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Lesson Description

For this lesson students will be exploring equations of circles. The teacher will place a giant graph in the room and students will make a circular art design to put on the graph. Students will then create an equation to represent their creation on the graph and then create equations for their peers circular creations as well.

As an extension to the main activity students will report data on their individual creations. After sharing that data with their classmates they will analyze the data to analyze trends in what they as a class created.

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Notes for teachers

This lesson would come prior to introduction of equations of circles. If can be your lesson that you introduce this concept with but it works better as an application activity rather than an introduction activity.
This lessons takes place over a class period but can be taken home by students to complete.
Students can work individually or in groups for this assignment
You will need a wall to set up the giant graph
Creativity should be encouraged with student’s designs

List of Materials�FOR TEACHER

Butcher Paper
Marker
Tape

FOR STUDENTS:

Coloring Materials

Markers, Pens, Colors Pencils, Crayons

Colored Construction Paper
Scissors
Glue/Tape

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HS Geometry Standards

G.G-C.A Understand and apply theorems about circles.

G.G-C.A.1 Prove that all circles are similar.

G.G-CO.A.4 Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.

A1.S-ID.A.2 Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets.

Standards for mathematical practice

G.MP.1 Make sense of problems and persevere in solving them

G.MP.2 Reason abstractly and quantitatively

G.MP.4 Model with mathematics.

G.MP.6 Attend to precision.

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Objectives:

Students will be able to identify the components needed to create an equation of a circle

Students will be able to generate an equation of a circle given the circle on a graph

students will be able to create a circle on a graph given its equation

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Agenda [60-90 Minutes]

Prior to lesson [Set up] [10-15 minutes]

Review concept [or introduce if you need] [5-10 minutes] [25-30 if teaching initial lesson]

Introduction to activity [5 minutes]

Student creation time [20-30 minutes]

Students graph their circles [5-10 minutes]

Generate equations for their circle [10-15 minutes]

Wrap up Discussion [10-15 minutes]

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Prior to lesson [Set up]

On butcher paper, teacher will set up a giant graph to be put on a wall of the room. (You can also just put a graph on your white board or project one on the wall with your smartboard/projector if that’s easier.)

Domain and Range of [-20,20] works well but you can choose whatever scale you’d like

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Lesson review

Ideally this is an application activity that you use to reinforce/expand your student’s understanding of equation of circles. You can use this in conjunction with the introductory lesson if you wish. I also find that it works better as a stand alone activity (i.e. I teach equations of circles one day and then the next day we do this), but it works well either way!

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Intro/Driving Question/Opening

What components do we need to create an equation of a circle?

( ___ __ ___)^__ __ ( ___ __ ___)^__ = ___^__

Teacher Note: Have students reflect on prior knowledge and fill in the blanks (you can make this better on your board if you write it out the equation should end up like (x – h)² + (y – k)² = r²

Green is variables, yellow is functions, blue is exponents.

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Review concept

Teacher note: Can have students review what the purpose of each variable is. Can lead class discussion or have them go into desmos and explore what manipulating each variable does.

Discuss/review how you identify h, k and r on a graph, and how you would graph a circle given h, k and r.

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Hands-on Activity Instructions

Modifications

Suggested Group Structure: Individual
Introduce students to the giant graph
Tell students they will be creating a circular image that will be placed on the graph
Gives students time to create their image
Have students put their items on the graph
Tell students to create an equation of a circle that represents their item on the graph
Tell students to create an equation of a circle for 3-5 other student’s creations.
Have students discuss their thought process throughout creating equations for their circles and their classmates circles.

Could group students in groups and have them make equations for their group member’s creations
Have students through a paper or a dart at the graph to decide where they have to place it
Have a random number generator decide where students need to place it�
Prompting questions for students:

“What did you look for when making your equations?”
“How did you decide where to put your circle?”

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Hands-on Activity Instructions Pt. 2

Suggestions

Have students collect data on the coordinates, area, volume, color, material, theme, etc., about their individual creation and share that data with each other in a class google sheet/excel.�
From the data that is presented in the google sheets group students into groups and have them go through and generate trends, data and other analytics for the materials that they created. �
Students can choose what aspect of the data they want to analyze and present that to the class

This component can be a good introduction for students to learn and view data analytics in excel/google sheets and starting them on using these materials to function within math.

You can present students with basic formulas “=Average”, “=StDev”, “=frequency” and have them explore those functions along with how to turn the data into graphs and images for their presentation on what they analyzed.�
There are several videos that students can explore to analyze the data as well and ChatGPT can provide additional assistance to help students with what they want to do.

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Assessment

Students should be able to generate an equation for their circle that they put on the graph and then an equation for 3-5 other classmate’s circles as well. Students can also be assessed on their effort for their creations as well as their contributions to class/group discussion regarding their process and thoughts for the equations of the circles that they generated.

For the data analytic portion students will be graded on their analysis and presentation of the material.

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Rubric

Criteria	Exceptional (5) - Excellent	Proficient (4) - Good	Basic (3) - Fair	Limited (2) - Needs Improvement	Inadequate (1) - Poor
1. Creativity and Artistic Expression	Highly creative, visually engaging, deep understanding of artistic principles	Creative, visually appealing, good understanding of artistic concepts	Somewhat creative but lacks depth, visual appeal	Lacks creativity and is visually uninteresting	Lacking in creativity
2. Equation of the Student's Image	Accurately represents the circular image, well-explained	Mostly represents the circular image, minor errors or incomplete explanations	Partially represents the circular image, significant errors or lacks proper explanation	Inaccurate representation, major errors, insufficient explanation	No equation or unrelated equation
3. Equations of Classmates' Images	Equations for all classmates' circular images accurately derived and well-explained	Equations for most classmates' circular images accurately derived with minor errors and adequate explanations	Equations for some classmates' circular images derived with significant errors or incomplete explanations	Equations inaccurately derived, major errors, insufficient explanation	No equations or unrelated equations
4. Overall Communication and Engagement [Discussion]	Actively engages in the discussion, effectively communicates ideas	Participates in the discussion and communicates ideas reasonably well	Participates but struggles to communicate effectively	Participates minimally and has difficulty expressing ideas	Does not engage in the discussion

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Differentiation

Students can work with a partner to generate equations of circles.

Can take additional time during the creation process to review with students who may need it.

Assist students with the placement of their circle to give them easy to work with coordinates

Break the activity up into smaller chunks for students who may need more scaffolding.

Remediation

Extension/Enrichment

Can give students equations of circles and have them try to guess which one you made the equation for.

Ask students to find the area of circles that were given

Ask students to find the distance between circles