For an ellipse, it's clear which part is the "inside" and which part is the "outside". For a hyperbola, which part is the "inside"? Which part is the "outside"? #iTeachMath

Take a number where the digits are increasing from left to right, and digits can repeat, like 134446689. If we multiply it by 9, then we get 1210020201. The sum of the digits of the new number is 1 + 2 + 1 + 0 + 0 + 2 + 0 + 2 + 0 + 1 = 9. Does this always happen? #iTeachMath

For a positive integer n, let f(n) = sin(1) + sin(2) + ... + sin(n), where sin is measured in radians. Plotting the values of f gives us a graph with some interesting structure. www.desmos.com/calculator/l... As a challenge, try to find the best bounds on f. @desmos.com #iTeachMath

I was gifted a gorgeous origami sculpture of five intersecting tetrahedra. This inspired me to recreate it using Desmos. www.desmos.com/3d/m7hqyu2gpr en.wikipedia.org/wiki/Compoun... @desmos.com #iTeachMath

An ellipse rotates so that it stays in contact with the coordinate axes. What curve does the center of the ellipse trace? (If you are feeling ambitious, you can try finding the curve the foci trace, and its area.) www.desmos.com/calculator/v... #iTeachMath @desmos.com

Some people need more help than others. #iTeachMath

Given triangle ABC, we construct squares AB B1 B2 and AC C1 C2 on the sides. Let M be the midpoint of B1 and C1. As point A varies, what do you notice about the position of point M? www.desmos.com/calculator/3... @desmos.com #iTeachMath

Suppose we want to arrange N points on a sphere, so that the smallest distance between any two points is maximized. For N = 4, 6, 8, we can take a regular tetrahedron, regular octahedron, and cube, respectively. One of these is not the optimal solution. Which one? #iTeachMath

While out on a walk, I noticed that the hubcaps on tires have very different kinds of symmetry. What kind of symmetry does your car have? What about the cars around you? #iTeachMath

Table of Lissajous Curves www.desmos.com/calculator/z... I've seen this animation online, so I thought I'd try building it myself. @desmos.com #iTeachMath

Trig is like an acquaintance who introduces themselves as being about triangles. When you really get to know them, their real identity is uncovered: they are about circles (more specifically about triangles inside circles w/base a radius from center edge) #iTeachMath

In the diagram below, triangle ABC is equilateral. Given a and b, can you find p and q? Given p and q, can you find a and b? #iTeachMath

Three congruent circles are inscribed in a triangle, as shown below. Can you compute the common radius? www.desmos.com/calculator/p... @desmos.com #iTeachMath

What is the sum of all the numbers in the 9 times 9 table below? #iTeachMath

I have been thinking about how Desmos can show multiple representations on the same graph. For example, suppose we cut a sector out of a circle with radius 1. We then roll the sector into a cone. What is the maximum volume of the cone? teacher.desmos.com/activitybuil... @desmos.com #iTeachMath

Apollonian Gasket www.desmos.com/calculator/e... @desmos.com

The Simson Line www.desmos.com/calculator/x... @desmos.com

Nested Hypocycloids www.desmos.com/calculator/l... @desmos.com

Poncelet's Theorem www.desmos.com/calculator/g... @desmos.com