Peeps are in stores everywhere at Eastertime. They come in many different colors and flavors, as well as in several not-to-be-eaten forms. I decided to make some peeps in Desmos and add some painted eggs for good measure. I’ve never seen green peeps before, but there will be green ones here.
I hope you find the best treats this year.
Not every topic in my mathematics education was covered equally well. I don’t recall learning anything about geometric transformations when I was in school. Many years after I graduated from college, when I was teaching mathematics, I learned how to make a shape rotate around the origin:
For each original point (x, y),
(-y, x) maps it 90° counterclockwise around the origin,
(y, -x) maps it 90° clockwise around the origin, and
(-x, -y) maps it 180° around the origin.
This year, the 9th-graders I work with at school need to know how to rotate a shape around a point that is NOT the origin. This is a topic I had never thought about before. To patch up this hole in my math knowledge, I decided to play with rotations in Desmos.
I began with
the point (-1, 2) and
the endpoints (2, 3) and (4, 7), and
the polygon function in Desmos to connect the endpoints.
I noticed that the point and the line segment had the same relationship as
(0, 0) and
the line segment with endpoints (3, 1) and (5, 5).
I noted that the following coordinates worked beautifully to rotate the line segment:
(x-1, y+2) maps it onto the line segment’s original endpoints, (2, 3) and (4, 7).
(-y-1, x+2) maps it 90° counterclockwise around (-1, 2),
(y-1, -x+2) maps it 90° clockwise around (-1, 2), and
(-x-1, -y+2) maps it 180° around (-1, 2).
I was quite pleased with the symmetry of those relationships, so I decided to add a few more ordered pairs to my table in Desmos: (4,-3), (6, 3), and (-2, 4). Desmos’s polygon function automatically rotated the line segments produced by those ordered pairs, and this lovely symmetrical design was produced:
Next, I wondered what would happen if I changed the center of rotation. I made an ordered pair, (a, b) and used sliders to move the point around. Now the point reminded me of a ball, and I got the idea to make the rotated shape look like a cat playing with the ball. I made the original cat a little darker than the rotated ones. This was the result:
That cat took a lot of ordered pairs to make. I got to thinking about how rotations are often used in tessellations. I found a simpler-looking kitten that tessellates and recreated it in this Desmos graph. Again, the original cat is darker than the rotated ones.
No cats were harmed in the production of these graphs.
Which of the cats do you like the best, or would you have used a different animal to play with the ball?
I made some Christmas ornaments in Desmos that I hope you will enjoy. If you click on and off the circles on the left of the descriptions, you can see all eight ornaments in one Desmos graph, or you can find them all pictured below in this post. If you click the arrow next to each description in Desmos, you can also see the equations used to produce each ornament. However, the snowflake and Rudolf’s face required many ordered pairs, which I put into a separate folder.
1. Decorated half red and half green:
2. Decorated with diagonal stripes:
3. Decorated with sines and secants:
4. Decorated with a snowflake:
5. Decorated with a checkerboard design: (This was a pleasant surprise that required only one equation!)
6. Decorated with a spiral
7. Decorated with ellipses for a 3D look:
8. Decorated with Rudolf’s face:
Perhaps you will choose to make an ornament yourself in Desmos. If so, I’d love to see it.
I hope you all have a very merry Christmas!
Merry Christmas, everybody! Can you make a Christmas design in Desmos?
Here’s how I solved this Desmos Christmas puzzle: A few weeks ago, I saw this post on Bluesky and was inspired by the climbing sine curves on the featured Desmos Christmas tree:
#mathstoday I began thinking about a Desmos activity for my year 11 in which they could make a Christmas tree. Then I got carried away, thought about climbing sine curves (tinsel) and translating polar graphs. I’m not sure it’s suitable for year 11 anymore… Oops
— over-drawn.bsky.social (@over-drawn.bsky.social) November 28, 2024 at 12:34 PM
What is a climbing sine curve, and could I use one to decorate the plain Desmos Christmas tree I made last year? I had to google “climbing sine” to proceed, but I learned that it is a function such as y = x + sin(x). That’s a familiar function; I just didn’t know it had a cutesy name.
I multiplied that function by a constant. Can you figure out what that constant was?
Later, I embellished the tree even more with lights and falling snow. I hope you enjoy it!
