GEOMETRY LESSON 8 Perimeter and Area - Wikispacesmrclarke.wikispaces.com/file/view/L8+Area+and+Perimeter.pdfGEOMETRY LESSON 8 Perimeter and Area ... Beckmann Activity - Class activity

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February 01, 2012

Dec 29­10:55 AM

GEOMETRY LESSON 8

Perimeter and Area

Dec 29­10:57 AM

AGENDAHomework, Review, Questions??Objectives

1. Define Perimeter and Area2. Perimeter 3. How are Perimeter and Area Related? 4. Area 5. Formulas for Triangles, Parallelograms and Trapezoids6. Homework

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February 01, 2012

Dec 29­10:59 AM

Define area and perimeter in a small group.

Perimeter =

Area =

Feb 25­12:36 PM

Perimeter

The distance around a geometric figure is called the perimeter.

Perimeter of a Rectangle:P = l + l + w + wP = 2l + 2w or 2(l + w)

l

ww

l

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February 01, 2012

Mar 2­3:00 PM

Find the perimeter of each rectangle.

11 in

5 in

15 cm

20 cm

Mar 3­12:03 PM

Find the missing length of each rectangle below.

a) P = 60 ft; w = 9 ft b) P = 28 m; w = 8 m; l = _____ ft. l = _____ m.

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February 01, 2012

Apr 13­8:03 AM

You TryFind the perimeter or circumference of each figure.

#1

4 m

4 m

#2

10 in

5 in

#3

4 ft

3 ft5 ft 6 in

#4

ANSW

ERS

Jan 27­7:03 AM

Objective 2: Perimeter

Finding and Measuring Perimeters: VDW p. 377

Choose 5 objects in the classroom that have perimeters you can measure, such as a bulletin board, the top of a table, or the side of the teacher's desk. Measure their perimeters and record your work.

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February 01, 2012

Jan 27­7:06 AM

Object Drawing of What I am Measuring Perimeter

Dec 29­11:02 AM

Beckmann Activity - Class activity 12T and 12U pp. 299-300

1. Imagine you are working with elementary school children. Tell the children what we mean by the term perimeter. You amy wish to use string to help you. If you didn't have string, how would you calculate the perimeter of the polygon in the next figure? Why does it make sense that we can calculate the perimeter by adding? The grid lines are 1 cm apart.

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February 01, 2012

Jan 31­7:01 AM

2. There are several different methods for calculating the perimeter of a rectangle. Describe at least two different methods. For each method, write the corresponding expression for the perimeter of an A-cm-by-B-cm rectangle.

A un

its

B units

A units

B units

Dec 29­11:03 AM

How do we measure perimeter?

What are some common misconceptions?

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February 01, 2012

Dec 29­11:05 AM

Objective 3: How are Perimeter and Area Related? Part 1: Suppose surveyors have determined that a forest on flat terrain has a perimeter of 20 kilometers. What is the area of the forest? Can you answer this question without additional information, or is more than one answer possible? Now suppose that you learn that the forest is in the shape of a rectangle. Can you determine the area now, or is more than one answer possible? Contemplate these questions for a few minutes, then move on to the next part.

Beckmann Activity 12V Page 301­303

Jan 28­9:04 AM

How are Perimeter and Area related?

Part 2: (Similar to VDW activity)Draw at least 5 different rectangles of perimeter 20 units on graph paper. Include some rectangles that have sides whose lengths aren't whole numbers of units.

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February 01, 2012

Jan 28­9:07 AM

Part 3: Determine the areas of the rectangles you just drew. In a table, list the areas of your rectangles in decreasing order. Below each area, draw a small sketch showing the approximate shape of the corresponding rectangle.

Area

Shape

Jan 31­7:10 AM

How are the larger­area rectangles qualitatively different in shape from the smaller­area ones?

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February 01, 2012

Jan 28­9:11 AM

Part 4: Show how two people can use a loop of string and 4 fingers to represent all rectangles of a certain fixed perimeter.Now consider all the rectangles of perimeter 20 cm, including those whose side lengths aren't whole numbers of centimeters.

What are all the theoretical possibilities for the areas of rectangles of perimeter 20 cm?

What is the largest possible area, and what is the smallest possible area (if there is one)?

Jan 28­9:08 AM

Suppose that a forest on flat terrain has perimeter 20 kilometers, but there is no information on the shape of the forest. What can you say about the forest's area? Use a loop of string to help you think about this question.

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February 01, 2012

Jan 28­9:09 AM

Class Activity 12W: Can We Determine Area by Measuring Perimeter?

