Similar polygons- Using ratios

14 Jan

Good group work problems are really hard to come by. Especially in math when looking for problems with multiple pathways to solving them. Here is one that seems quite simple, but kids definitely will struggle nicely with.

I ran it for the first time last year and I wasn’t quite ready for the struggle that students would have. This year went a  LOT better (which is why I’m ready to share). I gave them this activity after doing an activity on The Golden Ratio. I talked a little more in depth about setting up proportions and what that means better than I did last year too.

Anyways, the problem starts like this:

Before digital cameras became popular, 35 mm films used to take pictures in a width-length ratio of 2:3. Most standard picture frames were also created back then to be 4” x 6”. Why does this make sense? Explain.

 Nowadays, however, digital cameras take pictures in a 3:4 ratio, creating a need to crop pictures before they are printed if wanting to fit them in a 4” x 6” frame.

Below is an example of a digital photo the way it prints when sent to a printer without cropping. What is the problem when trying to fit it in to a standard 4” x 6” frame?


What will the new dimensions be if the digital photo were to be enlarged to fit the longer length of 6 inches? Then how much will you have to crop off? Explain with words and drawings.

Larger picture frames are typically 5” x 7” and 8” x 10”. In the same way, if a digital photo were to be enlarged so that there is no gap in the frame, how much of the picture would need to be cropped in order to fit each of the frames?

It’s a bit prettier and roomier on the actual worksheet, but if you work it out, it does a nice little twist by the time they have to fit the digital print into an 8″ x 10″ frame so that it tests their “monkey status,” as I like to call it (basically, how robotic they can get in solving a problem procedurally without understanding the concept of what they are doing). The last two problems in the worksheet are also nice and challenging.

A bit about the execution 

Misconceptions to understand in advance:

  • Students think that the proportions are additive, instead of multiplicative. (I wish I had an app/ applet to tackle that misconception! Anyone know of one?)
  • Students will try to double the 3:4 ratio so that the short side of the photo matches the long, 6″ side of the frame. Might want to have props ready to explain that one.
  • The picture I provide where the cloud picture does not fit the frame will throw them off. They have a hard time understanding that a 3:4 photo can have a short length of 4 inches to start. I’ve considered changing this picture, but instead turned it into a great way to start explaining the problem without giving too much away.

Other things noteworthy:

  • Order-wise, I had it mandatory that they finish pages 1 and 2.
  • Pg 4 came next as optional for those finished early. **They must read page 3 in order to understand page 4!
  • Pg 3 is a great example to do in getting them to use a variable to represent all 16:9 rectangles. I think it’s a great example for a lecture also because it reinforces the multiplicative idea. OR, it can be for your super SUPER kids.
  • Be ready to listen to what they are thinking and encouraging their chosen route! Be open to being surprised by their outcomes!For example: as I was explaining the problem itself, one student yelled out the right answer by simply enlarging and shrinking the photo in his head.
    3:4 became 6:8 or 9:12, for example, which became 4.5 : 6, which was the ratio we needed. He doubled the ratio, tried tripling the ratio, then he halved that tripled ratio… I hope that makes sense.

Anyways, you can find the full packet here.

One Response to “Similar polygons- Using ratios”

  1. Leatrice May 14, 2017 at 9:57 pm #

    I use a micro SD card between my laptop and netbook as an external hardrive for my class files. I often download files on one computer or the other without the drive in that computer. Is it possible to set my computer up so that when I insert my drive my computer auctoatimally updates the folder on my removable drive with the folder on my computer?.

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