Living Math with Angles

Taking a break from our intense multiplication study, we’ve moved on to a new topic — angles.  With living math, I do use our Singapore textbook. I browse through it to find another topic to study. (grin) Actually, I do have Sprite do a workbook exercise now and then.

Angles was one such topic that we hadn’t touched on this year, and it seemed easy enough to learn in a hands-on approach. (This is the face she wanted on the blog.)

Above and below you can see our most important tools:

1. a home made protractor in 360°
2. an angle maker
3. a store bought protractor

I find that in planning living math activities, I often have to reverse engineer the workbook exercises. I look at the problems and ask myself, “How can I convert this skill or concept into a hands-on or real problem solving activity?” And that’s where these activities came from.

First of all, I knew that the entire concept of degrees was important. Sprite needs to understand what an angle is and that it can be big or little and that the size is measured in degrees.

[The technical terms such as acute and obtuse are not vital at this point, so I didn’t use them.  Vocabulary can come later after the concepts are cemented. However, we had already learned about right angles (90°) with Pythagoras, so it was easy to talk about smaller than and larger than a right angle.]

So I had Sprite make a 360° protractor by simply marking degrees on a large circle. (Got to sneak in some x9 facts too — 90°, 180°, 270°, and 360°.)

The angle maker is simply two strips of cardstock connected at one end with a paper fastener. Using these two tools together, we arranged the angle maker in various positions and measured the resulting angle, approximately, with our protractor. We did this for a couple of lessons so that Sprite would get a feel for the sizes of angles — a 20° angle is tiny and a 280° angle is big. A 100 ° angle is a tad larger than a right angle, and so on.

Then come the bits of paper. (It seems we can’t do a math activity without loads of cut up paper.) I used some scrap paper to make circles. I made sure to mark the center with a pen so Sprite could cut out angled sections. Some circles were cut in two pieces, others in three.

Then she measured. First I asked her to estimate visually the degree of angle. Her skill at estimating got better and better. Then I had her use the homemade protractor and then the official one.

I had her label each angle with its measurement. And then we verified the rule that each circle has 360° by adding up the total of the angles. It worked each time!  Of course. But letting a child discover rules rather than dictating them is a halmark of living math.

It all made its way onto a notebooking page.

Next approach — triangles.

I gave Sprite some triangles and asked her what she thought we’d find when we measured the angles and totaled them. She wasn’t sure, but she felt certain it wouldn’t be 360° like the circles.

Oops… she ended up with several triangles whose angles totaled over 180°. She somehow sensed that 181° and 182° must be wrong and that the triangles that were measured at 180° must be correct. It actually was easy enough to go back and remeasure and find her small errors. It ended up that some of our “right angles” were not actually 90°.

We corrected the measurements on the backsides, and put a few sample triangles onto a notebooking page.

Just to be clear, these angle lessons were spread over about a two week period. Interspersed in with these hands-on activities were some workbook exercises and at the end I used this BBC worksheet as an overall wrap up.