An angle on circles

Teaching Pre K-8, Jan 1995 by Hatchett, Mary Alice

Writing is an invaluable teaching strategy for anyone who wishes to embrace the NCTM communication standard. In writing, students organize their thoughts and reflect on how to communicate them.

Math experiences can thus be "captured" for later recall in making logical connections between strands of mathematics, connections to other content areas, or as a springboard to content extensions.

With this in mind, let me share an activity, "An Angle on Circles," that includes writing. The objectives for this activity (I've used the activity in grades 3-10) involve concepts in geometry.

Materials. For each student: one 3-4" diameter circle cut from plain writing paper; one set of angle cards (Figure 1); writing paper ; and two 3 x 4" mylar mirrors, hinged with tape on the 3" side. (Figure 1 omitted) The mirrors should be able to open and close.

For the teacher: two or three circles cut from waxed paper; a large circle drawn on a transparency or chart tablet; transparency angle cards; one set of hinged mirrors; overhead projector; and timer.

The angle card needs to be on a flat surface. Place the mirror on the angle card so that the hinge is on the vertex of the angle and each mirror is on one of the legs of the angle. You should be able to see reflected images of the stickers.

Distribute one paper circle per student. Ask the students to examine this figure by running a finger around the edge, looking at it from several sides and silently reflecting on what they know about the figure. Ask, the name of the figure? And ask if everyone agrees with the name before the next step into free writing.

Part One: Free writing. Inform the students that they are going to engage in "free writing." At the signal, on the paper circle they were given earlier, they are going to write about circles continuously for one minute. After one minute, have them orally share their writing with their neighbor. Then ask student pairs to select two or three ideas about circles that they think probably everyone has and two or three ideas that they think are unique to their observations.

As student pairs orally share an idea they think everyone will have, record it on the large circle transparency or class chart. If their idea has already been used, go to their second or third choice. Discuss any ideas that may need explanation. Lead the discussion to the fact that a circle has 360deg.

Place a waxed paper circle on the overhead. Have students follow your movements as you run your finger around the edge of the circle. Ask students for the name and definition of this concept. Fold the circle in half and run your finger along the fold; then fold the semi-circle in half and point to the center. Each time, ask for the name and definition of the concept.

Open the circle. Ask how the two folded lines are related to each other. (perpendicular) Draw a curve from one radius to the adjacent radius to the left. What size angle is this? (90deg) How many 90deg angles are formed on this circle? (four) Draw a curve from one radius to the adjacent radius two to the left. What size angle is this? (180deg) How many degrees around the center point of a circle? (360deg)

Part Two: Counting images. Distribute the angle cards and hinged mirrors. Ask the students to select Card A. When they have positioned the mirrors on the card, ask them to determine how many images, including the original, they see? They then select any two other cards and repeat the procedure.

Next, have the students prepare, with their partners, a written statement about what they've observed and what conclusions they can draw. (A wider angle shows fewer images.) Ask if anyone has seen something different (Look for the converse of the first statement: smaller angles show more images.)

Ask the students to pick another card they have not examined. See if they can predict whether this card will show more images, fewer images, or the same number of images as card A. (Cards B and C will have more images; cards D, E and F will have fewer.)

Have students draw and label two overlapping circles like the ones shown here. Ask the students to place each of the cards in the appropriate positions in the diagram. After they've done this, ask what cards went in the circle on the left? (Card F) In the circle on the right? (Cards A, B, C and D) And where does card E go? Why? (Because it has exactly four images.)

Can the students determine how many degrees in the angle for card E? (90) Tell them angles with 90deg are called right angles; angles with less than 90deg are called acute angles. Now have them find which cards have acute angles? (A, B, C and D) Angles with more than 90deg are called obtuse angles. (F)

Working the angles. Suppose angles could read, write and talk. You'd like to hire 12 angles to work in your polygon factory. Angles will learn of this employment opportunity from reading your classified ad. You'll have to hire these angles without meeting them, but you don't want to hire all the same size angle. Because of angle discrimination laws, you cannot use angle names like acute, obtuse or right. Write your ad in 50 words or less.