Wednesday, April 28, 2010

Learning and Teaching Anchor Charts

One of the things that seems to be referenced on this blog are the anchor charts.  My K-8 geometry class this semester really took to them.  I'm posting the public class ones here, but they also made several interesting personal ones in their portfolios.  Several remarked on how they saw new things about what they had learned by putting it all together.  A great example of synthesis.

Deb Smith (who has shared a number of the resources at describes it as follows, quoting from Harvey and Goudvis' Strategies That Work, which is a great comprehension text.
“Synthesizing is the most complex of the comprehension strategies.  Synthesizing lies on a continuum of evolving thinking.  Synthesizing runs the gamut from taking stock of meaning while reading to achieving new insight.  Introducing the strategy of synthesizing in reading, then primarily involves teaching the reader to stop every so often and think about what she has read.  Each piece of additional information enhances the reader’s understanding and allows her to better construct meaning .” (page 144)

“We need to explicitly teach our students to take stock of meaning while they read and use
it to help their thinking evolve, perhaps leading to new insight, perhaps not, but enhancing
understanding in the process.  To nudge readers toward synthesis, we encourage them to
interact personally with the text.  Personal response gives readers an opportunity to
explore their evolving thinking.  Synthesizing information integrates the words and ideas in the text with the reader’s personal thoughts and questions and gives the reader the best
shot at achieving new insight.” (page 144-145)

Synthesizing modes of instruction and the Conditions of Learning.

Looking at the Conditions for Learning against the backdrop of the Teaching-Learning Cycle.

Really trying to put together teacher and learner roles, with conditions, learning theories, instructional modes ... and a flower. 

Boy, they were a good class.  I'm going to miss them!

Friday, April 23, 2010

Solid Unit

As our semester in geometry drew to a close, we investigated solids.  Rather than dispense formulas (Go, Kate!), we tried to follow the Van Hiele levels.  Play and touch to get some understanding and visual recognition, sorting to start thinking about characteristics, and summarizing findings in definition-like descriptions.  Then we started thinking about measures.  Surface area is so natural, especially combined with the idea of a net.  But what about volume.  It's very interesting to have students sort the power solids by volume.  The sphere and hemisphere are very subtle.  I find it's also very common for students (college math majors) to be unable to remember formulae.  "Isn't there one with a 4/3?"

We filled them with water, and with no instructions from me, they immediately set out to try and verify their conjectured order.  (It's not as messy as you'd think.  The top of the solids container makes a pretty good tray.)  Two methods come up:  adopting a unit, and measuring each of the shapes in terms of the smallest, and filling one and trying to pour it into the next.  The different methods lead to noticing different things.  It seems like the groups that adopt a unit notice more numerical relationships from the data, and the groups that directly compare notice more of the geometric properties of the solids themselves.  ("Where does the water go?")

Usually from this data, you can suggest the idea of comparing solids with similar relationships.  Cone, Sphere, Hemisphere, Cylinder; Triangular Prism-Pyramid, Cube-Square Pyramid, Cylinder-Cone; Square Prism-Rectangular Prism-Cube, Small Triangular Prism-Large Triangular Prism or Hexagonal Prism.  Brilliantly designed little set.  (Although it does get into the experimental error that Dan Meyer cautioned about (was excited by?) in his TEDx talk.)  We compare the exterior of the solids and the water compares the interior.  Significantly different for the smallest objects.  Students have suggested measuring then by immersion, but we have yet to try it.

So this brings us to the boundary of informal and formal argument/reasoning.  How can we relate the volume of the prism and pyramid.  I do like models that fit together, but then that's just one example.  Of course, then, I tried to model it in geogebra.

Webpage or geogebra file.

It didn't help most of the students.

So I tried again:

Webpage or geogebra file.

This was helpful.  Or far more helpful, anyway.

Both sketches make use of Cavalieri's Principle to show equivalent volume.  We got at that in class by doing some block building, where each student had the same number of blocks per level.  This we extended into understanding the volume of a generalized cylinder.

