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GPS Alignment: Instructor Page
Week 14: Data Analysis - Probability


Adding Marbles (Printable PDF)

A bag contains five red marbles, three green marbles, and four blue marbles. How many green marbles must be added so that the probability of drawing a green marble is 3/4?

Sample Spreadsheet - click here to view entire spreadsheet.

Green
Marbles
Added
to Bag

# of
Green
Marbles

Total # of
Marbels

Ratio of
Green
to Total

Decimal
Approximations

Percentage
Approximations

0

3

12

1/4

0.25

25%

1

4

13

4/13

0.307692308

31%

2

5

14

5/14

0.357142857

36%

3

6

15

2/5

0.4

40%

4

7

16

7/16

0.4375

44%

 

GPS

As seen in Problem Exploration

M6N1: Students will understand the meaning of the four arithmetic operations as related to positive rational numbers and will use these concepts to solve problems.

a. Use factors and multiples

f. Use fractions, decimals, and percents interchangeably
 

a. The student should examine multiples of 3 and 4 in solving this investigation (the number of green marbles and the total number of marbles where the numbers come from the final probability).

f. Using a spreadsheet, the probability can be expressed as a ratio, a fraction, a decimal, and/or a percent.

M6D2: Students will use experimental and simple theoretical probability and will understand the nature of sampling. They will also make predictions from investigations.

a. Predict the probability of a given event through trials/simulations (experimental probability), and represent the probability as a ratio.

b. Determine, and use a ratio to represent, the theoretical probability of a given event.

c. Discover that experimental probability approaches theoretical probability when the number of trials is large.
 

a. The student can use real marbles or a simulator (like on TI graphing calculators) to try out their predictions to see how they can get close to a probability of 3/4.

b. The student will determine the theoretical probability of drawing a marble(s) of a given color.

c. If a simulator is used, this will be easier for students to see the Law of Large Numbers.

M6A1:  Students will understand the concept of ratio and use it to represent quantitative relationships.
 

The probabilities can be expressed as a ratio of the # of times an event could occur (drawing a certain color) to the # of possible outcomes.
 

M6A2:  Students will consider relationships between varying quantities.

a. Analyze and describe patterns arising from mathematical rules, tables, and graphs.
 

As green marbles are added to the bag, a spreadsheet reveals a pattern in the probabilities - they get closer to 1. This is a good place to talk about probabilities of 0 & 1.
 

M7N1:  Students will understand the meaning of positive and negative rational numbers and use them in computation.

b. Compare and order rational numbers, including repeating decimals.
 

As the student adds green marbles to the bag using a simulator, the probabilities of drawing a green marble will increase. Therefore, the students can look at these probabilities in decimal or fraction form and see them in ascending or descending order.

M7A1:  Students will represent and evaluate quantities using algebraic expressions.

b. Simplify and evaluate algebraic expressions, using commutative, associative, and distributive properties as appropriate.
 

The student can generalize the number of green marbles added to the bag and set up a proportion and solve.

M7A2:  Students will understand and apply linear equations in one variable.

a. Given a problem, define a variable, write an equation, solve the equation, and interpret the solution.

b. Use the addition and multiplication properties of equality to solve one- and two-step linear equations.
 

a. The problem can be set up by the equation (3+g)/(12+g) = 3/4 and solving for g.

b. In determining the equation above, the property of equality could be discussed.

M8A1:  Students will use algebra to represent, analyze, and solve problems.

b. Simplify and evaluate algebraic expressions.

c. Solve algebraic equations in one variable, including equations involving absolute values.
 

a. The student can generalize the number of green marbles added to the bag and set up the proportion and solve.

c. The problem can be set up the equation (3+g)/(12+g) = 3/4 and solved for g.
 

M8D3:  Students will use the basic laws of probability.

a. Find the probability of simple independent events.

b. Find the probability of compound independent events.
 

a. The students can determine the probability of drawing a green, red, or blue marble from the bag with replacement.

b. The students should examine the probability of drawing a green or a  blue marble from the bag with replacement and examine the probability of drawing a green and a blue from the bag with replacement.