This summer, Marc and I created a series of videos designed to help parents support their children in Math 8 and 9. As best we could, we tried to have parents actively “do the math” rather than passively consume content. The explorations were meant to simulate the classroom experiences of their children. Here’s one of my favourites…
Display the original photo and five enlargements.
Ask “Which of these photos look the same as the original?” This phrasing is intentionally vague. Have students talk about what it means to “look the same.” Introduce labels — it’ll make conversations easier.
At this stage, no numbers are given. I want learners to use their intuition and get a “feel” for the problem. Tell them not to worry about making an incorrect choice — they’ll get a chance to revise their thinking later on. Likely, they’ll rule out photos B and D. Photo B looks like a square; it looks like photo D has been stretched more horizontally than vertically. Photos A, C, and E are contenders. For example, students might suspect that the dimensions of E are double those of the original. Ask “How confident are you?”
Now is the time for numbers.
Ask “Would you like to revise your thinking? How confident are you now?” The numbers confirm this hunch about photo E (and C). They can also determine close calls, like photo A. Here, scale factors of 0.75 (height_original : width_original) versus 0.8 (height_A : width_A) or 1.25 (width_A : width_original) versus 1.33 (height_A : height_original) prove that photo A is not a true enlargement of the original. (Note that this might surface if students are making absolute rather than relative comparisons: after all, adding 1″ to both the width and height of the original gets us photo A.)
This context can also be used to explore strategies for determining a missing value in a proportion. What if the photo were “posterized”?
Although these videos were designed for parents, we’re hopeful that teachers find them helpful.
Recommended reading: Tracy Zager’s Becoming the Math Teacher You Wish You’d Had (Chapter 9: Mathematicians Use Intuition)
Recommended activity: Desmos’ Marcellus the Giant