MthEd 377 Lesson Plan - Brigham Young University



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 demonstrate thoughtful |10 Met with Dr. Leatham and made appropriate revisions based on this |

|reflection and are clearly articulated |discussion |

| |30 3-5 page reflection paper demonstrates thoughtful reflection and |

| |is clearly articulated |

|Lesson Sequence: Learning activities, tasks |Time |Anticipated Student Thinking |Your response to student |Formative Assessment, |

|and key questions (what you will do and say,| |and Responses |responses and thinking |Miscellaneous things to |

|what you will ask the students to do) | | | |remember |

|Launching the Lesson |

|Draw two similar triangles on the board, |5 min | | |Students should remember|

|labeling with sides ABC and abc. | | | |these properties from |

| | | | |previous lessons. |

|“What do we know about these two triangles’ | | | | |

|sides and angles?” | |Similar triangles have |So, we know that A/a = B/b = | |

| | |congruent proportions and |C/c for ∆ABC and ∆abc to be | |

|Write [pic] on the board. | |angles. |similar. | |

| | | | | |

|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 | | | |Give half the class |

|(two pairs in each group, two worksheets | | | |worksheet one. And the |

|total.) | | | |other half worksheet |

| | | | |two. Each group of four|

|TASK 1: “Using a ruler and a protractor, |15 min |Student should have no problem |Encourage groups to make |will have the same |

|draw right triangles from the angle | |using a protractor and a ruler |whatever size triangle they |worksheet. |

|measurements given. When you are finished, | |to make right triangles. Then |would like. | |

|move onto the second part of the | |again, some may not know how to|Ask what degree a right |Provide a ruler and a |

|exploration.” | |draw a right angle. |triangle is. Show where 90 |protractor for each |

| | | |is located on the protractor.|pair. |

| | | |Connect the point of the | |

| | | |degree mark with the line to |Draw a table on the |

| | | |create a triangle. |board that will organize|

| | | | |all the information |

| | |Students may not know how to | |(rows=trig ratios and |

| | |describe their ratios. |Write on the board |columns=angle values.) |

|Once the class has finished the first part, | | |“Opposite,” “Adjacent” and | |

|tell the class the instructions for the | | |“Hypotenuse” on a triangle |While walking around, |

|second half, as written on the worksheets. | | |with a marked angle. Bring |assign groups one or two|

|Assign groups their angles/triangles to use.| | |this to the attention of the |angles each to find the |

| | | |entire class. |average of all the |

| | | | |ratios given. Write on |

|Draw a chart on the board to organize all | | | |an overhead |

|the information the groups will be | | | |transparency. |

|presenting. Only use the sine, cosine, and | | | | |

|tangent ratios. |10 min | | | |

|Move onto the first discussion. | | | | |

| | |They may draw a picture with a | | |

| | |triangle and fill in lengths |Ask “How did you know it was | |

|TASK 2: “Now that we know the ratios of the| |and angle measures they know |a triangle?... Well done.” | |

|angles, let’s find out how they can be | |for each question. In each | | |

|useful.” | |case they will notice a right | | |

| | |triangle. | | |

|Pass out the second worksheet to the same | |They may not think to draw a | | |

|pairs as previously assigned. | |picture of a triangle. |Lead them by asking them if | |

| | | |they could draw a picture of | |

| | |They may not realize it is a |the scenario. | |

| | |right triangle. | | |

| | | |Say, “If a flag pole is | |

| | | |standing straight up what | |

| | |Problem #1 |kind angle does it make with | |

| | |From the previous task they |the ground? | |

| | |know the tangent of an angle is| | |

| | |equal to the ratio of the |Ask, “Why did you use | |

| | |opposite leg to the adjacent |tangent?... Well done.” | |

| | |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. | | |

| | | | | |

| | |Problem #2 |Ask, “What are you trying to | |

| | |Their responses will be much |solve?” |This question is similar|

| | |like the previous questions. |“Is there anything you can |to problem 1, but |

| | | |use that we just learned |instead they will use |

| | |Problem #3 |about today?” “How does |the sine and cosine |

| | |In this question they will |knowing the angle measure |ratios rather than |

| | |notice in the picture that they|help?” |tangent. |

| | |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 |Ask, “How did you know to use| |

