## Thursday, February 25, 2010

### Motion Sketches

These Geogebra sketches are meant to serve as an introduction to the four Euclidean motions.  (I am a proud supporter of the Glide Reflection.)  Probably for middle school and above.  If you try them out, I am always interested in feedback.  The Motion Intro does not actually use a lot of the dynamic nature, but generated a lot of connections with my preservice teachers.  The Motion Control led to a nice discussion of what information is needed for each motion, as well as how to find that information (like the center of rotation) for a given motion.  The webpages have the questions that I asked my students.

MotionIntro: webpage and Geogebra file.

Motion Control: webpage and Geogebra file.

## Tuesday, February 23, 2010

### Math in Action 2010

Grand Valley State University sponsors a terrific little math conference each year called Math in Action.  I think started by Jan Shroyer back in the whereupon.  It has 30ish workshops for K-12 math teachers, mostly very practical.

I'm hosting 6 wonderful preservice teachers presenting geometry games for K-8 teachers, so I'm posting the electronic versions here for people to be able to download.  If you were a participant and wanted a Word file to edit instead of a pdf, just email me.  The address is available on my workpage, linked on the right.

Emily Trybus: Area Block (link to a previous post)
Rebecca Sochacki and Brynne O’Connell: Polygon Capture
Jill Dzierwa: Triangle Detective (link to a previous post)

As a bonus, here are the two bonus games from the Quadrilateral Concentration sheet.  Both are other uses of the Quadrilateral Cards.  Not included here is Quadrilateral Euchre, which is Euchre for a partially ordered card set.  Verrrry geeky in a midwest sort of way.

Materials:  Deck of Quadrilateral cards. Best with 3-5 players.

Setup:  Deal 5 cards to each player.  Put the rest face down in the middle, either in a neat stack, or mixed up in a big pond.

Gameplay
:  Start to the left of the dealer.  On a player’s turn they can ask a particular player for a specific property.  For example:  “Do you have a shape with opposite angles congruent?”  You can not ask for a shape by name.  (“Do you have a rectangle?”)  If the player has a card like that, they have to give it over.  If they have more than one, they get to choose which card to give away.  If you have a matched pair of the same type, you can play them down.

Winner:  First winner is the first player to go out.  Second winner is the player with the most pairs.

Variations:
•    Allow players to play cards on other people’s pairs.  (If you have a pair of rectangles I can play a rectangle.)

Materials:  Quadrilateral card deck.  2 players.

Setup:  Sort the quads by type.  Each player puts one quadrilateral of each type face up in front of them, and the others go face down in the middle.  Each player draws a card from the middle and keeps it hidden from the other player.

Winner:  first player to guess the other player’s card.

## Thursday, February 18, 2010

### Math Teachers at Play 23

Matht Eacher Sat Play 23 is up at Math Recreation.  Nice 23 theme and a couple cool 23 facts.  My entry is the recent money post.  But you will also find:

• Very cool post about young children learning number with a deceptive title: Infinity Plus One at republic of math.
• Sue's most recent Math Salon.   Great culture building.
• Two related posts on binder checks as assessment.  I wish more people wrote about assessment and evaluation.  (Don't think I have either, though.)  The second is at one of my favorite blogs, the unflinchingly honest f(t).
• Lots more good stuff, of course.

## Wednesday, February 10, 2010

### More Money

My preservice elementary teachers are preparing to do some 2nd grade tutoring, and the teacher asked for money and time.  (Really, who couldn't do with more of both?)  Those are always challenge areas, in my experience, so I thought I'd share our resources.  Here is a collection of some money activities.  It includes Change for the Better and Make It Take It which were described in an old post, which are two of my favorites.   It's been constructive to use games for practice, interesting problems to help with concept development, and explore multiple representations.  One thing that's often lacking is a visual representation to help understand the relative value of coins.

A quick easy game:  Monopoly Money Madness

Materials:  play money, 2 dice.
Math content:  addition, money recognition, unitizing (grouping into new amounts.)
Game play:  Very simple – roll two dice, and take that much money.  If you can group your money into a larger bill (for example, a five and five ones into a ten dollar bill).  First player to \$100 wins.
Variation:  have players “shop” a catalog or the web for something they would like.  Play until they have enough to buy the item they would like.

Money as a context offers some nice practice for developing unitizing, the understanding that allows learners to flexibly exchange between a groups and individual members, or the ability to switch what you are considering as a unit.  (I.e. switching from dollars to quarters or cents.)  It is such a key concept for 2nd grade, as students move from single digit arithmetic to multidigit arithmetic.  Lack of understanding in this will follow them for the rest of elementary school.

From Mathematics in the City: Measuring Teacher Change in Facilitating Mathematizing, Catherine Twomey Fosnot, Maarten Dolk, et al. (link goes to a pdf of the article)
Unitizing requires that children use number to count not only objects, but also
groups—and to do them both simultaneously.  The whole is thus seen as a group of a
number of objects. The parts together become the new whole, and the parts (the objects in
the group) and the whole (the group) can be considered simultaneously. For learners,
unitizing is a shift in perspective.  Children have just learned to count ten objects, one by
one. Unitizing these ten things as one thing—one group, requires almost a negating of the
original idea of number.  It is a huge shift in thinking for children, and in fact, was a huge
shift in mathematics, taking centuries to develop. Understanding that a square in a tiled
array can represent a column and a row simultaneously also involves a construction of
part/whole relations (Battista et. al., 1998), as does the relationship between
multiplication and division. There are many more. Because “big ideas” involve
part/whole relations, they require a shift in perspective by learners.
As we look at state content expectations, it's pretty clear they are focusing on skills.  Fosnot and Dolk (in Math in the City, their books on Young Mathematicians at Work, and their curriculum Contexts for Learning) have a nice way of organizing content into skills, ideas, and models, and then representing them on a landscape.  The preservice teachers took their information and tried to do the same for money.  Here's what they got: (as a pdf)

## Thursday, February 4, 2010

### Quick Triangle Sum & Pythagorean Proof

Just a quick, pretty unoriginal sketch to help secondary/tertiary students think through the justification for the sum of the angles in a triangle.

As a dynamic webpage or the original geogebra file.

EDIT:  As my students investigated (see these sketches elsewhere), they got interested in proofs of the Pythagorean Theorem.  Since they were investigating so nicely on their own, with interesting results, I had time to draw up a familiar proof in Geogebra.  Webpage or geogebra file.  The webpage has some additional hints to help towards a proof.