## #MathArtChallenge 9: Comparing Origami Butterflies

The #MathArtChallenge is just a fun, simple way to engage our brains during this time of unease. All tasks are low tech: paper, pencil, maybe string. Nothing fancy. I would LOVE to see what you come up with. Post on social media with the hashtag #MathArtChallenge!

THE CHALLENGE: Make two different sizes of origami butterflies and then see what you notice and wonder about your pieces!

MATERIALS NEEDED: Two different sizes of square paper. (Can be obtained from non-square paper if you ask nicely.)

Instructions below.

## #MathArtChallenge Day 8: Apollonian Gaskets

THE CHALLENGE: Draw a large shape. Then place a large circle inside that shape, touching the edges of the original shape. Then draw the next largest circle you can, and repeat drawing the next largest circle you can. (See video below for examples.)

Materials Needed: Writing utensil, paper.
Math Concepts: Ratios, radius, tangency (tangent circles), proportions, area, perimeter

If you’re interested in some of the math behind this, this is a great article, featuring the work of mathematician Hee Oh.

Vi Hart also has a great video along these lines, demonstrating some serious doodle skills.

Depending on how you use this activity, you may engage with different standards. Here are a few suggestions for how you might integrate the 8 mathematical practices. Feel free to add your own suggestions in the comments!

5.) Use appropriate tools strategically. What tools make this easier or more difficult? Is it necessary or better to use a compass and be completely precise, or is it better to “eye-ball” it?

7.) Look for and make use of structure. How does the radius of the largest inner circle affect the radii of the other circles?

8.) Look for and express regularity in repeated reasoning Compare multiple drawings. How does the choice of largest inner circle affect the final outcome? The amount of positive to negative area?

## #MathArtChallenge Day 7: Cardioids!

THE CHALLENGE:

Materials Needed: Paper, circle to trace (yogurt or oatmeal lid?), writing utensil, straight edge (doesn’t have to be a ruler, could just be a piece of cardboard cut straight or any other number of things.
Math Concepts: Sequences, Modular arithemetic, angles, geometric construction, ratios

Depending on how you use this activity, you may engage with different standards. Here are a few suggestions for how you might integrate the 8 mathematical practices. Feel free to add your own suggestions in the comments!

2.) Reason abstractly and quantitatively. What shapes/patterns appear in each cardioid? How can you predict what the results will be before you create one based on the ratio between the “skips”?

3.) Construct viable arguments and critique the reasoning of others. Create an hypothesis to predict how any new cardioid will appear.

6.) Attend to precision. What happens if you make a mistake? How likely is it to “ruin” the final outcome? How can you avoid mistakes?

## #MathArtChallenge Day 6: Circle Toruses!

THE CHALLENGE: Find a smaller circle you can trace. Then trace large circle to use as a guide. Finally, trace a bunch of smaller circles in a ring to create a torus (more commonly known as a donut).

Materials Needed: Paper, writing utensil(s), circles/compass. The circles can be whatever, but rigid is helpful and even better if they’re empty (masking tape is great!)

Here are mine!

Depending on how you use this activity, you may engage with different standards. Here are a few suggestions for how you might integrate the 8 mathematical practices. Feel free to add your own suggestions in the comments!

3.) Construct viable arguments and critique the reasoning of others. What kinds of interactions between two tori can be made? What can you deduce about the relative radii of tori that “intersect”?

5.) Use appropriate tools strategically. What tools work best for creating these tori? How might you make an image like this best without a compass? Is there a tool that is actually better than a compass? Why or why not?

7.) Look for and make use of structure. Let’s say you start with a circle of radius 5 units. What’s the largest radius you can use for your circles centered on the torus? What’s the minimum?

## #MathArtChallenge Day 5: Probability designs!

THE CHALLENGE: Use something like a die or a coin to get random outputs. The probabilities don’t need to be equally spread! Assign a design to each output, and then get to designing. I have two examples for you below.

Materials Needed: Honestly, whatever you want. There are endless possibilities on this one. Some examples: paper & pencil (like above and in the video below), yarn (friend ship bracelets or crochet), legos… See the examples of other people’s work below!
Math Concepts: Probability, randomness

Huge thanks to Emily Lynch Victory for the inspiration for this challenge. She’s a fantastic mathematical artist that I met a few years ago at a math teacher conference, who had a piece like (but far superior!) to my first one below.

Example 1: Roll a die, design a grid!

I assigned a design to each of the 6 outputs of a regular die, and then rolled the die to figure out how I should color in a 6×6 grid.

Example 2: Crochet!

