• Evaluate group presentations on the accuracy of content, quality of communication within the group, and fair distribution of tasks.

• Journal responses can reveal more subjective observations in addition to factual or quantitative reporting.

• Lesson 2 Exit Ticket (M-G-2-2_Lesson 2 Exit Ticket.doc) requires students to document specific facts learned about polygons.

• Evaluation of the Bridge Activity should include specific comparisons between predictions and outcomes.

This activity provides an outline for students to learn and practice some basic attributes of polygons. In finding different polygons, students will identify polygons by their sides and classify them as irregular or regular polygons. They will use basic formulas to determine perimeter, area, diagonals, and interior triangles.

Distribute a photo of your face or one from a magazine to each student. Instruct students to find and draw as many polygons on the face as they can, using their straightedge. You may need to model one or two before students understand the task. Ask them to try to find polygons with different numbers of sides. [IS.5 - All Students] You can show them the example provided (M-G-2-2_Polygon Face.doc).

After 10 minutes, post one picture on the wall and ask whether one of the students has a polygon s/he would like to share with the class. Ask him/her to approach the wall and draw the polygon on the face. This can be done randomly or in order according to the number of sides (triangle, quadrilateral, pentagon, hexagon, heptagon, octagon, etc.) Ask students to come up and add their own polygon to the picture until each student has had an opportunity to participate.

As a class, ask students to identify polygons with different sides, beginning with triangles or tri-gons and ending with decagons. Group students and assign each group a different polygon. Have the group cut their polygon out of the larger photo and tape it to a blank sheet of paper.

Ask the group to identify whether the polygon is a convex or concave polygon. [IS.6 - Struggling Learners] Have the group draw the diagonals of the polygon. (Note: if the polygon is concave, some of the diagonals will be outside the shape.) Students can determine the number of diagonals for their polygon through this formula:

number of distinct diagonals =

(where n = number of sides or vertices) [IS.7 - Struggling Learners]

Have students use their rules to measure the sides of their polygons and find the perimeters of the polygons.

Have a few students present their findings to the rest of the class in detail and discuss whether these findings are correct. Discuss the differences between the different polygons that students presented.

Hand out shape sheet sets at random; each student should get only one set (either three- and four-sided, five- and six-sided, seven- and eight-sided, or nine- and ten-sided) (M-G-2-2_Three- and Four-Sided Regular Polygons.doc, M-G-2-2_Five- and Six-Sided Regular Polygons.doc, M-G-2-2_Seven- and Eight-Sided Regular Polygons.doc, and M-G-2-2_Nine- and Ten-Sided Regular Polygons.doc). Ask, “What do you notice about these shapes as compared to the shapes you were looking at before?” (These are all regular shapes. The previous shapes should have been irregular.)

Each shape is labeled with its apothem and the length of one side. Have students find the perimeter of that polygon using basic measurements and/or the formula (write on board):

Perimeter = ns

(where n = number of sides and s = length of each side) [IS.8 - Struggling Learners]

Ask students to divide their polygons into triangles, [IS.9 - Struggling Learners] connecting each vertex back to one single vertex. Point out that the number of triangles in a polygon can be found through the formula (write on board):

number of triangles = n − 2

(where n = the number of sides) [IS.10 - Struggling Learners]

Using the apothem and the length of one side, students can find the area of the regular polygon using this formula (write on board):

Area =

(where A = apothem and P = perimeter) [IS.11 - Struggling Learners]

Draw the regular hexagon and its apothem on the board and ask students where this area formula comes from. If they need prompting, next draw the isosceles triangle of which the apothem is the altitude. Students should be able to see the six isosceles (also equilateral for the hexagon) partitions that make up the total area. The perimeter of the regular polygon is the sum of all the bases of the component triangles. Since the area of each triangle is half the base multiplied by the height (apothem), the area of any regular polygon will be the sum of the bases (perimeter) times apothem (height) divided by half the number of sides.

Group students by the polygon they are using. Have students check their work with their peers to see whether they all came up with the same solution.

Have students write about what they learned regarding polygons. [IS.12 - All Students] The journal page should be turned in so that you can check on the progress of each student.

Activity 2: Angles of Polygons

Students will be introduced to exterior and interior angle sums of polygons.

“In this activity, we will investigate the size of interior angles and exterior angles of different polygons and the sums of those angle measures. By knowing the sum of the interior or exterior angle measures, we can determine different angle measures of various polygons.”

Before the class period in which you use this activity, use masking tape to create a large pentagon (similar to the one below) with exterior angles on the floor.

For this activity, have a student start at one intersection and walk along the tape lines with the goal of returning to the exact same spot (the lines could be described as city streets or paths). Discuss with the class what this means:

“How much did s/he turn in total?” (540 degrees)

“What degree turn is it when you turn all the way around?” (360 degrees)

“Which angles on the taped figure represent each turn?” (exterior angles)

“What does that tell us about the sum of the exterior angles?” (they equal 360 degrees)

“If the pentagon was a different shape (sides are different lengths, interior angles are different sizes), would that change the sum of the exterior angles?” (no)

“If it had a different number of sides, nine sides for example, would the sum of the exterior angles change?” (No, you would still turn all the way around.)

