- What's My Angle? (Grades 1-3)
- Making an Astrolabe. (Grades 3 and up)
- Adding polygon angles. (Grades 5 and up)
- The Pythagorean Theorem. (Grades 5 and up)

For this lesson you will need an angle poster and a giant protractor. The angle poster is a picture of a circle cut into 360 slices with every 15 degrees labelled. The poster prints as nine 8 1/2 by 11 inch sheets of paper. You will need to trim the edges and tape the pieces of paper together. You can download it as a single postscript file, or as the following nine GIF files:

A one-page picture is myangle.gifTo make a giant protractor you will need a piece of cardboard approximately a foot (30 centimeters) square, two 4-foot-long pieces of 2-inch-wide lath, a 1/4-20 bolt at least 1 1/2 inches long, and a wing nut. On the cardboard draw a circle and mark angles at 15 degree intervals. Cut a 1/4 inch hole in the center of the circle. Drill a 1/4 inch hole close to the end of each piece of lath. Bolt the pieces of lath to the cardboard.

**Introductory remarks: explain the concept of angle.**Begin by asking students if they know what an angle is. If any of them do, ask them to explain in their own words. Define angle for them: the space between two lines that cross. Draw a picture on the chalkboard. Tell students that angles can be big or small.Ask students what the word angle sounds like. Hint: It's part of your body. Invariably, they will figure out that you're referring to the ankle. Explain that the word

*angle*comes from the word*ankle,*and that your foot makes an angle with your leg.**Introductory remarks: discuss the use of units in measurement.**Ask students if they know how to measure things. They will all say yes. Ask them how they would measure the length of something. Ask them what the marks on the ruler stand for. (If I see puzzled looks, I'll ask them if the numbers stand for pink elephants, or something equally silly.) Explain that we measure length with inches, centimeters, feet, or meters, and these are called*units*of length.Ask students how they measure weight. Ask what the numbers on the scale stand for. Explain that pounds or kilograms are units of weight.

Tell the students that we also have units for measuring angles, and that these units are called

*degrees.*Explain that these are not the same as temperature degrees, even though we use the same word. Tell them that if you take a circle and cut it into 360 slices, each of those slices is one degree.**Measure students' angles.**Show the students how to make an angle with your arms. Have each student come up and hold his or her arms out to make an angle. Use your giant protractor to measure each angle, and announce it to the class. Typically the students figure out that if they hold their arms down, they can make a large angle, and frequently the activity becomes a competition to see who can make the biggest one. It can be a little tricky finding the vertex of the angle because of the width of their shoulders.**Look for right angles.**Explain to the students what a right angle is. Have them all stand up and make right angles with their arms. Ask them to identify right angles in their classroom. If this is getting stale, ask them to point out angles that are NOT right angles. Keep going until each student has pointed out at least one angle.

A long time ago, people used astrolabes to measure the positions of stars in the sky. Sailors used this information to navigate.

- An astrolabe pattern, available as a postscript file or a GIF file.
- A piece of posterboard four inches square.
- A piece of string about 6 inches long.
- A small metal washer.
- A 4-inch long section of soda straw.
- Scissors.
- Cellophane tape.
- Glue.

Make sure the students are all familiar with the definition of an angle, and know how to measure angles.

- Cut out the astrolabe pattern on the heavy black outline.
- Glue the astrolabe pattern to the posterboard, being sure to match the corner of the pattern to one corner of the posterboard.
- Tie the washer to one end of the string.
- Punch a hole through the black dot.
- Poke the other end of the string through the hole. Adjust it so the washer hangs in front just below the curved edge of the pattern. Tape the end of the string to the back of the pattern.
- Tape the straw to the edge of the cardboard as indicated on the pattern.

Putting your eye at the end of the straw nearest the curved edge of the pattern, look through the straw until you can see the object whose elevation angle you wish to measure. The washer will swing so that the string is vertical. Still looking at the object through the straw, grab the washer and hold it in position. Now remove the astrolabe from your eye, holding the washer. Read the angle at which the string crosses the scale.

