• Evaluate group presentations on the accuracy of content, quality of communication within the group, and fair distribution of tasks.

• Lesson 3 Exit Ticket (M-G-2-3_Lesson 3 Exit Ticket.doc) requires students to make a plan for finding the areas of irregular shapes and detail the calculations.

Activity 1: Bull’s Eye

In this lesson, the concepts of probability will be reviewed and students will learn how to use the areas of different shapes to find geometric probability. It is expected that students have already learned how to find the area of regular polygons using this formula from Lesson 2:

Area =

(where A = apothem and P = perimeter) [IS.5 - Struggling Learners]

Place students into groups of three or four students. Give each group a “bull’s eye” (M-G-2-3_Bulls Eyes.doc), chosen randomly between the two possible bull’s eyes. Also give each group a handful of small cotton balls. Instruct groups to place the bull’s eye on the floor and take turns throwing the cotton balls one at a time onto the center of the bull’s eye. Each group should then record how many cotton balls landed on the center shape and how many landed outside of the center shape. Repeat this experiment until each group member has had a chance to toss the cotton balls at least once. Make sure the cotton balls are small enough and the targets are large enough to reliably measure the outcomes. Practice a few times before assigning students to do the activity.

Remind students that the definition of probability is the number of desired outcomes divided by the number of possible outcomes. Have groups find the experimental probabilities for each of their trials (as percents) and the combined probability for all the trials. Have each group present their bull’s eye and the results of their experiment to the class.

“Why did certain groups have higher probabilities and other groups have lower probabilities?” (There are many possible answers: different throwing techniques or different number of cotton balls, but the key answer is that the center shapes are not all the same shape or the same size.)

Ask students to describe their understanding of the differences between throwing the objects randomly and throwing them with the intention of landing in preferred locations. Extend the idea by asking how they could measure the degree to which tosses were random or intentional.

State the conclusion that the probability of one cotton ball landing at the center of the bull’s eye is determined by the area of the bull’s eye.

Have students discuss the difference between probability and odds. Clarify any misconceptions. Guide the class to calculate the probability and odds of landing in the bull’s eye. [IS.6 - All Students]

Have students repeat the process of calculating the probability and odds of landing at the center of the bull’s eye they used for their experiment (use the version with numbers). Have each group present its work to the rest of the class. Ask students to compare their experimental probability to their theoretical probability. This is an opportunity to review the law of large numbers.

Give students a geometric probability problem to complete before they leave class. For example, ask them to find the probability of a cotton ball landing on a bull’s eye, where the bull’s eye is a circle of radius 1 inch, which is in the middle of an 8 inch by 8 inch rectangle. You can quickly review the responses to determine which students may need more practice and which students have mastered the skill. (0.03 for 1.5-inch; 0.11 for 3-inch) [IS.7 - Struggling Learners]

Activity 2: Checkerboard

This activity will involve looking at geometric probability. Using an irregular checkerboard-like canvas and multiple colors, students will learn how to use the area of different polygons to calculate probability. Practice the activity with cotton balls and the targets before assigning to students.

Have students get into groups of three to five. Hand out one irregular checkerboard and one cotton ball to each group (M-G-2-3_Irregular Checkerboard.doc). Each group should then tape their checkerboard to the floor.

One at a time, students should take turns closing their eyes or blindfolding each other. The student who is blindfolded will then drop a cotton ball onto the checkerboard and the group will see where it lands. Students should record the color for the landing of each cotton ball.

Each group’s results will later be used to verify or contrast with the calculated geometric probability.

Next, assign each group a color. As a group, ask them to find the probability of the cotton ball landing on their color.

KEY: The approximate area of the irregular checkerboard is 93 square units.

The approximate total area, in square units of each color is

Yellow 22

Red 10

Green 9

Blue 17

Purple 19

Gold 16

Students should calculate the probabilities of the cotton ball landing on each color using their calculators (by dividing the area of the color by the total area, 93 square units). [IS.8 - Struggling Learners]

Key:

= 0.24 yellow

= 0.11 red

= 0.10 green

= 0.18 blue

= 0.20 purple

= 0.17 gold

Once students have calculated a probability, ask each group to present their findings to the whole class. For the teams that have triangles within their color, ask them how they arrived at the calculations for those triangles. Note: each color has two triangles, so by adding them together they become a square, creating a shortcut for the calculation. Have students discuss their own results from the irregular checkerboard activity and whether they are consistent or inconsistent with their calculated findings.

Next, hand out the following calculations for students to discover.

Ask students, “What is the probability of the one cotton ball landing on the following combinations?”

Possible combinations to ask: Blue and Yellow, Blue or Yellow, Red and Blue, Red or Blue, Red and Green or Purple, etc.

There are many options, and students can be given more than one option if their calculations go quickly.

Alternate Activity for Differentiated Instruction: The polygons on the grid can be made more complicated in order to give more proficient students more to work on. You can also begin with a simpler grid by covering one or more smaller, more complex polygons with simpler squares or triangles and removing them when students move into individual work to increase the level of the activity. Also, you can have students calculate the odds for or against landing on a specific colored portion as a bonus. If you want more hands-on activity, use two cotton balls and drop two at a time to find the probability of hitting two colors at the same time.

Have students turn in their calculations as a form of evaluation. Also, hand out Lesson 3 Exit Tickets to students at the end of the class (M-G-2-3_Lesson 3 Exit Ticket.doc).

Extension:

• Students could work on a “bull’s eye” problem with three areas (e.g., a circle inside of a triangle, inside of a square), where they need to find the probability of the middle section (the triangle, less the circle). Students could also work on a multishaped area problem where they need to find the probability of two different shapes (e.g., the circle or the pentagon).

• Show the calculation for finding the probability of randomly selecting a point inside a circle within a circle. This construction is called an annulus (Latin, little ring).The area of the ring, A, formed by two concentric circles is , where R is the radius of the larger circle and r is the radius of the smaller circle.