Draw the pentagon on the board and ask students to draw it on their paper. “We know that the sum of the exterior angles of a pentagon equals 360 degrees, but what about the interior angles?” Have each student connect two non-adjacent vertices of the pentagon and then connect the remaining two. [IS.13 - Struggling Learners]

“What is the sum of the interior angles of a triangle?” (180 degrees)

“If I put two triangles together,” (place 2 triangles together, leg to leg), “what shape do I have?” (quadrilateral)

“What is the sum of the interior angles?” (360 degrees)

Have students repeat the process with three triangles to make a pentagon. Ask students to create a table of results. The table should include the number of sides (three to ten), sum of interior angles, measure of each interior angle of equiangular polygon, measure of each exterior angle of equiangular polygon. Note: Begin by filling in the number of sides and sum of interior angles for the triangle–pentagon. Assist students as necessary in filling out the table. Samples have been provided below.

Sides	3	4	5	6	7	8	9	10
Sum of interior angles
Interior angle measure if equiangular
Exterior angle measure if equiangular

Sides	3	4	5	6	7	8	9	10
Sum of interior	180	360	540	720	900	1080	1260	1440
Interior angle measure if equiangular	60	90	108	120	128	135	140	144
Exterior angle measure if equiangular	120	90	72	60	51	45	40	36

“Do you notice a pattern in the sum of the interior angles?” (It goes up by 180 degrees each time.)

“What does 180 degrees have to do with each shape?” (It is the sum of the angles of the triangle and we add a triangle each time we need one more side.)

“If I had a regular pentagon, how could I find the measure of each angle?” (Divide the sum of the interior angles by 5. Remind students to put this answer into their table.)

“What if I knew that two of the angles of a pentagon were each 75 degrees and that the other three angles were equal to each other? How would I find the measure of one of the other angles?” (possible solution: )

Have students work independently or in groups to complete the table. Allow them to use more triangles if they would like. Walk around the room, helping students who need additional practice. Verify that their results are correct.

“What if I needed to find the sum of the interior or exterior angles of a polygon with 25 sides?”

“What if I also needed to find the measure of each interior or each exterior angle of an equiangular polygon with 25 sides?”

“Would I want to make a chart, or is there a better way?” (Find a rule.)

Have students work in groups to develop rules for:

1. The sum of the interior angles of an n-gon.

2. The measure of each interior angle of an n-gon.

3. The measure of each exterior angle of an n-gon. [IS.14 - All Students]
   

Have a few groups present their rules (and their process for developing the rule) to the class. Use the shape sheets to illustrate this.

Examples of each might be:

The next activity is optional but should be done if the time allows and if the materials for the activity are present. If you choose to not do the next activity, distribute the Lesson 2 Exit Ticket (see M-G-2-2_Lesson 2 Exit Ticket in the Resources folder).

Activity 3: Bridges

In this activity, students will study the practical application of polygons.

The properties of squares, triangles, pentagons, and hexagons will be reviewed, and students will learn how to prove which shape is the most stable when put under stress.

Begin by asking students what makes a good bridge. Good answers include: solid materials, quality construction, use of strong design.

Ask what some common structures of bridges are. Show examples of famous bridges.

Ask students what shapes they think will make the strongest bridge: triangle, square, or hexagon.

Divide students into six groups so that one group at a time may work on each pattern. Distribute the activity’s materials to each group, and hand each group the bridge diagrams (M-G-2-2_Bridge Diagrams.doc). Each group will be in charge of building a bridge based on one of the patterns. Assign one pattern to each group. Instruct students to build the bridges according to the diagram using the balsa sticks and glue.

Allow the bridges to dry overnight.

Before the next class, place cardboard tops on top of each bridge structure, poke holes into the sides of a small paper cup, and string twine through the holes. The other end of the string should be tied around one of the bridge tops (cardboard piece) so that the cup is hanging down the center of the bridge structure. This process will need to be done for each bridge as each one’s solidity is tested. To test each bridge structure, have each bridge set up over a gap between two desks with the paper cup hanging down the middle. Add one battery (weight) at a time to the cup until the bridge collapses. Record the weight at which each bridge collapses.

“The bridge with the most strength should be the bridge using the isosceles triangle with the one point of the triangle at the center of the top of the bridge.”

After watching the destruction of the bridges, have each group discuss the positives and negatives of their assigned bridge structure. Have each group come up with one solution to make their bridge stronger.

Have all of the groups present their positives, negatives, and solutions to the rest of the class. Also discuss the order in which the bridges failed (which collapses under the least amount of weight and which collapses under the greatest amount of weight) and the possible or probable reasons for this order. [IS.15 - All Students] Discuss the principles of the shapes that made the bridges stronger or weaker. Discuss the relationship of the length of the sides of a triangle to the angles of the triangle—how when one length changes, the angles must also change. As a result of this principle, the rigidity of the triangle structure makes it the strongest and therefore the best load-bearing form. Discuss other ideas of what could have gone wrong, construction errors, weakness of materials, etc.

Once the class discussion concludes, hand out the Lesson 2 Exit Ticket (M-G-2-2_Lesson 2 Exit Ticket.doc). (Note: This is the same Exit Ticket the class would do if you choose not to do Activity 3.)

Extension:

• Students can investigate patterns in interior/exterior angle sums when the polygon is not concave.

• Small Groups: Groups can be made larger in order to make things easier for students who might need additional practice or smaller to give proficient students the opportunity to have increased participation.

• Students who might need additional learning opportunities can be given the simpler shapes (like squares and triangles) in each of the activities as a way for them to understand the principles with fewer steps. While students are getting into groups to discuss their findings, you can check with students who might need more practice to make sure they understand everything.