Resources:  One of my favorite resources for this kind of classical problem is David Joyce's Java implementation of Euclid's ElementsBook XII is the one you need for these problems, especially Proposition 7 and 10.  Our department's java wiz David Austin is the one who connected us to those.  David A's visualization work is literally inspiring, and worth checking out.

We finished this all by building with polydron tiles a plethora of polyhedra.  I love how, left to their own devices, students invent regular polyhedra, antiprisms, various truncations, and completely original solids.  (Unfortunately these are pretty expensive, but they are durable and usable by kids as young as 2nd grade.  Cf.  ETA Cuisinaire.) I set them the challenge of building a polyhedron with volume between 1 and 2 liters as a fancy new container for a boutique.  The need for actual measurement and estimation as well as decomposition and formula use makes this quite a challenging problem.

Extension:  as I was thinking about this and looking for resources, I came across Archimedes' proof that a sphere is 2/3 of the circumscribed cylinder.  Famously, this is the relationship that Archimedes wanted put on his tomb.  I took the translation from the Archimedes' Palimpsest that was posted at Cut the Knot (an invaluable geometry site), added some clarifying comments and made it into a handout with an accompanying geogebra sketch.  The sketch isn't really for visualization, but allows the reader to experimentally test some of Archimedes' unjustified claims.  (All correct, though.  Were it today, the justification of the steps would be left as an exercise for the reader.  Pretty good exercise.)

Monday, April 19, 2010

Math Book Reviews Wanted

Sol at Wild About Math has been doing a couple of neat things to help build or support the math/math ed blogging community.

First, he offered to promote your website/math efforts through guest posts.

Now he's offering to help you connect with publishers to review books of interest to you.  I've taken him up on it already... against his better judgment?  Surely you will do better than I could.  Drop him a line - his address is on the About page.

From classic Peanuts at

Web Roundup

After sending links to a few people, I seem to always come to the conclusion that I should just be posting them.

Dan Meyer's excellent TEDx talk on Math Curriculum (and problem solving, and tech enhancement...):

What do you care enough about to make the subject of your 10-20 min TEDx talk?  You can find Mr. Meyer's excellent blog at, called dy/dan.  I saw this video first via Michael Paul Goldenberg

There's a kind of interesting web of blog posts on what interest is and how to generate it.  I'd start at wehrintheworld, which I found via Ben Casochna. (an entrepeneurial site.  Hmm.)  I'm interested because of how interest relates to engagement.  Which is Holy Grailish for me.

On an almost total aside, I've also been deeply interested in the fantastic images from Iceland's volcano, from two different educational sites:
 Both came up in What's Hot in Google Reader, so hat tip: internet.

Tuesday, April 13, 2010

I'm a Learner

Great send up of the "I'm a mac" ads.  Via David Coffey via Nick Ceglarek, super superintendent of Hudsonville Schools.  And proud GVSU alumnus.  They are just too good not to share.  Spread them around!

And the piece de resistance:

The makers, 21st Century Learning, or C21L, are a Colorado cooperative.   They have a wiki, which has more info than their homepage

What is it about CO?  That's where  the Public Education & Business Coalition is, which launched the whole Mosaic of Thought comprehension movement.

Thursday, April 8, 2010

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 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.)

Wednesday, April 7, 2010

Joke's On You

New resource at NCTM's Illuminations, an activity that parallels proof structure with the structure of jokes.

But the main reason for this post is to spread the news of their 2010 Illuminations institute.  A group of teachers will come together to write new and interesting lessons.  Only open to K-12 teachers, so I'm out.  But you could be in!  Find out more.  There's a stipend...

From the always entertaining sometimes profane xkcd.

"Well, the telling of jokes is an art of its own, and it always rises from some emotional threat. The best jokes are dangerous, and dangerous because they are in some way truthful."  - Kurt Vonnegut

"The love of truth lies at the root of much humor." - Robertson Davies