This is the third in a series about a Day of Maths I did in my daughter’s Year 7 classroom. I started back with The Best! Day! Ever! and then talked about Quarter the Cross. This post is about a puzzle I gave to students to fill in some time while their classmates finished off their good copies of their Quartered Crosses. I showed the puzzle on the big screen and asked any students who had finished their other tasks to have a go at it.
The puzzle is called “The Zero Zeros”, and it goes like this:
Use only the five numbers
10, 100, 1000, 10 000, 100 000,
with each appearing exactly once,
and as many of +, – , ×, ÷ (and brackets) as you like
to make numbers
with no zeros in their digits.
I created it only a month ago while trying to come up with alternatives to the Four Fours. I’ve tried it at One Hundred Factorial with university staff and students, and it was most interesting seeing how 12-year-olds responded to it. It was indeed most interesting, because there were a lot of responses I really didn’t expect.
The first thing I didn’t expect was the sheer number of clarifying questions the students asked! “Do I have to use all of them?”, “Am I allowed to use any other numbers?”, “Am I allowed to use them more than once”, “Do I have to use all of +, -, *, /?”, “Are brackets ok?”, “Are you sure I have to use all of them?” With each question I went up to the document on the screen and added a bit extra, which is why you may have noticed that this version is slightly different to the one I presented in the previous post. It’s worth mentioning that there were several times when I did just point to what was already written there and confirm that yes, that is exactly what I meant. If I were to give this as a whole-class activity, I would probably get the students to ask whatever questions they might have about it in a whole-class discussion to make sure we all understood it before I got them to work on it by themselves or in groups.
The second thing I didn’t expect was that pretty much every student who tried it declared it was impossible within thirty seconds of working on it. One particular student was adamant that there was absolutely no way at all to do it. On reflection, I am glad they thought it was ok to do such a thing — it takes a certain kind of bravery to say that an authority figure’s problem is wrong! Of course, I had to tell them that actually, yes, it was possible because I had done it myself, and that I thought they could figure it out too.
The third thing I didn’t expect was how many of the students fixated on calculations that included one number at a time. That is, they’d start with one number, combine another number with it to get a result, then combine another number with that result to get a new result, and so on. For example, one student did 10+100 = 110, 110*1000 = 110 000, 110 000/10 000 = 11. (At this point they couldn’t see a way of including the 100 000.) Of course they didn’t write it like I did just then; they wrote it like this: 10+100=110*1000=110 000/10 000=11. Really, this is a direct copy of the keys they pressed on their four-function calculator to get the result.
Now I did expect that to happen — even my university students do that sort of writing which ignores the meaning of the = sign! What I didn’t expect was the influence that this sort of thinking had on their problem-solving, and even their understanding of other students’ solutions. One student came up with this very cool solution: ( (100/1000) + (10 000/100 000))*10 = 2. But several other students had trouble understanding it, because they tried to evaluate it left-to-right. The idea of replacing the things in brackets with the number they become and then doing the next calculation was quite alien to them. Still, the cool thing was that the puzzle afforded a wonderful opportunity to support these students’ learning of this way of doing things and the order of operations to boot.
The final thing that I didn’t expect was the lack of inclination to come up with more than one solution. That student before had absolutely no desire to search for another solution. Several students ran up to him to be shown “the” solution, and stopped looking for their own. The student who had been so adamant that it was impossible found a way to produce 189.99, which I was suitably impressed by. Yet he just looked at me strangely when I wondered aloud if it was possible to produce a whole-number answer. I found this very surprising especially considering we had just spent all morning coming up with tens of solutions to the Quarter the Cross puzzle. Perhaps in their minds drawing-puzzles are allowed to have multiple solutions, but number puzzles must have only have one solution? Perhaps they didn’t want to diminish the triumph of finding their solution by looking for more? I’m not sure.
What I did expect (and did happen), was that we had multiple opportunities to discuss the size of numbers, and the effect different operations have on the digits of numbers, and the order of operations. This was of course what I designed the puzzle for, so I was very pleased when it did just that. And I’m glad about all the unexpected things too — I’ve had a golden opportunity to learn about how students think. I’ll be pondering the meaning of what I observed here for some time.
PS: Here are the rest of the posts in the series:
- How I felt about the day
- Quarter the Cross
- Zero Zeros
- Spotless Dice
- Mathematical art appreciation
- Hotel Infinity
- Thanks for coming