Lesson Plan

Lesson Plan (inquiry)

Name: Elizabeth Berlinger

Title of Lesson: Investigating Law of Sines and Cosines

Date of Lesson: Tuesday of Week 3 of the Unit

Length of Lesson: 50 minutes

Description of the class:

Name of course: Pre-Calculus

Grade Level: 11^th or 12^th

Honors or regular: Either

Source of the lesson:

http://www.univie.ac.at/future.media/moe/galerie/trig/trig.html

http://www.pen.k12.va.us/Div/Winchester/jhhs/math/lessons/trig.html

http://projects.cbe.ab.ca/sss/science/physics/map_south/applets/vector_addition_numerical/applethelp/lesson/lesson.html

TEKS addressed:

§111.35. Precalculus

(3) The student uses functions and their properties to model and solve real-life problems. The student is expected to:

(D) solve problems from physical situations using trigonometry, including the use of Law of Sines, Law of Cosines, and area formulas.

I. Overview:

I want students to understand how to calculate angle measurements and corresponding side lengths of scalene triangles. This concept is important for the students to learn, because for the Sundial unit, students must use law of sines to find the length of the gnomon.

II. Performance or learner outcome

Students will be able to:

1). Calculate, using the law of sines, an angle of a scalene triangle if given two sides and the angle opposite one of them.

2). Explain the relationship between sides and angles of scalene triangles when some sides and angles remain fixed.

3). Calculate, using the law of cosines, the length of one side of a triangle when the angle opposite it and the other two sides are known.

4). Explain how the concepts of law of sines and cosines can be used in their sundial projects.

III. Resources, materials and supplies needed:

1). Computers with internet access, preferably one per student

IV. Supplementary materials, handouts:

1). Worksheet on overview of when to use the law of sines vs. the law of cosines. (Handed out after lesson for future reference)

http://www.pen.k12.va.us/Div/Winchester/jhhs/math/lessons/trig/flowchrt.html

Five –E Organization

ENGAGEMENT:

Teacher Does Probing Questions Student Does

Discuss with students their sundial projects and brainstorm the kinds of things they think they will need to construct their own.

Using what you already know about sundials and how they are used, what are some things that you think will be useful when constructing your own?

What concepts from geometry do you think you will need? Do you think you will need to use any trig concepts? If so, which ones?

Students will respond with some of the information they gathered when doing their sundial research. They will probably mention the geometry of circles, triangles, measuring angles ect. If they fully understand how the sundial works, then they will most likely answer that they will in fact need to know trigonometry when constructing their sundials.

Establish prior knowledge.

What kinds of triangles do we know how to find angle measurement and side lengths for?

Right triangles.

What is used to find this kind of information for right triangles?

The Pythagorean Theorem

EXPLORE:

Teacher Does Probing Questions Student Does

Today we are going to learn how to find angle measurements and side lengths for scalene triangles instead of right triangles. This information will help you later with your sundial constructions.	What are some of the differences between a scalene and a right triangle?	Scalene triangles do not have a 90 degree angle, ect.
I want everyone to get a computer and go to the following website. (http://www.univie.ac.at/future.media/moe/galerie/trig/trig.html We are going to explore a little with finding missing angles and sides. There are some exercises and questions that follow. I want everyone to play around with the applet and see if they notice any patterns forming between the side length and the angle measures.		Students get computers and go the given website.
I will be walking around helping those of you that are stuck or need questions answered. Feel free to ask each other questions too, and be thinking about how this relates to your sundial project.	Does anyone have any ideas as to what the Law of Sines is? What kind of relationship does it describe? What happens on the applet when you push the Law of Sines 1 button? What do those calculations mean?	It is a relationship between a side length and the sin of its corresponding angle. When you push the Law of Sines 1 button, 3 calculations are given: a/sin(A), b/sin(B), and c/sin(C).
	What do you notice about these 3 numbers?	They are always equal to each other no matter how you change the size of the triangle.
When given a scalene triangle of any size, if the length of two sides and the angle opposite one of those sides is known, then you can use the *Law of Sines* to find the angle opposite the other side. The Law of Sines states that a/sin(A) = b/sin(B) = c/sin(C). You can use the information you know and this equation to find missing side lengths and angle measures.
Let’s discuss some of the answers that you guys came up with for the exercises.		Several students explain how they changed their triangle and what effect that had on the triangles’ measurements.
	Now, what do you think would happen if we didn’t know two side lengths and one of the opposite angles? What if instead we were given two side lengths and the enclosed angle? Do you think we would be able to complete the rest of the triangle? Would we be able to do it using Law of Sines, or would we have to come up with some other method?	We would probably have to use a different formula, because we would not have enough information to use the Law of Sines.
The Law of Sines does not work in cases like these, because you are not given enough information to work with. When this happens, the use of the *Law of* *Cosines* is helpful. The Law of Cosines helps you calculate one side of a triangle when the angle opposite and the other two sides are known.
The Law of Cosines says, that given a triangle a,b,c, with angle measures A,B,C, a² = b² + c² – 2bc(cos(A)).	Does anyone have any ideas as to why this is true and where parts of this formula were derived?	Part of it looks like the Pythagorean Theorem.
We are going to explore the Law of Cosines and it’s applications using a simulation of vector addition. I would like everyone to go to the following website: http://projects.cbe.ab.ca/sss/science/physics/map_south/applets/vector_addition_numerical/applethelp/lesson/lesson.html		Students access the applet.
I would like you to complete Exercise 4 and 5 using Method 1 as described. Please show me how you got your answers.		Students complete the exercises.
	How is vector addition an application of the Law of Cosines? How is the Law of Sines incorporated also?	It is used to calculate the magnitude of the added vector. The Law of Sines is used to calculate the angle alpha.

EXPLAIN:

Teacher Does Probing Questions Student Does

Ask questions to see how well the students understood the concepts.	Given two fixed length sides in a triangle, if the length of one side increases, what happens to the angle opposite it? Does it increase as well, or does it decrease?	It increases.
	Why do we use the Law of Sines and the Law of Cosines on scalene triangles instead of right triangles?	Because with right triangles, we already know one of the angles is 90 degrees, so we only need the Pythagorean Theorem and SOHCAHTOA to solve for missing side lengths and angle measures.
	How many pieces of information must you be given for any triangle to solve for the missing sides and angle measures?	Three
	How do you know when to use the Law of Sines, and when to use the Law of Cosines?	You can decide that when you know what pieces of the triangle you already know.

EXTEND/ELABORATE:

Teacher Does Probing Questions Student Does

	Now that you know the Law of Sines and the Law of Cosines, in what ways do you think this will be useful when constructing your group sundials?	Maybe we will use it to calculate the angles between the “time increments” on our sundial face. Answers will vary.
Explain that in fact the Law of Sines will be very helpful to them when they want to figure out how long to make their gnomon.
	In what other real-life situations do you think that this information might be useful?	When calculating the angle of view from our eyes to some object a set distance away from us? When calculating vectors.

EVALUATE:

Teacher Does Probing Questions Student Does

I would end the class with a wrap up discussion and possibly a journal for the students to write describing what they’ve learned.	What is the formula for Law of Sines?	a/sin(A) = b/sin(B) = c/(sinC)
	What pieces of information do you need to be given in order to use this equation accurately?	The length of two sides of a triangle and an angle opposite one of the sides.
	What kind of triangles can the Law of Sines and the Law of Cosines be used for?	Scalene triangles.
	What is the equation for the Law of Cosines, and what pieces of information do you need to know to use it?	a² = b² + c² – 2bc(cos(A)) The sides of the triangle and the enclosed angle must be known to calculate the third side.