Here I flipped 2 coins and assigned each of the permutations a type of crochet stitch. I made each row 10 stitches long, and switched stitches each row. In the end, I made it into a mÃ¶bius strip, because that’s just cooler.

HH: single crochet

HT: half double rochet

TH: double crochet

TT: Triple crochet

Great example here from Joel Bezaire

Depending on how you use this activity, you may engage with different standards. Here are a few suggestions for how you might integrate the 8 mathematical practices. Feel free to add your own suggestions in the comments!

2.) Reason abstractly and quantitatively. Once you’ve finished a piece, what do you notice about the distribution of values? How does that compare with the relative size of your piece?

5.) Use appropriate tools strategically. What methods make it easiest to explore the probability of values? What are the benefits or drawbacks of using a square grid? Would a 3D model make the chosen values easier or less easy to see?

8.) Look for and express regularity in repeated reasoning When completing with a class, what patterns do you see as you look over all the designs created by the class? How does the distribution of number change?

## #MathArtChallenge Day 4: Hyperbolic Geometry

THE CHALLENGE: Fold your very own Hyperbolic Plane from a simple piece of paper!

Materials Needed: A square piece of paper. Youtube instructional video below!
Math Concepts: Hyperbolic planes, vertices, opposites, exponential functions, reflections, symmetry

Depending on how you use this activity, you may engage with different standards. Here are a few suggestions for how you might integrate the 8 mathematical practices. Feel free to add your own suggestions in the comments!

4.) Model with mathematics Compare this model to a surface with positive curvature (a ball) or zero curvature (a piece of paper) surfaces. This surface has negative curvature. What do you notice about the 3 types of curvature? What’s the same? What’s different? What can you do on this surface that you can/cannot do with the others?

5.) Use appropriate tools strategically. As you’re folding this, discuss with students what tools make the folding easier or less easy. What kinds of paper works best? What sizes?

6.) Attend to precision. Discuss with students how successful your models are depending on the precision of your folds.

## #MathArtChallenge Day 3: Isometric Cube Cutouts

Draw a cube by connecting 3 rhombi (see below). Then use only parallel lines to those in the rhombi to embellish the cube.

Depending on how you use this activity, you may engage with different standards. Here are a few suggestions for how you might integrate the 8 mathematical practices. Feel free to add your own suggestions in the comments!

5.) Use appropriate tools strategically. How precise does one have to be in order for this to appear convincing? What tools make that more apparent? How can you make use of color?

6.) Attend to precision. Without attention to detail here, the visual falls apart.

7.) Look for and make use of structure. Can you create an “impossible” illusion? How so? What structure do you need to make use of to succeed in that?

## #MathArtChallenge: Day 2 Looping Colors

Materials Needed: writing utensil, writing surface
Math Concepts: intersections, knot theory, structure, functions

Draw a long, looping, self-intersecting line that meets back with itself at the start. Avoid having any 3 lines cross at the same intersection (although after a bit it may be fun to play with this). Then, select a color, and start coloring in the spaces created by your line’s intersections.

Can you color it such that you end with every section alternating colors?

Depending on how you use this activity, you may engage with different standards. Here are a few suggestions for how you might integrate the 8 mathematical practices. Feel free to add your own suggestions in the comments!

1.) Make sense of problems and persevere in solving them. Why can any graph be 2 colored? As you add intersections, how many new spaces are created? Can you create a function to describe the relationship between the number of intersections and the number of enclosed spaces?
2.) Reason abstractly and quantitatively. How many new spaces does each crossing create?
6.) Attend to precision. Ensuring that a 2-coloring is correct by methodically working through your design.

## #MathArtChallenge Day 1: Tons of triangles

THE CHALLENGE: Draw as many connected triangles as you can. Goal is to have as many vertices with 7 triangles as possible.

Materials Required: Writing surface, writing utensil
Math Concepts: Angles, vertices, triangles, graph theory, hyperbolic geometry, counting

Here’s a quick video tutorial after lots of requests for help in the comments.

Depending on how you use this activity, you may engage with different standards. Here are a few suggestions for how you might integrate the 8 mathematical practices. Feel free to add your own suggestions in the comments!

5.) Use appropriate tools strategically. What tools are best for this activity? How might your materials (sharpie vs fine tip pencil) change your level of success?

6.) Attend to precision. Mistakes will happen (6 lines or 8 rather than 7) when creating these. How can you minimize them, and what planning can you implement to minimize them?

7.) Look for and make use of structure. Is there a “best way” to grow this? How is your success altered when you alter the length of the lines?