When I was a kid we used to take road trips. We weren’t allowed to listen to our Walkmans because my dad thought it would ruin our hearing, so the only way we could amuse ourselves was by playing car games. Unfortunately, we only knew two. The first was the alphabet game, which involved finding all the letters of the alphabet, in order, on signs, bumper stickers, license plates, and billboards. This game was amusing until you hit Q and X, at which point it felt like a pointless frustration. The other was the mystery game, where one person makes an enigmatic claim, such as, ‘when the music stops, the lady dies’, and the others have to discover what it means and how it came to be by asking yes or no questions. (Spoiler alert: she was a tight rope walker. She had this act where she would walk blindfolded. Her husband would play music, and, when the music stopped, she knew she was at the end of the rope and could step down. However, her husband cut out the music early in order to kill her.) Sadly, we only had a few of those stories, and each story can only be played once. So we eventually got bored with our games and started fighting.
However, I have since then come up with new car games with a geeky, mathematical bent to them. They start off simple, but, as kids get weary of them, they can grow more and more complex.
To start with, you review with your kids the rule for divisibility by three (if the digits of the number sum to a multiple of three, then the number itself is divisible by three). Have your kids look around for numbers: on license plates, road signs, anywhere. Have them, as quickly as they can, tell whether a number is divisible by three.
Next remind them of the rules for divisibility by 2, 5, and 10. For any given number, have them tell which of these numbers it is divisible by.
When this gets easy, add in the rules for 4, 6, 8, 9 and 11. (Eleven is my favorite divisibility trick. Make the first digit positive. Then make the second digit negative, Then the third positive and the fourth negative, continuing this pattern to the end of the number. If the sum of the digits is a multiple of 11, then the number itself is divisible by 11. For example, 7348 is divisible by 11 because 7-3+4-8=0, and 0 is a multiple of 11.) The only number under 12 that doesn’t have an easy trick is 7, so you can encourage your kids to do mental long division. For every number you come across, have a race to see how many factors each person can find.
If the car ride is long and the road signs are few, start asking your kids why these number tricks work. They will likely be able to quickly tell you the rules for 2,5, and 10, where the only digit that matters is the last. Four and 8 require testing only the last two and three digits of the number respectively. Ask them leading questions to get them to recognize why this is the case. (Why are only the last two digits important for testing divisibility by four? What number do the last two digits represent? Is one hundred divisible by four? Why would the digits in higher places be irrelevant?)
Go through every rule and see who can come up with an explanation for why the rule works. You may want to bring some paper and pencils along with you if the car ride is long enough to get to this point. Also, you may want to remind them of modular arithmetic so they can deduce the rules for three and nine.
Once they have explained the rules and become experts at factoring numbers, they may get tired of this game. (But maybe not. When I’m stuck in traffic, I play the factorization game with the license plates and advertisements around me.)
At this point you can play another trick with modular arithmetic. Have them look at the clock in the car, assuming you have a digital clock that is visible from the backseat. We didn’t when I was a kid, but you probably aren’t riding around in a station wagon with faux wood paneling on the side. Have them add up the digits on the clock. Ask what the sum is. If it is greater than 9, have them add the digits of the sum until they get to a number less than 10. (Example 2:39 => 2+3+9= 14 => 1+4=5) Then ask them what the sum will be for the next minute, then the minute after that. Ask them why the sums will be consecutive. Then ask if that will be true for every time shown on the clock face. See if they can find an example of when it will not be true. The answer, of course, is changing from minute 59 to minute 00. Ask your kids why the pattern breaks here.
Turning any number seen along the highway into an exercise in factorization and number theory serves three purposes: it gets your kids thinking about math, it keeps them occupied, and it stops you from having to hear them whine about how you should be playing their favorite CD over and over again, all the way down I-80, watching as endless rows of corn sway to the beat of some bubblegum pop music that is as simplistic as the landscape.