Steven Bernstein
Valeri Kisiel
bern4990@fredonia.edu
kisi9051@fredonia.edu
The Equation of a Snowman
This lesson is meant to reinforce and allow 10th grade students to practice equations of circles by using GeoGebra
as a learning tool.
Professional Standards Addressed:

G.CM.2 Use mathematical representations to communicate with appropriate accuracy, including numerical
tables, formulas, functions, equations, charts, graphs, and diagrams.

G.CM.11 Understand and use appropriate language, representations, and terminology when describing
objects, relationships, mathematical solutions, and geometric diagrams.

G.G.71 Write the equation of a circle, given its center and radius or given the endpoints of a diameter.

G.G.73 Find the center and radius of a circle, given the equation of the circle in center-radius form.

G.G.74 Graph circles of the form
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Instructional Objectives:
At the end of this lesson students will be able to:

Understand how to use the basic functions of GeoGebra, such as plotting equations and points.

Understand the standard form for the equation of a circle,

Understanding how to manipulate circles by changing the values of h, k, and r .

Know how to find the center and radius of a circle when the equation is in standard form.
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Materials:



A copy of “The Equation of a Snowman” for every student.
Students should work individually on a computer; if necessary they may pair up but no more than 2 in a
group.
Students will need internet access and the ability to print their work.
Instructional Protocol/Itinerary:
First, do a quick review of what the equation of a circle is. During this review the teacher should explain standard
form for a circle and what the variables h, k, and r represent, as well as what happens when you change the values
of these variables and how they manipulate that circle. This should take no more than 5 minutes. You will first need
to go to www.geogebra.org, click “Start GeoGebra” then click “GeoGebra WebStart”. Then begin the snowman lab
with the students and complete the first four steps together. If you have the ability to do this over a projector so
students can see your screen this would be ideal. After that allow students to work alone. It is important that you
are free to answer any of the students’ questions or concerns. When they have finished the steps laid out they
should still have enough time to add their own creations so that they explore other parts of geometry in the
program. It’s also important to show them your Frosty creation or any other you have done. When students have
finished have them turn in their lab materials and a printed copy of their Frosty. If you wish you can turn it into a
contest for the most unique Frosty.
Some GeoGebra Fast Facts
For a beginner on GeoGebra, here is some help to find your way around.
At the top of the screen, you’ll see ten different functions you can click on to use. You’ll also see
that each one of these functions also has a drop down menu with other abilities.
To plot equations or points, simply type them on the bottom of GeoGebra where it says, “Input:”.
Example: “A= (2,3)” plots point A at (2,3)
The 1st function on the top of GeoGebra is the cursor. You’ll want it on the cursor for clicking
objects and to move around anything you’ve created.
To manually plot points, click on the 2nd function to the right of the cursor.
If you wanted to connect points, the 3rd function can create lines, segments, and rays.
The 4th function helps you form special types of lines. Some include:
 Parallel Lines
 Perpendicular lines
 Tangent lines
 Angle and line bisectors
The 5th function will help you create polygons. You can either manually create your own by clicking
on Polygon, or you can create a regular polygon where you input the exact number of points in the
polygon.
The 6th function has to do with circles. You can create them three different ways based on the
information you have.
 Circle with center through point
 Circle with center and radius
 Circle three points (Because any 3 points create a unique circle)
The 7th function is the measuring tool. You use this function if you want to measure lengths, angles,
or even create an angle by inputting the measure of angle desired.
The 8th function is used for translations, reflections, and dilations.
The 9th function has a useful tool that allows you to input any text.
The 10th function is very helpful, and that you click on it and you can move around your view of
your creation.
Last but not least, to turn on/off your axis or grid, go to View and they are the 1st two options.
Name: ______________________
The Equation of a Snowman
Frosty the snowman, was a jolly happy soul. He was made of points and circles though and he’ll come to
life with math…
Steps: (Check off each box as you go.)
 Click View and choose Grid.
 Click Options then go to Labeling and click on No new objects.
