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Kaitlin Siefke
Teaching Geometry Project
Stage 1 – Desired Results
Established Key Goals:
o Explore what an equilateral triangle is and the important properties that the equilateral
triangle has that make it an equilateral triangle.
o Work with the Geogebra Program to construct an equilateral triangle with the important
properties
Understandings:
Students will understand that…
 There are specific properties of a
triangle that make it an equilateral
triangle.
 Two circles with intersection points
and radiuses of certain lengths
create an equilateral triangle.
Essential Questions:
 What properties are specific to an equilateral triangle?
 Can a triangle have different properties?
 Are there any objects that are equilateral triangles in your life?
Students will know…
 The properties of an equilateral
triangle have equal side lengths
and equal angle measures.
Students will be able to…
 Construct an equilateral triangle with two circles using the Geogebra
program.
 Label correctly an equilateral triangle amongst other regular polygons.
Stage 2 – Assessment Evidence
Performance Tasks:
 Entrance Slip- labeling the
polygons that are given on the
worksheet
 Exploration worksheet and
answers to questions
throughout exploration
Other Evidence:
 Observation of students while using the Geogebra program to
construct the equilateral triangle
 Students ability to choose the equilateral triangle from the
entrance slip
Stage 3 – Learning Plan
Learning Activities:
-Pass out student entrance slip-Name the Polygons. Students will have to label the polygons (triangles, square, rectangle,
trapezoid, hexagon, octagon etc). (Students would label the triangles as triangles not specific types)
-Have students complete the entrance slip and turn in to keep for grading and further assessment.
-Introduce the topic of the day and the exploration that will be occurring. (Key Goal:Exploring how to construct an equilateral
triangle and what the definition of an equilateral triangle is)
-Pass out exploration worksheet. Explain that the students will follow the directions on the paper to construct an equilateral
triangle on Geogebra. Each student will create their own triangle and answer the questions on the worksheet.
-Use Geogebra diagram worksheet to review what each icon represents and to use for further reference while working on the
exploration.
-At the end of the exploration, have students turn in their worksheet for completion and grading.
-Have discussion with what the students discovered about the equilateral triangle.
-Ask each student up to the desk towards the end of class which of the triangles on their entrance slip resembles an
equilateral triangle. Ask why that one.
Kaitlin Siefke
Teaching Geometry Project
Description of topic: Triangles can be named differently based on the properties that are a part of that
specific triangle. Regular polygons are those shapes that have equal angles throughout and equal side
lengths. Equilateral triangles, specifically, have equal angle measures of sixty degrees and have equal side
lengths. Students explore how equilateral triangles are formed using two circles and the Geogebra program.
Step-by-step instructions lead students to being able to describe in their own words what an equilateral triangle
is and what it looks like. They can also explain important ideas of the equilateral triangle construction with
certain segments as the radiuses of the circles.
Common Core Standards Addressed: (specifically)
Fourth Grade- Geometry
4.G.1. Draw points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular and parallel
lines. Identify these in two dimensional figures.
How it’s related: Students in the fourth grade have begun to draw these different aspects of figures.
They are able to distinguish between regular polygons and are able to label correctly the polygons. Equilateral
triangles, specifically, include points (vertices), line segments as well as angles that can be drawn to form the
triangle.
Fifth Grade- Geometry
5.G.3. Understand that attributes belonging to a category of two-dimensional figures also belong to all
subcategories of that category. For example, all rectangles have four right angles and squares are rectangles,
so all squares have four right angles.
5.G.4. Classify two-dimensional figures in a hierarchy based on properties.
How it’s related: Students in the fifth grade understand that there are specific properties to certain
geometric figures. These include: the number of angles, the number of vertices, the number of sides etc.
Triangles can be labeled as just that, shapes with three sides, but can be in subcategories based on the
number of equal sides or angles (examples: equilateral triangles, scalene triangles, isosceles triangles).
Seventh Grade- Geometry
7.G.2. Draw (freehand, with ruler and protractor, and with technology) geometric shapes with given conditions.
Focus on constructing triangles from three measures of angles or sides, noticing when the conditions
determine a unique triangle, more than one triangle, or no triangle.
How it’s related: Students in the seventh grade should be able to understand properties of equilateral
triangles. The conditions of three equal side lengths and three equal angles create a unique triangle. Using
Geogebra, students can explore how the intersection of two circles and radiuses equal, can create an
equilateral triangle. They are able to make further connections and deeper their understanding of equilateral
triangles.
Van Hiele Levels of Geometric Thought:
Van Hiele Level 1: Analysis: During this level of geometric thought, students are able to begin identifying
properties of certain shapes and can explain those properties using specific vocabulary. Students are able to
focus on properties of an equilateral triangle and can classify those types of triangles based on their properties.
Van Hiele Level 2: Informal Deduction: During this level of geometric thought, students are able to begin
thinking about properties of shapes and their relationship among each other. During this level, students also
can begin logically thinking about the properties and “argue” why certain properties yield to a certain figure
name. For example: students will be able to create “if-then” statement; like “if a triangle has three equal side
lengths and three equal angle measures, then the triangle is an equilateral triangle”.
Kaitlin Siefke
Teaching Geometry Project
Name: ________________________ Date: ___________
Name the Polygons
1._________
2. _____________
3. ____________
4. _________
5. ____________
6.________
7.___________
8._____________
9.__________
Kaitlin Siefke
Teaching Geometry Project
Name: ___________________ Date:________
Exploring Equilateral Triangles
Follow the steps using the Geogebra program to explore the properties of equilateral triangles.
