<p>Students learn how to use the area of different shapes to compute the theoretical probability of an event. Student should be able to:&nbsp;<br>- calculate the area of polygons and other regular shapes.<br>- use subtraction to determine the remaining area.&nbsp;<br>- calculate probability.</p>

<p>- How can we represent an event's probability using geometric representations and the concept of length or area?</p>

<p>- Probability: A real number between zero and one that represents the likelihood of the occurrence of an event, where zero represents the impossibility and one represents the certainty of the event. If <i>p</i> is the probability of an event, then 0 ≤ <i>p</i> ≤ 1 .<br>- Odds: In probability, the ratio of the probability of the number of favorable outcomes to the probability of the number of unfavorable outcomes, specifically \(p \over (1 - p)\), where <i>p</i> is the probability of the event.<br>- Geometric Probability: The study of outcomes related to models that emphasize defined geometric objects and random points within and outside of those objects. For example, what is the probability of randomly selecting a point within a circle bounded by a square?</p>

<p>- multiple copies of bull’s-eye handouts, one bull’s eye per four students in your class (M-G-2-3_Bulls Eyes)<br>- copies of Irregular Checkerboard (printed in color) (M-G-2-3_Irregular Checkerboard)<br>- tape; masking, gaff, or duct<br>- small cotton balls<br>- bandana (for blindfold)<br>- copies of Lesson 3 Exit Ticket and KEY (M-G-2-3_Lesson 3 Exit Ticket and KEY)</p>

<p>- Evaluate group presentations based on accuracy of content, quality of communication within the group, and fair distribution of tasks.&nbsp;<br>- Lesson 3 Exit Ticket (M-G-2-3_Lesson 3 Exit Ticket) requires students to devise a strategy for calculating the areas of irregular shapes and detail their calculations.</p>

<p>Active Engagement, Modeling, Explicit Instruction<br>W: In this lesson, students will learn how to represent probabilistic outcomes using relationships between the areas of two-dimensional geometric objects. Understanding geometric probability necessitates proficiency in geometric reasoning and computing the probability of specific events.&nbsp;<br>H: Throwing objects at specific targets is a stimulating activity. However, in the bull's eye activity, students will toss small cotton balls at random onto targets that have subdivisions of varying areas.&nbsp;<br>E: To calculate probabilities, students first examine the relationship between probability and odds. Students will use the relative areas of the targeted regions to create conceptual and computational skills for understanding the relationships between theoretical probability based on area magnitude and actual outcomes based on the number of objects landing in the target.&nbsp;<br>R: To present their findings of probability to the class, students should provide supporting materials, describe procedures, and compute results. Collaboration with members of their group will also be required, encouraging the type of reflection and rethinking that will improve individual and group understanding.&nbsp;<br>E: The Exit Ticket evaluates students' comprehension of area calculations for landing on any color. These computations are difficult due to the irregular shapes and limited available dimensions. Students might utilize estimation tools to assess the reasonableness of their findings.&nbsp;<br>T: For students who might need additional practice generalizing quantitative outcomes, use numbers small enough to work with mental arithmetic. Use parallel examples with varying quantities to convey the same concept. For example, demonstrating that flipping one fair coin 100 times is the same as flipping 100 coins all at once. Students that are skilled at processing computations will require various representations of the same outcomes. For example, alternate between fractions, decimals, and percentages.&nbsp;<br>O: This lesson teaches students about probability and the areas of various geometric shapes, specifically polygons. Hands-on experience allows students to examine the differences between their experimental data and the actual theoretical probabilities.</p>

