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Computational Adventures

Make the Square Go Round

In this module, students will explore how the number of sides in a regular polygon relates to different properties of the polygon. They will also be introduced to limits by turning a regular polygon into a circle.

Appropriate for ages 10+.

Allow 20 minutes to complete the module.

Important note: This module should be led by an instructor with basic Wolfram Language knowledge. If you would like to learn the language, please try this free online introduction. If you would like a Computational Thinking Initiative ambassador or volunteer to help you run an adventure, please contact us.

Learning Objective

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Students will be able to approximate a circle using regular polygons

Computational Thinking Principles and Practices

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Making visualizations to think through ideas (not just as final output)

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Simulating things that are hard or impossible to do by performing real-world experiments

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Using code, graphics and text to communicate ideas while working on the project and after finishing

Standards Alignment

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Common Core:

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CCSS.Math.Content.HSG.GMD.A.1: Give an informal argument for the formulas for the circumference of a circle, area of a circle and volume of a cylinder, pyramid and cone. Use dissection arguments, Cavalieri’s principle and informal limit arguments.

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CCSS.Math.Content.HSG.CO.A.3: Given a rectangle, parallelogram, trapezoid or regular polygon, describe the rotations and reflections that carry it onto itself.

Helpful Background

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Information on circles: http://mathworld.wolfram.com/Circle.html

STARTING POINT

“We’re going to make pictures of regular polygons. Here is a square.”

In[1]:=

Graphics[RegularPolygon[4]]

Out[1]=

◼ What do you think the 4 represents?

◼ Why is this called RegularPolygon?

◼ Why do we need Graphics[] around it?

◼ How might you make a triangle?

◼ Can you make the shape of a stop sign?

CHECKPOINT

Check that the whole class understands that the argument represents the number of sides of the shape.

“Try making many polygons, but be sure to keep them on the page. Put each input in a different ‘cell.’”

You may need to explain what a “cell” is. Use the brackets on the right side of the notebook to help students see where cells separate, and mention that cells can come in many formats (e.g. input, text, etc.).

In[2]:=

Graphics[RegularPolygon[3]]

Out[2]=

“By putting Graphics[RegularPolygon[4]] and Graphics[RegularPolygon[3]] in different cells, we are able to keep track of our previous work.”

Encourage students to increase the number of sides.

“What shape is it starting to look like?”

◼ What if you have 2, 1 or 0 sides?

◼ For what types of numbers does the polygon have parallel sides?

◼ For what types of numbers is there symmetry in the polygon? How many lines of symmetry does it have?

◼ How many sides do you think we have to have before it really looks like a circle?

CHECKPOINT

Check that the class understands that a many-sided polygon approximates a circle.

FINAL POINT

Five minutes before the end of the module time.

Summarize

Summarize what was done in the module and talk about findings.

Refer

Refer back to the learning objective and summarize how you have reached it.

Extend

Extend the module to the future. For example, "If you have time at home, try using Manipulate to increase the number of sides in steps of 1."

Possible Additional Relevant Functions

Style

Possible Pitfalls

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Graphics[RegularPolygon[Infinity]]] may not work, but it is an accurate way to approximate a circle and is in the process of being fixed.

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RegularPolygon[4] with no Graphics wrapping comes back unevaluated. We need to tell the computer we want a graphic as an output.

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RegularPolygon[0], RegularPolygon[1] and RegularPolygon[2] will all return unevaluated expressions.