Right Triangles and Trigonometry Construc.on of the unit circle - - PDF document

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ì

Right Triangles and Trigonometry

Construc.on of the unit circle

Introduction

ì David Berger

ì HS teacher in Menomonie, WI ì Finishing my 16th year at MHS

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Ice Breaker

ì Introduce yourself to your table.

ì Where are you from? ì Where do you work? ì Favorite math joke?

Understanding the Unit Circle

ì Original presenta.on was November 2015 at NCTM Regional in

Minneapolis with Jake Leibold.

ì Built from previously understood concepts ì Allows students to experience the unit circle through

construc.on – great for visual and kinesthe.c learners

ì Doesn’t rely on memoriza.on tricks ì Developed by a team of teachers during a lesson study at the

Park City Math Ins.tute in 2014

ì Depending on the level of your students – could take between

60 and 120 minutes to complete

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Today’s Agenda

• 1. Pre-requisite Overview
• 2. Unit Circle Construc.on
• 3. What we learned…
• 4. Radian Explora.on
• 5. What we learned…
• 6. Reflec.ons, Logis.cs, Student Work Samples
• 7. Ques.on and Answer with Exit Ticket

Student Prerequisite Knowledge

ì Special Right Triangle Measurements ì Scaling Shapes (to scale down special right triangles) ì Right Triangle Trigonometry ì Ra.onalizing Denominators with single square roots

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Students should not know

ì Conversion between radians and degrees ì Reference angles ì Unit Circle

Goal: Create the unit circle by using an understanding of special right triangles

Common Core Standards

ì F-TF.3 (+) Use special triangles to determine

geometrically the values of sine, cosine, tangent for π/ 3, π/4 and π/6, and use the unit circle to express the values of sine, cosines, and tangent for x, π + x, and 2π – x in terms of their values for x, where x is any real number.

ì F-TF.4 (+)Use the unit circle to explain symmetry (odd

and even) and periodicity of trigonometric func.ons.

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Unit Circle Construction Materials

ì Picture of Graph

Unit Circle Construction Materials

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Unit Circle Construction Labeling the Triangles Unit Circle Construction –Plotting Points

1.

Grab one triangle

2.

Place the labeled angle at the origin and the darkened base along the x-axis

3.

Plot the point at the vertex of the other acute angle

4.

Label the coordinates of the point

5.

Flip the triangle along the y- and x-axis to plot more points

6.

Repeat the steps with the other 3 triangles

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Unit Circle Construction –Plotting Points

1.

Grab one triangle

2.

Place the labeled angle at the origin and the darkened base along the x-axis

3.

Plot the point at the vertex of the other acute angle

4.

Label the coordinates of the point

5.

Flip the triangle along the y- and x-axis to plot more points

6.

Repeat the steps with the other 3 triangles

Unit Circle Construction Plotting Points

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Unit Circle Construction Plotting Points

ì Once you have finished

plojng your points, grab 3 more of 1 color and glue them down they way you ploked them

ì Glue your other three

triangles on the bokom of the sheet

ì Turn the sheet over and

work on the ques.ons

Unit Circle Construction Degrees

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Unit Circle Construction Connections Unit Circle Construction Connections

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Unit Circle Construction Connections Unit Circle Construction Connections

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Unit Circle Construction Degrees

1.

Why is the circle you created called the unit circle?

2.

Where can you look on the unit circle to find the value

• f cos(30°) or cos(60°)? Why? Be specific

3.

What about sin(45°) or sin(30°)? Be specific

4.

Another ques.on asks you to find the sin(135°). Where do you think you would look to find that? Why?

5.

Finally, you are asked to find cos(-60°). Where do you think this would be located? Why?

Unit Circle Construction Recap

1.

Special right triangles put up on the board

2.

Scale hypotenuse to 1 (independent, group, or as a class) for both triangles

3.

Label given triangles cut-outs with this informa.on

4.

Use triangles to plot points

5.

Use triangles to label coordinates at the points

6.

Glue set of 4 of one color on the coordinate grid. Glue other 3 on bokom of sheet

7.

Answer ques.ons on the back (independent, group, or as a class)

8.

Walk through degree extension/connec.on

9.

Label degrees on created unit circle

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What We Learned

ì What are some things you no.ced about this

ac.vity?

ì Think about your students – how would this ac.vity

go for them? What might be some of their challenges or misconcep.ons?

ì What lingering ques.ons do you have?

Radian Exploration Introduction

ì What are some common misconcep.ons about

Radians?

ì What exactly is a Radian? ì How were you taught Radians? ì How do you teach Radians?

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Radian Exploration Standards

ì F-TF.1. Understand radian measure of an angle as the

length of the arc on the unit circle subtended by the angle.

ì F-TF.2. Explain how the unit circle in the coordinate plane

enables the extension of trigonometric func.ons to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle.

ì G.C.5 Derive using similarity the fact that the length of the

arc intercepted by an angle is propor.onal to the radius, and define the radian measure of the angle as the constant

• f propor.onality.

Radian Exploration Circle Circumference

ì Calculate the circumference of the unit circle ì C=2πr, r=1 -> C=2π ì There are Wikki S.cks at your table – wrap them around

the circle having the two tapped ends line up. How long is the Wikki S.ck?

ì Label each end of the pipe Wikki S.ck as 0 and 2π with

the marker at your table

ì No.ce the other taped intervals – label these with the

correct measurements

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Radian Exploration Wikki Stick Rulers

ì Aper the Wikki S.x/pipe cleaners are labeled, wrap

a Wikki S.x/pipe cleaner around the unit circle – no.ce how each interval basically matches up with

• ne of the points ploked

ì Label these measurements with a blue pencil

Radian Exploration Radians

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Radian Exploration Radian Connection

ì Through this ac.vity resist the urge to use “radian”

– use “arc length” instead

ì Near the end of the ac.vity, use the Common Core

Standards to define radian: “the length of the arc

• n the unit circle subtended by the angle”

ì Students experience the physical nature of radians,

not just another way of measuring an angle

What We Learned

ì What are some things you no.ced about this

ac.vity?

ì Think about your students – how would this ac.vity

go for them?

ì What might be some of their challenges or

misconcep.ons?

ì What lingering ques.ons do you have?

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Reflections

ì Students that struggle with the pre-requisite

knowledge will have difficulty

ì Can easily become teacher centered, not teacher

facilitated

ì Time consuming – hard to complete quickly ì Students that miss a day will have a hard .me

catching up

ì Supply prep can take awhile

Logistics

ì Digital Resources

ì

hkp://www.wismath.org/2017-WMC-Annual- Conference-Speaker-Materials

ì

Print-ready versions of today’s classwork sheet, triangles, exit .ckets, and addi.onal ques.oning ideas (for higher level students)

ì

Lesson Plans, Classroom PowerPoint slides ì Supplies

ì

Mul.-Color Card Stock Paper

ì

19’’/20’’ pipe cleaners or Wikki S.x

ì

Tape/glue s.cks/dry erase markers

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Questions and Answers Exit Ticket

ì

Did you come up with any neat ideas during the session? If so, I would love to hear from you!

ì

What revisions would you suggest to either por.on of the lesson?

• ì

Digital Copies of all resources from today, as well as detailed lesson plans, classroom PowerPoints, print-ready materials (including triangle sheets):

ì

hkp://www.wismath.org/2017-WMC-Annual-Conference- Speaker-Materials

ì

Contact: berger.dave.m@gmail.com