“How did you solve that math problem?”

The last few issues I’ve talked about mathacademy.com (no less than 7 readers are now doing it for themselves and/or for their kids!).

My mother was visiting this week and was doing the diagnostic over my shoulder while I was working on it. It really bugged her to realize how out of practice she was in elementary math so we went through some refreshers.

We reviewed a bunch of exponents stuff, for example, why 1/2 of 2²⁰ is 2¹⁹.

This is apparent when you think about it. But one of the things I noticed about how she and I do math is how methodical she is with trying to find the formula and how that’s not my first instinct at all. My first reach is always “what’s a simpler analogy and then extrapolate”. If that doesn’t work then get the pencil. I mean a lot of my motivation for retaking math ed is because my only mode is ‘trader math’. Formulaically, I reminded her that multiplying by 1/2 is the same as 2⁻¹ which is how she relates to the problem — she knows the rule for multiplying exponents with the same base is to add the exponents.

[My mom reads moontower believe it or not so it’s nice to share this in print even if a bit corny— we’ve always bonded over math. She went back to school in her 50s to get a college degree. She even took Java and C++. She is a determined learner at heart even if formal education took a backseat to more urgent pragmatism. She cut her college days short to work and get married back in the 70s. I was born the week she turned 24. Meanwhile, my eldest was born hours after my 35th birthday. Just acknowledging the change in norms in a single generation makes me feel like a flea in the sweep of time — no need to invoke cosmic proportion or even geographic birth lottery to think of how lucky I am to feel even remotely resourced while my kids are still kids.]

 

If you want a similar math problem to practice I shared Barclays quant question back in July:

Lily pad

You start with a single lily pad sitting on an otherwise empty pond. You are told that the surface area of the lily pad doubles every day and that it will take 30 days for the single lily pad to cover the surface of the pond.

If instead of one lily pad you start with eight lily pads (each identical in characteristics to the original single lily pad), how many days will it take for the surface of the pond to become covered?

A thought on the Lily Pad question and more:

[My son Zak solved it just like I did — by realizing the answer is the same as if you started after Day 3. My mother preferred the 2³⁰ / 8 = 2³⁰ / 2³= 2²⁷. The different ways we reason through a problem show up yet again.

I suspect my son is railroaded into my method because it wasn’t natural for him to see that representing 8 as 2³ was desirable for the purpose of doing exponent division (which follows a mechanical rule of subtracting exponents).

But getting to the formulaic version is what my mom searches for first.

Even when I was on the trading floor where you had to do mental math quickly to make markets, I enjoyed asking the people standing next to me how they priced the structure. There was a lot of variation. It’s a fun thing to ask others and, as I discovered, people usually like explaining how they mental math so it’s an all-around feel-good exercise.

One of the things I like about common core math is the emphasis on seeing numbers in different ways. My 8-year-old reflexively turns numbers into “friendly numbers” ie ending in 0s before doing operations, then undoing the adjustments before finalizing his answer. They are taught to do this. People my age usually landed on this method organically. But it’s good to teach it.

That said, Nate Bargatze owns the best common core bit:

 

 

Money Angle

Here’s a question I made for my mother to drill the exponent stuff that doubles as an investment problem.

For a fixed tax rate and rate of return is it better to have your return taxed every year or wait to be taxed on the gains all at once at the end?

Knowing the answer to the question is useful in itself but I also want to mention a collateral benefit. The meta-process for approaching the question can help organize your numerical intuition.

Think of what is required to answer:

1) recognition

What kind of problem is this?

Well, it’s a compounding problem.

What does that tell us about the function?

It’s exponential. It takes the form y = abˣ

2) ask yourself where the variable in question (in this case the tax rate) makes the largest impact

Is it as part of the a or the b?

Since the b gets exponentiated (the historical term for this is “involution” or “involuted”) the tax term will have its largest impact there.