Alg2 Lesson 4T.1: Measuring an Angle’s Openness

Lesson 4T.1: Measuring an

Angle’s Openness

Obj: I can find arc length.

EQ: How can I determine which arc is longer?

​

Roles:
Facilitator
Scribe
Resourcer
Includer

Lesson Goals:
● Creative Thinking
● Talk through controversies and conflict
● Recognize and reduce ambiguity
● Encourage thinking based on formulas and prior info
● Help explain ideas to each other
● Own your ideas and work
● Record ideas in your journal
● Answer Questions on Slides
● Follow your team roles

​

Facilitator

​

• Make sure that all peers are staying on task.

• Give advice or suggestions to resolve the problem.

• Be sure everyone is able to explain.

Scribe

​

• Make sure peers organize their results on their own papers.

• Remind peers to use color, arrows, and other math tools to
communicate your mathematics, reasons, and connections.

• Be ready to join the teacher for a huddle.

Resourcer

​

• Make sure peers are getting the materials needed.

• Make sure that all materials are put away neatly.

• Make sure that peers are logged in to the needed site.

• Help troubleshoot any technology difficulties that may arise.

Includer

​

• Make sure that all peers are talking about their work.

• Helps keep peers’ voice volume low.

• Encourages everyone to ask questions.

• Communicates conflicts or questions to the teacher.

● Check off tasks & skills on calendar.

● Select skills to work on.

● Work on Deltamath.

Remember to work on the following too…

Part 1: Comparing Attributes

of Central Angles

Terms and Definitions

Central Angle: the angle measurement of a sector of a circle

Radius: the distance between the center and a point on a circle

Circumference: the distance around a circle

Arc Length: the distance around a sector of a circle

Record your Measurements

Use the given circle to
find various things
such as the radius
and circumference.
Check with a partner
and compare results.

Task: Handout 4T.1.A: Comparing Attributes
of Central Angles

A satellite is a moon, planet, or machine that orbits a planet or star. For
example, Earth is a satellite because it orbits the sun. Also, thousands of
artificial, or manufactured, satellites orbit Earth. Some take pictures of the
planet that help meteorologists predict weather and track hurricanes. Most
artificial satellites are used for communications, such as beaming TV
signals and phone calls around the world. A group of more than 20
satellites make up the Global Positioning System (GPS). If you have a
GPS receiver, these satellites can help figure out your exact location. So,
if you’ve ever used an app to navigate to a specific location—you can
thank a satellite!

How are the
orbits similar?
How are the
orbits different?

​

​Insights from Handout (add to your handout)

• ​The fraction of the orbit traveled is the ratio of the arc length and the circumference.

• Suppose you know that two angles have the same measure. Which of the other quantities in the table will have the same value?
  If two angles have the same measure, then they open the same fraction of the circle and the ratio of the arc length to the radius will also be the same. The radius length, arc length, and circumference might all be different.

• The angles appear to be congruent.

Multiple Choice

For each of your angles, what is the ratio of the arc length to the radius?

• The ratio of the arc length to the radius is 1 because we constructed an angle that
  cuts off an arc length of 1 radius.

• The ratio of the arc length to the radius is 3 because we constructed an angle that
  cuts off an arc length of 3 radii.

• The ratio of the arc length to the radius is 2 because we constructed an angle that
  cuts off an arc length of 2 radii.

• The ratio of the arc length to the radius is 0.5 because we constructed an angle that
  cuts off an arc length of 0.5 radii.

Multiple Choice

The angle measures 1 radian or a little less than ___ degrees.

• 60

• 45

• 15

• 30

Radian

One radian is the measure of the
openness of the central angle that cuts
off an arc whose length is equal to the
length of the radius.

Match

Match each degree with its equivalent radian.

• 180º

• 30º

• 6º

• 15º

• 45º

π

π/6

π/30

π/12

π/4

• π

• π/6

• π/30

• π/12

• π/4

Part 2: Creating a Radian

Protractor

Task

How many radians do you think
are in a full circle? Create a
protractor for a full circle with
radian measures in 0.5 radian
increments.

preap.org/Desmos-Radian

Multiple Choice

About how many radians does an angle need to rotate to open a full circle?

• 6

• 12

• 24

• 15

• 344

​

• Why isn’t the angle measure around a circle exactly 6 radians?
  Because the circumference of a circle is equal to the product 2πr, there are 2π radius lengths in the arc that opens a full circle. Therefore, an angle that opens a full circle has a measure of 2π radians.

• At the beginning of the lesson, we saw that angles with equal measure sweep out equal fractions of a circle. What fraction of a circle does an angle with a measure of 1 radian sweep out?
  Since the fraction of the circle is the ratio of the arc length to the radius, if a circle has a radius of r, then an angle that has a measure of 1 radian sweeps out a fraction of r/2πr or 1/2π of the circle.

Insights and stuff to add to journal

Multiple Choice

What is a reasonable decimal approximation of 2π?

• 6.00
• 3.14
• 2.72

• 6.28
• 1.62

Part 3: Defining the

Relationship Among Angle
Measure, Radius Length,

and Arc Length

Task: Handout 4T.1.B Defining the
Relationship Among Angle Measure,
Radius Length, and Arc Length

Multiple Choice

If you know the radius length and the angle measure in radians, how can you determine the arc length that a satellite travels?

• I can set up the proportion degrees = arc length/radius length

• I can set up the proportion radians = arc length/radius length

• I can set up the proportion arc length = radians/radius length

• I can set up the proportion arc length = degrees/radius length

Word Cloud

How is this process related to how you created your radian protractor?

Drag and Drop

Would your method work if the angle measure was given in degrees? ​

Placeholder

, because the degree measure of an angle ​

Placeholder

related to the arc length or the radius of the circle. Just like percentages need to be ​

Placeholder

, degrees need to be ​

Placeholder

to be inputted into an equation.

Drag these tiles and drop them in the correct blank above

No

Yes

is not

is

converted to decimals

converted to radians

Insights & Stuff to add to your Handout

Handout 4T.1.C: Practice with Radian Measure

Random Question of the Day Time

https://wheelofnames.com/4ke-epz We’ll spin
the wheel as a class and spend a minute or so
discussing our answers.