## All Systems Go

I noticed the boys having an intense conversation in front of this sign at our local pizza shop: When I asked what they were doing, they said, “We’re trying to figure out how much one slice and a beer would cost.”

As you read that, there were likely two thoughts that crossed your mind:

• Why can’t these poor boys look at a pizza menu without perceiving it as a system of equations?
• Why are eight-year-olds concerned with the price of beer?

The answer to both, of course, is that I’m a terrible father, and both beer and math are prominent in our daily lives.

But you have to admit that it’s pretty cool that my sons recognized, and then solved, the following system: $\begin{array}{rcl} 2p + s & = & 6.00 \\ 2p + b & = & 8.00 \\ p + s & = & 3.50 \end{array}$

They didn’t use substitution or elimination because they didn’t have to — and, perhaps, because they don’t know either of those methods yet. But mental math was sufficient. If two slices and a soda cost $6.00, and one slice and a soda cost$3.50, then one slice must be 6.00 – 3.50 = $2.50. Consequently, two slices cost$5.00, so a beer must be 8.00 – 5.00 = $3.00. A beer and a slice will set you back$5.50.

I remember once visiting a classroom in Somerville, MA, and the teacher was reviewing the substitution method. My memory is a bit fuzzy, but the problem she solved on the chalkboard was something like this:

Mrs. Butterworth’s math test has 10 questions and is worth 100 points. The test has some true/false questions worth 8 points each and some multiple-choice questions worth 12 points each. How many multiple-choice questions are on the test?

The teacher then used elimination to solve the resulting system: $\begin{array}{rcl} t + m & = & 15 \\ 5t + 10m & = & 100 \end{array}$

The math chairperson was standing next to me as I watched. “Why is she doing that?” I asked. “You don’t need elimination. It’s clear there have to be 8 or fewer multiple-choice questions (8 × 12 = 96), so why not just guess-and-check?”

“Because on the MCAS [Massachusetts Comprehensive Assessment System], if they tell you to use elimination but you solve the problem a different way, it’ll be marked wrong.”

So much for CCSS.Math.Practice.MP1. Although most of us would like students to “plan a solution pathway rather than simply jumping into a solution attempt,” apparently students in Massachusetts need to blindly follow algorithms and not think for themselves.

The following is my favorite system of equations problem:

I counted 34 legs after dropping some insects into my spider tank. How many spiders and how many insects?

Why is this my favorite system of equations problem? Because there is a unique solution, even though it results in just one equation with two unknowns. Traditional methods won’t work, and students have to think to solve it. Blind algorithms lead nowhere.

Other things that lead nowhere are spending your leisure hours reading a math jokes blog. But since you’re here…

Why did the student put his homework in a fish bowl?
He was trying to dissolve an equation.

An engineer thinks that his equations are an approximation to reality. A physicist thinks reality is an approximation to his equations. A mathematician doesn’t care.

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