Objective
Construct an equilateral triangle with only a straight-edge and a compass. Copy a line segment.
Common Core Standards
Core Standards
The core standards covered in this lesson
G.CO.A.1— Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.
Congruence
G.CO.A.1— Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.
G.CO.D.12— Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.
Congruence
G.CO.D.12— Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.
G.CO.D.13— Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.
Congruence
G.CO.D.13— Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.
Foundational Standards
The foundational standards covered in this lesson
7.G.B.4
Geometry
7.G.B.4— Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle.
Criteria for Success
The essential concepts students need to demonstrate or understand to achieve the lesson objective
- Describe constructions as instructions you can use to create geometric figures with only a compass and a straightedge.
- Define circles as closed figures (but not polygons) where each point on the circle is equidistant from a center point.
- Define a radius as the line segment that has the circle center as an endpoint and a point on the circle as the other endpoint.
- Explore with constructions and identify resultant shapes formed from constructions with circles.
- Use the properties of circles to construct congruent line segments.
- Construct an equilateral triangle by using the properties of a circle. Describe constructions.
Tips for Teachers
Suggestions for teachers to help them teach this lesson
- Math Open Reference “Constructions” is helpful to get a visual representation and a written description of constructions.
- This online activity is helpful for students to get excited about constructions in a game format. (This is an online activity that supports critical thinking and constructions.) There is a mobile app similar to this “Euclidea.”
Students will need compasses and straight edges for this lesson.
Lesson Materials
- Compass
- Straightedge
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Anchor Problems
Problems designed to teach key points of the lesson and guiding questions to help draw out student understanding
Problem 1
All the circles below have congruent radii, but different centers.
- What polygons can you create from the marked points in the following figure?
- Describe the features of each polygon, and use the properties of circles to justify your reasoning.
Guiding Questions
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Problem 2
Margi has three cats. She has heard that cats in a room position themselves at equal distances from one another and wants to test that theory. Margie notices that Simon, her tabby cat, is in the center of her bed (at S), while JoJo, her Siamese, is lying on her desk chair (at J). If the theory is true, where will she find Mack, her calico cat? Use the scale drawing of Margie’s room shown below, together with (only) a compass and straight edge. Place an M where Mack will be if the theory is true.
Guiding Questions
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References
EngageNY Mathematics Geometry > Module 1 > Topic A > Lesson 1—Example 1 "Sitting Cats"
Geometry > Module 1 > Topic A > Lesson 1 of the New York State Common Core Mathematics Curriculum from EngageNY and Great Minds. © 2015 Great Minds. Licensed by EngageNY of the New York State Education Department under the CC BY-NC-SA 3.0 USlicense.Accessed Dec. 2, 2016, 5:15 p.m..
Problem 3
The cats changed positions. Mack left the room. Simon and JoJo also moved but stayed the same distance apart. Draw two new locations for Simon and JoJo.
Guiding Questions
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References
EngageNY Mathematics Geometry > Module 1 > Topic A > Lesson 1—Example 1 "Sitting Cats"
Geometry > Module 1 > Topic A > Lesson 1 of the New York State Common Core Mathematics Curriculum from EngageNY and Great Minds. © 2015 Great Minds. Licensed by EngageNY of the New York State Education Department under the CC BY-NC-SA 3.0 USlicense.Accessed Dec. 2, 2016, 5:15 p.m..
Target Task
A task that represents the peak thinking of the lesson - mastery will indicate whether or not objective was achieved
Below is an equilateral triangle. Show the constructions that could have been used to create this equilateral triangle.
Additional Practice
The following resources include problems and activities aligned to the objective of the lesson that can be used for additional practice or to create your own problem set.
- Include problems:
- Practice constructing triangles, given one length of the triangle using only the compass and a straight edge.
- Identify missing steps in a construction.
- Follow a set of construction steps to create another geometric figure (like an isosceles triangle).
- “Why do we use a compass for constructions? What is the benefit of a circle being the primary tool for constructions?”
- EngageNY Mathematics Geometry > Module 1 > Topic A > Lesson 1—Independent Practice
- EngageNY Mathematics Geometry > Module 1 > Topic A > Lesson 2—Exit Ticket (Include a few problems that mimic these problems.)
Lesson 1
Lesson 3
