## Look-Alike Photos

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 favouritesâ€¦

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

## Don’t mean to burst your bubble

via Colossal, an art, design, and photography blog:

While waiting for a train, commuters can help themselves to square sheets of bubble wrap labelled with how long it would take to pop them.

I love this idea. The world is a better place because of it. I hestitate to bring this up, but â€¦

the math is wrong.

It looks like the approximate times are based on length.Â Above, the ratio of side lengthsÂ is 3 to 5 to 10, or 1 to 1.67 to 3.33. Let’s assume that the small sheet does, in fact, take 3 minutes to pop, one bubble at a time. The large sheet does not have 3.33 times more bubbles; it has 3.33 times as many rows and 3.33 times as many columns. Therefore, it has 3.33^2, or 11.11, times as many bubbles. A better approximation for the large sheet would be 30 minutes. If we base the approximate times on area, the ratio of sides lengths would be 3 toÂ âˆš(5/3) toÂ âˆš(10/3), or 1 to 1.29 to 1.83, as shown below.

I’m thinking about how I could use this image or idea in class. Some possibilities:

1. As-Is

Display the photos. Ask students, “Are the times accurate?” Have students apply their understanding of the relationship between scale factor and area. M’eh.

2. Hands-On

Display the photos. In pairs, have students record how long it takes to pop a small square sheet of bubble wrap. Pose the problem,Â “A square sheet takes twice as long. What are its dimensions?” Have students test their predictions. In this activity, students develop their understanding between scale factor and area. They poke holes in the common misconception that when dimensions are doubled, area is doubled, too.

3. Three-Act

Play a video of a small square sheet of bubble wrap being popped. Include a timer. Maybe a soundtrack, too. Play the beginning of a video showing a large square sheet of bubble wrap being popped.Â Have students guess how long it will take. Ask, “What information would be useful?” Show the dimensions of the squares.Â Play the answer video.

I see this task being similar to Dan’s Penny Circle. Dan filmed himself filling a circle with 663 pennies so that the rest of us wouldn’t have to. I have a roll of bubble wrap measuring 24″ by 30′. Before I take one for the team and spend a ridiculous amount of time enjoying bubble wrap, any suggestions?