## Triangles and Parallelograms

I. Section Objectives

- Understand the basic concepts of the meaning of area.
- Use formulas to find the area of specific types of polygons.

II. Cross- curricular-Reflecting Pool dimensions

- Use the following image from Wikipedia of the Reflecting Pool in Washington, DC.
- This is Figure 10.01.01
- www.en.wikipedia.org/wiki/File:Reflecting_pool.jpg
- Here is the problem.
- According to Wikipedia, the dimensions of the Reflecting Pool are long and wide.
- Given this information, what shape is the Reflecting Pool?
- What is the perimeter of the pool?
- What is the area of the pool?
- Draw a diagram to explain your answer.
- Solution:
- The shape is a rectangle.
- The perimeter is
- The area is

III. Technology Integration

- There are two great short videos on this website for area and length.
- One is an architect and one is on an apartment design.
- www.thefutureschannel.com/hands-on_math/apartment.php
- Have students watch the videos.
- Then you can expand on this by having the students draw a design of their room at home.
- Students will need to go home and do some measurements and then come back with the area and perimeter of their room.
- Rooms with unconventional shapes will be the most fun and challenge.
- Allow time for students to share their work.

IV. Notes on Assessment

- Assess student work on the Reflecting Pool problem.
- Is the diagram accurate?
- Did the students calculate the area correctly?
- Did the students calculate the perimeter correctly?
- Provide students with feedback on their work.

## Trapezoids, Rhombi and Kites

I. Section Objectives

- Understand the relationships between the areas of two categories of quadrilaterals: basic quadrilaterals and special quadrilaterals.
- Derive area formulas for trapezoids, rhombi and kites.
- Apply the area formula for these special quadrilaterals.

II. Cross- curricular-Room Design

- Tell students that they are going to design a room that has a trapezoidal shape.
- Students can complete this in connection with the Technology Integration if you choose.
- If not, have the students use the dimensions of their own bedroom (they did this in the last lesson), or the classroom or a standard size bedroom for example.
- Students are going to redesign this area as a trapezoid.
- They want to come as close to the original area as possible.
- So if the room was , the area is
- How can you come close to the same area if the shape of the room is a trapezoid?
- Students should draw their design on grid paper and explain their thinking.
- Allow time for students to share their work when finished.

III. Technology Integration

- This is a website that shows a house designed as a trapezoid.
- www.momoy.com/2009/04/02/l-house-beautiful-trapezoid-house-design-by-philippe-steubi-architekten-gmbh/
- Students can look at the trapezoid shape of the house and the floor plan is also included.
- There are views of the inside of the house and the outside of the house as well as some of the rooms.
- Conduct a discussion about the house. What would be the challenges of designing and building such a house?

IV. Notes on Assessment

- Assessment will come with student presentations and work product.
- What did students learn about the relationship between rectangles and trapezoids?
- Were they able to come up with a room with an area close to the original?
- Who got the closest?
- Provide students with feedback on their work.

## Area of Similar Polygons

I. Section Objectives

- Understand the relationship between the scale factor of similar polygons and their areas.
- Apply scale factors to solve problems about areas of similar polygons.
- Use scale models or scale drawings.

II. Cross- curricular-National Mall Mapping

- Ask students to use the Wikipedia image of the National Mall to create a map of it.
- This is Figure 10.03.01
- www.en.wikipedia.org/wiki/National_Mall
- Then tell the students that the mall is
- They are going to use what they have learned about scale and measurement to create their own map of the mall.
- They need to choose a scale to work with.
- Then they use grid paper to design the mall.
- When students have the area of the mall correct, they can draw in as many different museums and monuments as they can.
- Extra details add extra credit to their work.
- When finished, allow time for students to share their work.

III. Technology Integration

- Use the following website on the National Mall in Washington DC.
- www.en.wikipedia.org/wiki/National_Mall
- Have students complete some research about the mall.
- Possible questions include:
- Who designed it?
- When was it built?
- What is at the North end?
- What is at the South end?
- How many different museums can you visit there?
- Have you been to the mall?
- Which museum would you most like to visit or did you enjoy and why?

IV. Notes on Assessment

- Assess each student map.
- Is the use of scale done correctly?
- Are the measurements correct?
- Is the map accurate?
- Has the student take the time to add in details?
- Provide students with feedback on their work.

## Circumference and Arc Length

I. Section Objectives

- Understand the basic idea of a limit.
- Calculate the circumference of a circle.
- Calculate the length of an arc of a circle.

