Counting Without Counting: An Introduction

On the second day of Precalculus class, before embarking on our starting unit on Combinations and Permutations, I put students in groups and had them work on this packet. I’m including it below with my teacher notes in the margin, and also without my teacher notes in the margin:

(Also the .docx file so you can modify, cut, paste, hack apart! To see with teacher notes, go to the “Review” tab and click on “Final: Show Markup” and to see it without teacher notes, click on “Final.”)

This was a two day activity, with the groups working together, and every so often, I would stop and we would go over some of the problems. Some problems were assigned for nightly homework, and because of time issues, I had to cut out the Applebee’s problem completely. I’m quite proud of some of the problems… namely the Applebee’s problems, the / QR code problems, and the Mozart’s Minuet problem (largely taken from here, and modified with some extensions).

The goal of the packet was fourfold:

1) I wanted problems which promoted thinking, conversation, etc., before students were introduced to formulas, notation, etc.

2) I wanted students to understand that “counting without counting” means that instead of listing all possibilities and counting, there are often other faster ways to get answers. This is the essential understanding that I would hope for in any unit on combinations and permutations. In order to do this, students will need to organize your information in some special, logical way. Usually this requires students to multiply numbers. But students need to really understand why multiplication (rather than another operation, like addition).

3) I wanted students to work in groups, so the problems were designed to be conducive for groupwork.

4) I wanted students to get some sense of what huge numbers mean.

How did it work? Overall, I think it worked pretty well. I gave groups “hint tokens” and most didn’t rely on them for a hint. Most students were able to see that you have to multiply for most of the problems, but most had trouble explaining why. Finally, most had never seen a tree diagram. In the future, I honestly think having students draw a complete tree diagram and explain what each leaf means would be useful. Adding two questions like the following would help:

1. If you have the letters A, B, C, D, E, and F and you want to write a three letter code and you are allowed repetition, what would the tree diagram look like. Make all the branches. Then pick a single “leaf” of the tree and explain what that leaf means.

2. If you have the letters A, B, C, D, E, and F and you want to write a three letter code and you are not allowed repetition, what would the tree diagram look like. Make all the branches. Then pick a single “leaf” of the tree and explain what that leaf means.

The reason I say this is that I’ve been collecting homework problems, and the tree diagrams some of the kids are constructing are just nonsense.

I probably should write more about this, but I’m exhausted and all I want to do is sleep. (I’ve been sick since Monday.) I suppose I should end adding that I’m teaching a Precalculus Advanced class. I don’t think I would have these problems for a standard Precalculus class… I would use fewer of them, and I would scaffold them more, and build in more “listing” of things, rather than go straight into “how many different ways…?”



  1. I used a modified version of this in my class this year. Here are some of my suggestions:
    On #0, make there be only 23 bolts. That way you can say that when you count with the boxes it’s easier, but you actually counted TWO things (first the boxes, then the empty boxes, then subtracted) and that was still easier than counting the bolts without organization.
    I added questions scaffolding in the tree diagram like you suggested around number 1 (dinosaurs). I had them draw a complete one for a smaller number of things and then next to each leaf draw (and color) what that dinosaur would look like.
    On #8 you mention it’s “only $6.99!” so I included a question about how much money you’d have to pay for all those combinations.

    Overall it went ok. I didn’t give out “tokens” but I was liberal with my hints and they caught on.

      1. Well, mine was 99% the same as yours. I just retyped it (since I couldn’t find the docx). I do think I’ll use it again next year. They weren’t as excited about the minuet as I thought they might be, but maybe that’s just my all-boys school or something?

  2. Love the introduction! I stole it, but unfortunately our Precalculus curriculum doesn’t start doing probability and combinatorics until mid-spring. Hopefully all the other notation and formulas we talk about until then won’t get in the way of their exploring by the time we get to this unit!

  3. Yo, just wanted to say I think the two tree diagrams you mentioned are a great idea! This sounds like such a fun topic to teach. Also, I love that this whole post really brought me back to KSI.

  4. I realize this is an old post, but we started our probability unit today in Algebra 2 and I used lots of this packet in my class. Next year I’d like to begin the unit with the question and have it more open ended – have the students begin class just thinking about if there are enough’s for all the websites…what information do we need to know? How can we find it? How do we calculate these numbers? This might make the class more about the problem and less about the procedure.

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