## Thursday, September 17, 2009

### 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?

1. I'm just now reading Seymour Papert's book, Mindstorms (written in 1980), for the first time ever. He talks about how using LOGO (a programming language) and Turtle Geometry helps students understand angles. When they 'play turtle', they turn the same amount they want to make the turtle on the screen turn. He talks about the knowledge becoming body knowledge, and going much deeper than the usual school math.

Here's another benefit Papert describes: Programming something they're interested in helps them see mistakes as time for debugging, instead of something to feel bad about and run from.

The answer to your first challenge question depends on whether the clock is truly analog or, like most clocks these days, a digital (quartz) clock pretending to be analog. Do the hands truly move smoothly, or do they jump, one second or minute at a time?

2. I love that book. I've used LOGO quite a bit, and there was some good research that supported its effect with kids learning. StarLogo, http://education.mit.edu/starlogo/, is a nice free implementation of it, and there's a pretty fair sized group that works on lessons and applications.

Nowadays I just have the kids do the motions and turning. Easy assessment!

Re: challenge problem, feel free to assume a truly analog - and rare - clock!