Area of a Sector (Lesson 8.8)
Unit 1: Reasoning in Geometry
Day 1: Creating Definitions
Day 2: Inductive Reasoning
Day 3: Conditional Statements
Day 4: Quiz 1.1 to 1.3
Day 5: What is Deductive Reasoning?
Day 6: Using Deductive Reasoning
Day 7: Visual Reasoning
Day 8: Unit 1 Review
Day 9: Unit 1 Test
Unit 2: Building Blocks of Geometry
Day 1: Points, Lines, Segments, and Rays
Day 2: Coordinate Connection: Midpoint
Day 3: Naming and Classifying Angles
Day 4: Vertical Angles and Linear Pairs
Day 5: Quiz 2.1 to 2.4
Day 6: Angles on Parallel Lines
Day 7: Coordinate Connection: Parallel vs. Perpendicular
Day 8: Coordinate Connection: Parallel vs. Perpendicular
Day 9: Quiz 2.5 to 2.6
Day 10: Unit 2 Review
Day 11: Unit 2 Test
Unit 3: Congruence Transformations
Day 1: Introduction to Transformations
Day 2: Translations
Day 3: Reflections
Day 4: Rotations
Day 5: Quiz 3.1 to 3.4
Day 6: Compositions of Transformations
Day 7: Compositions of Transformations
Day 8: Definition of Congruence
Day 9: Coordinate Connection: Transformations of Equations
Day 10: Quiz 3.5 to 3.7
Day 11: Unit 3 Review
Day 12: Unit 3 Test
Unit 4: Triangles and Proof
Day 1: What Makes a Triangle?
Day 2: Triangle Properties
Day 3: Proving the Exterior Angle Conjecture
Day 4: Angle Side Relationships in Triangles
Day 5: Right Triangles & Pythagorean Theorem
Day 6: Coordinate Connection: Distance
Day 7: Review 4.1-4.6
Day 8: Quiz 4.1to 4.6
Day 9: Establishing Congruent Parts in Triangles
Day 10: Triangle Congruence Shortcuts
Day 11: More Triangle Congruence Shortcuts
Day 12: More Triangle Congruence Shortcuts
Day 13: Triangle Congruence Proofs
Day 14: Triangle Congruence Proofs
Day 15: Quiz 4.7 to 4.10
Day 16: Unit 4 Review
Day 17: Unit 4 Test
Unit 5: Quadrilaterals and Other Polygons
Day 1: Quadrilateral Hierarchy
Day 2: Proving Parallelogram Properties
Day 3: Properties of Special Parallelograms
Day 4: Coordinate Connection: Quadrilaterals on the Plane
Day 5: Review 5.1-5.4
Day 6: Quiz 5.1 to 5.4
Day 7: Areas of Quadrilaterals
Day 8: Polygon Interior and Exterior Angle Sums
Day 9: Regular Polygons and their Areas
Day 10: Quiz 5.5 to 5.7
Day 11: Unit 5 Review
Day 12: Unit 5 Test
Unit 6: Similarity
Day 1: Dilations, Scale Factor, and Similarity
Day 2: Coordinate Connection: Dilations on the Plane
Day 3: Proving Similar Figures
Day 4: Quiz 6.1 to 6.3
Day 5: Triangle Similarity Shortcuts
Day 6: Proportional Segments between Parallel Lines
Day 7: Area and Perimeter of Similar Figures
Day 8: Quiz 6.4 to 6.6
Day 9: Unit 6 Review
Day 10: Unit 6 Test
Unit 7: Special Right Triangles & Trigonometry
Day 1: 45˚, 45˚, 90˚ Triangles
Day 2: 30˚, 60˚, 90˚ Triangles
Day 3: Trigonometric Ratios
Day 4: Using Trig Ratios to Solve for Missing Sides
Day 5: Review 7.1-7.4
Day 6: Quiz 7.1 to 7.4
Day 7: Inverse Trig Ratios
Day 8: Applications of Trigonometry
Day 9: Quiz 7.5 to 7.6
Day 10: Unit 7 Review
Day 11: Unit 7 Test
Unit 8: Circles
Day 1: Coordinate Connection: Equation of a Circle
Day 2: Circle Vocabulary
Day 3: Tangents to Circles
Day 4: Chords and Arcs
Day 5: Perpendicular Bisectors of Chords
Day 6: Inscribed Angles and Quadrilaterals
Day 7: Review 8.1-8.6
Day 8: Quiz 8.1 to 8.6
Day 9: Area and Circumference of a Circle
Day 10: Area of a Sector
Day 11: Arc Length
Day 12: Quiz 8.7 to 8.9
Day 13: Unit 8 Review
Day 14: Unit 8 Test
Unit 9: Surface Area and Volume
Day 1: Introducing Volume with Prisms and Cylinders
Day 2: Surface Area and Volume of Prisms and Cylinders
Day 3: Volume of Pyramids and Cones
Day 4: Surface Area of Pyramids and Cones
Day 5: Review 9.1-9.4
Day 6: Quiz 9.1 to 9.4
Day 7: Volume of Spheres
Day 8: Surface Area of Spheres
Day 9: Problem Solving with Volume
Day 10: Volume of Similar Solids
Day 11: Quiz 9.5 to 9.8
Day 12: Unit 9 Review
Day 13: Unit 9 Test
Unit 10: Statistics and Probability
Day 1: Categorical Data and Displays
Day 2: Measures of Center for Quantitative Data
Day 3: Measures of Spread for Quantitative Data
Day 4: Quiz Review (10.1 to 10.3)
Day 5: Quiz 10.1 to 10.3
Day 6: Scatterplots and Line of Best Fit