 On the bottom of the screen next to Input type x^2 + y^2 = 9 and hit Enter. This creates a circle
centered at the origin with a radius of 3.
 To plot, type the equation of a circle centered at (0,-7) with a radius of 4.
o
Equation:
 Type the equation of a circle centered at (0,-16) with a radius of 5.
o
Equation:
 Type the equation of a circle centered at (-1, 1) with a radius of 0.8.
o
Equation:
 Type the equation of a circle centered at (1, 1) with a radius of 0.8.
o
Equation:
 Type the equation of a circle centered at (0, 0) with a radius of 0.5.
o
Equation:
 Type the equation of a circle centered at (-1.5,-1.5) with a radius of 0.4.
o
Equation:
 Type the equation of a circle centered at (1.5,-1.5) with a radius of 0.4.
 Type the equation of a circle centered at (-.75,-2) with a radius of 0.4.
 Type the equation of a circle centered at (.75,-2) with a radius of 0.4.
 Type the equation of a circle centered at (0,-2) with a radius of 0.4.
 Type the equation of a circle centered at (0,-5) with a radius of 0.6.
 Type the equation of a circle centered at (0,-7) with a radius of 0.6.
 Type the equation of a circle centered at (0,-9) with a radius of 0.6.
 To plot the points for the arms type (4,-7)
 Type (-4,-7)
 Type (10,0)
 Type ( -10,-14)
 Type (-11,-14)
 Type (-11,-15)
 Type (-9,-16)
 Type ( 10, 1)
 Type (11.5,1.5)
 Type (11.5,0)
 To create line segments for the arms type (so that it is written as)
segment[(-4,-7),(-10,-14)].
 Type segment[(-10,-14),(-11,-15)].
 Type segment[(-10,-14),(-9,-16)].
 Type segment[(-10,-14),(-11,-14)].
 Type segment[(4,-7),(10,0)].
 Type segment[(10,0),(10,1)].
 Type segment[(10,0),(11.5,1.5)].
 Type segment[(10,0),(11.5,0)].
Now be creative and unique. Give Frosty a little something special and really make this your creation.
Feel free to use any of the functions in GeoGebra!
 Click View and choose Grid.
 Click View and choose Axis.
And then we have our friend Frosty!
Frosty Wrap Up:
The general equation of a circle is:
Assume that we have the unit circle centered at the origin [
], then
1.) if it is shifted 4 units to the right the equation becomes:
2.) if it is shifted 3 units down the equation becomes:
3.) if it’s radius is increased by 6 the equation becomes:
4.) if it is shifted 2 units up, 7 units left, and it’s radius is increased by 3 the equation becomes:
Name:
KEY
The Equation of a Snowman
Frosty the snowman, was a jolly happy soul. He was made of points and circles though and he’ll come to
life with math…
Steps: (Check off each box as you go and fill in the equations where indicated.)
 Click View and choose Grid.
 Click Options then go to Labeling and click on No new objects.
 On the bottom of the screen next to Input, type x^2 + y^2 = 9 and hit Enter. This creates a
circle centered at the origin with a radius of 3.
 To plot the next part of his body, type the equation of a circle centered at (0,-7) with a radius of
4.
o
Equation:
 Type the equation of a circle centered at (0,-16) with a radius of 5.
o
Equation:
 Type the equation of a circle centered at (-1, 1) with a radius of 0.8.
o
Equation:
 Type the equation of a circle centered at (1, 1) with a radius of 0.8.
o
Equation:
 Type the equation of a circle centered at (0, 0) with a radius of 0.5.
o
Equation:
 Type the equation of a circle centered at (-1.5,-1.5) with a radius of 0.4.
o
Equation:
 Type the equation of a circle centered at (1.5,-1.5) with a radius of 0.4.
 Type the equation of a circle centered at (-.75,-2) with a radius of 0.4.
 Type the equation of a circle centered at (.75,-2) with a radius of 0.4.
 Type the equation of a circle centered at (0,-2) with a radius of 0.4.
 Type the equation of a circle centered at (0,-5) with a radius of 0.6.
 Type the equation of a circle centered at (0,-7) with a radius of 0.6.
 Type the equation of a circle centered at (0,-9) with a radius of 0.6.
 To plot the points for the arms type (4,-7)
 Type (-4,-7)
 Type (10,0)
 Type ( -10,-14)
 Type (-11,-14)
 Type (-11,-15)
 Type (-9,-16)
 Type ( 10, 1)
 Type (11.5,1.5)
 Type (11.5,0)
 To create line segments for the arms type (so that it is written as)
segment[(-4,-7),(-10,-14)].
 Type segment[(-10,-14),(-11,-15)].
 Type segment[(-10,-14),(-9,-16)].
 Type segment[(-10,-14),(-11,-14)].
 Type segment[(4,-7),(10,0)].
 Type segment[(10,0),(10,1)].
 Type segment[(10,0),(11.5,1.5)].
 Type segment[(10,0),(11.5,0)].
Now be creative and unique. Give Frosty a little something special and really make this your creation.
Feel free to use any functions in GeoGebra!
 Click View and choose Grid.
 Click View and choose Axis.
And then we have our friend Frosty!
Frosty Wrap Up:
The general equation of a circle is:
Assume that we have the unit circle centered at the origin [
], then
1.) if it is shifted 4 units to the right the equation becomes:
2.) if it is shifted 3 units down the equation becomes:
3.) if it’s radius is increased by 6 the equation becomes:
4.) if it is shifted 2 units up, 7 units left, and it’s radius is increased by 3 the equation becomes:
Snowy Night
Teacher’s Creation
Students will originally create: