Geogebra

I know I've gone post crazy. I will stop soon, really.

I'm trying Geogebra this semester with my Geometry K-8 class, first computer session tomorrow. It's a freeware java implementation of dynamic geometry quite similar to Geometer's Sketchpad, with an even stronger algebraic interface. In addition to playing with it to get a sense of what's going on, I've got two sketches to look at:

Square Not Square Where you adjust quadrilaterals by vertices
to check the 'real' type, as all 9 look like squares.

Quad By Angle Where you determine a quadrilateral

by the angles rather than by edges or diagonals.

Links lead to webpages with the java sketches, as geogebra has a great export to webpage feature. They are java, so some browsers don't handle them as well as Firefox. If you prefer the actual geogebra sketches, it's SquareNotSquare.ggb and QuadByAngle.ggb. Most of the sketches I've seen so far have been for HS or university, so if you're interested in K-8 uses, please let me know!

EDIT: Followup - the students were mostly positive, and some extremely positive. Definitely a better reaction than to the old computer labs with sketchpad. There was more interest in the program because it is so accessible; many more of them saw themselves as potential users of this time. It also connected with people's desire for visual representation. Makes me wonder, have I been underusing manipulatives this semester?

Book Review: Babymouse

Read a very cute book recently, Babymouse: Dragonslayer, by Jennifer Holm and her brother Matthew. I had never read Babymouse before, but it was recommended by Jen Robinson (who herself was recommended by Kathy Coffey, my guru in all things about teaching reading). Jen Robinson's blog is a great resource for anyone interested in kid's lit as a parent, teacher or reader.

Babymouse is a math-hater, because she is bad at it. Her teacher, shockingly, starts her on her adventure by sending her to be a mathlete. While there is no math per se in the book, it is all about immersion and expectations by the Mathlete coach, and employment and approximation on Babymouse's part. And all with excellent connections and fantasies of our great fantasies, Narnia, Lord of the Rings and Potter. A homerun all around. My only unrequited wish - more actual math. All that for only $6 at Amazon.

The italicized terms here are from one of the best frameworks for teaching and learning I've ever seen. It's really the lens through which I evaluate and work on my teaching. Brian Cambourne's article "Toward an educationally relevant theory of literacy learning: Twenty years of inquiry" is available online here.


My daughter Ysabela's review is here. She says the title misspelling is intentional...

Math Teachers at Play 15?!

The new carnival is up at mathfuture. One of the most interesting things about it is the host site, which looks at Web 2.0 math applications. For example, their GameGroup, which I may be seeking to join.

Other than that the 2 posts that were most interesting to me are a description of a role-playing site for math-centric careers and 25 uses of Wolfram Alpha (which I love).

Denise has a good starter on Mental Math, but I don't see these as techniques you teach as much as ways of thinking that you demonstrate and can grow out of kids' number sense. I was saying to Xavier (my 4th grader) last night, knowing 6x9 is not as important as being able to find it efficiently, and knowing things like it's connected to 6x10 - 6 or 6x5+6x4 is much more important.

Angles

What do you see as the big ideas with respect to teaching angles?

To me:

• filling around a point, no gaps or overlaps between two boundaries
  
• connection with a circle (filling all the way around a point) - important for units
• size of the angle corresponds to how open

So I love to begin teaching angle with pattern blocks. The activity I start with is adapted from one taught at GVSU by Jan Shroyer, don't know where she got it or whether she wrote it. The activity as a Word .doc is here. If printing from the web, try to size the pattern blocks so they are life size. (Doesn't affect the angles, of course, but makes it much more natural.)

A Very Special Blossom

A blossom is a special pattern in mathematics when copies of the same shape are arranged to fit together all the way around a point. Try to blossom the following shapes. Record how many fit around a point. Sketch either the shapes or the edges that meet at the point.

What do you notice?

Would the blue or red pattern blocks blossom?

Teaching notes: the white rhombus is chosen especially, since the wide angle doesn't blossom. The narrow angle will provoke a little discussion, 11 or 12 to blossom, because if they're tracing one block, the thickness of the drawn lines add up. The wide angle will draw responses of 1, 2, 3, 4, and 8. 4 will usually be two wide and two narrow angles (tessellating the rhombus) and the 8 is from filling the rest with the narrow angle. The red trapezoid and blue rhombus will sometimes have the students seeing blossoms with the narrow angle but not the wide.

After the connection of 360 degrees with filling all the way around, these blossoms can be used to deduce the measures of the pattern block angles. This is nice in conjunction with measuring practice with a protractor or angle ruler.



Filling Time

Use pattern blocks to measure the following clockwise angles. (Start at 12, and then measure in clockwise direction to the other edge.) Use all of the same unit for each angle. Measure each angle twice using different units, if possible.

Teaching Notes: You will see students make a lot of connections with congruent angles doing this. Also, there will probably develop an appreciation for the smaller angles as units. There is a natural tendency to measure the smallest direction, so that will bring up the clockwise/counter-clockwise thing, which is a nice connection with rotations, which will be a great way to teach angle to kinesthetic learners. The middle left angle brings up the idea of partial units, as it is not a whole number of pattern blocks for any of the shapes. The scientific standard is to measure to half of the smallest unit, so a good answer is 1 1/2 white rhombus (small angle) or 1/2 square. How many green triangles is a nice discussion.

The next activity I'm including the way I work it for preservice or inservice teachers. Easy to adapt for 5-12th grade students, though. The Word .doc version is here.

Telling Angles

Objective: TLW expand their understanding of angles, connect with the angles on a clock face, and use reasoning to find angle measures by comparison with known angles.

Schema Activation: What do you know about angles, measuring them and degrees? List your top 3 facts or bits of relevant knowledge.







Activity:
1) Forget the time, what angle is it? Record the angles made by the clock hands below. Add one sentence of justification for how you know.

2) Teacher question: why might I have sequenced the clocks the way I did?




3) Record the angles made by the clock hands below. Add one sentence of justification for how you know. Notice the hour hands are no longer pointing directly at a number.

4) 11:50, 1:10, 3:20, 7:40. For each time, draw in the hands precisely, and then determine the angle between the hands. Describe your process for each time. Start with the one you think would be easiest.

Reflection: What 3 ideas do you most want your students to understand about angles and angle measure?







Extensions: Challenge questions:
a. Is there a time for any angle?
b. Is the clock more likely to have an acute, right, or obtuse angle when you look at it?
c. How many degrees does the angle change in 1 minute? 5 minutes? 10 minutes? Does it depend on what time it is at the start?

Sorting and Math Teachers at Play 15

The new carnival is up.

Some of my favorite items: Kindergarten sequences (good for older kids, too) and the math skits idea.

I'm not crazy about the fractions or quadrilaterals activities, as I think both fall into some common pitfalls of the traditional thinking on these topics.

Sorting
As we started class this week, one activity we did in both my classes (geometry for K-8 and math student teacher assisting) is a Piece of Me. Students come up with one question about the class and one question about the teacher(s). They then get together in pairs/groups and decide on one of each to ask. They never ask the sort of thing you'd expect, or the sort of thing I used to share in my own introductions. It's an excellent quick pre-assessment of where they are with the class and what they care about. And, they get to discuss with other students what makes a good question. I got this activity from Dave Coffey, but I can't remember from where he got it.

The first activity I had the geometry students do was a game, that I've played with many levels of students.

In or Out
Materials: rope or chalk circle (For 20ish students a 40-50 ft rope will suffice. Of course, divide by 6 to estimate the diameter of the circle formed.)

Students stand outside of the circle. To demonstrate how a circle sorts, have a few different rules, and students who fit the rule step in. I usually use the old riddle: how many sides does a circle have? (Highlight for the answer: Two - inside and outside)

The game is a guess the rule game. Someone has a rule for the circle and people guess whether they are in or out. If they are correct, they can guess at the rule. If you guess the rule, you can make up the next rule.

I usually demonstrate the first three rules. One that is obvious, one a bit more subtle, and then a hard rule. The hardest rules are ones that change: in or out depends on body position, or on what the person before says. (Almost a guaranteed rule stumper is - you're out or in based on what the person before you guessed.) This can also be used to demonstrate the concept of function. (That's when I'm using the game in a college algebra class.)

Mostly the point is to get students thinking about characteristics. With upper elementary students or middle school students you could ask the questions I asked this time, after playing the game for awhile:

Activity: Work your way through some of the following questions.
1) What were some of the rules used?
2) Was there a rule that was easy to guess? Why?
3) Was there a rule that was difficult to guess? Why?
4) What is a rule that would divide our class into two groups of roughly the same size?
5) What are two rules that would divide our class into 4 groups of roughly the same size?
6) Why do two rules divide a population into 4 groups? Give an example.
Extension: Into how many groups would 5 rules divide a population? N Rules?

Reflection:
• Whole class: Share some of the ideas from (5). Try them out.
• Analysis: What are you doing when you’re sorting?
• Inference: What does this lesson have to do with geometry?