Day 7: Predictions and Residuals
Day 8: Models for Nonlinear Data
Day 9: Quiz Review (10.4 to 10.6)
Day 10: Quiz 10.4 to 10.6
Day 11: Probability Models and Rules
Day 12: Probability using Two-Way Tables
Day 13: Probability using Tree Diagrams
Day 14: Quiz Review (10.7 to 10.9)
Day 15: Quiz 10.7 to 10.9
Day 16: Random Sampling
Day 17: Margin of Error
Day 18: Observational Studies and Experiments
Day 19: Random Sample and Random Assignment
Day 20: Quiz Review (10.10 to 10.13)
Day 21: Quiz 10.10 to 10.13
Learning Targets
Define sectors as slices of circles that contain a fraction of the total area.
Understand that the area of a sector is proportional to the central angle and use this to calculate area.
Tasks/Activity | Time |
---|---|
Activity | 20 minutes |
Debrief Activity with Margin Notes | 10 minutes |
Quick Notes | 5 minutes |
Check Your Understanding | 15 minutes |
Activity: How Much Pizza Did You Eat?
Lesson Handouts
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Answer Key
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Homework
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Experience First
The final two lessons in unit 8 have students thinking about fractions of the total area and circumference (i.e. sector area and arc length). It is critical that students are using proportional reasoning and thinking about parts of the whole when doing this activity. Note that students going into this lesson do not know the word “sector” yet, nor do they have any formulas for the area of a sector. First we have students thinking about unit fractions, ½ of the pizza, ¼ of the pizza, and 1/12 of the pizza. Finding the area of the slice is intuitive for students, since they can divide the total area by 2, 4, or 12. The bigger goal is to connect this to the measure of the central angle. This is done in question 3 by saying the slice makes a right angle, instead of saying your brother eats a quarter of a pizza. In question 5, students reason about a 20˚ slice. Most likely, they will figure out how many 20˚ slices are in the pizza in order to determine the unit fraction (1/18) and portion of the total pizza. Question 6 puts these ideas together by having students make comparisons between the number of slices and the degree of each slice. It may not be immediately obvious to them that less slices actually means a bigger portion of pizza. Finally in question 7, students write out in words their findings about how the central angle relates to the area of the slice. This is then used to link to the generalized formula in the margin notes during the debrief. Make sure it is the students generating the formula through your line of questioning, not that you are just presenting the formula as an alternate way to find the answer. Guiding Questions
Formalize Later
While there are multiple ways to write the formula for the area of a sector we like to teach it by using the “fraction of the whole” method. This is similar to what students did in previous units when talking about sides of similar figures and right triangle trigonometry. While setting up a proportion also works well, students tend to just use the cross multiplication algorithm instead of reasoning proportionally. Tomorrow’s lesson on arc length will use the same approach, so it will be helpful for students to get familiar with this line of reasoning. You may notice that students don’t have problems finding areas of slices but then struggle in a more abstract context. One of the issues may be that for pizza, students tend to first figure out how many slices are in the pizza if given the degree of a slice and then divide the total by the number of slices. To students, dividing by a whole number feels different than thinking fractionally. This approach will actually work for all problems, but the number of slices may not always be a whole number. For example, if the central angle of a sector is 100˚, students could say that there are 3.6 slices in the circle, and thus we can take the total area of the circle and divide by 3.6.