Pyth On

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.

Christmas Lights

aMacHan @ Flickr

Our family put up a Christmas tree this weekend, and it reminded me of one of my favorite problems ever! Students did great work on it, and developed a number of techniques.

On the weekend after Thanksgiving Karen put up a 2 meter tall Christmas tree, 170 cm wide at the base. She put an astounding 1500 lights on the tree. Write a story problem using this information.

One thing you should know about Karen's tree decorating is that she believes that lights should be distributed uniformly throughout the tree, not just on the surface. It's like the tree is full of lights. She started at the base, and put on 750 lights. Then she asked me how many will she need to finish? Now that we know she needed 1500, how far up from the ground did she get with those first 750 lights?

EDIT:
I saw that Abstruse Goose, an funny, edgy cartoon that's hip to math & physics also had some nice Christmas tree problems...

Pi in June

In honor of 2pi day (June 28th)...

From the awesome Dinosaur Comics, of course.

Why celebrate half the holiday?  Get the whole circle!  That old pi day is for squares.  Er, semicircles, at least.

Seriously, we did this activity in the preservice elementary class and I thought I'd share.  It culminates in the usual measure a bunch of circles activity, but tries to motivate it.  Sure it's amazing that all those ratios are close, but why would you do it in the first place?  And why are they all the same anyway?  For me the answer is similarity.  This activity was motivated by my students wanting to know more about pi, as it came up when doing volume of solids (someone remembered a formula), and that got people wondering.  In addition, our geometry class comes before our number class, and lots of students had said that fractions were what was most confusing.  And pi practically lives in ratio city.

Similarity

Two objects are similar in geometry if one is an enlargement of the other.  Mathematicians often use ratios to investigate them.

Consider a 2x3 rectangle.  A 4x6 rectangle is an enlargement.  But a 4x5 rectangle is not.  Can you tell by looking?  Describe what you see.


People discussed how the 4x5 should be similar because its 2 wider and 2 longer, but in the 4x6, which looks the same, you can see 4 or the original rectangle.


What ratios can you make with the two similar rectangles that are the same?

Students made both 2/3 to 4/6 and 2/4 to 3/6.


If you wanted to make a rectangle that was similar and 5 squares wide, how long should it be?  Can you prove your answer is right?

Some students set up a proportion and solved with cross multiplying... although they confessed that they didn't know why cross multiplication worked.  One student saw 5 as 2 + 2 + half a 2, and calculated 3 times 2.5.  We talked about how that was excellent proportional thinking because instead of 2+2+1, she relatedit back to the original as half of 2.  I showed how in the picture you could see the 7 as 2.5 of the 3's, or 3+3+half of 3.

A 24x36 poster is supposed to be an enlargement of an 18 x 27 poster.  Is that possible or not?  Explain?

Saw quickly with ratios.

Find the lengths of the diagonal of each poster using the Pythagorean Theorem. 

Refresh, connect question.

Sum up what you see about ratios and enlargements/similarity:

Jotted down and discussed at their tables.  Enlargement and proportional were heard in a lot of the conversations.

Common photograph sizes are 3x5, 4x6, 5x7, 8x10, 11x14 and 16x20 and 20x30.  Sketch, outline or shade in the ones you can on graph paper.  Which are similar?  Which are closest to similar?


The sketched them separately, so I did a nested version on the board to show another way.  (Often a student will have done it that way.)


Most cameras record in a ratio of 3:2, movie camcorders in 4:3 and hi-def in 16:9.  How do these compare to common print sizes?  What sizes would you use for your regular prints and for an enlargement?

General agreement on the 4x6.



Circular Arguments

Looking at these circles, they look pretty similar in the usual English sense.  Are they similar in the mathematical sense?  How do you know?



This was an unexpectedly interesting discussion.  Most people thought yes, they are all similar,  and then one student made them hesitate.  That led to proportional talk and someone brought ovals into it.  I asked if people knoew what made a circle a circle and they didn't.  They discussed how the only things you could measure are the radius/diameter or the circumference.


Give three different pairs of similar rectangles.  Compute the ratios of their perimeter to their longest side.  What do you notice?

Students saw the constant ratios quickly when they shared examples.  One student made the jump to the idea of scale.  Pretty cool.

What might that mean about circles?  Give your reasons.  (This is basically asking for an educated guess.)

Time was tight, so I was a bit leading here.  What ratio would the rectangle example be like for circles?  "Circumference to radius."  So if circles are all similar... "Then the ratios should be the same."


Divide and Conquer

•    Data collection:  in your group measure the circumference and diameter of at least 10 different circles. (details to follow)
•    Add your data to the class stem and leaf.
•    Which is the best choice for typical (mean, median, mode) and why?

 We discussed how to measure diameter and circumference, and agreed on precision.  (Two decimal places for the ratio.)  We got over 50 data points, and it made a pretty nice bell curve between 2.8 and 3.8.  There were two measurements of 6.00, but she realized she had divided by the radius.


The mode was 3.20 (5) followed by 3.18 (4).  The median was 3.19.  The mean about 3.22.  Curiously, the majority of students felt that the spread was too big to be explained by measurement error, so circles must not all be similar!  I did let them know that they were, but it would take a more theoretical argument to prove it to them.

