Unit Plan

## Two-Dimensional Figures

• Assessment Anchors
• Eligible Content
• Big Ideas
• Mathematical statements can be justified through deductive and inductive reasoning and proof.
• Numbers, measures, expressions, equations, and inequalities can represent mathematical situations and structures in many equivalent forms.
• Objects can be transformed in an infinite number of ways. Transformations can be described and analyzed mathematically.
• Patterns exhibit relationships that can be extended, described, and generalized.
• Relations and functions are mathematical relationships that can be represented and analyzed using words, tables, graphs, and equations.
• Similarity relationships between objects are a form of proportional relationships. Congruence describes a special similarity relationship between objects and is a form of equivalence.
• Some geometric relationships can be described and explored as functional relationships.
• Spatial reasoning and visualization are ways to orient thinking about the physical world.
• There are some mathematical relationships that are always true and these relationships are used as the rules of arithmetic and algebra and are useful for writing equivalent forms of expressions and solving equations and inequalities.
• Concepts
• 2- and 3-dimensional figures
• Analytic Geometry
• Geometric probability
• Geometric Relations: Congruence and Similarity
• Geometric Representations
• Reasoning and Proof
• Trigonometric Ratios
• Competencies
• Apply geometric properties of length or area to represent and calculate probabilities.
• Define and describe types of geometrical reasoning and proof, using them to verify valid conjectures as they surface in the study of geometry; develop a counter example to refute an invalid conjecture.
• Use concepts of congruence and similarity to relate and compare 2- and 3-dimensional figures, including trigonometric ratios.
• Use coordinates and algebraic techniques to interpret, represent, and verify geometric relationships.

### Objectives

Students will learn about the classification, properties, and formulas regarding two-dimensional geometric figures. They will:

• examine properties of triangles and other polygons.

• develop rules representing angle measures in polygons.

• study perimeter and area.

• use area in probability situations.

#### Essential Questions

• What characteristics of triangles and other polygons can guide us in making useful generalizations about plane geometry and how do those generalizations give us tools to solve real-world problems?

### Related Materials & Resources

The possible inclusion of commercial websites below is not an implied endorsement of their products, which are not free, and are not required for this lesson plan.

• View

# Multiple Choice Items:

1. What is the significance of the centroid (the point of concurrency of the medians)?

 A It is the center of the inscribed circle. B It is the center of the circumscribed circle. C It is the center of gravity of the triangle. D It is at the midpoint of the longest side.

2. Betty has a scalene triangular flower bed and is covered uniformly with flowers. She needs to place a circular sprinkler so that it will water the entire bed. Which triangle concurrency will locate the sprinkler so the water most efficiently covers the flower bed and why?

 A Medians, because it is closest to each side B Perpendicular Bisectors, because it is the center of the largest circle C Angle Bisectors because it is the center of the smallest circle D Altitudes, because it is closest to each vertex

3. Sid has three trees in his backyard. He wants to place a bench equal distance from all three trees. Which point of concurrency should Sid find?

 A Circumcenter B Orthocenter C Incenter D Centroid

4. Francis wants to build a pentagonal garden. The garden already has one 90 degree angle. Francis wants the other angles to be equal. What will be the measure of each of the other angles?

 A 90 degrees B 108 degrees C 112.5 degrees D 135 degrees

5. What is the measure of an interior angle of a regular 12-sided polygon?

 A 1800 degrees B 360 degrees C 150 degrees D 30 degrees

6. What is the measure of an exterior angle of an equiangular octagon?

 A 45 degrees B 135 degrees C 360 degrees D 1080 degrees

7. What is the probability of a chip that lands randomly in the large triangle also lands in the shaded area?

 A 25% B 37.5% C 75% D 150%

8. There is a rectangular park with a road running diagonally through it. When it rains, what is the probability that a raindrop randomly hits the road rather than grass within the park boundary?

 A 20% B 30% C 40% D 80%

9. Dan launched a model rocket in a field that has two circular ponds. What is the probability that his model rocket that lands randomly within the field also lands in one of the ponds?

 A 4.2% B 16.7% C 20.9% D 79.1%

 1. C 2. B 3. A 4. C 5. C 6. A 7. A 8. A 9. D

10. Amber is making a table with a triangular top. She needs to find the center of the top so she knows where to attach the pedestal. Explain what steps Amber should take to find this center.

11. What is the sum of the interior angles for a polygon with 28 sides? Show your work.

12. A circular bull’s eye is made up of four circles, each with a radius 2 inches larger than the previous circle. Find the probability that a randomly thrown dart lands in the shaded area. Explain your reasoning.

# Short-Answer Key and Scoring Rubric:

10. Amber is making a table with a triangular top. She needs to find the center of the top so she knows where to attach the pedestal. Explain what steps Amber should take to find this center.

Amber should find the median of each side. The median is the line connecting the midpoint of each side to the opposite vertex.