Nick wants to find the area of an irregular shape. Nick cuts a piece of string so that it goes all the way around the outside of the shape. Then Nick forms his piece of string into a square on top of graph paper. Using the graph paper, Nick gets a good estimate for the area of his string square. Nick then uses the square's area as his estimate for the area of the original irregular shape.

Jan 31­7:17 AM

Is Nick's method a legitimate way to estimate the areas of the irregular shapes.

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February 01, 2012

Dec 29­11:06 AM

What did you observe?

What conjectures did you make?

What conclusions can be drawn?

Dec 29­11:07 AM

Objective 4: Area Activity 19.10 - vdw p. 379

(Teacher ­ Draw two rectangles and a blob shape on a sheet of paper Make it so that the three areas are not the same but with no area that is clearly largest or smallest.)

Students: 1st make an estimate about which is the smallest and the largest of the three shapes.Use a filler of their choice to decide. Explain in writing what they found out.

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February 01, 2012

Dec 29­11:09 AM

Note that squares are the most common unit of area measurement - but any tile that conveniently fills up the plan region can be used.

(Even filling the shape with beans or pennies would help with the meaning of area.)

Jan 31­3:07 PM

Fill with blocks to find the area

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February 01, 2012

Jan 31­3:08 PM

Fill with blocks to find the area

Jan 31­3:08 PM

Fill with blocks to find the area

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February 01, 2012

Dec 29­11:10 AM

Can different shapes have the same area?

Class activity 12B - Beckmann p. 282 or VDW p. 377-378 - 19.8 Two Piece Shapes 20.8

Dec 29­11:11 AM

Beckmann p. 507 - some problems for participants to practice finding area. Use prob. on p. 509 for those in need of more of a challenge.

1. Make a shape that has area 25 in2, but that has no (or almost no) straight edges. Explain how you know that your shape has area 25 in2.

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February 01, 2012

Jan 31­3:13 PM

2. The figure below shows the floor plan for a one-story house. Calculate the area of the floor of the house. Explain your method clearly.

24 ft

32 ft

40 ft

24 ft

40 ft

40 ft

Jan 31­3:17 PM

Use the moving and additivity principles about area to determine the are, in square inches, of the shaded design in the figure. The shape is a 2-inch-by-2-inch square, with a square, placed diagonally inside, removed from the middle. In determining the area of the shape, use no formulas.

1 in

2 in

2 in

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February 01, 2012

Dec 29­11:12 AM

Objective 5: Formulas for Triangles, Parallelograms and Trapezoids

You will derive the formula for the area of Triangles, Parallelograms and Trapezoids.

Dec 29­11:14 AM

http://illuminations.nctm.org/LessonDetail.aspx?ID=L580

Triangles lesson 1Parallelograms lesson 3Trapezoids lesson 2

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February 01, 2012

Apr 2­4:51 PM

Area of squares, rectangles, parallelograms, and rhombus

Use the following formula to find the area of a parallelogram

A = b * h

Find the area of the following

7 m

7 m 2 cm

10 cm15 in

7 in6 in

Remember that base and height will always form a right angle!

Apr 13­8:40 AM

To find the area of a Parallelogram

A = bh

6 in

10 in

5 in

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February 01, 2012

Apr 13­8:18 AM

Area = 84

Find the perimeter and area of the following shape.

3

3

2

4

4

12

12

8

Perimeter = 48

Apr 13­8:03 AM

You TryFind the area of each figure.

#1

4 m

4 m

#2

10 in

5 in

16 m2

50 in2

#3 #4

6in

4in3in

5in4in

5in

5in

5in

18in2 20in2

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February 01, 2012

Dec 29­11:14 AM

http://illuminations.nctm.org/LessonDetail.aspx?ID=L580

Triangles lesson 1Parallelograms lesson 3Trapezoids lesson 2

Apr 2­8:50 AM

Area of a Triangle

Area of a triangle is found by using the following formula:

A = 1/2 * b * h

Find the area of each of the following triangles.

Remember that base and height must form a right angle!

18 in

24 in

28 in 28 in

8 ft

15 ft17 fy

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February 01, 2012

May 9­8:44 AM

Find the area of the triangle.

A = 1/2 * b * h

16 in

15 in25 in

10 in

Apr 13­8:24 AM

Make sure you know the difference between area and perimeter.

Area is space inside Perimeter is distance around the outside

15 in17 in

8 in

Find the perimeter and area for the triangle.

