Prokop

2. Algebra 3. Geometry 4. Measurement 5. Data Analysis and Probability 6. Problem Solving 7. Reasoning and Proof 8. Communication 9. Connections 10. Representation ||
 * Understanding By Design Lesson Template **
 * ** Title of Lesson ** |||| Area of Polygons and Circles || ** Grade Level ** |||| 9th ||
 * ** Curriculum Area ** |||| Geometry (HN) || ** Time Frame ** |||| 7 days ||
 * ** Developed By ** |||||||||| Kristie Prokop ||
 * ** Identify Desired Results (Stage 1) ** ||
 * ** Content Standards (from NCTM) ** ||
 * 1. Number and Operations
 * ** Understandings ** |||||| ** Essential Question(s) ** ||
 * ** Overarching Understanding ** |||| ** Overarching ** || ** Topical ** ||
 * ●Knowing about angle measures in a polygon can assist you in determining other characteristics of the polygon.

●Polygons divided into triangles can reveal certain characteristics of the polygon.

●Knowing 2 polygons are similar, what can that reveal about their side lengths, perimeter, and area?

●Arc Length is related to the circumference of a circle.

●The area of a sector is related to the area of a circle.

●Probability is a ratio that represents the chance that an event will occur. |||| ●How do mathematical representations assist you in problem solving?

●How are 1-D, 2-D, and 3-D shapes related?

●How can you apply your knowledge about shapes to real-life situations?

●What is the purpose of proofs? || ●Using 2 representations, how can you find the sum of the measures of the interior angles of a regular polygon?

●How do you find the area of a regular polygon? Why does it need to be regular?

●How can you find the area of a polygon if you know its perimeter and the perimeter and area of a similar polygon?

●Where would you use geometric probability?

●If an expert archer shoots an arrow, what is the geometric probability involved? What about a novice teacher? ||
 * ** Related Misconceptions ** ||
 * ●Students may make algebraic errors when solving equations.

●Students may misunderstand the difference between exact and approximate answers.

●Students may confuse the value of pi, both with and without a calculator when finding both exact and approximate solutions.

●Students may fail to see the three representations of a ratio: a to b, a/b, a:b

●Students may not order the polygons correctly when they are layered to find the probability of the shaded region. || Students will know… |||||| ** Skills ** Students will be able to… ||
 * ** Knowledge **
 * ●Formulas for (1) finding the sum of the interior angles in a polygon, (2) one interior angle in a regular polygon, (3) one exterior angle in a regular polygon, (4) area of a regular polygon, (5) circumference of a circle, (6) arc length, (7) area of circle, (8) area of a sector

●The relationship between the perimeters and areas of similar polygons.

●How to find geometric probability involving length and area.

●Definition of (1) center of polygon, (2) radius of circle, (3) apothem, (4) central angle of a regular polygon, (5) circumference, (6) arc length, (7) sector of a circle, (8) probability, (9) geometric probability |||||| ●Find the measure of interior and exterior angles of polygons

