Math Hombre

Mel Bochner, Pythagoras (4) from wikipaintings

Arithmetical Design (quite a fun tumblr) posted this beauty today...

I thought that this was something that screamed to be dynamic. Off to the GeoGebra Cave, old chum!
















The sketch started with a right triangle, and then the regular polygon tool to make the squares on the side. I wanted the triangle connecting the next squares to be similar to the original, so I made the side of a square to be the new hypotenuse, rotated it by one of the non-right angles then used the perpendicular tool to make the similar right triangle. Finally, I constructed  the first two additional squares.

Clearly too much work to repeat in the dozens. To use the Create New Tool command you select item or items in the sketch. Then select the command from the tools menu. My first try I forgot that I would need the points to make subsequent squares. Delete the bad tool from the Tool Manager. (Can also rename there if you're trying for something more pythy than Tool 1.)



When I had the squares and vertices selected, the second step of the Create New Tool dialogue was to determine the inputs. GeoGebra will select some ancestors to start, but you can modify the inputs. In this case, GeoGebra selected my first two free points, which doesn't suit. I wanted the inputs to be the the endpoints of the hypotenuse. At the last step you select a name and can attach a custom icon if you're being tricksy.

Once I had the tool it was quick to construct the spirals, and then aesthetics like a coloring scheme and positioning. From the GeoGebra color dialogue you can click the plus, which brings up an RGB color input. (For those times when you need beige, 255-245-235.)

















I was going to stop there, but decided that people needed to be able to make their own spirals how they wanted, so added a checkbox to go back to the beginning. (If you make something send me the pic and I'll add it to the post.) Sadly the new points show up with labels - I don't know how to turn that off. Maybe if the labels are off before I make the tool? Tried that and it works!

Here's the finished sketch at GeoGebraTube: teacher page or applet. Sadly, the custom tools don't seem to show up in the HTML5 mobile applets yet.

Bochner has several mathematically influenced paintings, as well as the first three Pythagoras painitings. Check them out at wikipaintings.

Today my students are investigating the Pythagorean triangle relationships.  The first question is can we tell the angle type of a triangle just by the side lengths?  (A previous post shares that investigation.)  Then they'll look at problems using the Pythagorean Theorem (same post - it was a long one!).  Finally, they'll look at developing some reasons that the theorem is true.  One visual geogebra proof is here - that's the easiest to extend to an algebraic proof.  But my favorite visual is the puzzle proof, where the medium square is cut up to make the square on the hypotenuse with the smallest square.  Here's the sketch to explore that:


Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (Click here to install Java now)

I was going for a feltboard look - what do you think?  It's also available as a standalone webpage or the original Geogebra file.

My notes on embedding geogebra in a webpage are here, based on Kate Nowak's instructions.

What can we deduce from the side lengths of a triangle?

This is a preservice teacher activity, easily adaptable to middle or high school use. GeoGebra is a free dynamic geometry program available as an online applet or downloadable program, from geogebra.org.

Objective: TLW explore triangle properties relating type and side length.

Schema Activation: What are the 7 triangle types? Fill them in in the ‘Type’ column below. You'll fill in the other columns as you go through the activity.

.....Type ....................... Examples [Like (3,5,5) ] .......................What do you notice?
1.

2.

3.

4.

5.

6.

7.

Focus: One of the reasons dynamic geometry is so powerful is the support it allows the teacher (or curriculum designer) to give students for finding examples. Lots of examples. As we’ve discussed in class, the natural way we reason is to go from lots of specific examples to the general.

Activity:
1) Open the sketch TriangleBySide.ggb. (Online at faculty.gvsu.edu/goldenj/TriangleBySide.html)
a. Collect at least 2 examples of each type of triangle. Where possible, try not to have the triangles be similar, where all the sides are multiples of another triangle.
b. Which were hardest to find? Was it something to do with the type or how you were looking?

2) Open the sketch PythagoreanData.ggb. (Online at faculty.gvsu.edu/goldenj/PythagoreanData.html)
a. Look at your right triangle examples on this triangle. How do you think ancient mathematicians noticed this cool pattern with the squares

b. Check your other examples on this sketch. For each, record whether the sum of the areas of the smaller squares is <, =, or > the area of the large square.

c. What pattern(s) do you notice?

d. Why do you think your pattern(s) might be true?

3) Go to http://teachers.henrico.k12.va.us/math/GeoGebra_Site/pythagoras/pythaPrblm1.html (link on Blackboard) for the Ladder sketch. Answer the questions there.
a. How high will he get if he places the ladder 3 m off the wall? Drag the ladder point and find a solution. Sketch the solution with all its lengths (s, r, h) on paper.

b. Now, calculate the solution of task (a) on paper. Which lengths are given, which are sought? Do you get the same solution?

c. At what distance of the wall should Pythagoras place his ladder in order to reach the window 4.50 m above the ground? Sketch the solution with all its lengths (s, r, h) on paper.

d. Now, calculate the solution of task (c) on paper. Compare your solution to your sketch?

e. As a teacher, what do you notice about the difference between doing the tasks in sketch and on the paper?

4) Open the sketch AdjustableLadder.ggb (Online at faculty.gvsu.edu/goldenj/AdjustableLadder.html)

a. What young Pythagoras might not have known is that a safe ladder ratio is 4:1 for the height to distance from the wall. What length ladder does he need to reach the window safely? Is there one solution or more?

b. How would you solve this problem on paper?


Reflection:
a. What confirmed, new or deepened understanding did you develop regarding triangles through these activities?

b. How did the dynamic environment affect your understanding?

EDIT:
Be sure to look at the comments. Scott Farrar has some notes on using this with 9th grade geometry, and has added a complimentary activity.

• http://scottfarrar.googlepages.com/triangleineq.html
• The worksheet http://scottfarrar.googlepages.com/TriangleCategories.pdf