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MthEd 377 Lesson Plan Cover Sheet Name: John Jett and Edda Date: November 7, 2005 Section Title: 13.6 The Tangent of an Angle; 13.7 The Sine and Cosine Ratios Big Mathematical Idea(s): Congruent angles in similar right triangles have corresponding ratios of the sides. These ratios will always be equal to each other for a chosen acute angle of that right triangle. In a right triangle, if a side and an angle (other than the right angle) are given, the other side lengths can be found. Similarly, if two sides are given, any angle other than the right angle can be found as well. This happens because all congruent angles have the same corresponding ratios of the sides, as stated above. Why are these BMIs important mathematically? While similarities have mainly dealt with two triangles having equal proportions, trig functions extend this idea to infinitely many triangles. Now we can know the ratio of any triangle’s side lengths by just choosing one angle. In other words, the tangent, cosine, or sine of an angle will always stay the same, no matter what the side lengths are. The only criteria is if the triangles are similar, or right triangles in this instance. Knowing this relationship also leads to important problem-solving techniques. For example, students can find any side of a right triangle by knowing one of its non-right (acute) angles. How does this lesson fit in to the overall unit? (i.e., How does this lesson build mathematically on the previous lessons and how do subsequent lessons build mathematically on it?) Students have just learned the side lengths and angles of special right triangles. As such, they have some previous understanding of the minimal requirements to solve for the remaining asides of a special right triangle. They generalized these side lengths using variables, which mean any sized special right triangle will have the same angles. Students have dealt extensively with similar triangles. They know that similar triangles have the same side proportions and congruent angles. This lesson builds on this idea by introducing infinitely many triangles. These triangles are produced by creating right triangles from the same angle. Because the triangles all have congruent angles, they are similar. This implies that they will have the same proportions one with another. However, we can now choose any acute angle in our right triangle and know the proportions of its own sides due to the trigonometric functions. On a simpler note, trigonometric functions allow people to solve for any side or angle of a triangle. This comes particularly handy in elementary physics. On a broader note, this lesson will extend into the mathematical branch of trigonometry. This will lead to theorems such as the Law of Cosines, Law of Sines, and eventually identities. We left some sections open-ended to lead into our inverse and reciprocal trig functions. Pretty soon, a simple trig function will turn into an Euler’s formula or be used to find polar coordinates. Let the games begin… Grading rubric (for Keith’s use) 5 The Big Mathematical Idea addresses core mathematical concepts and is clearly articulated 5 Description of the importance of the topic is well thought out and relevant 5 There is a clear, insightful discussion of how this lesson fits in to the mathematical content of the overall unit 5 Lesson sequence is well thought out and detailed 5 Students' thinking is anticipated with forethought and detail 5 Reactions to students' thinking is mathematically oriented, insightful and detailed 10 3-5 reflection paragraphs 10 Met with Dr. Leatham and made demonstrate thoughtful appropriate revisions based on this reflection and are clearly discussion OR articulated 30 3-5 page reflection paper demonstrates thoughtful reflection and is clearly articulated Lesson Sequence: Learning activities, tasks and key questions (what you will do and say, what you will ask the students to do) Anticipated Student Time Thinking and Responses Your response to student responses and thinking Formative Assessment, Miscellaneous things to remember Launching the Lesson Draw two similar triangles on the board, labeling with sides ABC and abc. 5 min “What do we know about these two triangles’ sides and angles?” Write A B C a b c Students should remember these properties from previous lessons. Similar triangles have congruent proportions and angles. So, we know that A/a = B/b = C/c for ∆ABC and ∆abc to be similar. on the board. Using algebra, show that A/B=a/b from the previous review. “We know that these two proportions are equal for these two triangles, but what if we had infinitely many similar triangles? Would it still work?” Move onto Task 1. Orchestrating the Task Divide class into pairs in groups of four (two pairs in each group, two worksheets total.) TASK 1: “Using a ruler and a protractor, draw right triangles from the angle measurements given. When you are finished, move onto the second part of the exploration.” 15 min Once the class has finished the first part, tell the class the instructions for the second half, as written on the worksheets. Assign groups their angles/triangles to use. Draw a chart on the board to organize all the information the groups will be presenting. Only use the sine, cosine, and tangent ratios. Move onto the first discussion. TASK 2: “Now that we know the ratios of the angles, let’s find out how they can be useful.” Pass out the second worksheet to the same pairs as previously assigned. Student should have no problem using a protractor and a ruler to make right triangles. Then again, some may not know how to draw a right angle. Students may not know how to describe their ratios. Encourage groups to make whatever size triangle they would like. Ask what degree a right triangle is. Show where 90 is located on the protractor. Connect the point of the degree mark with the line to create a triangle. Write on the board “Opposite,” “Adjacent” and “Hypotenuse” on a triangle with a marked angle. Bring this to the attention of the entire class. 10 min They may draw a picture with a triangle and fill in lengths and angle measures they know for each question. In each case they will notice a Ask “How did you know it was a triangle?... Well done.” Give half the class worksheet one. And the other half worksheet two. Each group of four will have the same worksheet. Provide a ruler and a protractor for each pair. Draw a table on the board that will organize all the information (rows=trig ratios and columns=angle values.) While walking around, assign groups one or two angles each to find the average of all the ratios given. Write on an overhead transparency. Lesson Sequence: Learning activities, tasks and key questions (what you will do and say, what you will ask the students to do) Anticipated Student Time Thinking and Responses right triangle. They may not think to draw a picture of a triangle. They may not realize it is a right triangle. Problem #1 From the previous task they know the tangent of an angle is equal to the ratio of the opposite leg to the adjacent leg in a right triangle. They may set up the equation: Tan (40) = height/20. To solve this equation they will probably multiply both sides of the equation by 20. In order to multiply 20 by Tan (40), they will have to know a number or ratio of Tan (40). They will know this from the previous task. After multiplying they will find the height of the flag pole. They may not know how to set up the equation. They may not see the relation with tangent. 