So, after the crappy lesson on Friday, I was due for a better one.
I’ve taught rates (and the introduction of dimensions & units) for the past 4 years, and it’s always been a slow start. I eventually get the kids going, but there is a good 6-7 minutes of muddling as I try to get them to figure out what my questions mean. And then, as I try to transition them from the concept of units to combining them as ratios, I’d lose a bunch of the class to disinterest.
Thanks to some recent nudges re: design, I decided to rework my approach.
Here’s the core of the resulting lesson:
“What are we measuring?”
For the first two, the students pretty quickly answered with “weight” and “time”. It took a bit of prompting to get them to list units for weights, because (a) they were not warmed into the game yet, and (b) because they presumed pounds to be the default weight, as in “I weigh 145”. I think I may rearrange the pictures to do time first, because they were more inclined to include units when actually stating times.
On the white board next to the screen I listed the dimension they yelled out, followed by whatever units I heard. It quickly became a game of trying to come up with odder, unusual units. Some students yelled out “4 pounds” which almost automatically lead to a student discussion with little need for my input. Eventually things petered out, and I moved onto the next dimension, or added a couple of esoteric ones of my own to the list (stone, lightyears, furlongs, fortnights).
It is notable that once we got to distance (which I used as a starting point years ago) that they didn’t have the sense of abstraction to deal with distance – they searched for a more concrete term like height, or width, or even size, and gave up, jumping straight to units rather than dimensions. They also didn’t have the vocabulary to come up with volume (except for one class), and jumped straight to listing units for that as well.
I then let them try to come up with some dimensions of their own. Common responses were speed or density (which work perfectly into rates, so I listed those on the board to return to later), though one kid mentioned sound (measured in watts, and I provided dB) and I prompted them for money. You’d think immigrant kids would be quicker to catch on that both dollars and pesos are units of money, but they always seem to get stuck on that cultural shift.
I ran through a couple of minimalist slides relating the word “per”, and division, and fractions, with discussion on division being a binary operator, and fractions needing both a numerator and denominator, so that the word per (and by extension rates) need two different units. The last slide (showing a fraction) led into this set:
The slides all represent common rates that the students are familiar with, but the images gave me a chance to reinforce the idea of units vs. dimension, as well as introduce the concept of dimensional analysis when they gave me dimensions rather than units for the pictures.
Oddly enough, adults had more difficulty finding the context for the last rate than the kids did, even though the kids don’t drive yet (and yes, that’s a hint).
Today was PD day, with shortened classes, and therefore I didn’t get a chance to try out my dragon word problems involving flying speed, gold per square foot, and farms burned per day. I also owe a thank you to whoever’s blog I read about the dragon word problems on – I immediately resolved to do fewer questions about road trips or pet frogs and more about zombies, pirates, ninjas, and velociraptors.