Here are some other delightful Christmas Desmos designs I saw on Bluesky. this first one rotates in 3-D.
Happy Holidays! 🎄
http://www.desmos.com/3d/p5t7m4kh4s
#iTeachMath— Raj Raizada (@rajraizada.bsky.social) December 10, 2024 at 10:46 AM
Enjoyed re-creating this visual in the @desmos.com Geometry tool: http://www.desmos.com/geometry/lx7… #mathsky
— Tim Guindon (@tguindon.bsky.social) December 11, 2024 at 1:08 PM
More snowflake fun in @desmos.com
I don’t think it can show text mirror-flipped yet (?), so for this, you type your word, screenshot it, then load it as an image.
I’m hoping to have students load in pics of their names, then snowflake-ify them.
http://www.desmos.com/geometry/afo…
#iTeachMath #MathSky— Raj Raizada (@rajraizada.bsky.social) December 17, 2024 at 11:17 AM
This next one isn’t a Desmos design, but I enjoyed its playful nature just the same. Do you recognize the number pattern?
Inspired by @studymaths.bsky.social – #MathPlay 🧮 via Pascal’s Dice 🎲🔺
#ITeachMath #MTBoS #STEM #Maths #ElemMathChat #Math #MathSky #MathsToday #EduSky
— Libo Valencia 🧮 MathPlay (@mrvalencia24.bsky.social) December 12, 2024 at 4:00 AM
I know 1804 is divisible by four because the last two digits are divisible by 4.
1804 ÷ 4 = 451. Oh, and 4 + 1 = 5, so 451 is divisible by eleven and forty-one! Here’s a factor tree for 1804:
1804 is the hypotenuse of one Pythagorean triple:
396-1760-1804, which is (9-40-41) times 44.
1804 looks interesting in some other bases:
It’s A8A in base 13 because 10(13²) + 8(13) + 10(1) = 1804.
It’s 4A4 in base 20 because 4(20²) +10(20) + 4(1) = 1804.
I wanted to create a Dot-to-Dot in Desmos for my students that wouldn’t require them to type in many ordered pairs. I concluded that if most points could be reflected over the x or y-axis, I could eliminate the need to type in about half the points. With that in mind, I recently created this mystery dot-to-dot you can enjoy over the Thanksgiving weekend.
What will this unfinished dot-to-dot become when the dots are connected, and 90% of the image is reflected over the y-axis?
My sister guessed it was a cat. The image reminds me of a snowman. What did you think it might be?
You can discover what it is by clicking on this pdf and following the instructions: Desmos Mystery Ordered Pair Dot-to-Dot
The instruction will look like this:
Depending on your device, you may be able to click on the lower right-hand corner of the Desmos image below to see how much fun I had transforming it four different ways: I made the image slide along the x-axis, rotated it 90 degrees, reflected it over the x-axis, and dilated it. (The location of the turkey’s wattle can help you determine if an image is a reflection, a rotation, or a combination of both.) If clicking the lower right-hand corner does not work on your device, click this link. These transformations are all essential concepts for students to learn, and Desmos can make the process quite enjoyable.
Did you guess right? Have a very happy Thanksgiving!
I created this polygonal Christmas tree with a polygonal star on Desmos, and it looks like it is living and breathing to me!
Later I saw this Christmas tree post and decided to share it here:
For anyone looking for a Christmas-themed Maths lesson which practises plotting quadratics and other non-linear graphs, check out these at https://t.co/Aamu6gBYQU
🎄🎅#mathsresources #mathschat #christmaths pic.twitter.com/r5HvOxra6l— Amanda Austin (@draustinmaths) December 13, 2023
Can you find the factors that belong on this Christmas factor tree for 1768?
1768 is the sum of two squares in two different ways:
38² + 18² = 1768, and
42² + 2² = 1768.
1768 is the hypotenuse of FOUR Pythagorean triples:
168-1760-1768, calculated from 2(42)(2), 42² – 2², 42² + 2²,
680-1632-1768, which is 136 times (5-12-13),
832-1560-1768, which is 104 times (8-15-17),
1120-1368-1768, calculated from 38² – 18², 2(38)(18), 38² + 18².
The first triple is also 8 times (21-220-221), and
the last triple is also 8 times (140-171-221).
