Using Polygons and Circles to Problem Solve
Using Polygons and Circles to Problem Solve
Objectives
In this unit, students will use formulas to solve for perimeter, circumference, and area of circles and polygons. Students will confirm the relationships between radius, diameter, circumference, and area of circles. Students will extend their knowledge of area and perimeter of polygons to compound figures. The concepts of congruence and similarity will be explored and extended to a coordinate plane. Students will use these relationships to solve real-world problems involving a variety of geometric figures. Students will:
- identify the parts of a circle and a polygon.
- understand the relationship between radius, diameter, circumference, and area of a circle.
- understand the relationship between perimeter of a polygon and circumference of a circle.
- calculate the circumference of a circle and perimeter of a polygon.
- calculate the area of a circle, triangle, or quadrilateral.
- calculate the perimeter and area of compound figures involving quadrilaterals and triangles.
- identify the parts of a coordinate graph.
- plot a variety of points and geometric figures in the four quadrants of a coordinate plane.
- apply circumference, perimeter, area, congruence, and similarity concepts to solve a variety of real-world problems.
Essential Questions
- How can the characteristics of circles and polygons and their relationships with each other be used to solve problems?
Related Unit and Lesson Plans
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.
- Explore changing areas of rectangles and irregular figures given a specific perimeter
http://www.shodor.org/interactivate/activities/AreaExplorer/
- Calculate perimeter and area of rectangles and irregular figures generated by the computer
http://www.shodor.org/interactivate/activities/ShapeExplorer/
- Enter any value for radius to have circumference and area of a circle instantly calculated
http://www.calculatorsoup.com/calculators/geometry-plane/circle.php
- First 1000 digits of pi
http://www.factmonster.com/ipka/A0876705.html
- Song about parts of a circle and the formulas for area and circumference, use as introduction, review, or memory tool (1:35 min)
https://www.youtube.com/watch?v=lWDha0wqbcI&feature=related
- Geometry videos section to find a variety of surface area and volume lessons including cylinders, cones, prisms, and pyramids
http://www.mathplayground.com/mathvideos.html
- Game: Student names the coordinate to locate aliens http://www.mathplayground.com/locate_aliens.html
- Game: Student selects a coordinate pair from four choices to catch a mole, multiple difficulty levels
http://funbasedlearning.com/algebra/graphing/points3/
- Video clip reviewing definition of similarity and corresponding parts
- Requires Adobe Flash Player
http://illuminations.nctm.org/ActivityDetail.aspx?ID=176
- Video clip reviewing definition of similarity and corresponding parts
https://www.youtube.com/watch?v=9IUI3jtSEWU
- Calculate the surface area and volumes of a variety of rectangular and triangular prisms by adjusting the dimensions
http://www.shodor.org/interactive/activities/SurfaceAreaAndVolume/
- Interactive practice for parts of a circle
http://www.ixl.com/math/practice/grade-7-parts-of-a-circle
- Interactive circle word problems
http://www.ixl.com/math/practice/grade-7-circle-word-problems
- Interactive circumference and area questions
http://www.ixl.com/math/practice/grade-7-circles-calculate-area-circumference-radius-and-diameter
- Interactive question session on similar and congruent figures
http://www.ixl.com/math/practice/grade-7-similar-and-congruent-figures
- Short tutorial on polygons
http://www.eduplace.com/cgi-bin/schtemplate.cgi?template=/kids/mw/help/eh_popup.thtml&grade=4&chapter=16&lesson=4&title=Quadrilaterals+and+Other+Polygons&tm=tmfe1604e Requires Adobe Flash Player
Formative Assessment
-
View
Multiple-Choice Items:
1. Which of the following is a chord in circle A?
A
BC
B
GA and AD
C
HD and EF
D
DF
2. Edwin is trying to determine the distance he can travel with one rotation of his bicycle wheel. If each spoke on the wheel is 13 inches, what is the circumference of the wheel? Use 3.14 for π. Round to the nearest hundredth.
A
40.82 inches
B
78 inches
C
81.64 inches
D
530.66 inches
3. Jillaina is planning to make a flower garden. She began by marking off a circular area in the grass. She measured the distance across the circle and found that it was 9 yards. What will be the area of her garden? Use 3.14 for p. Round to the nearest hundredth.
