This blog is about Simon, a young gifted mathematician and programmer, who had to move from Amsterdam to Antwerp to be able to study at the level that fits his talent, i.e. homeschool. Visit https://simontiger.com
Simon has been pondering a lot about various ways to visualize or prove the quadratic formula.
He eventually came up with a 4-meter-long quiz sheet, slowly revealing the logic behind the quadratic formula as one solves the 9 problems one by one. Simon borrowed the actual problems from Brilliant.org but reworded some of them to match his personal style, writing all of them down in his beautiful handwriting on large sheets of paper taped together to form a road to the quadratic formula. The answers were hidden under crafty paper flaps. We had a lot of fun traveling down this rabbit hole as a family, Neva stuck around solving the tasks until half-way through.
Simon has come up with an equation to solve the Too many Twos, the puzzle mode of the Add ‘Em Up game:
x is the number of twos I used to clear out just a single two at a time
y is the number of twos I used to clear out six twos at once.
We have two pieces of information. At the beginning, the twos are arranged in a pattern with 40 twos in it. And the number of twos I can use to clear out the whole grid is 25.
x + 6y = 40
x + y = 25
We thought we solved it, but no! The reason why is because of the way the twos are arranged there were spots where there were exactly 6 twos neighbouring an empty cell. And there was only one spot where there wee more than 6. Our equation says that there must be 3 of those. the way I solved this problem was by considering a third variable, z = the number of twos that I place without clearing any twos in the grid. So now our two equations look like this:
x + 6y – z = 40
x + y + z = 25
With a little bit of cleverness though, we know that these are all integers. You don’t have 2.7 twos! That doesn’t exist! Which means that we can use some number theory to narrow it doen. After solving these equations we get: x = 25 – y – z and y = 3 + 2z/5
We’ve got a fraction. We need to carefully choose the z for this to result in an integer! This is only true if z is divisible by 5.
I don’t want to check infinitely many solutions. Luckily, we know one more quite obvious thing: all of our variables must be positive. So if z gets too large, x will become negative. How large? Let’s just be lazy and use trial and error. Let’s draw a table. In our table we now only have four solutions that we need to check. The first one, with 0 z‘s, clearly doesn’t work.
For his math class Simon programmed a scratch pad (well, he didn’t actually invent the code, but found it in Daniel Shiffman’s coding challenges and built the scratch pad on his laptop). It’s just more fun to solve the equations in a self-programmed scratch pad than on paper, don’t you think?
It’s been great watching Simon turn a word problem into an equation today. He was busy with “Sums of consecutive integers” (Practice finding the nth number in a sequence of consecutive numbers based on the sum) on Khan Academy.
Equations have definitely become our friends now. Simon got really creative during his math lesson and came up with an “Algebra Visualization Kit” – creating an abstract representation of an equation using nuts and sticks and solving it:
Yesterday was a break-through day! Simon finally cracked those equations with variables on both sides, even the ones including parentheses! And this is what helped: just like with programming, he switched to English and tried to imitate the Khan Academy video’s he had watched. He also invented using different colour markers for the variables remaining on the same side of the equations and the variables he had to move to the opposite side. But what really catalysed the learning process was discovering a scratch pad on the Khan Academy interface: Simon, who normally hated writing things down when solving an equation, suddenly got a passion for carefully sketching all the steps, in different colours:
Equations with variables on both sides remain a difficult subject to grasp. Well, all right, we only tried it twice so far. On Monday evening I saw Simon sit down to practice these for himself, attempting to solve an equation by drawing scales. Dad spent an hour trying to explain the concept of bringing all the variables to one side and the “numbers” to the other, and how to do that. I think we’re almost there.