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.

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.)

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

Coordinate Connect

This is a modification of a game that I can not remember where I saw. EDIT:  Yeah, remembered!  It was at NRICH, the game of the month.  They have a java implementation.  Give it a go.

Modifications by Jill, Leah, Lauren, Ashley, Megan, Nick, Jenn, Alyssa (Preservice geometry K-8 class) and myself.

Game:  Coordinate Connection

Materials:  10x10 grid with no axes, pen (or colored writing utensils)

Gameplay:  The idea of the game is players take turns naming a coordinate pair and then filling in the point.  The first player to make four in a row, wins.

The first point of the first player is the origin, (0,0).  It still counts as their point and can be one of the four in a row.  Later points are given by naming a coordinate pair.  For example, (2,-3), is two to the right of the origin and 3 below it.

Players take turns naming a point and then marking it.  If it's not where you thought, too bad, so sad, that's where it goes.  (Unless another point is already there - lucky.)  If players are having a tough time just verbalizing it, they can be required to write the coordinates down before drawing the point.

The winner is the first player to get four in a row, vertically, horizontally or diagonally.  The loser decides whether they want to go first or second in the next game.

Hints:

• consider blocking your opponent from getting three in a row. 
• get and stay on the offensive
• use a special mark

Variations:

• Play with three or four people.  (This was surprisingly fun and complex.)
• Allow any 4 consecutive collinear points to win.  For example, (-3,-3), (-2,-1),(-1,1), and (0,3)
• Play with younger students by giving directions from home, such as 2 left and 3 up.
• In the original game, you marked the point then said the coordinates, losing it if you were incorrect.

Give it a go, and let us know what you think!

Paper for playing - 2 sided, 4 10x10 grids each side.

Sample game:

Throw Out Your Lesson Plans

Preamble:  the Common Core Standards for K12 Mathematics are up and available for comment.  See http://www.corestandards.org/.  My two two word reviews: too much and too little.  They just couldn't focus.  And, there's very little attention to the processes.  In related news, as Congress considers the revamping of the Elementary and Secondary Education Act, the Forum for Education and Democracy put together a pretty nice little manifesto, er, recommendations.  I glossed over it once, but Dave Coffey repointed it out to me.  Worth a look.

We were talking about lesson planning this week, and I enjoyed the think aloud enough that I thought I'd post it.

It's easy and common to confuse lesson planning with a lesson plan.  When you ask preservice teachers about planning they invariably talk about lesson plans, and most usually, particular lesson plan formats.  Then they get more classroom experience and 95/100 supervising teachers tell them that they don't use lesson plans any more and the novice teachers decide they don't them either. 

Of course, they're right.  They probably don't need lesson plans the way we often teach them.  I used to require awful things.  Huge four column Japanese style lesson plans with loads of information.  Then I started just using those to capture a lesson.  Finally, I gave up making any kind of stink about the format.  I still share those, as a way to capture a lesson.  But I make no pretense that you would use them to prepare a lesson.  As a department (okay, Dave, Rebecca and I) we are trying to get away from planning without students in mind.  Real students.  Talk about sending a bad message!


But the lesson plans are the bathwater.  Helpful.  Bubbly?  The baby is the planning.  That is absolutely essential.  And every intentional teacher I know spends time, thought and energy planning.  We may not have enough time for it, and would probably like to do more of it, but it's a crucial part of the teaching.  That's why it gets a spot of its own on the Teaching Learning Cycle.  (Adapted from the Learning Network model.)

So to try and capture the difference, this semester, I talked about both as the questions I ask myself while thinking about them.  This feels more authentic to me, because I'm constantly adding to and changing the planning questions, which feels like when I'm planning. 

Plan vs Planning:  Essential Questions

Lesson Plan

• What will help you organize your thinking?
• What physical record would be a good reference while teaching?  What details, sequencing or answers would be handy to have available.
• What record will help you keep track of what was done and what you learned from it?

Lesson Planning
What do you want students to learn?

• What do they already know about it?
• Why do they need to know this?  Or, what’s important about this?
• Consider big, long term goals and specific lesson objectives.
• Consider process goals as well as content goals.
• What do they already know about this?
• How will you be able to tell when they’ve learned it?

What experiences will move students forward towards the objectives?

• What lesson structure will be good for this?
• What mode (individual, cooperative) is good for this?
• Have you tried it?
• What’s engaging about this?

What support would help students?

• How will you equip diverse students?
• What are possible student responses or questions? Your responses to that?
• What representations will you be using?
• Would a demonstration help?
• Are there math or life connections to this?

What data will you collect?

• What does understanding look like?
• When and how will you observe the students?
• What record will you make of the data?
• How will students consolidate or reflect on their work?

What other questions do you ask yourself while planning?  What's most important to you?  I'd love to hear about it.

Tessellations and Geogebra

In my Geometry K-8 class we've been study transformations.  Which always leads one place for me ... my love, my joy... tessellations.  Arty, playful, deep underlying structure, corner cases that require thought even still; they're perfect.  To me.  I understand how others have dabbled and grown tired, but for me they are ever fresh.

There's a probably a few too many sketches here, but let's have a look.

First - Look at a tessellation, identify the motions, and consider what properties allow it to tile that way:

Quadrilaterals:  webpage and geogebra file

Hexagons1:  webpage and geogebra file

Hexagons 2:  webpage and geogebra file










Second - Look at a tessellation, identify the motions, and then alter the tile Escher-style!

Isosceles Triangles:  webpage and geogebra file

Quadrilaterals:  webpage and geogebra file








Third - Control the properties of the tile so that it will tessellate with the given motions:

Pentagons:  webpage and geogebra file

 (A midpoint rotation and 2 side to side rotations.)

