Lesson Title: Measures of Arcs and Radians
Date: _____________ Teacher(s): ____________________
Course: Common Core Geometry, Unit 3
Start/end times: _________________________
Lesson Standards/Objective(s): What mathematical skill(s) and understanding(s) will be developed? Which
Mathematical Practices do you expect students to engage in during the lesson?
G.C.B.5 Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the
radius, and define the radian measure of the angle as the constant of proportionality.
MP2:
MP3:
MP5:
MP6:
MP7:
Reason abstractly and quantitatively.
Construct viable arguments and critique the reasoning of others.
Use appropriate tools strategically.
Attend to precision.
Look for and make use of structure.
Common Core Geometry, Unit 3
Lesson Launch Notes: Exactly how will you use the
first five minutes of the lesson?
Ask students to convert the following fluid ounce
measure to inches or vice versus using this conversion:
10 fl oz = 296 ml..
If necessary, do an example using multiplication by 10/296
or 296/10
16 oz (answer: 473.6 ml)
20 oz (answer: 592 ml)
100 ml (answer: approx 3.38 oz )
250 ml (answer: approx 8.45 oz )
Lesson Closure Notes: Exactly what summary activity,
questions, and discussion will close the lesson and
connect big ideas? List the questions. Provide a
foreshadowing of tomorrow.
Have all students write a short description of what a radian
is. Circulate around the room and use the responses as a
formative assessment of their conceptual understanding of
the basis for the radian as a unit of angle/arc measure.
Allow students to read answers. Allow for critique and
amendment.
Ask students to give answers and to justify their
answers. Allow for discussion of approaches and
encourage alternatives. Highlight the use of the ratio
multiplication cited above. (Look for evidence of MP2,
MP6, MP7)
Lesson Tasks, Problems, and Activities (attach resource sheets): What specific activities, investigations,
problems, questions, or tasks will students be working on during the lesson? Be sure to indicate strategic
connections to appropriate mathematical practices.
1. Display or distribute the Activity 1: Find the Arc Length. Have students, in pairs, find the arc length indicated
on the given circles. Have randomly selected pairs share answers and methods. Highlight answers that focus on
a proportion of the circumference and that the circumference is 2πr. (Answers: 2 units, approx 3.06 units, 12
units) (Look for evidence of MP6, MP7)
2. Assign students to groups of 3 or 4. Give each group the task of drawing a circle using a compass and indicting
the arc length that is exactly the length of one radius of their circle. Make string or thread available, but do not
require use of it. (Look for evidence of MP1 and MP5)
3. Have groups display their circles to the class by holding them up. Have randomly selected groups describe how
they located the length on their circle. Highlight any answers that focus on the relationship between the
circumference and the length of the radius. (Look for evidence of MP3)
4. Ask, “What do you notice that arcs have in common, without regard to the size of the circle?” “What do you
notice is different about arcs on different size circles. Use the Geometer’s Sketchpad file to show a circle with
an arc of approximately one radian on a circle. By grabbing point B, the size of the circle can be changed,
changing the literal length of the arc, but not the relative measure of the arc to that of the circumference of the
HCPSS Secondary Mathematics Office (v2.1); adapted from: Leinwand, S. (2009). Accessible mathematics: 10 instructional shifts that raise student
achievement. Portsmouth, NH: Heinemann.
Lesson Title: Measures of Arcs and Radians
Course: Common Core Geometry, Unit 3
Date: _____________ Teacher(s): ____________________
Start/end times: _________________________
circle. This may help student pairs to generate answers to the questions. Highlight answers that focus on the
proportion being the same (while the literal length changes).
5. Have the groups draw the central angle that intercepts their arc. Give groups the task of finding the measure of
the central angle (and, therefore, the measure of the arc) in degrees. (Answer is approx 57.3 degrees). Have
randomly selected groups give their measure and explain how they got it. (This can also be demonstrated, with
some accuracy, on the sketchpad file by constructing the central angle and measuring it.)
6. Explain that the measure of this arc and angle can be given in a different unit call a radian. The measure of this
angle/arc is one (1) radian, as the arc has a length of one radius or the central angle captures an arc of length one
radius (for any size circle). Like degrees, radians are measures of arc length relative to the size of the circle
7. Ask “How many of these arcs, one radian long, would fit in the circle?” Let groups work on this answer.
(Answer is 2π, but answers may be an estimate a little greater than 6.)
