowd at a rally or a football game is estimated
by aerial photographs because there is no other practical way to count all in attendance. The crowd is estimated by sampling. A sample in probability is used to make predictions about an entire group.

Display the How Large is the Crowd transparency.
Say: Imagine that this illustration is an aerial view of a crowd at a rally.
Each dot represents one person. Let's estimate how many people attended the rally. 

Allow students to work in pairs to devise a solution. Distribute a How Large Is The Crowd worksheet to each pair. (Give each pair a ruler and an index card also.) 
Note: Students should not count all dots. Encourage them to devise another method of predicting. Some students may measure a square unit and count the number of people in a unit, then multiply that number by the number of units in the entire crowd.

Upon completion, the pairs will present their solutions and explain their processes to the class. 
Note: Most answers should be close to each other and as long as the numbers are reasonable, all processes should be valid. (The purpose of this activity to model a real-world situation.)



Class Activity: 

Capture-Recapture Activity. 

How can you estimate what you cannot see?

Materials: 
340 clear marbles
60 blue marbles
Clear container with top 
Paper plates for counting

1. Tell students that 400 marbles are in the container, some clear, some blue.

2. Pass unopened container around.

3. Ask students to guess how many marbles are blue. Students should write their estimate on a sheet of paper.

4. Allow each student to take a sample (a handful) of marbles. Students will record number of marbles in sample, including how many are blue, and return marbles to container.

5. Each student will write a proportion of blue marbles to total marbles in sample (handful). Use proportions to estimate the total number of blue marbles in container of 400.

  
6. Write original guess and activity prediction for each student on overhead.

7. How do original predictions compare to predictions after the activity? Students will respond.

8. Ask students to look at all data and predict how many blue marbles are in the container. 

9. Tell students there are 60 blue marbles in container.

10. Teacher will ask students to evaluate and summarize the Capture-Recapture Activity. 

Ask: What do you think will happen to the experimental probability if we repeat this activity 1000 times? (Predictions will be closer to actual number.)


Ticket to Leave: (Summarizer) 

Whole class oral discussion:

What are some real-life examples of sampling? (Election polls, surveys, etc.)
Why do businesses use sampling? (To determine where to build a store, etc.)



LESSON FOUR

How do you determine outcomes using lists, tree diagrams, sample spaces, and the counting principle?


Activator: 

Write Lunch Menu on overhead transparency or board.

The choice for today's lunch are posted in the cafeteria:

Ask students to name the different combinations of main dish and dessert. Write combinations in grid as follows:

How many different lunches consisting of one main dish and one dessert can be selected? (9) These are called outcomes.

Ask? What is the probability of choosing the pizza/cake outcome? (1/9)

Say: In this activity a grid was made to organize and list all the possible outcomes when choosing one main dish and one dessert from the cafeteria menu. This grid is called a "sample space". Another visual method for organizing outcomes is called a "tree diagram".


A penny and a nickel are tossed. How many possible outcomes are there for heads and tails?

There are four possible outcomes.

Let students construct a tree diagram for three coins at their seats. Monitor and give assistance, if needed. Allow a student to display his tree diagram on the overhead. (HHH, TTT, HHT, TTH, HTT, THH, HTH, THT)
There are 8 possible outcomes.

Ask: What is P (exactly 2 tails)? (3/8)
You can see that as the choices increase, the outcomes increase, and it may become inconvenient to list all outcomes by using a tree diagram.

Another method for determining the number of outcomes is the counting principle. The counting principle uses multiplication to determine the number of outcomes. The number of outcomes is found by multiplying the number of choices at each stage by each other.

Example: Benita is configuring a computer system for her home office. She must choose from 4 different computers, 2 types of monitors, and 3 different brands of printers. How many different configurations can she choose from?

Do another example. Discuss and answer questions.



Guided Practice: 

Allow students to complete worksheet, Class Ring. Circulate and offer assistance, if needed.

Allow students to display results on overhead. Check by using the counting principle.

Discuss and answer questions.

Independent Practice: 

Students will complete and worksheet, Let's Make A Meal.
Teacher will circulate and assist, if needed. Teacher will correct worksheets later.

Ticket to Leave: (Summarizer) 

Each student will write:

1. One question about today's content-something that has left you puzzled.

2. What did you learn today that was new to you?

All responses are acceptable. Teacher will read tickets later to check for understanding and to determine if re-teaching is needed.




LESSON FIVE

Return corrected worksheets and tickets from previous lesson and discuss. Clarify any misconceptions. 


What is meant by a fair game?


Activator: 

Twelve Number Board Activity: 

Materials: 
Number Board Sheet for each student
Twelve markers or coins per student
Pair of dice 

1. Tell students to place the twelve coins on the board. The only restrictions are all coins must be used and the coins must cover numbers. 

2. Tell students you will roll a pair of dice and add the numbers. The sum will match a number on the board. Any coins covering a sum should be removed when called. Continue to roll the dice until someone wins. (The student who clears his/her board first wins.)

3. Ask students to place coins on board again. Repeat two additional times.

4. Ask: What have you learned? What patterns have you noticed?

• Number 1 is impossible.

• Numbers as 7 and 11 appear more often.

• 2, 3, and 12 do not appear often.

• You can place more than one coin on a number thus increasing you chances of winning.

5. Ask: Is the game fair? Discuss. (The game is not fair because all sums do no have the same probability of appearing. Example: 7 can be 3+4, 4+3, 6+1, 1+6, 2+5, 5+2) 

6. Ask: How can you make the game fair? Discuss.

7. Ask: Is the Georgia Lottery fair? Was it designed to be fair? Why?


Collaborative Activity: 

Read the Story of 100 Blue Balls, 100 Red Balls, and 3 Urns. 

Many years ago in the far off land of Richlandmania, a prisoner was to be executed. The King of Richlandmania was a sporting fellow who thought highly of good thinkers. 

The King offered a prisoner a chance for freedom. He was given 100 blue balls and 100 red balls, and was told to distribute them into three similar urns in any way he liked. He was then blindfolded and told to draw a ball at random from one of the urns. If he drew a blue ball he would be executed. If he drew a red ball, he would have his freedom. (The urns were rearranged.)

The prisoner was smart (and lucky) so he was set free! How would you have distributed the balls in the three urns?

In pairs, students will develop a solution. Students will share solutions with class. NOTE: The solution is to place one blue ball in two urns, and 98 blue and 100 red balls in the third urn. 

Discuss: Why is this the best solution? (Urn 1 and 2 are sure things and urn 3 has the greatest mix possible.)




Ticket to Leave: (Whole class review)

The unit is named Let's Take a Chance. Display the Concept graphic organizer transparency. Ask cooperative pairs to give information for organizer. Write definition of probability in the square. Write the details and additional information in the circles. Use a round-robin technique until the class feels that they have listed all important details. (You may draw additional circles, if needed.) Students will copy the organizer and use the information to assist in completing the culminating activity and to study for the teacher-made assessment.

If time permits, allow students to form culminating activity groups to develop strategies for next class. 


LESSON SIX/SEVEN

Let's Take a Chance (Culminating Activity)


Allow students to work in groups on culminating activity, Throwing Dice.
Students should be provided access to computers with word processor and graphics. This lesson may require two days. Teacher will circulate and give assistance, if needed. Projects will be displayed in class after they have been assessed.

Note: Remind students to refer to rubric at all times. 

Check all that apply:

9. Signature of teacher(s) submitting the lesson/unit:

(A signature verifies that all copyright laws have been observed.)