Crowd Stuffing Revisited (Introduction to Area)


After much dithering, I have revised and attempted a fixed version of this lesson as the start of my area segment in geometry.

Day One:

This was my opener. I should note I have a gadget on my software where I can slide a screen, so the entire slide above was not visible at once. First I showed the two pictures, and asked what was going on (with starting background that these were members of a fraternity and enough fishing both classes were able to figure it out). Then I slid down to the story section, and only at the end did I reveal the question.

The primary motive for starting this slide was not to teach unit conversion, but to establish that units of measurement are arbitrary and can be even based on the height of a random person during a prank.

(Click on the image to download the document file.)

The first page is self-explanatory, but the second page deserves comment: I gave every student a piece of patty paper (6 in x 6 in) and established that it was an entirely arbitrary unit of measure, then challenged them to measure various things in the classroom, such as:

Two tiles of the floor
One brick
The face of another student
The bottom of a shoe
The circular vent of the air conditioner system hanging overhead

What’s interesting is that given a large enough object (like the front door of a classroom) quite of my few students without prompting invented their own shortcuts to finding the area: that is, they figured out how many squares go on the length and width, and then multiplied, essentially discovering the rectangle area formula on their own.

Also, some items (like the circular vent mentioned above) were difficult to reach so the lesson required standing on chairs. The students loved this.

Day Two:

I opened with the picture at the top of this post, and again asked what was going on. I wasn’t interested in any mathematics for this post, I was just using the picture as a hook and establishing safety boundries to get into this question:

How many people can we fit into this classroom?

(I originally tried writing this lesson around a piece of media, but eventually settled on the simple question above.)

I also added an extra question after:

How many people can we legally fit into this classroom?

The class went through questions in this sequence:

1. A journalist is asked to cover a major event in a large ballroom. She wants to include a note about how many people attended the event. How could she estimate the number of people without actually counting everyone?

2. Describe several different methods that could be used to estimate the size of a crowd at an outdoor political event. How might the estimate of the crowd size supplied by organizers of the event differ from estimates made by other groups? Why?

3. How many people fit in a 3×3 foot square?

4. Knowing that, estimate how many people fit in a 5×5 foot square:

5. Test your theory. Was your estimate accurate?

6. How many people could fit in this classroom?

7. How many people are allowed in this classroom within the law?

8. Why are there signs in some places that give a limit on the number of people?

Question #7 requires access to the fire code in your area, something like this document. (I’m not located in Scottsdale, but at least it’s in the same state.)

And yes, Question #5 requires the entire class.

Last comment:

I’ve brought up before on this blog that my students seem to know much less geometry coming in to my class (10th grade) than they should by the standards. If I recall correctly, introducing area is something like a 2nd grade standard.

A good half of my class had never worked with area before. Make of that what you will.


8 Responses

  1. I’d be willing to bet that they had, but it had been drowned out in algebra prep and whatever else such that they forgot.

    But I’m optimistic that way.

    • I know what you mean. I’d rather have the students not know concepts from previous classes, because they inevitably get misinterpreted (e.g. a negative and a negative is a positive), if they’re even correct to start with.

      Plus, I teach IMP, so if the kids know, for example, the Pythagorean Theorem, then that kind of spoils the “Tri-Square Rugs” game.

      It’s easier to teach fresh than it is to have students unlearn harmful learnings and then relearn them correctly.

  2. I have no doubt they worked with area before. It may have been back in 5th or 6th grade and they don’t recall a thing.

    I looped with my class twice from 4th to 5th. I can’t count the number of times I would refer to something we did in 4th grade and they would say, “We never learned that!” It was about as painful a comment as I could imagine.

    • I used to have confidence the students forgot, but I have been assessing more and more carefully and I do honestly believe a good chunk (in this case half) were never exposed at all.

      I mean, it could be worse. I had a student whose prior geometry teacher taught origami, and only origami, the entire year (this is before NCLB).

  3. Not sure if this is appropriate for your purposes, but if you use the Ruler tool in Google Earth, one of the units you can choose is Smoots.

    Here is a screenshot of measuring a sideline at a field at MIT:

    If you can project GE in class, you could measure the actual bridge in Smoots. Or your school. Anyway. Cool lessons.

  4. […] Crowd Stuffing Revisited (Introduction to Area) […]

  5. 5th grade teacher here. I am perplexed about the discussion on skill lose and whether concepts were taught or not taught. I use a lot of mnemonics in my teaching and am wondering if a high school student who seemed clueless when asked to complete an area application would perk up with the first word of an area chant that they repeated throughout their year in 5th grade along with a plethora of other concept connections, example “area array operation multiplication” Of course, I could be one of the teachers whose practices you try to erase. Thanks for thought provoking content.

    • It would certainly help to have access to exactly how a topic like area was done in their earlier grades — which mnemonic or whatnot — so it’s easier to access at the upper level.

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