| | |like this: |tangent? Why did you use |The ratio they find |

| | |Tan (angle) = 60/100. |that angle?” |(60/100) will be a ratio|

| | |They might realize that 60/100 | |that is already on the |

| | |= .6 or 3/5, which is | |board. They can look |

| | |approximately a number they | |there as a reference to |

| | |have used in the previous task.| |find which angle |

| | |So whatever angle they used to | |corresponds with this. |

| | |get that ratio is what their | | |

| | |answer will be. | |Bring up arc sin, arcos,|

| | |They may be confused about what| |and arc tan in the |

| | |to solve for, or in writing an | |discussion. |

| | |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?” | |

| | | | | |

| | | |Have them remember what we | |

| | | |did in the previous task. | |

|Facilitating the Discussion |

|TASK 1: By this time, all ratios are on the| |Some students may wonder why I |Acknowledge their |If there’s time, bring |

|board and overhead projector. Call the |15 min |didn’t write all the ratios on |observation. Explain the |up the fact that the |

|attention of the class | |the board. |names of the ratios at this |ratios we have come up |

| | | |point rather than ask the |with (and are given on |

|“What do you notice about all these ratios?”| | |questions. Make sure they |our calculators) are |

| | | |understand that these ratios |only for right |

|“Why are they all the same?” | |They are all the same, or very |are reciprocals of each other|triangles. Show an |

| | |close to each other |and that they have their own |example using an |

|We call the relationship between the sides | | |special names. Now ask the |equilateral triangle. |

|and angles of right triangles trigonometric | |The triangles are all similar, |questions! |We already know that the|

|functions. | |and will have equal proportions| |ratio of the sides are |

| | |of the sides. |Right. These values are |all 1, but the sin(60) ≠|

|Tangent θ = [pic] | | |approximations of the true |1. Why isn’t this |

|Sine θ = [pic] | | |values, since either out |working? IT’S NOT |

|Cosine θ = [pic] | | |instruments or our measuring |RIGHT!!! |

| | | |techniques are not always | |

|“Since the ratios are all the same for a | | |accurate. This is why we | |

|given angle, we can use calculators to find | | |wrote the averages on the | |

|the ratio without drawing the triangles.” | | |overhead. | |

| | | |Recall the launch if some | |

|Show them how to enter the trig functions | | |have forgotten. | |

|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 many ratios do we know? |While walking around the|

|How did you know to use the |10 min |We used tan/cos/sin. |Why don’t we know more? What|class,s look for |

|tangent/cosine/sine ratio? | |We took what we knew, (whether |about the tan/sin/cos buttons|students who have solved|

|Could you do this for any angle? | |it was legs, or hypotenuse) and|on your calculator? Anyone |the problems and |

|What minimum requirements do you need to | |by the definitions that we went|have an idea of where they |understand the concepts |

|solve for the remaining sides or angles? | |over before we knew that |got these numbers? |well. Ask them if they |

| | |tan/cos/sin was the ratio of | |would draw their |

|How did you figure out which angle to use on| |this side to this side. So if | |solutions on the board |

|problem three? | |we knew one side and the angle,| |and explain it to the |

| | |we could figure out the other | |class during the |

| | |side, because we know every | |discussion |

| | |angle has the same ratio. If | | |

| | |we know two of the sides we | |If there’s extra time |

| | |know a ratio, and hence can | |(haha), What can we say |

| | |compare it to a tan/cos/sin | |about scalene triangles?|

| | |ratio that we already know. | |(Draw a picture) Could |

| | | | |we figure out all the |

| | |Some students may have been | |sides from knowing only |

| | |stumped on this part. Others | |an angle and two sides |

| | |may have looked at eh given | | |

| | |ratios on the transparencies as| | |

| | |a reference. | | |

| | | | | |

| | | | | |

| | | | | |

| | | | | |

|Debriefing the Lesson |

|“What have we discovered about angles in a |5 min |There are different ratios |That’s right. This works |TASK 1: Discover that |

|right triangle?” | |corresponding to the sides that|because any right triangle |congruent angles in a |

| | |will always be the same, no |you choose with the given |right triangle give the |

| | |matter how big or small the |angle will be similar (by AA |same proportions of the |

| | |triangle is. |similarity,) and thus have |corresponding sides. |

| | | |equal ratios of sides. | |

| | | | | |

| | | | |TASK 2: Discover how to |

|“How are these ratios important?” | |If we know an angle measure and| |find other side lengths |

| | |a side in a right triangle we | |and angles using this |

| | |can figure out the remaining | |relationship. Discover |

| | |sides. Or if we know two of | |the minimum requirements|

| | |the sides of a right triangle | |in all these cases. |

| | |we can figure out each angle | | |

| | |measure, along with the third | | |

| | |side. | | |

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