This blog post is about a cardboard calculator.
But first, a math problem:
Two points, A and B, lie on the graph of y=x2 with x-coordinates of a and -b respectively. A line is drawn between these two points. In terms of a and b, where does this line intersect the y-axis?
If you want to give this a go, don’t scroll down. There are some pictures that might help you.
There are many ways to approach this problem.1 I will be presenting how I think is most intuitive: with triangles.
Start by drawing triangles:
I didn’t label them in the drawing, but angles BDP and BCA are both right.
We are solving for the y value of P. In our drawing, P (which has the same y value of D) is at b2 + x, so we are essentially solving for x.
The first thing to notice is that these two triangles are similar.2 This means that all ratios between corresponding sides are proportional.
Many math books will make you look back and forth and back and forth and back and forth between the algebra and the drawing,3 but I will try to minimize neck cramps by using words instead of “BD ∝ BC”
Ok so, they are portional. Short leg (of small triangle) to short leg (of big triangle) is the same as long leg (of small triangle) to long leg (of big triangle). This lets us set up a proportion:
Now all we have to do is fill it in.
Short leg of the small triangle is x.
Short leg of the big triangle is a2-b2. (explanation that might make it more confusing:) C is has the same y value as A. A is at a2. The distance between C and the x axis is the short leg of the big triangle plus the distance from B to the x axis, so to find the short leg of the big triangle, you subtract the distance from B to the x axis (which is b2) from C: a2-b2.
Long leg of the small triangle is b.
Long leg of the big triangle is a+b.
Now we just plug it all in:
To solve for x, we multiply both sides by (a2-b2)
a2-b2 is a difference of squares link, and has a special factorization: (a+b)(a-b)
The (a+b) on top and bottom cancel.
Yay! We did it! But we’re not done. Remember, we are solving for the y value of P, not x. As we said before, the y value of P is x plus b2.
Now why do I even mention this problem?
Well, most problems give answers like
Fairly uninteresting right? b squared minus 4 over five sounds exactly like something you wouldn’t remember. Furthermore, that’s not something you would ever use.
ab, on the other hand, screams “USE ME TO CALCULATE THE PRODUCT OF TWO NUMBERS!”
So I did.
I started by cutting off the face of a Fruit Loops box and (very carefully) drawing a parabola:
Especially astute readers will notice that I dilated the x axis. The real graph of y=x2 is far skinner. I wanted to use the entire board, so I stretched it out.
Bonus question: why can I do that without messing up the multiplication?4
Next, I used an exacto knife to cut along the parabola, then took some 22 gauge wire, threaded it through, and added some circles so you can slide the points around. BOOM. done.
I’ve enlarged the pictures so you can see the markings on the y axis. (ctrl + scroll wheel if you still can’t)
But we all know that. What about something slightly more complicated?
Well what’d you know
I was actually very impressed by the precision this thing gives. As long as you make sure the ends are actually on the two points, and that the wire is pulled taut, it will be within half a centimeter of where it should be. Which, in my book, is more than enough accuracy.
Pretty cool right?
In my research, I have not found a single person that has done this before. Plenty of people have shown this problem and demonstrated the solution, but I haven't faound a real, physical version. If I am the first (which I doubt), I am claiming exclusive naming rights to this device.5
“Parabola Calculator” is good, descriptive, but a bit boring, so I’m going to call it the….
1 Brownie points to anyone who can solve it with coordinate algebra.
2 By AAA, pedants.
3 If they are so kind as to give you a drawing.
4 As a bonus bonus question, why does it/does it not work with negative numbers?
5 For the record, if someone beat me to it, I’m still claiming naming rights. Just not exclusive naming rights
6 Not to be confuesd with it's distant cousin, the tabulator.