II. Cross- curricular-The Pantheon

- Have students use the image of the floor plan of the rotunda of the Pantheon to calculate the circumference of it.
- This is Figure 10.04.01
- www.en.wikipedia.org/wiki/Pantheon,_Rome
- The diameter of the dome is
- Given this measurement, what is the circumference?
- Have the students draw a diagram to explain their work.
- Allow time for students to share their diagrams in small groups.

III. Technology Integration

- Have students use the following Wikipedia site to research information on the Pantheon.
- www.en.wikipedia.org/wiki/Pantheon,_Rome
- Students can use this information to write a short essay.
- Students should hunt for mathematical information about the Pantheon for their essay.
- For example, height of the columns.
- What is a portico?
- What is a rotunda?
- Have the students complete this work and then collect it for your review.
- Extension on initial exercise- have students research the dimensions of the rectangle that connect the portico and the rotunda.
- What is the area of the rectangle?
- What is the perimeter?

IV. Notes on Assessment

- Look at student work.
- Is it accurate?
- Does the diagram represent student work?
- Provide students with feedback on their work.

## Circles and Sectors

I. Section Objectives

- Calculate the area of a circle.
- Calculate the area of a sector.
- Expand understanding of the limit concept.

II. Cross- curricular-History

- Use the following image from the round table used by King Arthur.
- This is Figure 10.05.01
- www.crystalinks.com/roundtable.gif
- The diameter of the round table was
- Given this measurement, calculate the area of the round table.
- If the table was divided between each of the knights evenly, what is the area of one of the sectors?
- Draw a diagram to explain your work.
- Allow students time to share their diagrams when finished.

III. Technology Integration

- Have students use the following website as a tutorial on area and circumference of circles.
- www.mathgoodies.com/lessons/vol2/circle_area.html
- Students can review already learned material.
- There is also a worksheet section for them to work with and practice solving problems.

IV. Notes on Assessment

- Examine student diagrams.
- Were they able to find the correct area of the table?
- How about the sectors?
- Does the diagram accurately show their work?
- Is there anything missing?
- Provide students with feedback/correction on their work.

## Regular Polygons

I. Section Objectives

- Recognize and use the terms involved in developing formulas for regular polygons.
- Calculate the area and perimeter of a regular polygon.
- Relate area and perimeter formulas for regular polygons to the limit process in prior lessons.

II. Cross- curricular-Architecture

- Use the following image of a roof in the shape of a hexagon.
- This is Figure 10.06.01
- www.space-frames.com/commercial buildings/xha28.htm
- Have the students use the dimensions of this design to figure out the area of the roof of this hexagon.
- Then have the students draw a diagram and explain how they figured out the area of the hexagon.
- Allow time for students to share their work when finished.

III. Technology Integration

- Have students complete some research on where to find hexagons and pentagons.
- Students can search architecture, nature or their own subject.
- Ask the students to keep track of the websites that they visit.
- Students should prepare a presentation of at least five examples of pentagons or hexagons in their given subject area.
- Students should include diagrams or images with their work.

IV. Notes on Assessment

- Assess student diagrams.
- How did the students figure out the area of the roof?
- Does their method make sense?
- Did they divide it into triangles?
- Did they divide it into trapezoids?
- Provide students with feedback on their work.

## Geometric Probability

I. Section Objectives

- Identify favorable outcomes and total outcomes.
- Express geometric situations in probability terms.
- Interpret probabilities in terms of lengths and areas.

II. Cross- curricular-Target Practice

- Use the following image of a dartboard.
- This is Figure 10.07.01.
- www.home.wlu.edu/~mcraea/GeometricProbabilityFolder/Introduction/Problem0/images/images/dartboard.gif
- Here is the problem.
- What is the geometric probability of hitting the center of the internal square of the dartboard?
- Use probability to figure this out.
- Show your work with a diagram and be prepared to explain your answer.
- Allow time for students to share their work when finished.

III. Technology Integration

- Visit the same website that the image came from and explore the solution to the problem.
- www.home.wlu.edu/~mcraea/GeometricProbabilityFolder/Introduction/Problem0/images/images/dartboard.gif
- The answer to the problem that the student solved above is there.
- Have students use this to correct their own work.
- Show any changes/corrections that they completed.
- Then explore the other problems on the site.

IV. Notes on Assessment

- Because students are going to correct their own work during the technology integration, use this as a time to assess student work through observation.
- Are the students able to apply the concepts of probability to geometry?
- Refer students back to the text if they are having difficulty.

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