●Find the area of a regular polygon

●Compare perimeters and areas of similar figures

●Find the circumference of a circle

●Find the length of a circular arc

●Find the area of a circle

●Find the area of a sector

●Find a geometric probability || 1. Find the probability that someone throws the beanbag through the left eye. 2. Find the probability that someone throws the beanbag through either eye. 3. Find the probability that someone throws the beanbag through the nose. 4. Find the probability that someone throws the beanbag through the mouth. 5. Find the probability that someone throws the beanbag through the circle at the far right of the mouth. 6. Find the probability that someone will win a prize. 7. Suppose that after the first night of the carnival, there are very few winners. A new target board is made with the lengths of the shapes on the target increased by two centimeters. The eyes are now 22 cm by 17 cm, the nose has a height of 17 cm and a base of 20 cm, and the circles for the mouth have a radius of 8 cm. Find the probability that someone will win a prize with the new target. || ● Daily Homework ● Unit Quiz ● Unit Test || Day 2: Explore Areas of Regular Polygons Day 3: Explore Perimeters and Areas of Similar Figures Day 4: Explore Circumference and Arc Length Day 5: Explore Areas of Circles and Sectors Day 6: Explore Geometric Probability || From: Wiggins, Grant and J. Mc Tighe. (1998). //__Understanding by Design__//, Association for Supervision and Curriculum Development ISBN # 0-87120-313-8 (ppk) This was a thought provoking assignment. It really made me think about why concepts are presented in a certain course. It definitely took a lot of time but now I have a better map in my head of where this unit is coming from and where it is going. Personally, I think that this format should be used when designing curriculum.   **I really like your performance task for this unit. It helps the students to see how you could use this mathematical concept in real life! As a suggestion for an extension activity- maybe you could have the students design a their own bean bag game board with a specific probability of winning?? It just turns the problem around but it would allow the students to incorporate their creativity into the problem. Just an idea!**
 * ** Assessment Evidence (Stage 2) ** ||
 * ** Performance Task Description ** ||
 * ** Goal ** |||||||||| To demonstrate the key concepts of the unit. ||
 * ** Role ** |||||||||| To assess student understanding of major concepts of the unit. ||
 * ** Audience ** |||||||||| Carnival management ||
 * ** Situation ** |||||||||| Students are at a carnival and they approach a booth which involves tossing bean bags at a target. A person wins a prize if the bean bag goes through the eyes, nose, or part of the mouth. ||
 * ** Product/Performance ** |||||||||| Students will answer the following questions assuming each person is throwing one bean-bag:
 * ** Standards ** |||||||||| 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 ||
 * ** Other Evidence ** ||
 * ● Class Starters
 * ** Learning Plan (Stage 3) ** ||
 * ** Where are your students headed? Where have they been? How will you make sure the students know where they are going? ** |||||||| This unit continues to transition the students from 1-D to 2-D shapes and their properties. Once the students have an understanding of 2-D shapes, 3-D shapes will be explored. ||
 * ** How will you hook students at the beginning of the unit? ** |||||||| I will open the unit by presenting a group of photos that illustrate polygons in the real world and hold a discussion about why we would want to know about their measurements. ||
 * ** What events will help students experience and explore the big idea and questions in the unit? How will you equip them with needed skills and knowledge? ** |||||||| Day 1: Explore Angle Measures in Polygons
 * ** How will you cause students to reflect and rethink? How will you guide them in rehearsing, revising, and refining their work? ** |||||||| I will present them with challenging problems that force students to integrate their prior understandings with the new ones they obtain through this unit. ||
 * ** How will you help students to exhibit and self-evaluate their growing skills, knowledge, and understanding throughout the unit? ** |||||||| The answers, not the work, of the homework assignments are given to the students at the beginning of the unit. They will use these answers to help guide them in their understanding of the concepts of the unit. Here is how the homework process works: students complete the assignment on their own and check their answers and review the ones that they did not understand. Then they rework the problems and turn to Hotmath.com for help. If, after that time a student does not understand a problem, then they bring up their concern in class. ||
 * ** How will you tailor and otherwise personalize the learning plan to optimize the engagement and effectiveness of ALL students, without compromising the goals of the unit? ** |||||||| Students will work in pairs and as a class to work through challenging and thought provoking problems. ||
 * ** How will you organize and sequence the learning activities to optimize the engagement and achievement of ALL students? ** |||||||| The individual class periods will review previous material learned and build on this knowledge and understandings to optimize their learning. ||
 * Kate

I like Kate's idea. Another way is to have students role play carnival management to determine what the chances are that people will win at certain games in order to decide how many prizes to buy. Or, have students go to a carnival and do some action research and reflect on the results. Nice job overall. BB **


 * I like the real world connection you made. Another thought is maybe the management teaming is noticing too many people are winning. What can the students do to decrease the probability of too many customers winning? liz **