I have a tricky Year 9 (13-14 yrs) class and was rather apprehensive as they don’t work well with open ended tasks where the structure is not immediately apparent. They were great! The maths talk & debate that came out of the lesson was brilliant. They lost their way when it came to the actual calculations but nevertheless their involvement in trying to solve the problem was fantastic. I’m looking forward to the next time I use this type of lesson.

>>who was closest, 5 million or 10,000?

I also picked up on this. As you point out, the arithmetic mean is nowhere near, but in fact the geometric mean is much closer:

sqrt(10,000*5,000,000)=223,606, not so far off 287,820.

I guess this is because when estimating, people calibrate their estimate by multiplying (e.g. by up to 10) rather than adding a few centimetres as might be the case if you were estimating someone’s height?

I don’t see any part of learning something “new” in math, including the 5 items on rational functions, as boring and nitty gritty at all. What is boring is having to repeatedly do procedural stuff. There is necessary procedural practice, then there is overkill procedural practice. I think kids would WANT to learn all that rational expression/equation stuff if there was a HOOK for them to learn it. “Work” problems might provide the hook to learn how to solve rational equations, although I don’t teach it using equations, I do it visually first (http://fawnnguyen.com/2012/12/11/20121211.aspx)

We were talking about irrational numbers recently in my algebra 1 class, and I started by asking them about the diagonal of a 1×1 square. The fact that its diagonal is sqr(2) is mind blowing — this shook the followers of Pythagoras, they HID this fact because they were scared of it (just as the Greeks banned the idea of zero)! Kids love to hear stories. I’d done a lesson on constructing irrational numbers on the real number line (without telling them how), but they remembered this and it made all the subsequent “boring” stuff not so boring because there was a reason for learning it.

We just did Penny Pyramid and the heart of our lesson was actually in trying to find the equation: http://fawnnguyen.com/2013/05/18/20130514.aspx

But I have 8th graders, we ventured in that direction and we learned what we learned.

I don’t mean to speak for @Dan, but I don’t think the penny modeling problem was intended to assist in deriving the formula for finding the sum of squares.

Right. It wasn’t my objective for the lesson. My objectives were for students to model with mathematics and to use summation notation to describe a given series.

I’m going to answer your question with reference to the unit I just completed, leaving the pyramid question to the side for a moment. I’ve just completed a unit on Rational Functions with my Algebra 2 students. Here is a list of some of the things students learned:

1. how to add and subtract rational expressions
2. how to simplify, multiply, and divide rational expressions
3. how to simplify complex fractions
4. how to solve equations with rational expresssions
5. how to analyze the graph of a (simple) rational function

My lessons were completely devoid of pyramids, pennies, photos, videos, context, pseudo-context, or anything remotely interesting. Just plain old, boring, “nitty gritty algebra stuff.” I am not proud of this, but it’s true. I would be interested to know what a Dan Meyer version of this unit would look like. (Part of the problem here is the limitations of my own knowledge of rational functions. I can do all of the above tasks in a pure math setting, but have little knowledge of why rational functions are useful for solving interesting problems.)

Back to my main point about your lesson on pyramids: for me, the mathematical high point of the task is *developing* the formula for the sum of squares. In theory, all that “hook” you invested — showing the video, getting them to generate questions and guesses, etc — is all working towards getting them invested in doing the deep mathematics. So as I said, I was disappointed that this didn’t receive any attention. Yes, you discussed the key features of sigma notation, but that misses the point I’m driving at. Analogy: you’re teaching a lesson on the area of a triangle. The teacher can stop to make sure the students understand the *features* of the formula — what does b stand for? what does h stand for? can you point these out in the figure? etc — but if the teacher never makes the students *understand* the origins of the formula, then that is a travesty.

Maybe this is a matter of taste. I tend to favor deriving *everything,* but I’ve learned over the years that this is not necessarily best.

Summary of math required of the participants in the video:

1. Lots of modeling stuff. All important. Asking questions. Figuring out what info is needed. Making estimates. etc.

2. Plug n = 40 into formula that came “from the clear blue sky,” provided by the teacher.

So, again, if the purpose of the lesson is to practice math modeling, then great! But if the lesson were given in a context of exploring and learning about sequences and series, then it leaves a bit to be desired in that area.

Looking forward to your response if you care to write one!

It’s not all bad though and may be useful for other purposes. It’s quicker.

That’s the “one upside” I referenced above. You’re getting a lot of extra skills and engagement in the three-act treatment but it doesn’t come for free. It costs time.

I agree with Karl… “when are you going to put together the digital “textbook” — common core aligned, of course — that includes all the “upsides” and mitigates any “downsides?”

How long until textbooks evolve into web-based / online / digital-only texts? A student who is actively engaged in a lesson like the one you presented in the video is so much better off than one who encounters the textbook homework problem. My best guess is 10 years.