Jimmy Clemson

Dr. Adu-Gyamfi

MATE 4001

Sum of Interior Angle of a Polygon

Statement of Mathematical Investigation: In this investigation the students will explore the concept of an interior angle of a polygon and figure out the sum of the interior angles of different sides of polygons and the angle measures of their regular polygons.

An auditorium manager is building an auditorium and needs to know what angle brackets he needs for the junctions of the wall. His building is going to have 72 sides and it will be regular. What angle brackets does he need?

Exploration: Start by constructing a triangle and measuring the interior angles and their sum.

Is this true of all triangles? Test it out.

A regular polygon is a polygon where all the angle measure and side lengths are the same.

Knowing that a regular triangle has to have equal angles, what angle does it have to be in order to make it a regular polygon? 60°. How did you come up with that.

Make a chart that shows the number of sides and sum of the interior angles and the angle of a side of the regular polygon for that many sides.

Now move up to quadrilaterals.

mDAB + mABC + mDCB + mADC = 360.00°

mADC = 44.07°

mDCB = 127.52°

mABC = 45.85°

mDAB = 142.57°

A

B

CD

Is this always the case? Test it out.

mDAB + mABC + mDCB + mADC = 279.14°

mADC = 4.46°

mDCB = 127.52°

mABC = 7.60°

mDAB = 139.57°

A

B

CD

Be Careful! Here it would seem that our quadrilateral would not fit our pattern but look what happened to angle A. It switched from an interior angle to an exterior angle. If you made it add the interior angle it would still add up to 360°.

What angle measure for each side would make a quadrilateral a regular quadrilateral? Use what you learned about finding a regular triangle to help find the angle for a regular quadrilateral.

Record your results in your chart and move onto a pentagon.

mCDE + mEAB + mDEA + mABC + mBCD = 540.00°

mBCD = 134.42°

mABC = 127.09°

mEAB = 83.96°

mDEA = 121.71°

mCDE = 72.82°

C

B

A

E

D

The sum of the interior angles of a pentagon adds up to 540°. Does this work for all pentagons?

What angle measure for each angle would make a pentagon a regular pentagon?

Record your results in the table. Move onto the hexagon.

mFAB + mABC + mBCD + mCDE + mDEF + mEFA = 720.00°

mEFA = 109.23°

mDEF = 132.95°

mCDE = 139.34°

mBCD = 109.54°

mABC = 133.24°

mFAB = 95.69°

A

B

C

D

EF

The sum of the interior angles of a hexagon are 720°. Find the individual angle measure for a regular hexagon. Record this in your chart.

Make a heptagon. What do you notice about the sum of the interior angles of a heptagon?

mGAB + mABC + mCDE + mBCD + mDEF + mGFE + mAGF = 900.00°

mAGF = 105.91°

mGFE = 148.26°

mDEF = 132.95°

mCDE = 139.34°

mBCD = 109.54°

mABC = 133.24°

mGAB = 130.75°

A

B

C

D

EF

G

The sum of the interior angles of a heptagon is 900°. Find the interior angle measure of a heptagon.

The chart should look something like this:

Number of sides Sum of the Measure of Interior Angles

Angle measure of Regular Polygon

3 180° 60°4 360° 90°5 540° 108°6 720° 120°7 900° 128.57°

What is the formula for coming up with the sum of the measure of the interior angles?

(n-2)*180

An auditorium manager is building an auditorium and needs to know what angle brackets he needs for the junctions of the wall. His building is going to have 72 sides and it will be regular. What angle brackets does he need?

Angle=((n−2 )∗180)

n

Angle=(72−2 )∗180

72

Angle=175°

IDP TPACK TEMPLATE (INSTRUCTIONAL DESIGN PROJECT TEMPLATE)

NAME: ______Jimmy Clemson______ DATE:_____11/17/13___________

Content. Describe: content here. (COMMON CORE STANDARDS)

CCSS.Math.Content.HSG-MG.A.3 Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios).★

Describe: Standards of mathematical Practice

CCSS.Math.Practice.MP1 Make sense of problems and persevere in solving them.

CCSS.Math.Practice.MP3 Construct viable arguments and critique the reasoning of others.

CCSS.Math.Practice.MP4 Model with mathematics.

CCSS.Math.Practice.MP5 Use appropriate tools strategically.

CCSS.Math.Practice.MP8 Look for and express regularity in repeated reasoning.

Pedagogy. Pedagogy includes both what the teacher does and what the student does. It includes where, what, and how learning takes place. It is about what works best for a particular content with the needs of the learner.

3. Describe instructional strategy (method) appropriate for the content, the learning environment, and students. This is what the teacher will plan and implement.

This is a guided exploratory lesson. I will pose the question to them at the beginning of class. Since they most likely won’t have an understanding of regular polygons, this will lead to a discussion of regular polygons and their properties.

I will guide the students through the first shape, a triangle. Have them draw a triangle of sketchpad and measure the interior angles of it. Once they figure out that all triangles add up to 180 we can discuss a regular triangle.

2. Describe what learner will be able to do, say, write, calculate, or solve as the learning objective. This is what the student does.

From there the student works through the rest of the shapes and figure out the sum of their interior angles and the angle measure of the regular polygon.

The students will create a way to find the sum of the interior angles based on the number of sides and be able to solve the initial problem.