1768 looks interesting in some bases you probably would never care about:
It’s 404 in base 21 because 4(21²) + 0(21) + 1(1) = 1768.
It’s 1Q1 in base 31,
YY in base 51, and
QQ in base 67.
Can you solve for Q and Y?
An equation of a unit circle centered at the origin is x² + y² = 1.
If we change just the “y” part of that equation, we can get a lovely heart just in time for Valentine’s Day. Try it yourself by typing the equations into Desmos.
There are other mathematical equations for a heart, but this is the one I’m exploring in this post.
I was puzzled over how I could transform that heart. Can I make it bigger, or dilate it? Can I slide it away from the origin or translate it? Can I rotate it? Can I reflect it across the x or y-axis?
These are questions I’d like you to explore as well.
In this first graphic, I was able to make my heart bigger. What kind of math let me do that? Also, how did I color the inside of some of the hearts? Look at the equations next to the heart and try to figure it out. The concentric hearts are evenly spaced. Do you recognize a pattern in the numbers that made that happen?
If we changed the center of a circle to (a, -b) instead of the origin, we would slide the whole circle. Here’s how we change the equation of the circle to give it a new center:
(x-a)²+(y+b)² = 1.
Similarly, in the next graphic, I was able to slide my heart away from the origin. How did I do that? Look at the equations to see how.
A circle looks the same no matter how it is rotated, but the same isn’t true for a heart. Look at the equations below. How was I able to rotate my heart around the origin?
Since a heart is symmetric, its reflection across the x-axis doesn’t look that interesting to me. Instead, I created a double heart that I reflected across both the x-axis and the y-axis:
Next, I was curious about what would happen if I changed the exponents on the outside of the parenthesis, so I changed a 2 from my original equation to an 8 in a couple of different places as I moved the heart from left to right. How did changing the exponent affect my heart? I found that as long as the exponent stays even, it still looks a little like a heart.
I was also curious about what would happen to my heart if I changed the “2/3” to a different fraction. I used fractions less than one as well as fractions greater than one. For many of my fractions, I used the post number, 1743, as the denominator. As long as the numerator was even and the denominator was odd, the graph still looked mostly like a heart. However, the closer the fraction was to zero, the more it looked like a circle.
Finally, I created this lovely flower using some of what I learned by making these transformations:
And for just a little bit more fun, I created a simple but chaotic-looking animation that I’ve titled Hearts in Motion. Enjoy!
I had so much fun exploring this heart in Desmos. Thank you for allowing me to share my excitement with you. Here’s a different Desmos heart created by a reader and shared with me. Click on it to make the heart beat:
— سیّــد (@Seyed_ARA) June 29, 2023
1743 is the difference of two squares in four different ways:
872² – 871² = 1743,
292² – 289² = 1743,
128² – 121² = 1743, and
52² – 31² = 1743.
A teacher at my school had his students graph some polynomials and their inverses. I got to help some of his students with their graphs. After seeing the beautiful symmetry of the graphs together, I excitedly exclaimed to a few of the students, “Isn’t this a cool assignment?”
During my lunch, I put one of the graphs, its inverse, and some of their translations on Desmos and made a simple but lovely piece of art in the process.
Before I was done, I showed it to a couple of students. One of them asked, “Are you saying that math can create art?” I loved replying, “Yes, it can!” Now that student wants to create some works of art, too. It was a privilege to show her how to use Desmos.
These are the inequalities I used to make my work of art:
MANY teachers have figured out that students could learn a lot about functions and their graphs by using Desmos to create drawings, pictures, or artwork. For example, look at this tweet and link shared by Chris Bolognese:
Here's the final @desmos coloring book from my amazing Precalculus students. We are headed to the lower school next week to join them in some coloring fun. Pictures to come! https://t.co/iejf2RYVWX @ColumbusAcademy @CAVikesSTEAM @mathequalslove #iteachmath
— Chris Bolognese (@EulersNephew) December 8, 2018
Now I’ll share some facts about the number 1314:
1314 is the sum of two squares:
33² + 15² = 1314
1314 is the hypotenuse of a Pythagorean triple:
864-990-1314 which is 18 times (48-55-73) and
can also be calculated from 33² – 15², 2(33)(15), 33² + 15²