A
28.26 yd2
B
63.59 yd2
C
88.74 yd2
D
254.34yd2
4. Calculate the perimeter and area of parallelogram EFGH.
A
P = 22.4 inches; A = 44.8 square inches
B
P = 29.4 inches; A = 100.4 square inches
C
P = 40 inches; A = 122.2 square inches
D
P = 44.8 inches; A = 91 square inches
5. Find the perimeter and area of the triangle below.
A
P = 60 cm; A = 133 cm2
B
P = 50 cm; A = 266 cm2
C
P = 266 cm; A = 60 cm2
D
P = 133 cm; A = 266 cm2
6. Vang needs to find the perimeter and area of the figure below. The formulas he knows for geometric figures do not include this shape. He tries several methods for finding these values. Which set of his answers is correct?
A
P = 284 feet; A = 2090 square feet
B
P = 209 feet; A = 2720 square feet
C
P = 209 feet; A = 3840 square feet
D
P = 284 feet; A = 2720 square feet
7. Select the statement below which correctly compares rectangle PQRS to rectangle ABCD.
A
Rectangle PQRS is similar to rectangle ABCD.
B
Rectangle PQRS is congruent to rectangle ABCD.
C
Rectangle PQRS is both congruent and similar to rectangle ABCD.
D
Rectangle PQRS is neither congruent nor similar to rectangle ABCD.
8. What is the length of the unknown side in the two similar figures below?
A
10 in.
B
12 in.
C
16 in.
D
22.6 in.
9. A local diner has a large sign with its name displayed above the building. The owner would like to have a smaller sign just like it hanging next to the door of the building. Find the missing measurements needed to be sure the sign is the right shape and size.
A
XY = 6 m; YZ = 5 m
B
XY = 2 m; YZ = 1.6 m
C
XY = 5 m; YZ = 4 m
D
XY = 3 m; YZ = 2.5 m
Multiple-Choice Answer Key:
1. C
2. C
3. B
4. D
5. A
6. D
7. A
8. C
9. D
Short-Answer Items:
10. Reggie is going to buy a pizza. He has three choices that are all the same price. He believes the best deal will be the pizza with the greatest area. Which of these pizzas would be the best deal for him? Explain why.
- The Huge Hitter with a diameter of 16 inches
- The Jumbo Jack with a radius of 7.4 inches
- The Massive Muncher with a circumference of 50 inches
11. Order the figures below least to greatest using their areas. Show all your work.
12. Describe the steps necessary to graph any ordered pair (x, y). Graph points A, B, C, and D on the coordinate grid.
A
(−3, 2)
B
(5, 2)
C
(6, −4)
D
(−3, −4)
Short-Answer Key and Scoring Rubrics:
10. Reggie is going to buy a pizza. He has 3 choices that are all the same price. He believes the best deal will be the pizza with the greatest area. Which of these pizzas would be the best deal for him? Explain why.
- The Huge Hitter with a diameter of 16 inches
A = 8 x 8 x 3.14 = 200.96 square inches
- The Jumbo Jack with a radius of 7.4 inches
A = 7.4 x 7.4 x 3.14 = 171.95 square inches
- The Massive Muncher with a circumference of 50 inches
D = 50 ÷ 3.14 = 15.92, r = 15.92 ÷ 2 = 7.96 in.
A = 7.96 x 7.96 x 3.14 = 198.95 square inches
The Huge Hitter is the best deal with 200.96 square inches of pizza. This is only slightly larger than the Massive Muncher, whose area is 198.95 square inches. The Jumbo Jack is smaller than both of these with an area of 171.95 square inches.
Points
Description
3
- Comparison statement is complete, correct, and clearly uses area values to compare.
- All three pizza area calculations are correct with work shown.
- Student demonstrates thorough understanding of the circumference and area of a circle.
2
- Written explanation is understandable but brief or simplistic, limited use of area values to compare.
- Two pizza area calculations are correct, but the third is missing or incorrect.
- Student demonstrates good understanding of the circumference and area of a circle.
1
- Written explanation difficult to understand or partially incorrect, may not have included any reference to the pizza areas.
- Two or all three pizza area calculations are incorrect or missing. Another calculation such as circumference may have been attempted.
- Student demonstrates limited understanding of the circumference and area of a circle.
0
- Written explanation is incorrect or missing.
- The area calculations were not attempted.
- Student demonstrates no understanding of circumference and area of a circle.
