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Math 2 | Phase 2 Week 5 Days 1&3 | Measuring Angles With Radians

Author:Todd Fogdall
Topic:Angles, Circle

Lesson Overview

This week's lesson is shorter than last week's. We will introduce one new concept (radian measure), and then show how radian measures simplify the process of finding arc length and sector area. (If you didn't participate in last week's lessons, you will likely be confused. Go back and look at the Week 4 materials first.) (You should have already downloaded this week's notes so that you can fill them in as you go through this activity.) To set the stage, you should watch the video below.

Introductory Video (3:01)

Origin of Degrees: So where does 360 degrees come from? (3:50)

So here are three important facts. Fact #1. The Babylonians used base 60 because it's the smallest number divisible by 1, 2, 3, 4, 5, and 6, and that made calculations easier. Of course, 60 is a factor of 360, which is close to the number of days in one Earth year. Fact #2. The previous fact should tell you that it really was a random choice. To this day, we use that number because an ancient civilization decided we should. Fact #3. Here's the most important fact: This choice of 360 for the number of degrees in a circle is not connected in any way to the size of the circle. Consequence: As a result of these three facts, modern science and mathematics use a different unit of measure that is directly connected to the size of the circle. Below is a sketch that works similar to the previous week's sketches. When it loads, all you should see is a central angle of in a circle with a radius of 3.5 and an arc length of 1.222. (You should definitely check that number for accuracy with what you learned from last week. Now for your task: Modify the central angle to change the arc length so that it is almost the same length as the radius. When you get close, the sketch will tell you you're getting warmer, and when you're super close, it will help you by giving you a button to make it perfect. NOTE TO MOBILE USERS: The first time you load this applet and try to use the first button that appears, you'll get an error message, and then you'll be able to use it. If you're using a desktop, this shouldn't happen to you.

Question #1 - Number of degrees that make the arc length the same as the radius.

How many degrees, approximately, make the radius and the arc length the same? (Remember, you're not marked right or wrong; you're just checking your answer.)

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Question #2 - What if the radius changes?

Change the radius of the circle. Does the size of the angle change?

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Question #3 - How many radians in a circle?

When the radius and the arc length became the same, you should notice that the sketch now labels the angle as being "1 radian". This is a new unit of measure that you are being introduced to. You should also notice that a button appears that allows you to continue to add radians until there's no more room. How many radians can fit inside a circle? (If the number isn't a whole number, just state the whole number it is close to.)

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How to convert between degrees and radians.

Saying there is a little bit more than six radians in a circle isn't going to be as accurate as we need. Watch the video below to see how we convert.

Converting Between Radians and Degrees (5:04)

Question #4 - Converting between radians and degrees

Saying there are six-ish radians in a circle isn't good enough. What's the conversion factor that you learned from the video?

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Converting Between Radians and Degrees – Guided Practice

Now we'll continue to work through the guided practice on your notes. Check out the video below.

Using Radians to Calculate Arc Length - Guided Practice

Questions #5 - Arc Length

What's the new formula for arc length with radians that you learned from the video?

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Now we'll finish the last guided practice in your notes about calculating sector area with radians. Check out the video below.

Using Radians to Calculate Sector Area - Guided Practice

Questions #6 - Sector Area

What's the new formula for sector area with radians that you learned from the video?

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That's it, folks! It's time to try out the independent practice on the last two pages of your practice. Be sure to check your work against the key, and don't forget to ask your teacher for the check for understanding. Have a great week!

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