Missing Square Puzzle

Starter of The Day Transum Mathematics Topics

David, Kawartha Pine Ridge District School Board
My main concern is that this references the shape as a right-triangle which neither is in actuality. I think this throws the students off a possible thinking path. If instead it referred to the shape less specifically, then students might more readily venture down this path.
Transum,
Good point David. The phrase 'right-angled triangles' has now been replaced with the word 'shapes' in the text above. Thanks very much for the suggestion.
Tara, Brisbane
Was trying to plan a good lesson for ratio. As a substitute teacher, putting together an engaging lesson is paramount to avert discipline issues. So thought it just perfect, in keeping with the first step of arousing the interest of the class. Thank you.
H Rollo, Brechin High
Could be used for introducing gradient.

[Transum: That is a very good idea. I too use the notion of gradient when getting students to analyse the situation.]
Brandon, The Bronx
Cheating. The top triangle, measured as one unit has an area of 32.5. If we assume each cut up piece touches the grid, then we get get triangles of area: 12 (1/2* 8*3), 8, 7 and 5 (1/2 * 5*2) = 32. It doesn't matter what the bottom triangle looks like, the top triangle is lying to us and the bottom could be anything marginally different by not quite, but almost touching the grid lines anyway you'd like. Tricky. I like it becait tests our tacit assumption that the sub-shapes all touch the grid lines, when they don't.
Berten Stan, Oxford High
Good activity,would like to see more.
Ben Sparks, Twitter
Here's a classic 'Missing Square' dissection. The pieces don't change area. Where does the square come from? Attributed to Mitsunobu Matsuyama. Interactive @geogebra file here: https://t.co/ffBYXQYFYy pic.twitter.com/ZFLvsxzdda
— Ben Sparks (@SparksMaths) February 27, 2021

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Chess Board Paradox

Sam Loyd presented this Chessboard Paradox at the American Chess congress in 1858. Notice the Fibonacci numbers which can be found in both of these diagrams.

Parallel or not?

As you probably guessed, even though the red lines don't look parallel they actually are.

More Mathematical Optical illusions

Transum.org/go/?to=illusions

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Answers


	Teacher, do your students have access to computers? Do they have iPads or Laptops in Lessons? Whether your students each have a TabletPC, a Surface or a Mac, this activity lends itself to eLearning (Engaged Learning).

Here a concise URL for a version of this page without the comments.

Transum.org/go/?Start=July26

Here is the URL which will take them to a related student activity.

Transum.org/go/?to=areatri

Students can create their own presentation of the Missing Square Puzzle to show to other classes or in an assembly. Here are some guidelines for using PowerPoint

PowerPoint 2007:

On the Home tab, in the Drawing group, click Arrange, point to Align, and then click Grid Settings.
Tick the Snap objects to grid and the display grid on screen boxes. Select from the dropdown box a spacing of 1cm.

Snap to grid

PowerPoint 2013:

Snap to grid

The red and blue right-angled triangles can be made using the "Right Triangle" tool which can be found in the Home tab, in the Drawing group.

The green and yellow shapes can be created by putting together a number of 1cm by 1cm squares. Upon completion of the shape drag over the shape to select all of the squares then select "Group" from the Format tab, Arrange group.

Turn the Snap To Grid option off an add custom animations to each of the shapes to make the first arrangement of shapes transform into the second.

The images on this page are from the Wikimedia Commons. The descriptions of the licences can be found on the following pages: Missing Square Puzzle and Sam Lloyd Image.