11. Order the figures least to greatest using their areas. Show all your work.
The area of A = (11.3)(12.7)/2 = 143.51 ÷ 2 = 71.755 square centimeters
The area of B = (12 + 5) ÷ 2 = 8.5(8) = 68 square centimeters
The area of C = (7)(7) + (3)(7) = 49 + 21 = 70 square centimeters
The areas in order least to greatest are: B (68 cm2), C (70 cm2), A (71.755 cm2).
Points
Description
3
- Written explanation and area calculations are complete, correct, and detailed.
- Figure order is correctly stated (B, C, A) with clear reasoning in the work and explanation.
- Student demonstrates thorough understanding of the area of polygons concept.
2
- Written explanation and area calculations are correct but brief or lacking detail.
- Figure order is correctly stated (B, C, A). Reasoning for the order may not be completely evident in the calculations or explanation.
- Student demonstrates good understanding of the area of polygons concept.
1
- Written explanation and area calculations are partially incorrect or partially missing.
- Figure order is stated with 1 error.
- Student demonstrates limited understanding of the area of polygons concept.
0
- Written explanation and area calculations are mostly incorrect or missing.
- Figure order is completely incorrect or not stated.
- Student demonstrates no understanding of the area of polygons concept.
12. Describe the steps necessary to graph any ordered pair (x, y). Graph points A, B, C, and D on the coordinate grid.
A
(−3, 2)
B
(5, 2)
C
(6, −4)
D
(−3, −4)
The first number, or x value, in the ordered pair tells you how far to the left or right to move. You place your pencil at the origin (0, 0) and count to the right for positive numbers or to the left for negative numbers. Then, from this location you move up or down the number of spaces given by the second number, or y value, in the ordered pair. You move up if the value is positive and down if it is negative. Make your point at that intersection on the graph.
Points
Description
3
- Written description for plotting each point is accurate and detailed.
- All four points are graphed correctly.
- Student demonstrates thorough understanding of coordinate graphing.
2
- Written description for plotting each point is accurate.
- Three points are graphed correctly.
- Student demonstrates solid understanding of coordinate graphing.
1
- Written description for plotting each point is only partially accurate or missing important details.
- Only 1 or 2 points are graphed correctly.
- Student demonstrates limited understanding of coordinate graphing.
0
- Written description for plotting each point is completely inaccurate or missing.
- No points are graphed correctly or they are missing.
- Student demonstrates no understanding of coordinate graphing.
Performance Assessment
Materials:
- student copies of The Growing Hexagon handout (M-7-6_Performance Assessment and KEY.docx)
The Growing Hexagon
Find the perimeter and area of the given hexagon.
P = ________
A = ________
A scale factor was used to enlarge the hexagon. Find the scale factor. Find all unknown side lengths. Then find the perimeter and area of the large hexagon.
Performance Assessment: The Growing Hexagon―Answer KEY
Scale factor = 18/15 = 1.2
Performance Assessment Scoring Rubric:
Points
Description
4
- Work with scale factor is clear and all enlargements are correct.
- Student demonstrates a substantial understanding of the mathematical concepts of similarity and scale factor.
- Student demonstrates thorough understanding of breaking down compound figures to find area and perimeter.
- Areas and perimeters are all correct.
- Completely meets all problem requirements.
3
- Work with scale factor is shown, but not clearly explained for all three objects. Answers are present but one may not be reasonable.
- Student demonstrates a solid understanding of the mathematical concepts of similarity and scale factor.
- Student demonstrates adequate understanding of breaking down compound figures to find area and perimeter.
- Areas and perimeters are nearly all correct.
- Meets or partially meets most of the problem requirements.
2
- Work with scale factor is shown but only partially correct. Student demonstrates a fair understanding of the mathematical concepts of similarity and scale factor, but has some misconceptions.
- Student demonstrates some understanding of breaking down compound figures to find area and perimeter.
- Area and perimeter calculations are partially correct.
- Meets approximately half of the problem requirements.
1
- Work with scale factor is present for one or two dimensions.
- Student demonstrates a very limited understanding of the concepts of similarity and scale factor.
- Student demonstrates extremely limited understanding of breaking down compound figures to find area and perimeter.
- Areas and perimeters are nearly all incorrect.
- Does not meet several of the problem requirements.
0
- Work with scale factor is not shown or is completely incorrect.
- Student demonstrates no understanding of similarity and scale factor concepts.
- Student demonstrates no understanding of breaking down compound figures to find area or perimeter.
- Areas and perimeters are all incorrect or not attempted.
- Did not meet any problem requirements.
DRAFT 10/10/2011
