Math Hombre

Quick idea for a math game on angles, hopefully I get to try it this week.

Materials: deck of angle vocabulary cards, blank paper, ruler, pens, protractor.

Set up: (make if necessary and) shuffle angle vocabulary cards.

Draw phase: teams take turns

• add a point, and 
• connect to one, two or three other points from your new point.
• each team adds right angle mark or congruent length if that's their intent
• both teams make 5 points.

An example:
Play phase: on your team's turn

• roll a die (that's this turn's points)
• flip a card. Claim an angle or a set of angles that fit the condition. You can only claim unused angles.
• score that many points for each angle you claimed that fits the condition.
• check: if you can't find one or find one mistakenly, the other team can catch you for 2 points per angle.
• game is to 20 points, run out of cards, or all angles are claimed.

 Example: red scored an acute, a right and a pair of vertical angles. Green scored a pair of congruent angles and a set of supplementary angles.

Design reflection:
Could use a context, but the only thing that comes to mind is shooting metaphors. Maybe bird watching? You know how the kids love bird watching!

Foxtrot, of course, has angle games covered. They even get triggy with it.

Lots of nice bits here, I hope. Constructing the board, using notation, eventually even making the cards. Some classic interaction (catch the opponents out in a mistake), but could be more. The thing I like the best is how the game will change in between playing. What angles were you unable to find, what combinations can you make, etc.

Possible starting card set:

• an acute angle
• a right angle
• an obtuse angle
• a pair of congruent angles
• a set of congruent angles that are not right angles
• a pair of complementary angles
• a pair of adjacent angles
• a pair of cute adjacent angles
• a pair of acute obtuse angles
• a pair of vertical angles 
• a set of angles that add to 180 degrees
• a set of angles that 
• a pair of corresponding angles
• a set of interior angles
• a pair of congruent exterior angles
• a pair of angles that add to 180 degrees
• a straight angle

What else would you add? I'd want a set of cards a playing field to start, then introduce the making aspects when the students know how to play. Warning: only roughing out playtesting so far.

What do you think?

I've never seen a spiral board... wow!

Since Vi Hart released her Christmas time spiral celebration, I've been digging spirals again. I made a GeoGebra sketch to go with her video (link includes a link to her video), that I quite like. My tumblog is where I post one-off math and reblog other Tumblr math. So when I got the word from Mr. Schiller that this gameday the "Topic will be polygons/angles/rotational symmetry," it didn't take me too long to get to the idea of a spiral game.  The way it worked out, though, has me wondering: how good does an educational game have to be?

Click for full size

There are lots of things to recommend it: kids think spirals are cool, it makes a nice race track, it allows you to see circle connections to angle, and that angles have the same measure whether small or big in size. I like race games for practicing with quantities, because it gives some repeated experience with a variety of the quantity, and gives you a reason to talk about the quantities. The GeoGebra I used to make the Archimedean Spiral (as opposed to Vi's logarithmic spirals) is posted on Tumblr, too. Then I just used GeoGebra's export as image to get the track into Word.

The problem with race games is that many of them devolve into chutes and ladders (American; snakes and ladders elsewhere).  This one definitely did. I thought a one die game might be easiest, and after some practice settled on moving 15º times the die roll. It included right angles and gave some nice opportunity for mental multiplication. I justified the simplicity of the game to myself by adding a game-design objective. The winner adds a rule; that rule has to help with the catch-up characteristic of the game.


In addition - since the game was simple - I wanted to have another option. I brought some triangle grid paper and an eightfold diagram (links to Google docs) to support the students in making an art project with rotational symmetry.  As I told the students, math art is as close to my heart as math games. I explained how to color a piece and then imagine it turning, or on the 8-fold to color in 1, 2, or 4 wedges and then copy it.

1. Goal(s) - good concrete objectives.
2. Structure - the spiral really fit the objectives well.My main question here is if the board should have angle measures on it. (Definitely, if polar coordinates are the objective.)
3. Strategy - no real strategy. Real room for improvement here.
4. Interaction - with no choices the interaction is limited to the racing.  It's a hook, but no way to effect your opponent.
5. Surprise - not really relevant to this game.
6. Catch-Up - this game has it, both through randomization of the die and the board structure; but it's of the candyland/chutes and ladders variety.
   
7. Inertia - main reason for shortening the game. Overstayed it's welcomed. I think race games, in particular, probably need to be mindful of pace.
8. Rules - clean, simple. The add-a-rule rule was a big hit. I'll be using that again.
9. Context: Fun-Flavor-Hook. The spiral is a start to this, but some context for the spiral might have helped here. With all the spirals in nature, it shouldn't be too hard to add something. Maybe birds flying to the eye of a tornado? Hurricane?

The warning sign for this game is being weak in the green characteristics. Mr. Schiller and I were excited about it because of the strength in the yellow areas. If I was thinking about a commercial game,  I think I'd make a deck of cards for movement, that would give more strategy and interaction. But that's a lot of printing for a one-off classroom game. Maybe you could simulate that with multiple dice? Make the rules a bit more complex, but worth it for gains in the green. Maybe roll three dice, pick one to use that you'll reroll next time. Trade for an opponent's die with one of yours that is higher.  Worth a try! It will even increase angle use.

The modified game is up at Google docs.


Snakes & Ladders Image: Smabs Sputzer @ Flickr

Trig Problem 2

For my preservice high school teachers' "final" (really a last Standards Based Grading opportunity), there were two problems that while similar in many respects were quite different in results. All of the problems were listed by one standard, but typically could be used for other standards. It's the student's responsibility to describe what standards they are demonstrating, though I will help if it demonstrates something well that they need.

Trig Problem 2. (Standard: Law of Sines, Law of Cosines and applications)

Figure out some of the missing information in the diagram.



The pictures were made in GeoGebra, which I highly recommend for mathematical image creation, as well as more active uses.




Geometry Problem 1. (Standard Lines: parallel, perpendicular, properties of angles)

Find more angles.

Geometry Problem 1

Similarities: visual, finding connections, geometry, students have previously done and been assessed on similar problems.

Have to love easy-to-draw memes.

Differences:  throughout the semester students saw trigonometry as something difficult, and had much less confidence on them.  Students were very successful with the angles problem, able to find all the angles, and be able to justify their results. Why vertical angles are congruent, why there are 180º in a triangle, etc. On the "trig" they quickly resorted to visual inference (like the angles at A were all 60º), supposition, and ignored contradictions (such as finding that the length of CD was less than 6 units), and did almost no extension to other standards from circle geometry.

It was fascinating to read their work, and I wish we had more class time to look at the results. It felt like direct confirmation of the Van Hiele levels, and convicted me that as much time as we devoted to trigonometry, I need to find more ways to increase their experience.  While I thought the circle diagram was more subtle, I didn't realize the great difference in how students would see it. Only one student realized CD must be 6 units, which is the entry to me for many of the possible values that can be determined.

Wrote this problem for my spring final and fell a little too in love with it.  Has some fun algebra behind it.

Jane glanced up at the clock and noticed that when the second hand was on the 12, the three clock hands divided up the clock into a right, acute and obtuse angle.  What time was it:

• 3:30, 
• 5: 43, 
• 1:22, 
• 9:15, 
• more than one possibility or 
• none of those  

Remember the hour hand moves during the hour.

When grading, many students ignored the idea of the hour hand moving in between, so I evaluated based on their assumptions.  In general on the test, I was trying to create the possibility of seeing some problem solving, where they could demonstrate understanding of ideas without necessarily having to get a right answer.  It was partially successful.

As I was trying to write the problem, I posed myself this question: 

If it is y o'clock and x minutes past the hour, what is the angle formed by the clock hands?

If you're considering either, I'd love to hear what you think in the comments.  How do you evaluate the first?  In the second, would you expect the equation to be linear?  Why?

Cartoon from xkcd, of course.



Some of the exam problems were pretty open-ended, like:

1.    Find an L-shaped figure with an area of 84 sq.cm and a perimeter of 44 cm.  Is there more than one?
2.    What kind of triangles can be made by connecting vertices on a regular octagon?  Specify the side-angle type.  Did you find all of the types? 

And some were more closed, but hopefully with multiple ways to do them.

5.    Sort the quadrilaterals into two overlapping Venn diagram circles: one for rotational symmetry, one for reflectional symmetry.  Quadrilaterals that don’t fit either should go outside.
6.    A Hershey’s chocolate bar is 43 g.  A kiss is 4.56g.  You remember that 1 pound is 454g and 1 pound is 16 oz.  How many ounces is a Hershey bar?  A Hershey kiss?  How many kisses in a bar?  (Make a joke if you want.)

Nobody made a joke.  How many kisses in a bar?  Come on!

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.

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.