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Area and Applications of Laws (Lesson 5.4)
Unit 0: Prerequisites
Day 1: The Cartesian Plane
Day 2: Equations of Circles
Day 3: Solving Equations in Multiple Representations
Day 4: Reasoning with Formulas
Day 5: Quiz 0.1 to 0.4
Day 6: Linear Relationships
Day 7: Reasoning with Slope
Day 8: Set Notation
Day 9: Quiz 0.5 to 0.7
Day 10: Unit 0 Review
Day 11: Unit 0 Test
Unit 1: Functions
Day 1: Functions and Function Notation
Day 2: Domain and Range
Day 3: Rates of Change and Graph Behavior
Day 4: Library of Parent Functions
Day 5: Transformations of Functions
Day 6: Transformations of Functions
Day 7: Even and Odd Functions
Day 8: Quiz 1.1 to 1.6
Day 9: Building Functions
Day 10: Compositions of Functions
Day 11: Inverse Functions
Day 12: Graphs of Inverse Functions
Day 13: Piecewise Functions
Day 14: Quiz 1.7 to 1.11
Day 15: Unit 1 Review
Day 16: Unit 1 Test
Unit 2: Polynomial and Rational Functions
Day 1: Connecting Quadratics
Day 2: Completing the Square
Day 3: Polynomials in the Short Run
Day 4: Polynomials in the Long Run
Day 5: Review 2.1-2.4
Day 6: Quiz 2.1 to 2.4
Day 7: Factor and Remainder Theorem
Day 8: Factor and Remainder Theorem
Day 9: Complex Zeros
Day 10: Connecting Zeros Across Multiple Representations
Day 11: Intro to Rational Functions
Day 12: Graphing Rational Functions
Day 13: Quiz 2.5 to 2.9
Day 14: Unit 2 Review
Day 15: Unit 2 Test
Unit 3: Exponential and Logarithmic Functions
Day 1: Exponential Functions
Day 2: Graphs of Exponential Functions
Day 3: Compound Interest and an Introduction to "e"
Day 4: Review 3.1-3.3
Day 5: Quiz 3.1 to 3.3
Day 6: Logarithmic Functions
Day 7: Graphs of Logarithmic Functions
Day 8: Logarithm Properties
Day 9: Solving Exponential and Logarithmic Equations
Day 10: Quiz 3.4 to 3.7
Day 11: Exponential and Logarithmic Modeling
Day 12: Unit 3 Review
Day 13: Unit 3 Test
Unit 4: Trigonometric Functions
Day 1: Right Triangle Trig
Day 2: Inverse Trig Ratios
Day 3: Radians and Degrees
Day 4: Unit Circle
Day 5: Unit Circle
Day 6: Other Trig Functions
Day 7: Review 4.1-4.6
Day 8: Quiz 4.1 to 4.6
Day 9: Graphing Sine and Cosine
Day 10: Transformations of Sine and Cosine Graphs
Day 11: Graphing Secant and Cosecant
Day 12: Graphing Tangent and Cotangent
Day 13: Quiz 4.7 to 4.10
Day 14: Inverse Trig Functions
Day 15: Trigonometric Modeling
Day 16: Trigonometric Identities
Day 17: Unit 4 Review
Day 18: Unit 4 Review
Day 19: Unit 4 Test
Unit 5: Applications of Trigonometry
Day 1: Law of Sines
Day 2: The Ambiguous Case (SSA)
Day 3: Law of Cosines
Day 4: Area and Applications of Laws
Day 5: Vectors
Day 6: Review 5.1-5.5
Day 7: Quiz 5.1 to 5.5
Day 8: Polar Coordinates
Day 9: Equations in Polar and Cartesian Form
Day 10: Polar Graphs Part 1
Day 11: Polar Graphs Part 2
Day 12: Review 5.6-5.9
Day 13: Quiz 5.6 to 5.9
Day 14: Parametric Equations
Day 15: Parametric Equations (With Trig)
Day 16: Unit 5 Review
Day 17: Unit 5 Test
Unit 6: Systems of Equations
Day 1: What is a Solution?
