Angles and Sides - University of Central Florida



Angles and Sides

Description

Students will discover for themselves the trigonometric ratios associated with the different acute angles of right triangles. They will also perceive a trigonometric decimal as a ratio and connect it to the prior knowledge of similar triangles and proportions. Finally, students will see the connection to the real world through an application problem.

Objectives (Lessons to be learned)

• Understand the characteristics of right triangles

• Feel comfortable with relevant triangular ratios and proportions

Sunshine State Standards/Benchmarks

• MA.C.3.4.1 Represents and applies geometric properties and relationships to solve real-world and mathematical problems including ratio, proportion, and properties of right triangle trigonometry.

• MA.B.2.4.1 Selects and uses direct (measured) or indirect (not measured) methods of measurement as appropriate.

Bodies of Knowledge (Approved September 2007)

• MA.912.G.5.3 Use special right triangles (30-60-90 and 45-45-45) to solve problems.

Relevance

A simple understanding of proportions can go a long way. Proportions and ratios are very important mathematical concepts that are used in the business, engineering, and even entertainment world. Being able to realize that two things are similar and scale them can prove to be a very useful skill in the real world.

Tools Needed

• Projector

• Pre-made sketch

• GeoLegs (or angle rulers)

• Flat end rulers

• Calculators

• White boards (optional)

• TI Smartview Emulator Software

Learning Challenges

Inquiry Questions

1. What do all right triangles have in common?

2. If the sides of my triangle get bigger, do the angles also get bigger?

3. If the angles get bigger, do the sides get bigger?

4. Can the sides get bigger while the angles stay the same?

5. If the legs get bigger, does the hypotenuse have to get bigger?

Conclusion Statement

An understanding of ratios and similarity in certain triangles should be gained. Also, a feeling of confidence should be gained in dealing with triangles and their angle and side measurements using the relationships of 60-60-60, 30-60-90, and 45-45-90 triangles.

The “Aha!” Moments

1. When the student realizes that in order to draw a triangle with side lengths proportional to x, 2x, x*sqrt(3), all they have to do is draw ANY triangle with angles 30, 60, and 90 degrees.

2. During the technology or even the extra activity when the student realizes that “hey, no matter how big the triangle gets on the calculator/projector, the ratios of the sides is still equal. This must mean that the relationships for triangles MUST BE TRUE!!”

Inquiry Procedure/Assignment

Part A

Introduce Geo-Legs:

Distribute one geo-leg per person. Instruct the students: Hold the geo-leg with the short-leg closest to you. Notice the “0” on the compass (round part) as the blue line is on top of the red line. Push the short leg up until the “30” lines up with the red line. Your blue and red lines should form a “30 degree” angle. (At this time, visually check to see if everyone has a 30 degree angle in their hands).

How would you create a triangle on the top of your paper that has sides proportional to x, 2x, and x*sqrt(3), where x is an arbitrary length? Do this.

How would you create two more triangles each increasing in size from the first one with the same proportional side lengths, one in the middle of your paper and one on the bottom? Do this.

What is the altitude of each of these triangles?

What type of triangles have you just created?

How could you find all three interior angles of each of these triangles if you didn’t already know the measurements and had no means of physical measurement? Do this.

Part B

We are now going to investigate some of these triangles. Think about these questions and make some predictions:

1. What do all right triangles have in common? (Name more than one thing).

2. If the sides of my triangle get bigger, do the angles also get bigger?

3. If the angles get bigger, do the sides get bigger?

4. Can the sides get bigger while the angles stay the same?

5. If the legs get bigger, does the hypotenuse have to get bigger?

Class Discussion:

1. Does everyone have different size triangles?

2. Why does everyone get the same ratio?

3. Do you think that this ratio will hold true for any right triangle with a 60 degree angle? Why?

Now, back to your paper. Draw another 60 degree angle at the bottom of your paper. When you draw the third side of the triangle, make sure that the short leg measures 0.5. It will be tiny. Without measuring the hypotenuse, set up your ratio of short side/hypotenuse and fill in what should go at the bottom of the ratio (the ratio should be .5/1, as in the unit triangle). Measure your hypotenuse now. If you drew very carefully, your hypotenuse should be one centimeter long, or very close to it.

Part C

Technology: (Use calculator instead)

Use a sketch (made earlier by you) on Geometers Sketchpad with an overhead projector to demonstrate some 60 degree right triangles. Have the points labeled and the sides measured. Construct the right angle using “construct a perpendicular”. This will allow you to change the bottom left angle without losing your right angle. Keep the angle at 60 degrees and let the students copy the short leg and hypotenuse measures (listed to the side of your sketch) and calculate their ratio again (this time getting exactly 0.5 without rounding. This is a good time to discuss inaccuracies in measurement.

Short Leg/Hypotenuse Ratios for several different size angles:

As you project the sketch, change the bottom left angle, allowing the students to record each angle and its ratio (the Sketchpad changes the side measures as you change the angle). The students change these ratios into decimals, rounding to the thousandths.

Now let them see a table of trigonometric ratios or use a scientific calculator. They are amazed to see that they have already “discovered” those strange looking numbers.

Part D

Group Discussion:

Discuss with your group how this discovery could be used to find an unknown length in a 60 degree triangle. (Prior knowledge includes using cross- products to solve unknown lengths in similar triangles).

Your group is in charge of ordering rolled steel cable as a stabilizing guy- wire for a the masts on a sailboat. Draw the mast. Add the guy-wire on your drawing, which must have an angle of elevation of 60 degrees from the floor. If the wire is anchored to the deck 3 feet away from the mast, how long should the wire be?

Extra Activity:

Put a triangle on the overhead and ask students to physically measure the sides of it. Pull the overhead back making the triangle larger and ask them to measure them again. Push the over head closer than it originally was to the wall and ask them to measure again. Finally, ask them to calculate a specific ratio for each triangle and let them discover that they are all the same.

Grading Rubric for Student Work

I. Triangle Exploration (2 points each for a total of 20 points maximum).

Complete: First three triangles plus the small unit triangle at the bottom.

All sides measured and labeled.

All four triangles have a ratio listed.

All four triangles have a decimal listed.

Measuring: Measures are accurate to the millimeter.

Angles are labeled and accurate.

Neat: Lines are ruler drawn.

Triangles are closed.

Right angles are exact.

Numbers are readable.

II Group Project: Guy Wire for Sailboat (10 points maximum)

Attractive, accurate drawing.

Sides and Angles labeled with measures.

Shows full proportion (2 ratios).

Correct Answer

................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download