When we first introduce angles, we can immediately draw the connection between angles and circles. This is a rich and interesting relationship, and because we are drawing circles and angles, and measuring them numerically at the same time, we can clearly see the relationship between the numbers and the drawings.
The following is taken from a presentation I gave in 2015, which is noted in more detail in my discussion of line designs.
1. The first regular shape visible to people before writing was a circle: the sun, perhaps?
2. There are 2 perspectives from which we see circles: from the outside, and from the center. Looking at the sun, we see a circle from the outside.
3. “If I stand in place and turn completely around, my body and view have turned a circle with me at the center. Now I am looking at the circle from the inside. The circle outside of me is the horizon: the circumference.”
At this point, I turn completely around. I then ask all the students to get up and do the same. When I ask, “Who can describe what we just did?,” some students will always reply, “Turned around in a circle.”
4. Now let’s see what happens when we only turn part of a circle. I put my right hand straight out in front of me, pointing at the chalkboard. Then I make a quarter-turn to the right. I ask the students to do the same.
5. Then I draw the figure to the right on the board. “What 3 things do we see in this drawing that show what we just did?”
6. Some students will say, “Two lines.” And what do the two lines make together? “An angle.” That’s one thing.
Usually after a minute, someone will say, “A slice of pizza.” That’s two.
Sometimes I have to prompt: “What forms the other edge of the slice of pizza?” It’s the arc…students may or may not have the word for it.
7. So right away we’ve established the connection between an angle, a part of a circle, and an arc.
8. To summarize:
a. We have 2 segments (rays) that intersect to form an angle. This is the textbook definition of an angle.
b. The rays cut out part of the circle (circumference) that the intersection is the center of. This part of the circle is an arc.
c. Together the angle and arc enclose a fraction of the circle.
d. Because angles cut out parts of circles, we use the same unit of measure to measure both angles and circles.
9. Units of Measure
Does anyone know what unit we use to measure angles/circles? In 4th grade, some students already know this.
How many degrees are in a full circle? Usually, someone will know this. Whether or not anyone knows, ask this: if we face north, turn to the right, and draw that angle, how many degrees will that angle have? Usually, some students know it’s a 90º angle, even if they don’t know what that means.
How many of these angles does it take to form a full circle?” Another way of asking this is, “What fraction of the circle is this?”
Start either with 360, or 90, and calculate the other measurement. Either: 4 × 90 = 360 or 360 ÷ 4 = 90
So students can derive the number of degrees in a full circle based on their acquaintance with a right, or 90º, angle.
10. What tool or instrument do we use to measure angles? Someone may know it’s a protractor.
11. Refer back to the right angle drawn earlier: “Let’s check and make sure it’s 90 degrees.”
12. Procedure for using the protractor:
- Place the baseline of the protractor along one side of the angle.
- Line up the center point of the protractor with the vertex of the angle.
- Make sure that the second side of the angle lies within the protractor.
- These 2 steps are difficult – many students don’t see how to align the protractor with the angle and will just line up the vertex without aligning the baseline, or will align the vertex and the baseline but will have the protractor “backwards” so the second side of the angle is outside the protractor. There’s not really any way to correct this from the board –you need to walk around and help students individually.
- Read the degree measure.
13. Leave off discussing which of the two numbers to read until you’ve measured the first 2 (right) angles. Then have students measure the 70 degree angle on page 2. Show it on the board. Now you point out that there are 2 possible numbers. “Which one is it?….Well, we know a right angle is 90…Is this angle narrower (smaller) or wider larger) than a right angle?….Smaller, so you read the lower number…”
14. Make the connection between the sums of each combination of angles on the protractor. Each sum is 180…notice that the degree measure of the baseline on the protractor is also 180. What fraction of a circle is this?
15. Here is a protractor image that you can paste into a Smartboard or text file – it’s transparent, so angles can be seen through it. It’s similar to most protractors in classroom use today.
16. And here’s an introductory activity for students in .pdf.