Hexagons:  webpage and geogebra file

 (Quite challenging!  3 side to side rotations.)









Bonus - Kaleidoscopes!  What's the connection between reflectional and rotational symmetry?

Control the number of Sectors:  webpage and geogebra file

Control the Angle:  webpage and geogebra file






The kaleidoscopes were to investigate an open conjecture we have.  My one disappointment with Geogebra in comparison with sketchpad is that animation isn't as easy.  It's nice to have an animate button on your kaleidoscopes. 

The Leah-Jill Conjecture:  If a shape or design has n lines of symmetry, then it will have n-fold rotational symmetry, for n > 1.  Having rotational symmetry does not imply reflectional symmetry for any n.

I don't think the sketches helped.  I can't decide if we should tackle it another way or if we should just move on.

If you have any ideas for a dynamic tessellation sketch, please let me know.  It doesn't take much of an excuse to dive right in.

Conquer and Divide

I've been working a lot on division lately.  Long division with my son in 4th grade, with his classmates in small groups and with my preservice teachers.

Xavier's teacher sent home a note before the unit, showing the four ways they would be approaching it.  From the traditional algorithm to the "forgiving algorithm" and a couple in between steps.  My wife couldn't make too much of the note, despite being bright, living with more math than anyone would think is reasonable, and being comfortable with her own computation.  Don't know what it was like in other homes.

Xavier was making pretty good sense of it.  The curriculum sensibly starts out with dividing by 5 only, which was a nice touch.  I think I might even start with 10s, then 5s, having seen how well this worked.  He transferred this pretty well to working with other single digit numbers.  For example, 61/4.  When working, he used appropriate language and responded to questions like "how many fours go in 61?"

I did the soda machine problem (from Contexts for Learning, my favorite curriculum, Exploring Soda Machines: a context for division) with a small group the week before, and they did amazing work.  In the problem, you describe a pop machine (in Michigan, soda=pop) which has 6 flavors.  When full, it has 156 cans of pop.  How many cans of each flavor, if there's the same of each?  But wait, that's a lot of pop.  When I buy pop it's usually in a six pack.  How many six packs will that be?

Students do the most amazing work wth this problem.  Most choose to tackle the 6 flavors question first, and drew their own pop machines.  Six columns, and fill in pop cans 1 per column until they have 156.  156 is an inspired amount - not so much as to be overwhelming, but enough so that they usually have to start some kind of record keeping to keep track, often multiples of 6 as they add cans.  Sometimes drawing a line, with the total amount under that line.  Some students draw neat stacks of cans in a row, and some draw crazy piles of circles of many sizes.

Not very many students saw the connection with the two problems.  Only one realized that both were 156/6.  Others started to think about it when they realized the answer to both questions was 26, and then saw a connection. Many used their pictures of the flavors and started circling groups of 6.  Which was interesting, becuase then there are 2 left over of each flavor.  Some had 24 six packs with 12 left over, and some had 26 six packs.  (Talking they agreed on 26.)  The student who made the division connection shared, but the other students didn't really seem to hear her.

The formal language for what's going on here is that there are two different division actions.  When students can solve some division stoies but not others, sometimes this is the underlying cause.  (Quotative and partitive division, from learner.org, with kid video, too.)  When explaining this to our preservice teachers, we often use the terms fair share (how many of each flavor) and measure (how many six packs) for the different actions.

The next week I wanted to bring another context.  I brought a bunch of play money that I wanted to sort into 6 bags for my preservice teachers.  We counted up the 74 quarters together by making stacks of 10. I wanted to have enough that they were counting up in 10s, so that when we were dividing we'd see the benefit of the 10s.

Do we have enough for 10 in each bag? "Yes."  How much left? "14."  (I kept the notes on the left.  10x6=60)  How much more can we put in each bag? 10? "No!!"  "Two," one of the kids suggested.  People agreed, so in went 2 each. 2x6=12.  2 left, not enough for even one more in each bag.  "That's the remainder."  Excellent!

We then divvied up the rest of the coins.  We had the most pennies, so I asked for a volunteer team.  Then the next team chose dimes over nickels.  96 nickels, it turned out.  But then they lumped all of their neat stacks of 10 together again!  149 dimes.  And 491 pennies.  (One of the kids even noticed the anagram.)  The nickel team was done pretty quickly, 10 in each, then 5 in each and 6 left over.  "Oh, that's enough for one more in each bag."  The dime team did 10 each, then 10 again.  5 each... not quite.  Get one out of each of the five bags with 5.  Remainder 5.  The 491 team did multiple rounds of 10 each.  Then just kind of scooped that last 11 into a bag.  All the bags had lots of pennies by that point.  Then each team made a number record like on the left, and nobody saw the connection with the division algorithm they have been doing.

Wow!

Okay, that's immediately a preservice teacher activity.  At GVSU we're blessed with a goodly pile of manipulatives.  So each table got 6 tubs of blocks (2 each with unifix cubes, wooden cubes, and snap cubes.)  5 (or so) minutes to play with them, because play is important.  Of course they made many mathematical designs and structures.

Then it was time for the task:

1)    How many of the object did you get?
2)    Physically divide them up into the 6 tubs evenly.  How did you do it?  How many in each tub?
3)    Show with a number record what you did.
4)    Use a sense-making method to do the associated division problem.  How would what you did make sense as physically dividing the objects?  Why does your method work?
_________________________________________

1)    ___________ blocks in each tub.


2)    Description of method:


3)    Number record:


4)    Sensible division problem:

And then to make a poster of the connections between their number record and their division work.  Here's what they did! (Click on the images for full scale.)

And the piece of least resistance:

Those are some beautiful connections!