1. Hide the coordinate axes by right clicking on the screen and clicking the axes button. Also,
click on the ‘x’ to close the algebra window on the left of the program.
2. Click the segment between two points tool and make two points on the drawing pad to
construct a segment AB.
3. Click the move button (or the pointer button) took. Use this to highlight over segment AB and
right click the segment. Click on object properties—show label—drop down bar to value—
close. (This should show the value of the length of the segment AB)
4. Now, you will create a circle with the center A and the circle passing through point B. Click the
Circle with Center through Point tool. Click point A then click on point B. Right click and click
on show label.
 What does line segment AB represent in the circle c that was just created?
__________________________________________________________
 How many of these line segments are there on this circle c? How do you know?
____________________________________________________
__________________________________________________________
5. Now, you will create another circle, d, with center B and the circle passing through point A.
Click the Circle with Center through Point tool. Click point B then click on point A.
 What does your picture now look like? Sketch your drawing here
6.
7.
8.
9.
 How many intersection points does your drawing have? _________
To intersect the two circles at the certain points, click on the New Point tool and then click on
Intersect Two Objects. Click on the circumferences of both circles and the points C and D will
appear at the intersections of both circles.
To form an equilateral triangle we will need only points A, B, and C. Click on the Polygon tool
then click on points A, B, C, and A again. The equilateral triangle will now be shaded.
 How many sides does this triangle have? ________________________
 How many vertices does this triangle have? ______________________
Click on the Move tool again. Right click on segment BC and click object properties—show
label—drop down bar to value—close. Repeat this step on segment AC.
 What do you notice about each of the lengths of the segments of the equilateral
triangle? ________________________________________
________________________________________________________
Right click on circle d. (The right circle) Click show object. The circle should disappear.
 What does segment AC represent?
________________________________________________________
Kaitlin Siefke
Teaching Geometry Project
Click Edit from the top bar and click Undo. Right click on circle c. (The left circle) Click
Show object. The other circle should disappear.
 What does segment BC represent?
_________________________________________________________
 What does each segment on the triangle have in common? What does this mean?
________________________________________________
_________________________________________________________
Right click the remaining circle. Click show object. The circle should disappear and the
triangle should remain.
10. Click on the Move tool. Click on point A or point B. You can now make the triangle smaller or
larger, or you can move the triangle around one point.
 What happens to the side lengths when you click on one of the vertices and make the
triangle smaller or larger? __________________________
__________________________________________________________
11. Click on the Angle tool. Then click the center of the triangle.
 What is the measure of Angle ABC? ____________________________
 What is the measure of Angle BCA? ____________________________
 What is the measure of Angle CAB? ____________________________
 What does all the measure of angles on an equilateral triangle have in common?
__________________________________________________
__________________________________________________________
 What is the sum of the angles in the equilateral triangle? _____________
Now that you have explored an equilateral triangle, in your own words, write a definition for what an
equilateral triangle is: _____________________________________________________________
_______________________________________________________________________________
_______________________________________________________________________________
How is this triangle different than a right triangle? _______________________________________
_______________________________________________________________________________
Draw the equilateral triangle that you found using Geogebra in the space provided. Be sure to label
all of the sides and the angle measures.
Kaitlin Siefke
Teaching Geometry Project
Geogebra Icon Diagram
MOVE Tool
Polygon Tool
Segment
between Two
Points Tool
Angle Tool
Circle with
Center
through Two
Points Tool
New Point Tool /
Intersect Two
Objects Tool
Object PropertiesShow Label-Value
Kaitlin Siefke
Teaching Geometry Project
Rubric for Equilateral Triangle Exploration
Student Name: _________________________
Category
Triangle Properties
(Definition)
Completion of
Questions
Geogebra
Completion
Polygon Entrance
Slip/Exit slip
Question
5 Points
Answers show
complete
understanding of the
properties of an
equilateral triangle.
Students are able to
describe an
equilateral triangle
with both properties
(side lengths and
angle measures)
All questions are
completed to the
best of the student’s
ability. With little or
no errors.
Students followed
each step to
correctly construct
an equilateral
triangle on the
Geogebra Program.
Students are able to
correctly label each
of the polygons on
the worksheet with
little or no error.
Students are able to
correctly label which
of the triangles is an
equilateral triangle.
3 Points
Answers show some
understanding of the
properties of an
equilateral triangle.
Students understand
only one of the
properties of what
an equilateral
triangle is (either
side lengths of angle
measure).
Some of the
questions are
completed. With
some errors to
answers.
Students followed
some steps correctly
but have a few
problems with the
equilateral triangle
(for example: when
moving vertices, the
side lengths don't
remain the same)
Students are able to
label only a few of
the polygons on the
worksheet correctly.
Students are able to
correctly label the
equilateral triangle
after one attempt.
0 Points
Answers show no
understanding of the
properties of an
equilateral triangle.
Students have no
understanding of
what an equilateral
triangle is.
No questions are
completed. No effort
was put forth. Many
errors to answers.
Students did not
follow the directions
to answer the
questions and
construct an
equilateral triangle.
No effort to complete
exploration was
shown.
Students have many
errors when labeling
the polygons.
Students are not able
to correctly label the
equilateral triangle
after 2 attempts.
Total: _____/20 points
Comments:_______________________________________________________________________
________________________________________________________________________________
________________________________________________________________________________