3. Describe how creative thinking—or, critical thinking, --or innovative problem solving is reflected in the content.

The students are not given the pattern for finding the sum of the interior angles, the students have to recognize the pattern and figure that out for themselves.

Technology.

1. Describe the technology

GSP is a microworld geometry software. GSP allows students to manipulate all kinds of geometry and algebra and explore to gain an understanding of mathematical concepts.

2. Describe how the technology enhances the lesson, transforms content, and/or supports pedagogy.

The technology allows the students to efficiently find the sum of the interior angles and see if it hold true for all polygons.

Without the use of sketchpad this lesson would be too long for one period and much more inaccurate.

3. Describe how the technology affects student’s thinking processes.

The technology will let the students at first see that triangles are always 180, but when they get to quadrilaterals they will be tricked into thinking that it doesn’t work for all quadrilaterals by the way sketchpad will measure the angles but once that point is clarified then they will see that it works for all.

Reflect—how did the lesson activity fit the content? How did the technology enhance both the content and the lesson activity?

Reflection

The lesson fit the content area fairly well because the students will have to use geometric principles to find the answer to a word problem.

The technology lets the students efficiently tally up the totals of the interior angles. Without the technology the students would have a hard time accurately computing the sum of the interior angles and therefore would have a hard time seeing the pattern that arises if they do not have the correct sums.

Lesson Plan Template MATE 4001 (2013)

Title: Polygons

Subject Area: Math 2

Grade Level: 9-11

Concept/Topic to teach: Sum of the Interior Angles of a Polygon

Learning Objectives:

Content objectives (students will be able to……….) Students should be able to figure out the sum of the interior angles of a polygon based on the number of sides the polygon has and the angle measures of the regular polygon of that many sides.

Essential Question

Given a polygon with a certain amount of sides can you figure out the angle measures of its regular polygon?

Standards addressed:

Common Core State Mathematics Standards:o CCSS.Math.Content.HSG-MG.A.3 Apply geometric methods to solve design problems

(e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios).★

Common Core State Mathematical Practice Standards: CCSS.Math.Practice.MP1  Make sense of problems and persevere in solving them.

CCSS.Math.Practice.MP3  Construct viable arguments and critique the reasoning of others.

CCSS.Math.Practice.MP4  Model with mathematics.

CCSS.Math.Practice.MP5  Use appropriate tools strategically.

CCSS.Math.Practice.MP8  Look for and express regularity in repeated reasoning.

o

Technology Standards: Copy and paste from NCDPIHS.TT.1.1 Use appropriate technology tools and other resources to access information

(multi-database search engines, online primary resources, virtual interviews with content experts).

HS.TT.1.2 Use appropriate technology tools and other resources to organize information (e.g. online note-taking tools, collaborative wikis).

Required Materials:

Computers

Sketchpad

Paper/pencil

Notes to the reader:

Write anything here that you think the reader needs to know before implementing this lesson. For example, if you are assuming that the students have already experienced a particular topic this is the place to note that.

Time: Assume 90 minutes ***

Time Teacher Actions Student Engagement

Before

20

Introduce problem, as students won’t know what a regular polygon is talk about a regular polygon and interior angles.

Guide students through triangle exploration and summarize what the students learn. Assist in construction if needed. Introduce chart for students to put down observations and results.

“You are to explore quadrilaterals, pentagons, hexagons, and heptagons. Record your results in the chart and write any observations you have underneath the chart.

Take notes on regular polygons and interior angles.

Explore triangles and record observations on interior angles.

During

45

Circulate room, probing students to keep them in the right direction

Have students fill in chart up front. Go over observations that students had about exploration.

Discuss how students found the angle of

Students will work through all of the shapes mentioned above and explore sum of interior angles and angles of regular polygons.

Students will present their findings to the rest of the class.

each regular polygon.

After

15

What if I gave you a polygon with s number of sides. How would I find the sum of the interior angles.

Show explanation for why that formula is.

Have students complete initial problem.

Students will attempt to figure out the formula for the sum of interior angles.

Answer initial question of the day.

*** Your lesson plan should ALL be included here (the reader shouldn’t have to go anywhere else to find the plans.) The teacher should be able to read it chronologically. The only things to be included at the end of the plan are supplemental artifacts (e.g. handouts, tech files, ppt). If you chose not to use the table then the time, teacher actions and student actions should be clearly noted throughout your plan.

Make sure that your lesson is detailed enough that someone else could teach from it. This is especially important during class discussion phases. For example, be sure to detail what the teacher should be sure to bring out in a whole class discussion, including questions to push students to build conceptual understanding, questions to assess student understanding, and transitions between portions of your lesson.

If students are working in pairs / small groups this should be noted (including how the groups are to be determined)

All tasks / examples should be worked out and included in the body of the lesson plan All HW should be worked out

Reflection

This technology helped my understanding of this concept because I was able to dynamically check whether it checks out for each situation. It allowed me to figure this stuff out with relative ease. It will be the same for the students although it is tricky at some points because if you measure the angles on the <180 side, if you move the polygon so that that particular angle is >180 the angle it measure is on the outside of the polygon, not the interior angle. It shows you on sketchpad but some people won’t notice it. That could lead to student

to thinking that their conjectures are incorrect even though they probably are since this formula is true for all polygons even non-convex ones. This could be a tricky part of the lesson.