Lesson Title: Constructing a square inscribed in a circle
Date: _____________ Teacher(s): ____________________
Course: Common Core Geometry
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.CO.13
MP2:
MP3:
MP5:
MP6:
Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.
Reason abstractly and quantitatively.
Construct viable arguments and critique the work of others.
Use appropriate tools strategically.
Attend to precision.
Lesson Launch Notes: Exactly how will you use the
first five minutes of the lesson?
Review with students the appropriate tools used in a
construction.
Conduct a class discussion using the prompts and
questions below:
1. Brainstorm a list of constructions (figures) that can
be created using only these tools.
2. Choose a tool(s) and list the procedure for
constructing a set of perpendicular lines.
3. Ask the class what the term “inscribed” means in
geometry? Discuss the requirements for a figure to
be inscribed within another figure.
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 the students complete the closure activity below. On
a half sheet of paper respond to the following:
1. What? (What did you learn today?)
2. So What? (What is relevant, important, and useful
from what you learned today?
3. Now What? (How does what you learned today relate
to this current unit? Can you predict where we are
going next in this unit?)
Lesson Tasks, Problems, and Activities (attach resource sheets): What specific activities, investigations,
problems, questions, or tasks will students are working on during the lesson? Be sure to indicate strategic
connections to appropriate mathematical practices.
Constructing a Square inscribed in a Circle
1. Have students work individually to construct a square inscribed in a circle.
2. Give directions, “Using dot paper or graph paper, straight edges and protractors, or dynamic geometry
software program, construct a square inscribed in a circle.”
3. Instruct students to record their steps while completing the construction.
4. Upon completion of the square inscribed in a circle, student will verify that the figure is a square using the
appropriate measuring tools. (Look for evidence of MP5 and MP6.)
Note: If students need prompting to begin the construction prompt, have them review the properties of a square and
a circle to look for similarities. If a student needs further scaffolding prompt them to start with the center of the
circle and square.
Group Analysis and Critique of Constructions
1. Upon successfully completing the construction and the procedure for the construction, divide students into
groups of threes.
2. Have the students share their constructions and procedures with their group members.
3. Have groups respond to the following:
a) Using all of your groups procedures develop the best possible procedure for constructing a square
inscribed in a circle. (Look for evidence of MP3, MP5 and MP6.)
b) Approximate the percentage of the area of the circle that is represented by the area of the square. (Look
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: Constructing a square inscribed in a circle
Course: Common Core Geometry
Date: _____________ Teacher(s): ____________________
Start/end times: _________________________
for evidence of MP2.)
c) Brainstorm a method for determining the actual percentage of the area of the circle represented by the
area of the square. Use this method to find the actual percentage. (Look for evidence of MP2)
4. Have groups share their procedure with the class.
5. Have the class determine which group developed the most comprehensive procedure that is also the easiest
to follow. (This procedure can be used as a model as students develop further procedures for constructing
geometric figure in the future.) (Look for evidence of MP3)
6. Have groups share the results of their investigation into the percentage of area.
7. Pose the question to the class “Does the percentage of area change when the size is changed?” (It may be
useful to demonstrate this using dynamic geometric software.)
Note: Refer to the document “Inscribing a square in a circle” for an example of the procedure to complete this
construction.
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 be able to develop accurate procedures that others could follow to construct a square inscribed in a
circle. This demonstrates a student’s understanding of the relationship between inscribed figures as well as a
student’s ability to make a construction.
Student success will be measured by the teacher circulating between the students to determine the level of
understanding. As no steps are provided in this activity, it will be important for the teacher to monitor student
progress and prompt students who are struggling. The procedures developed by the groups will be successful if they
are both comprehensive and easy to follow.
Notes and Nuances: Vocabulary, connections, anticipated misconceptions (and how they will be addressed), etc.
Be sure to properly explain the concept of inscribed. Students get easily confused between figures that are inscribed
and those that are circumscribed. The key difference is lies in which item is being examined.
The figure above is an example of a square inscribed in a circle or a circle circumscribed around a square. When
referring to the inscribed figure it will always be the figure on the inside.
Resources: What materials or resources are essential
for students to successfully complete the lesson tasks or
activities?
Homework: Exactly what follow-up homework tasks,
problems, and/or exercises will be assigned upon the
completion of the lesson?
Dot or graph paper
Straightedges, compasses, and protractors
Dynamic Geometry Software (such as Geometer’s
Sketchpad or GeoGebra)
Homework should involve problems that require students
to create constructions of other inscribed figures such as
an equilateral triangle inscribed in a circle, a regular
hexagon inscribed in a circle, or a circle inscribed in an
equilateral triangle.
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?
Do my students know the difference between inscribed and circumscribed figures? Can my students construct a
figure inscribed in a circle?
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: Constructing a square inscribed in a circle
Date: _____________ Teacher(s): ____________________
Course: Common Core Geometry
Start/end times: _________________________
Will my students be able to use constructions of inscribed figure to extend and enhance their study of circles?
Did my students demonstrate an understanding of the 4 SMPs that I tried to elicit in the lesson? To what extent did
they demonstrate proficiency in using and applying the 4 SMPs elicited in the lesson?
HCPSS Secondary Mathematics Office (v2.1); adapted from: Leinwand, S. (2009). Accessible mathematics: 10 instructional shifts that raise student
achievement. Portsmouth, NH: Heinemann.