2. Problem #2 Their responses will be much like the previous questions. Problem #3 In this question they will notice in the picture that they know the legs of a right triangle but not the angle. They may notice that the legs have to do with tangent ratio. So they may set up an equation like this: Tan (angle) = 60/100. They might realize that 60/100 = .6 or 3/5, which is approximately a number they have used in the previous task. So whatever angle they used to get that ratio is what Your response to student responses and thinking Formative Assessment, Miscellaneous things to remember Lead them by asking them if they could draw a picture of the scenario. Say, “If a flag pole is standing straight up what kind angle does it make with the ground? Ask, “Why did you use tangent?... Well done.” Ask, “What are you trying to solve?” “Is there anything you can use that we just learned about today?” “How does knowing the angle measure help?” This question is similar to problem 1, but instead they will use the sine and cosine ratios rather than tangent. Ask, “How did you know to use tangent? Why did you use that angle?” The ratio they find (60/100) will be a ratio that is already on the board. They can look there as a reference to find which angle corresponds with this. Bring up arc sin, arcos, and arc tan in the discussion. Lesson Sequence: Learning activities, tasks and key questions (what you will do and say, what you will ask the students to do) Anticipated Student Time Thinking and Responses Your response to student responses and thinking their answer will be. They may be confused about what to solve for, or in writing an equation. Or they may not know how they can figure out what angle gave them that ratio. Lead them in the right direction by asking, “What do we know about the legs in a right triangle?” “Can you use this information at all?” Formative Assessment, Miscellaneous things to remember Have them remember what we did in the previous task. Facilitating the Discussion TASK 1: By this time, all ratios are on the board and overhead projector. Call the attention of the class 15 min Some students may wonder why I didn’t write all the ratios on the board. “What do you notice about all these ratios?” They are all the same, or very close to each other “Why are they all the same?” We call the relationship between the sides and angles of right triangles trigonometric functions. The triangles are all similar, and will have equal proportions of the sides. opposite leg adjacent leg opposite leg Sine θ = hypotenuse adjacent leg Cosine θ = hypotenuse Tangent θ = Acknowledge their observation. Explain the names of the ratios at this point rather than ask the questions. Make sure they understand that these ratios are reciprocals of each other and that they have their own special names. Now ask the questions! Right. These values are approximations of the true values, since either out instruments or our measuring techniques are not always accurate. This is why we wrote the averages on the overhead. Recall the launch if some have forgotten. If there’s time, bring up the fact that the ratios we have come up with (and are given on our calculators) are only for right triangles. Show an example using an equilateral triangle. We already know that the ratio of the sides are all 1, but the sin(60) ≠ 1. Why isn’t this working? IT’S NOT RIGHT!!! “Since the ratios are all the same for a given angle, we can use calculators to find the ratio without drawing the triangles.” Show them how to enter the trig functions into their calculators. TASK 2: Call attention to the class. Have assigned students come up to the board and explain their solutions. What did you do to figure out the side or the angle measure? Which trig function did you use? How did you know to use the tangent/cosine/sine ratio? Could you do this for any angle? What minimum requirements do you 10 min We used tan/cos/sin. We took what we knew, (whether it was legs, or hypotenuse) and by the definitions that we went over before we knew that tan/cos/sin was the ratio of this side to this side. So if we knew one side and the angle, we could figure out the other side, because How many ratios do we know? Why don’t we know more? What about the tan/sin/cos buttons on your calculator? Anyone have an idea of where they got these numbers? While walking around the class,s look for students who have solved the problems and understand the concepts well. Ask them if they would draw their solutions on the board and explain it to the Lesson Sequence: Learning activities, tasks and key questions (what you will do and say, what you will ask the students to do) Anticipated Student Time Thinking and Responses need to solve for the remaining sides or angles? Your response to student responses and thinking we know every angle has the same ratio. If we know two of the sides we know a ratio, and hence can compare it to a tan/cos/sin ratio that we already know. How did you figure out which angle to use on problem three? Formative Assessment, Miscellaneous things to remember class during the discussion If there’s extra time (haha), What can we say about scalene triangles? (Draw a picture) Could we figure out all the sides from knowing only an angle and two sides Some students may have been stumped on this part. Others may have looked at eh given ratios on the transparencies as a reference. Debriefing the Lesson “What have we discovered about angles in a right triangle?” “How are these ratios important?” 5 min There are different ratios corresponding to the sides that will always be the same, no matter how big or small the triangle is. If we know an angle measure and a side in a right triangle we can figure out the remaining sides. Or if we know two of the sides of a right triangle we can figure out each angle measure, along with the third side. That’s right. This works because any right triangle you choose with the given angle will be similar (by AA similarity,) and thus have equal ratios of sides. TASK 1: Discover that congruent angles in a right triangle give the same proportions of the corresponding sides. TASK 2: Discover how to find other side lengths and angles using this relationship. Discover the minimum requirements in all these cases.