All in all it felt much more contextual to start with similarity, and students made a lot of sense about why pi might have been discovered.

The activity sheets are below.  But also be sure to check the comments, where Alexander Bogomolny has a couple of very relevant links about the idea of measuring one circle as accurately as possible.
Similar to Circles

Similarity Day

4 Similarity Stations

All work on extending 1 dimensional similarity (distances) to 2 or 3 dimensions (area and volume).  This is very counter-intuitive for students, and I believe they need multiple experiences to retrain their intuition.  Of these, (1) is probably the toughest because students jump to linear relationship for area and volume.  (3) is the best for countering that ill assumption, although (4) can help also.

1.  Big Trouble.

Finn Mac Cumhail, (pronounced Finn McCool; no, really) leader of the ancient Fianna warriors, and gifted with "magic, insight and the power of words" when he was the first to eat of the Salmon of Knowledge, and ended up a giant. (Only in Ireland do magic powers come with the gift of gab.)  One of his rival giants, Benandonner, lived across the sea in Scotland. Benandonner wasn't able to swim across the sea to Ireland for a proper gigantic challenge so Finn tore pieces of volcanic rock into columns to make the causeway to Scotland. 

Benandonner came across to Ireland and Finn's house, where Finn was dressed up as a baby. Yes, a baby over 15 feet long! The "baby" bit the Scottish giant's hand off and the Scot took off for Scotland, terrified at how big Finn himself must be if his baby was so big.

Draw a picture for each of these questions.  Label edges with dimensions.
a)    If Finn was really a 15 foot long baby, how tall would the father be? (State any assumptions clearly.)
b)    Say a typical 6-foot tall Celtic Warrior weighs 9 stone.  (Ancient weight measure.)  How much might the 15 foot tall Finn weigh?  (Weight, density being equal, corresponds roughly with volume.)
c)    If it takes three square yards of wolf pelt to make a fierce looking warrior garb for your typical 6 foot warrior, how many much material would Finn need to make a costume?  If that takes two wolves for 3 square yards, how many wolves for Finn?
d)    Give the measurements (dimensions, area, volume, weight, etc.) of a giant sized something you might find in Finn’s house.  (Iron cooking skillets feature heavily in the Benandonner story, but don’t feel limited by that.)

(Tomie DePaola did a version of this story, but he mixes up Finn, Benandonner an Cuchalain - pronounced 'Kuh-kullen' - another Irish hero of myth.)

2.  Tangram
Requires multiple tangram sets or copies of paper tangrams.  Can eliminate step (1) for time.

1)    Use all the Tangram pieces of one set to make a square.
2)    Since all squares are similar (and why is that?) this large square is similar to the small square in the set.  What is the scale factor? 
3)    If the small square has area = 1, what is the area of the large square?
4)    Use the tangram pieces to make a figure and two other figures that are similar to the first.  (Bigger and even bigger, or bigger and smaller, or...) 
5)    Prove the similarity of your figures in (4) by using ratios.

See also, the teacher.net Grandfather Tang lesson.










3.  3-D Similarity

Requires: 100 cubes or so

1)    Build the building with mat plan (also called a base plan): 

2)    Build a geometrically similar building twice as large in height, width and length.
3)    Prove your building is similar with ratios of corresponding sides.
4)    Build or design a building three times larger than the original.  Explain how you know what is needed.
5)    Find the volume and surface area of each building.  What relationship do the enlarged surface areas and volumes have with the original?  Why is it like that?
6)    Can you design a building which has a buildable enlargement of 125%?  Find their surface and volumes. What scale factor relationship do the buildings' area and volume have?  How does that compare to (5)?

4.  Dilation
Requires: computer access

Open the Hexagon Dilation geogebra sketch or webpage.

In this sketch, the blue hexagon is dilated from the red point by a scale factor of S. The sketch allows you to change S, and move the dilation point or any of the blue vertices. It also measures the area and perimeter of the hexagon and the dilation.

The check box lets you show a square with area equal to 1 square unit for comparison, and its dilation by a scale factor S also. 

1)    Try varying the scale factor S. What do you notice? What questions do you wonder about?
2)    Collect data on the areas and perimeters for a fixed blue hexagon and its dilation as you vary S.
3)    Can you find a pattern in your data? Can you find a formula for the purple area and perimeter in terms of the original measurement and S?
4)    Use your formula to make a prediction for a scale factor and original area of your choice. Use the sketch to check. Does your formula work for a scale factor that is a decimal? Does it work for a scale factor less than 1?
5)    Compare the edges of the original and the edges of the image. What do you notice as you vary S? As you move the center of dilation?
6)    Can you predict the coordinates of the image of a vertex if the center of dilation is at the origin? If it is not at the origin?

Extension:  Open the sketch gigantotron.ggb (or webpage) and investigate 3-D similarity.  What questions would you ask to investigate?
(Now also on GeoGebraTube and a mobile applet.)

A Bigger Hex

As a webpage, and as a geogebra file.

Made a simple hexagon dilation sketch in geogebra for my geometry class. Let's you vary the objects, measures area and perimeter, and control scale factor. Algebra view let's you see individual side lengths also.

It's part of a set of four similarity problems for stations. Here's the pdf.