<p><strong>Activity 1: Bull's Eye</strong></p><p>This lesson will revisit the concepts of probability and teach students how to use the areas of different shapes to calculate geometric probability. Students are expected to have already learnt how to find the area of regular polygons using this formula from Lesson 2.</p><p><strong>Area = \(AP \over 2\)</strong></p><p><strong>(where </strong><i><strong>A</strong></i><strong> = apothem and </strong><i><strong>P</strong></i><strong> = the perimeter)</strong></p><p>Form groups of three or four students. Give each group a "bull's eye" (M-G-2-3_Bulls Eyes) at random from the two possible bull's eyes. Give each group a handful of little cotton balls. Instruct groups to place the bull's eye on the floor and then take turns throwing the cotton balls one at a time into the center of the eye. Each group should then keep track of how many cotton balls fell on the center shape and how many cotton balls fell outside of the center shape. Repeat the experiment until each group member has tossed the cotton balls at least once. Make sure the cotton balls are small enough and the targets are large enough to accurately quantify the results. Before assigning the activity to students, practice it several times.</p><p>Remind students that the definition of probability is the number of desired outcomes divided by the number of possible outcomes. Ask groups to calculate the experimental probabilities for each of their trials (in percentages) as well as the combined probability for all trials. Allow each group to present their bull's eye and the findings of their experiment to the class.</p><p><strong>"Why did certain groups have higher probabilities and other groups have lower probabilities?"</strong> (There are other possible explanations, such as different throwing techniques or the quantity of cotton balls used, but the main point is that the center shapes are not all the same shape or size.)</p><p>Ask students to explain the differences between tossing objects randomly and throwing them with the purpose of landing in specific locations. Extend the idea by questioning how they could measure the degree to which tosses were random or planned.</p><p>State the conclusion that the area of the bull's eye determines the probability of one cotton ball landing in the center of it.</p><p>Encourage students to discuss the distinction between probability and odds. Clarify any misconceptions. Teach the class how to calculate the probability and odds of hitting the bull's eye.</p><p>Students should repeat the procedure of determining the probability and odds of landing in the center of the bull's eye they used in their experiment (the version with numbers). Let each group present their work to the rest of the class. Ask students to compare their experimental and theoretical probability. This presents an opportunity to revisit the law of large numbers.</p><p>Give students a geometric probability assignment to solve before they leave class. For example, ask them to calculate the probability of a cotton ball falling on a bull's eye, which is a circle with a radius of one inch located in the center of an 8 inch by 8 inch rectangle. You can quickly review the responses to see which students need additional practice and which have mastered the skill. (<i>0.03 for 1.5-inch; 0.11 for 3-inch</i>).</p><p><strong>Activity 2: Checkerboard</strong></p><p>This activity will involve exploring geometric probability. Students will learn how to calculate probability by measuring the area of different polygons using an irregular checkerboard-like canvas and multiple colors. Before assigning the activity to students, practice it using cotton balls and the targets.</p><p>Have students form groups of three to five. Give one irregular checkerboard and one cotton ball to each group (M-G-2-3_Irregular Checkerboard). Each group should then tape its checkerboard to the floor.</p><p>One at a time, students should take turns closing their eyes or blindfolding one another. The student who is blindfolded will next drop a cotton ball onto the checkerboard, and the group will see where it lands. Students should record the color of each cotton ball that lands.</p><p>The outcomes of each group will eventually be compared against the computed geometric probability.</p><p>Next, give each group a color. As a group, ask them to calculate the probability of the cotton ball landing on their color.</p><p>KEY: The irregular checkerboard measures around 93 square units.</p><p>The approximate total area, in square units, for each color is</p><p>Yellow 22<br>Red 10<br>Green 9<br>Blue 17<br>Purple 19<br>Gold 16</p><p>Students should use calculators to calculate the probabilities that the cotton ball will land on each color.</p><p>Key:</p><p>\(22 \over 93\) = 0.24 yellow&nbsp;<br>\(10 \over 93\) = 0.11 red<br>\(9 \over 93\) = 0.10 green<br>\(17 \over 93\) = 0.18 blue<br>\(19 \over 93\) = 0.20 purple<br>\(16 \over 93\) = 0.17 gold</p><p>Once students have computed a probability, have each group present their findings to the entire class. Ask the teams with triangles in their color how they calculated the triangles. Note that each color contains two triangles, therefore adding them together yields a square, providing a shortcut for the calculation. Encourage students to discuss their own results from the irregular checkerboard exercise and whether they are consistent or inconsistent with their computed findings.</p><p>Then, assign the following calculations for students to discover.</p><p>Ask your students: <strong>"What is the probability of the one cotton ball landing on the following combinations?"</strong></p><p>Possible combinations to ask: Blue and Yellow, Blue or Yellow, Red and Blue, Red or Blue, Red and Green or Purple, and so on.</p><p>There are numerous possibilities, and students may be given more than one if their calculations proceed quickly.</p><p><strong>Alternate Activity for Differentiated Instruction:</strong> The polygons on the grid can be made more difficult to give more advanced students more to work with. You can alternatively start with a simpler grid by covering one or more smaller, more complex polygons with basic squares or triangles, then removing them when students move on to individual work to increase the level of the activity. You can also have students calculate the probability of landing on a specific colored portion as a bonus. If you want a more hands-on activity, drop two cotton balls at a time to determine the probability of hitting two colors at the same time.</p><p>As part of the evaluation process, ask students to submit their computations. Also, at the end of class, distribute Lesson 3 Exit Tickets (M-G-2-3_Lesson 3 Exit Ticket) to students.</p><p><strong>Extension:</strong></p><p>Students could work on a "bull's eye" problem using three areas (e.g., a circle inside a triangle, inside a square) and determine the probability of the middle section (the triangle, less the circle). Students could also work on a multishaped area problem that requires them to calculate the probability of two different shapes (for example, the circle or the pentagon).</p><p>Show the calculation for determining the probability of randomly selecting a point within a circle. This is known as an annulus (Latin for "little ring").The area of the ring, A, created by two concentric circles is <i>A</i> = <span style="background-color:rgb(255,255,255);color:rgb(51,51,51);"><i>π</i></span>\(R^2\) – <span style="background-color:rgb(255,255,255);color:rgb(51,51,51);"><i>π</i></span>\(r^2\), where <i>R</i> is the radius of the larger circle and <i>r</i> is the radius of the smaller circle.</p>