Day 2: Solving Systems with Substitution
Day 3: Solving Systems with Elimination
Day 4: Review 6.1-6.3
Day 5: Quiz 6.1 to 6.3
Day 6: Solving Systems in 3 Variables
Day 7: Solving Systems in 3 Variables
Day 8: Partial Fractions
Day 9: Unit 6 Review
Day 10: Unit 6 Test
Unit 7: Sequences and Series
Day 1: Introducing Sequences
Day 2: Using Sequences and Series to Describe Patterns
Day 3: Arithmetic Sequences and Series
Day 4: Review 7.1-7.2
Day 5: Quiz 7.1 to 7.2
Day 6: Geometric Sequences and Finite Series
Day 7: Infinite Geometric Sequences and Series
Day 8: Proof by Induction
Day 9: Proof by Induction
Day 10: Quiz 7.3 to 7.5
Day 11: Unit 7 Review
Day 12: Unit 7 Test
Unit 8: Limits
Day 1: What is a Limit?
Day 2: Evaluating Limits Graphically
Day 3: Evaluating Limits with Direct Substitution
Day 4: Evaluating Limits Analytically
Day 5: Evaluating Limits Analytically
Day 6: Review 8.1-8.4
Day 7: Quiz 8.1 to 8.4
Day 8: Continuity
Day 9: Continuity
Day 10: Intermediate Value Theorem
Day 11: Intermediate Value Theorem
Day 12: Review 8.5-8.6
Day 13: Quiz 8.5 to 8.6
Day 14: Limits at Infinity
Day 15: Unit 8 Review
Day 16: Unit 8 Test
Unit 9: Derivatives
Day 1: Introduction to Derivatives
Day 2: Average versus Instantaneous Rates of Change
Day 3: Calculating Instantaneous Rate of Change
Day 4: Calculating Instantaneous Rate of Change
Day 5: The Derivative Function
Day 6: The Derivative Function
Day 7: Review 9.1-9.3
Day 8: Quiz 9.1 to 9.3
Day 9: Derivative Shortcuts
Day 10: Differentiability
Day 11: Connecting f and f’
Day 12: Connecting f and f’
Day 13: Review 9.4-9.6
Day 14: Quiz 9.4 to 9.6
Day 15: Derivatives of Sine and Cosine
Day 16: Product Rule
Day 17: Quotient Rule
Day 18: Review 9.7-9.9
Day 19: Quiz 9.7 to 9.9
Day 20: Unit 9 Review
Day 21: Unit 9 Test
Unit 10: (Optional) Conic Sections
Day 1: Intro to Conic Sections
Day 2: Defining Parabolas
Day 3: Working with Parabolas
Day 4: Quiz 10.1 to 10.3
Day 5: Defining Ellipses
Day 6: Working with Elllipses
Day 7: Defining Hyperbolas
Day 8: Working with Hyperbolas
Day 9: Quiz 10.4 to 10.7
Day 10: Unit 10 Review
Day 11: Unit 10 Test
Learning Targets
Apply Law of Sines and Law of Cosines to applied problems
Identify general area formulas for oblique triangles based on the given parts
Reason about the validity of a mathematical model
Tasks/Activity | Time |
---|---|
Activity | 20 minutes |
Debrief Activity | 5 minutes |
Important Ideas | 5 minutes |
Check Your Understanding | 20 minutes |
Activity: How Far is that Commute?
Lesson Handouts
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Answer Key
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Homework
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Experience First
The purpose of today’s activity is to get students meaningful practice with the Law of Sines and Law of Cosines in an applied context. Students must determine when each law should be used and must explore the validity of a model. In particular, how do actual driving distances compare to straight line distances on a map? Why might it be useful to find the area of a triangular region? (We look at population density as one possible reason). Students will need access to some kind of device to check Google Maps. At first, students thought that the calculated distance between UNC and NCSU should be greater than the Google Maps answer since driving doesn’t happen on perfectly direct routes. This is a great line of reasoning but students were challenged to think about the data they were given, which already represented driving distances, not necessarily the sides of that triangle. It’s important that students learn to model with mathematics (MP4) and be prepared to explain differences between a model’s prediction and the true reality. When finding the area of the triangle, students had no trouble drawing in the altitude that represented the height. Many used the Law of Sines instead of right triangle trig to find this value.
Formalize Later
Students look at one possible way of finding area during the activity by using right triangle trig to find the height of the triangle. This is then generalized into the area formula A=½(a)(b)sin C and its various forms. Discuss with students how each triangle has three heights and thus we get the three different forms. We tell students that there is no need to memorize this formula. They are welcome to first find the height of the triangle in any method they choose and then apply their traditional formula from Geometry for the area of a triangle. Heron’s formula is a way of finding the area when given three sides of a triangle, which does not allow for the use of right triangle trig to find the height. We discuss our knowledge of the prefix “semi” to conclude that semiperimeter means half the perimeter. Students may struggle to draw an accurate picture in question 3 of the Check Your Understanding. Encourage students to check each other’s drawings.