11. What is the sum of the interior angles for a polygon with 28 sides? Show your work.

4680 degrees; 180(n – 2); 180 (28 − 2)

12. A circular bull’s eye is made up of four circles, each with a radius 2 inches larger than the previous circle. Find the probability that a randomly thrown dart lands in the shaded area. Explain your reasoning.

 Points Description 2 Response is complete, correct, and detailed. Student demonstrates understanding of the concept. Response is justified with supporting work. 1 Response is partially correct or true, but does not answer the specific question or is correct but lacking detail. Student demonstrates partial understanding of the concept. Supporting work is incorrect or absent. 0 Response is incorrect. Student demonstrates no understanding of the concept. No example or other support is provided.

# Performance Assessment:

## Part 1

Cindy has been hired to build a fountain for a local park. Her design calls for a triangular base with a circular fountain head placed on the triangle base so that the water reaches to each vertex of the triangle.

1. Explain to Cindy how she can find where to place the fountain head and how you know that the placement of the fountain head will reach each vertex of the triangle.

Cindy has taken some measurements of the triangle. She finds that the base is 6 feet and the height is 9 feet. She also knows that the fountain head sprays water out to a maximum radius of 5 feet.

2. What is the probability that a water droplet leaves the fountain head and lands on the triangular base? Show how you found your answer.

Part 2

Tony was cleaning out his basement when he found a broken piece of a polygonal serving platter. Because all sides are congruent and all angles are congruent on the edge of the platter piece, Tony decides that the original platter must have been a regular polygon. He measures one of the interior angles to be 140 degrees. How many sides did the original platter have? Show how you found your answer.

Part 1

Cindy has been hired to build a fountain for a local park. Her design calls for a triangular base with a circular fountain head placed on the triangle base so that the water reaches to each vertex of the triangle.

1. Explain to Cindy how she can find where to place the fountain head and how you know that the placement of the fountain head will reach each vertex of the triangle.
Answers may vary. Cindy should find the perpendicular bisectors of each side. To do that, she should find the midpoint of a side and draw a line through that midpoint but at a right angle to the side. Where all of the perpendicular bisectors meet is where Cindy should place the fountain head. This placement will reach each vertex of the triangle because the perpendicular bisectors meet at the circumcenter. The circumcenter is the center of the circumscribed circle, which is the circle whose edge touches each vertex of the triangle.

Cindy has taken some measurements of the triangle. She finds that the base is 6 feet and the height is 9 feet. She also knows that the fountain head sprays water out to a maximum radius of 5 feet.

2. What is the probability that a water droplet leaves the fountain head and lands on the triangular base? Show how you found your answer.
The area of the triangle is:
The area of the circle is: π • r2, or π • 25 = approximately 78.5 sq ft
The probability is
The probability of a water drop landing on the triangle is approximately 34.4%.

Part 2

Tony was cleaning out his basement when he found a broken piece of a polygonal serving platter. Because all sides are congruent and all angles are congruent on the edge of the platter piece, Tony decides that the original platter must have been a regular polygon. He measures one of the interior angles to be 140 degrees. How many sides did the original platter have? Show how you found your answer.

Using the formula:

Multiply both sides by n: 180(n − 2) = 140n

Distribute 180: 180n − 360 = 140n

Subtract 180n: −360 = −40n

Multiply both sides by −1: 360 = 40n

Divide both sides by 40: n = 9

The serving platter had 9 sides.

# Question 1

 Points Description 4 Student correctly identifies the circumcenter, includes a complete explanation of how to find it, and explains that it will find the circumscribed circle, which is what Cindy is looking for. 3 Student correctly identifies the circumcenter, includes an adequate explanation of how to find it (has missing parts), and explains that it will find the circumscribed circle, which is what Cindy is looking for. 2 Student correctly identifies the circumcenter, but either fails to explain how to find the circumcenter or fails to explain why the circumcenter is the point Cindy is looking for. 1 Student correctly identifies the circumcenter, but provides no explanation for how to find the circumcenter or why the circumcenter is the point Cindy is looking for. 0 Student fails to find the circumcenter and fails to identify that the circumcenter is what Cindy is looking for.

# Question 2

 Points Description 4 Student correctly shows how to find the area of the triangle and the circle. Student also shows correctly how to find the probability. Student work is easy to follow. 3 Student correctly finds two of the three calculations (area of triangle, area of circle, probability). Student work is mostly easy to follow. 2 Student correctly finds one of the three calculations. Student work is somewhat easy to follow. 1 Student has incorrect calculations or fails to show his/her work. 0 Student has incorrect calculations and fails to show any supporting work.

# Part 2

 Points Description 4 Student correctly finds the number of sides and his/her work is easy to follow. 3 Student correctly finds the number of sides, but has some missing steps and explanations. 2 Student shows an understanding of the process and the necessary formula but has some miscalculations. 1 Student fails to show an understanding of the formula or fails to show any work. 0 Student fails to show an understanding of the formula and fails to show any supporting work.

DRAFT 08/31/2011