A project about math, education, and math education.
Look at the rectangle below with one vertex fixed at the origin. Assume this has a width of 1 and a length of 2. A mathematician, rather than caring about applications, they may wonder, how many whole number points exist on or within this rectangle?
What about a 2×3, 3×4, and 4×5, assuming one vertex is fixed at the origin? Is there a pattern unfolding? Experiment with a couple of your own.
Now, what if your width is 2.5 and your length is 4.3 with one vertex fixed at the origin? Does your pattern fall apart or not? If it does, can it be saved? If not, why do you think it still works?
Are rectangles the only shapes in the universe? I think not and neither do mathematicians, whom may go on to wonder about circles. How many whole number lattice points exist on or within a circle of radius 1 fixed at the origin like below?
The next thought may be to try a circle with radius 2, 3, 4, etc. Is there a pattern?
Now, nothing says the radius has to be a whole number. Try experimenting with some other radii.
Gauss defined the problem as finding the points (m, n) where m and n are integers for the equation below. Can you explain what this means to a non-mathematician?
What you have been experimenting with are known as Gauss Circles named after Carl Friedrich Gauss, a famous mathematician who first considered this problem and tried to come up with a satisfying answer. Other mathematicians have tried to find an explicit function for establishing how many integer points lie within a circle, but the best they can do so far is shrink the possible error bounds for any circle with radius r. One such mathematician was G.H. Hardy. You can start here, then use the references to guide you to other resources.
Just because Gauss was thinking about circles does not mean we have to stop our own imagination there. What about spheres and whole number points on or within a sphere? Do similar properties hold or do we have to treat this as an entirely unique problem?