For students who understand the concept of similar triangles, you can show them how to use the astrolabe to measure the heights of trees, or your school flagpole. Measure the distance on the ground from the object to the place where you are standing. With your astrolabe, measure the angle between the ground and the line to the top of the tree. Draw a right triangle with that same angle. Measure the height and base of that triangle. The ratio of the height to the base is the same as the ratio of the height of the tree to the distance from the tree to your position. (Don't forget to correct for your height.)

For this activity, each student will need paper, scissors, a ruler, a pencil, and a protractor.

- Make sure the students know what an angle is, and how to measure an angle with a protractor.
- Have the students draw a triangle on the piece of paper. Make sure they draw the lines with a ruler, and that their triangles are at least as large as half a sheet of paper.
- Draw a triangle on the chalkboard or overhead, and demonstrate how to label the angles. The name of each angle should be written INSIDE the angle for this activity.
- Have the students label the angles in their triangles. I usually use A, B, and C, but I tell the students they can use any letters, numbers, friends' names, et cetera, their imagination suggests. Make sure all of the students write the names near the vertices of the angles, and not in the middle of the edges of the triangles.
- Show the students how to mark their triangles for cutting. Put a dot in the center of the triangle, and mark the center of each edge. Connect the center dot with each edge by drawing a slightly curved line. Have the students mark their triangles.
- Have the students cut out their triangles, and then cut the triangle into three pieces on the lines they've drawn.
- Show the students how to reassemble their triangles by placing the vertices of the angles they labelled on the same part, and aligning the edges of the three pieces so that they fit together without overlapping. They are visually adding the three angles. Ask them to measure the angle formed by the three angles together. You may need to help the students assemble the angles.
- As students report their measurements, (all 180 degrees, of course) point out that they each drew a different triangle. Explain that it is true in general that the three angles of a triangle add up to 180 degrees. (Those of you who are familiar with non-Euclidean geometries know that this is only true in a Euclidean space, and you may decide whether or not to discuss this with your students.)
- Make sure the students know what a quadrilateral is. Ask them what they think the angles of a quadrilateral will add up to. Have them each draw a quadrilateral, label the angles, cut the quadrilateral, and assemble the angles. (If a student draws a concave quadrilateral, make sure she labels the large angle correctly INSIDE the outline.)
- When the students have reported their result (360 degrees), you may want to point out that they could divide a quadrilateral into two triangles, and thus the sum of the angles is the sum of the angles of the two triangles.
- You may want to continue with pentagons.

Each student will need copies of the two Pythagorean theorem patterns available as a postscript file or as GIF files: the Loomis projection and the Perigal projection. I suggest photocopying each of the patterns on a different color paper, so the students don't get the pieces mixed up. Students will also need scissors, and, if the patterns are on white paper, two different crayons.

It will make the activity less confusing if you only pass out one pattern at a time. Start with the one in which the small square is whole, and the medium-sized square is cut somewhat like a pinwheel. This pattern is called the "Perigal cut." The other is the "Loomis cut."

- Make sure the students know the definition of a
*right angle.* - Define a right triangle, and the
*hypotenuse.* - Make sure the students know how to calculate the area of a square.
- Explain the
*Pythagorean theorem:*the square of the hypotenuse is equal to the sum of the squares of the sides. Tell the students this means that if they cut up the two smaller squares in their pattern, they can reassemble the pieces to fit exactly into the largest square. - If you have not copied the patterns onto different colors of paper, have the students color the squares in the first pattern with a crayon. All three squares should be the same color.
- Have the students cut out the small and medium squares. Have them cut the medium square into pieces along the printed lines.
- Tell the students they now have a puzzle. They should try to assemble the five pieces to fit in the large square.
- When some students have solved this puzzle, pass out the other pattern. Have them color the squares a different color from the first puzzle. Have them again cut out the small and medium square, and cut these squares on the printed lines. This puzzle is a little harder to put together.

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