Crowd Stuffing Revisited (Introduction to Area)

[Source.]

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.