P = 40 in

A = 60 in2

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February 01, 2012

Apr 13­8:14 AM

Perimeter = 18in

Area = 13.4 in2

Area of Parallelograms = b*hArea of Triangles =

8 in

8 in

5 in5 in7 in 7 in

4 in

6.7 inPerimeter = 26in

Find the perimeter and area of both figures.

Area = 40in2

Sep 23­1:56 PM

Find the perimeter and area for each figure. Round to the nearest tenth, if necessary. *Don't forget to put the correct units after your answer.*

1. 2. 3.

ANSW

ERS

You Try

20 in.

21 in.18 m

31 m25 in.

17 m

21 m

44 ft.

19 ft.

30o

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February 01, 2012

Dec 29­11:14 AM

http://illuminations.nctm.org/LessonDetail.aspx?ID=L580

Triangles lesson 1Parallelograms lesson 3Trapezoids lesson 2

Sep 23­1:58 PM

Areas of Trapezoids, Rhombi, and Kites

Area of a TrapezoidThe area A of a trapezoid is one half the product of the height h and the sum of its bases, b1 and b2.

A = 1/2h(b1 + b2)

How does the area of a trapezoid relate to the area of a parallelgram?

Find the area and perimeter 1.of the trapezoid. Round tothe nearest tenth, if necessary.

1. P = 44.3cmA = 96 cm2

Click for answers

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February 01, 2012

Sep 23­1:58 PM

Click here for formula.

Area of a Rhombus or KiteThe area A of a rhombus or kite is one half the product of the lengths of its diagonals, d1 and d2.

A = 1/2d1d2

Find the area ONLY of each rhombus or kite. Round to the nearest tenth, if necessary.

2. 3.

2. A = 84 mm2

3. A = 72 in.2Click for answers

Mar 9­3:40 PM

You TryFind the perimeter and area of each figure. Use 3.14 for pi. Round answers to the nearest hundredth, if necessary.AN

SWER

S

1. 2.

3.

7 in.

7 in.

6 in. 16 in.

8 in.

8 in.

11 m

11.1 m 9.4 m

5.2 m

15 cm

34 cm

37.2 cm 37.2 cm

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February 01, 2012

Sep 23­1:58 PM

A C

E

P

D

B

Area of Regular Polygons

Remember that in a regular polygon ALL the sides are congruent and ALL the angles are congruent. If a regular polygon is inscribed in a circle, then the center of the regular polygon and the radius of the regular polygon are also the center and radius of the circle.

A C

E

P

D

B

A segment drawn perpendicular to a side of a regular polygon is called an apothem. Its length is the height of an isosceles triangle that has two radii as legs.

A central angle of a regular polygon has its vertex at the center of the polygon and its sides pass through consecutive vertices of the polygon. The measures of each central angle of a regular n-gon is 360/n.

A C

E

P

D

B

Sep 23­1:58 PM

There are a couple of different ways to find the area of a regular polygon. One method is to divide the polygon into congruent isosceles triangles. This is sometimes called decomposing the polygon into triangles.

The top of a table shown is a regular hexagon with a side length of 3 feet and an apothem of 1.7 feet. What is the area of the tabletop to the nearest tenth?

Step 1: Divide the triangle into congruent isosceles triangles.

Step 2: Find the area of ONE triangle.

Step 3: Multiple the area of one triangle by the total number of triangles.

3 ft.

1.7 ft.

15.3 ft.2

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February 01, 2012

Sep 23­1:58 PM

We can also use a formula to quickly find the area of a regular polygon.

Area of a Regular PolygonThe area A of a regular n-gon with side length s is one half the product of the apothem a and the perimeter P.

A = 1/2a(ns) or A = 1/2aP

Find the area of each regular polygon. Round to the nearest tenth, if necessary.

5 m

9 cm

65.0 m2 68.175 cm2

3.03 cm

Jan 31­6:50 AM

Test you skill at application.

You have purchased a home that needs some work. Find the cost for the following upgrades.

Tile the master bath. Cost = $ 1.99 sq. ft.

Carpet in the master bedroom.Cost = $3.99 sq. ft.

Hardwood Floor in the great room.Cost = $5.25 sq. ft.

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February 01, 2012

Jan 31­6:37 AM

Master Bath Tile = $253.84

Master Bedroom Carpet = $816.95

Great Room Hardwood = $1555.17

Dec 29­11:16 AM

Objective 6: Homework

Reading: Beckmann hardbound pp. 481-483; 555-557

Differentiated Lesson due NEXT WEEK!Bring a box no larger than a shoebox.