The Four Fours
In my first year of teaching I came across the Four Fours challenge. I loved it. I’ve always considered puzzles, brainteasers, and trivial challenges of that sort to be rather entertaining, so I went to town on the Four Fours right away. The problem gets better as you go: easier target numbers are out of the way and the real mental effort begins as you look to fill in the gaps.
Beyond the personal puzzle perspective, as a brand new teacher (my course load then was split 50/50 between 7-8 and 9-12) I considered this challenge to be a gem for my middle school students. Here’s a few reasons why:
- Mundane-yet-necessary practice is made more engaging in the context of a challenge (just like in this problem).
- Shifting the task from evaluating expressions to writing expressions ramps up the critical thinking component of the task. Students have to look for and make use of structure in order to bend the expressions to their will.
- The problem is flexible and can be presented to individuals, small groups, entire classes (class vs. class challenge, anyone?), or with a twist (find as many expressions as possible with a value of, say, 23).
And Then It Happened: The Internet Failed Me
I’m sure there are more reasons why the Four Fours is awesome. If you think of any, share them in the comments. But I know of one reason why the Four Fours is not awesome. In fact, I know of one reason why the Four Fours is worthless. That’s right, useless. Totally devoid of any value.
Wait? Just a minute ago I was singing the problem’s praises. Why the dramatic turn? One reason:
I typically love the Internet. Defender of the integrity and usefulness of Wikipedia and all that. But not this time. Here, the Internet failed me.
See, someone decided it woud be a good idea to ruin the Four Fours by posting entire solutions all over the place. Now any time I give this problem to students, they’re one Google search away from Four Fours glory.
So what do you do if the Internet robs you and your students of a great problem? Make another one. Keep the good stuff (see the bullets above) and tweak the parameters.
Flash back to my first year. When someone told me the full solution to the Four Fours was online, I rewrote the rules and called it Expressions Challenge. Once I got going, it was easy to make additional versions. I’ve included the directions (and handouts I mocked up this year) for the first three versions, as well as some comments on how to
tweak Internet-proof the problem further.
Here are the directions for the first three versions of the Expressions Challenge. Note that there is only one difference between the three: Use the numbers in (1) any order, (2) increasing order, or (3) decreasing order. Obviously, versions 2 and 3 are more challenging.
Just to be clear on the order element of the directions, here are examples for each version (1, 2, and 3, respectively):
And the handouts (nothing special):
- Expressions Challenge (Version 1)
- Expressions Challenge (Version 2)
- Expressions Challenge (Version 3)
Further Defense Against the Evils of the Internet
If the Internet destroys any of the challenges above, no problem. Just tweak the problem again. Use the numbers 1, 2, 3, and 4. Or 1 through 6. Or the first four odds (or evens). You get the idea.
Also, you could change the word “numbers” in the directions to “digits” to open up some additional possibilities. And for the record, whether your students should be allowed to use decimals, percent signs, radicals, ceiling/floor/rounding functions, etc., is totally up to you.
Let me know in the comments if you find any of this useful, or if you have a similar challenge you use with your students.