Tag: Pythagorean Theorem

Posted in Algebra, Geogebra, Geometry

The Pythagorean Theorem Puzzle

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Math puzzles are great activities for enjoying and learning mathematics. The following is an example of Tachiawase. Tachiawase is a popular puzzle in Japan which involve dissecting a geometrical figure into several parts and then recombining them to form another geometrical figure. The puzzle below is credited to Hikodate Nakane (1743). This was one of the puzzles distributed at the booth of Japan Society of Mathematical Education during the ICME 12 in Seoul this year.

Make a shape that is made from two different sizes of squares by dividing them into three parts  then recombine them into one square. [Reformulated version: Make two cuts in the shapes below to make shapes that can be recombined into a bigger square.]

two squares puzzle

Here’s how I figured out the puzzle: I know that it must have something to do with Pythagorean Theorem because it asks to make a bigger square from two smaller ones. But where should I make the cut? I was only able to figure it out after changing the condition of the puzzle to two squares with equal sizes. It reduced the difficulty significantly. This gave me the idea where I could make the cut for the side of the square I will form. The solution to this puzzle also gave me an idea on how to teach the Pythagorean theorem.

I made the following GeoGebra mathlet (a dynamic math applet) based on the solution of the puzzle. I think the two-square math puzzle is a little bit tough to start the lesson so my suggestion is to start the lesson with this mathlet and then give the puzzle later.  As always, the key to any lesson are the questions you ask. For the applet below, here’s my proposed sequence of questions:

  1. What are the areas of each of the square in the figure? Show at least two ways of finding the area.
  2. How are their areas related? Drag F to find out if your conjecture works for any size of the squares.
  3. Can you think of other ways of proving the relationships between the three squares without using the measures of the sides?
  4. If the two smaller squares BEDN and GFNH have sides p and q, how will you express the area of the biggest square LEJG in terms of the area of the smaller ones?
  5. Express the length of the sides LEJG in terms of the sides of BEDN and GFNH.

[iframe https://math4teaching.com/wp-content/uploads/2012/08/Pythagorean.html 550 450]

After this lesson on Pythagorean relation you can give the puzzle. Once they have the correct pieces, ask the students to move the pieces using transformation in the least possible moves. They should be able to do this in three moves using rotation. Click here to download the applet. Note: If you don’t see the applet, enable java in your browser

Use the comment sections to share your ideas for teaching the Pythagorean relation. If you like this post, share it to your network. Thank you.

Posted in Teaching mathematics

What is the role of visualization in mathematics?

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Like abstraction and generalization which I described in my earlier posts here and here,visualization is central to the learning and understanding of mathematics. (Note that these processes are also natural human mental dispositions and so we can claim that doing mathematics is a natural human activity.)

Visualization used to be considered only for illustrating otherwise abstract ideas of mathematics but now visualization has become a key component of mathematical processes such as reasoning, problem solving, and even proving.

What is visualization?

Synthesizing the definitions of visualization offered by Zimmermann and Cunningham (1991, p. 3) and Hershkowitz (1989, Abraham Arcavi proposes the following definition:

Visualization is the ability, the process and the product of, creation, interpretation, use of and reflection upon pictures, images, diagrams, in our minds, on paper or with technological tools, with the purpose of depicting and communicating information, thinking about and developing previously unknown ideas and advancing understandings– Abraham Arcavi, ESM, 2003

What are examples of use of visualization in mathematics?
  1. For communicating information, the graph is perhaps the most recognizable of the visual representations of mathematics.
  2. For proving, visual proofs are already accepted as legitimate proofs. For example, here’s a visual proof of the Pythagorean Theorem. Click here for source of movie. [iframe 350 500]
  3. Of course, visuals can also be used to challenge students to reason and explain in words and symbols. For example teachers can show the visual in #2 then ask the students what the visual is telling them about the relationships between the areas of the three squares and about the sides of right triangles. Students should be asked to support their claim.
  4. Visualization tasks also trains students mind to ‘think outside the box’. Click here for an example of a problem solving tasks which can be solved by visualizing possible arrangements. Patterning activity like Counting Hexagons are great activities not only for generating formulas and algebraic expressions but trains the mind to look for relationships, an important component in algebraic thinking.
  5. Because what we see usually depends on what we know, visuals can also be used as context for assessing students knowledge of mathematics. Click here for an example on how to assess understanding by asking students to construct test items.