8. Have groups explain how they arrived at their answers. Look for both physical explanations using thread or
some other equivalent of the radius as well as the use of the formula for circumference. Once 2π is established
as the answer, ask students to compare this to the total number of degrees in circle. Relevant to the lesson
launch, ask groups to establish conversion ratios for degrees to radians and vice versa. (Students may suggest
the use of the 57.30 = 1 radian, found previously, as the ratio. Point out that this is an estimate and to use the
number π in the ratio makes it accurate). With sharing and large group critique, the ratio π/180 or 180/π should
eventually be established. (Look for evidence of MP2, MP5, MP7)
9. Display these angle measures, given in degrees, and have students, in their groups, convert them to radian
measures. Have them give answers both as decimal approximations and in terms of π. (Highlight that radian
measures have no units written after them and degree measures have the degree (o) symbol. Any measure
without a degree symbol defaults to radians.)
90o =
60o=
120o=
75o=
(Answers: π/2≈1.57 , π/3≈ 1.05, 2π/3≈2.09 , 5π/12≈1.31)
Display these angle measures, given in radians, and have students, in their groups, convert them to angle
measures. Some are given as decimals and some in terms of π.
π/6 =
π/2=
3π/4=
2=
3=
(Answers 30o, 90o, 135o, ≈114.59o, ≈171.89o)
Have randomly selected pairs share answers and methods. (Look for evidence of MP5,MP6)
10. Distribute Activity 2 to each student. Have students work in groups to find the radian measures. (Be prepared
to field questions concerning angles/arc measures that are greater than 180˚). Have randomly selected pairs share
answers and methods. (Look for evidence of MP5, MP6)
Evidence of Success: What exactly do I expect students to be able to do by the end of the lesson, and how will I
measure student success? That is, deliberate consideration of what performances will convince you (and any outside
observer) that your students have developed a deepened and conceptual understanding.
Students will understand that there is more than one unit of measure for arcs and angles.
Students will understand that a radian is a unit of measure defined by the length of the radius.
Students will be able to convert between arc/angles measures in degrees to radians and vice versa.
Notes and Nuances: Vocabulary, connections, anticipated misconceptions (and how they will be addressed), etc.
Radians and degrees need to have their sameness as a measure of arcs/angles stressed. Some students see radians as
something completely different. The concept of a radian needs to be clearly shown as a relative measure on the
circle, just as angles are. All circles have similarity and therefore the relative measure of an arc or angle, as a
number of radii, will be the same portion of any circle. This is true of degrees as well, however the degree is not
directly based on some measurement on a circle (rather, it is the division of the circle into a convenient number of
equal pieces – 360 being a “nice” number for dividing into pieces).
Resources: What materials or resources are essential
for students to successfully complete the lesson tasks or
Homework: Exactly what follow-up homework tasks,
problems, and/or exercises will be assigned upon the
HCPSS Secondary Mathematics Office (v2.1); adapted from: Leinwand, S. (2009). Accessible mathematics: 10 instructional shifts that raise student
achievement. Portsmouth, NH: Heinemann.
Lesson Title: Measures of Arcs and Radians
Course: Common Core Geometry, Unit 3
Date: _____________ Teacher(s): ____________________
Start/end times: _________________________
activities?
completion of the lesson?
Compasses
String or thread
Protractors
Sketchpad File
Activity 2 can be given as homework or a textbook page
of radian to degree and degree to radian measure can be
given for skill practice
Lesson Reflections: How do you know that you were effective? What questions, connected to the lesson
standards/objectives and evidence of success, will you use to reflect on the effectiveness of this lesson?
How well did the students seem to connect the ideas of degrees and radians?
How well did students conceptualize the basis of radian measure?
How proficiently can they make the conversions?
Howard County Public Schools Office of Secondary Mathematics Curricular Projects has licensed this
product under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License.
HCPSS Secondary Mathematics Office (v2.1); adapted from: Leinwand, S. (2009). Accessible mathematics: 10 instructional shifts that raise student
achievement. Portsmouth, NH: Heinemann.