Solve for x: 3(x - 3) + 5x = 8x - 10
(a) 5
(b) -6
(c) 1
(d) no solution
Ah. The dreaded “no solution” option. Every student hates when “no solution” lurks among the multiple choices. The idea that a problem might not have a solution often rattles students. Instinctively, they push the possibility to the margins as they search for the answer anyway.
We may not realize it, but this impulse—to believe in real solutions—is an act of faith. It clings to an assumption that is foundational to every math class: The truth exists, and the truth can be found.
Truth is the lifeblood of mathematics. If there is no truth, our algebraic searching is a fool’s errand. And if there is truth, but it is beyond our comprehension or discovery, then our problem-solving is all for nought as well.
I can toss the kids a piece of candy every now and then, but practically speaking, truth is the only reward I have to offer my math students. It’s hard enough to motivate kids to be excited about the quadratic formula at 8:30 in the morning. Imagine if every problem I put before them had no solution . . . Good luck to me!
This is not the case, thank God! Every math problem promises the discovery of one of three things.
Most of the time, we discover truth. The majority of the problems we tackle in high school math have a solution—or a host of solutions. The trick is infecting students with the pioneer spirit. Once they’ve gotten up the courage to strike out in search of the truth and had minor success, they keep coming back for more. The discovery of a hard-earned solution brings a wholeness and completion that the human soul longs for.
Sometimes, we uncover falsehood. There are times when, no matter what scenario or extra bit of information we add, a statement is false on its head—like the problem above. If you simplify the equation, it ends up as the statement 0 = -1. No value for x can make that statement true. Still, there is benefit in uncovering falsehoods, because it keeps others from seeking an answer where there truly is no solution.
Occasionally, we find truth beyond our comprehension. What is the sum of the series 1 + 2 + 4 + 8 + . . . ? It keeps growing and growing forever. We have a symbol for truth that stretches beyond the reach of our finite minds: ∞ , infinity. These sorts of problems reveal to us the borders of our own human comprehension—there are really real things that will never fit into our little brains.
Some truths stretch into eternity. Which is good news for problem-solvers: we can seek and discover and delight in the Truth for 10,000 years . . . and there will still be infinitely more of Him to seek and discover and delight in.
Kitchen Solutions
Baking is all about problem solving. Whether you are trying to find a suitable substitute for a missing ingredient or trying to avoid burnt edges in an oven that cooks unevenly, we are always trying to anticipate problems and come up with solutions.
Some of my favorite chefs are my favorite not just because they have great recipes—but great kitchen solutions.
Alton Brown’s long-running series Good Eats is chock full of fantastic—and at times idiosyncratic—solutions to various problems you may encounter in the kitchen. It’s engaging, goofy, instructive, and a great watch for the whole family. The Ashbys highly recommend!