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LESSON 2: All About That Base . . . and Height • M1-15

WARM UPWrite 3 different expressions to describe the total area of this rectangle.

LEARNING GOALS• State and compare the attributes of different shapes.• Explain that the area of a parallelogram is the same as

that of a rectangle with the same base length and height.• Derive the formulas for the areas of triangles, parallelograms,

and trapezoids by composing or decomposing the various shapes into rectangles, triangles, and other shapes.

• Apply the techniques of composing and decomposing shapes to solve real-world and mathematical problems.

KEY TERMS• parallelogram• altitude• variable• trapezoid

All About That Base . . . and HeightArea of Triangles and Quadrilaterals

2

You can take a shape apart and put it back together in a different way without changing its area. How can you use composition and decomposition of shapes to reason about the areas of shapes and to derive formulas for the areas of common shapes?

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M1-16 • TOPIC 1: Factors and Area

Getting Started

In the 20s

Consider each figure.

1. Can you name each fi gure?

2. Describe the attributes of each shape. Are there any attributes that are shared across the different shapes?

3. Each shaded fi gure shown has an area of exactly 20 square units. Show how you know.

An attribute is a characteristic to describe a figure.



In this activity you will investigate the area of a parallelogram using what you know about the area of a rectangle. A parallelogram is a four-sided figure with two pairs of parallel sides and opposite sides that are equal in length.

1. Cut out a parallelogram from the grid at the end of the lesson.

2. Cut your parallelogram into pieces so that it can be reassembled to form a rectangle. Tape your rectangle in the space provided.

Parallelogram Rectangle

In a parallelogram, any of the four sides can be labeled as the base. The altitude of a parallelogram is another name for the height of a parallelogram. The altitude of a parallelogram is the perpendicular distance from the base of the parallelogram to the opposite side, represented by a line segment.

3. Label the base and height of the parallelogram and rectangle.

Investigating the Area

of a Parallelogram

ACTIVIT Y

2.1

A rectangle is a

special type of

parallelogram.

c



4. How does the height of the parallelogram relate to the height of the rectangle? How does the length of the base of the parallelogram relate to the length of the base of the rectangle? Explain your reasoning.

5. Describe the relationship between the areas of a parallelogram and rectangle that have the same base and height.

6. Use the terms base and height to describe how to calculate the area of a parallelogram.

In mathematics, one of the most powerful concepts is to use a letter to represent a quantity that varies, or changes. The use of letters, called variables, helps you write expressions to understand and represent problem situations.

7. Write the formulas to calculate the areas of a parallelogram and a rectangle. Use b for base and h for height.

A variable is a

letter that is used to

represent a number.

When you write a sentence to explain your reasoning, be sure to express a complete idea. Your sentence should make sense standing alone.



The base of a triangle, like the base of a parallelogram, can be any of its sides. The height, or altitude, of a triangle is the length of a line segment drawn from a vertex of the triangle to the opposite side so that it forms a right angle with the opposite side.

base

altitude

base

altitudealtitude

C A

B

A B

C

base

A

B C

Sailboat racecourses are often shaped like triangles. The course path is defined by buoys called marks. When the course is triangular, the marks are located at the corners, or vertices, of the triangle. Here is a sample course with the marks numbered.

MARK1

MARK3

MARK2

START/FINISH

Race officials need to know the area inside the course so that they can plan for the number of spectator boats that can anchor within.

Investigating the Area

of a Triangle

ACTIVIT Y

2.2NOTES



2

3 1

1. Estimate the area of the triangular course in square units. Justify your estimate.

2. Use two sides of the triangle to draw a parallelogram on the grid. How does the area of the parallelogram relate to the area of the triangle?

3. Calculate the area enclosed by the triangular course.

The triangle

represents a sailboat

racecourse. Each

square on the grid

represents 0.1 mile by

0.1 mile.

Use a straightedge to draw your parallelogram.



4. Label a base and height of the original triangle in the diagram. Describe how to calculate the area of any triangle in terms of the base and the height.

5. Suppose you create a parallelogram using a different side of the triangle. Does this change the area of the triangle? Explain how you know.



You have seen that taking apart, or decomposing, a parallelogram forms a rectangle. And putting together, or composing, two triangles also forms a parallelogram. Composing and decomposing can help you think about the shapes differently in order to determine their areas. In this activity you will take apart and put together shapes to determine the formula for calculating the area of a trapezoid.

A trapezoid is a quadrilateral with two bases, often labeled b1 and b2. The bases are parallel to each other. The other two sides of a trapezoid are called the legs of the trapezoid. An altitude of a trapezoid is the length of a line segment drawn from one base to the other and perpendicular to both.

1. Label the bases of each trapezoid as b1 and b2.

Cut out two of the trapezoids at the end of the lesson to show how to determine each area.

ACTIVIT Y

2.3Investigating the Area

of a Trapezoid

The variable b

represents a base,

but a trapezoid

has two bases. So,

subscripts are used to

distinguish between

the two different

bases; b1 and b2 are

not equal in length.



2. Marcus cut out and composed two trapezoids into a parallelogram to fi gure out the exact area of one trapezoid. Show what Marcus did to determine the area.

3. Zoe folded the trapezoid so the bases aligned, cut along the fold, and rearranged the parts to form a parallelogram. Show what Zoe did to determine the area.

4. Angela decomposed the trapezoid into two triangles to determine the exact area. Use this trapezoid to recreate Angela’s strategy.

5. Describe how to calculate the area of any trapezoid in terms of the two bases and the height.

Think about how the new shapes formed relate to the original trapezoid.



NOTESTALK the TALK

All Three Shapes

1. Draw each shape and then label a base and height. Next, write the formula to calculate the area of each. Use A for the area, b for the length of the base, and h for the height.

parallelogram triangle

trapezoid

2. Show that the two triangles have the same area.

R P

MG

3. Write a paragraph that will convince your readers that the two triangles have the same area.



Shape Cut Outs

Extra shapes are included.


ReviewAssign_para

1. Assign_num_list

2. Assign_num_list

3. Assign_num_list

1.2 A Using Tables to Represent Equivalent Ratios • M-27

Stretch1. Assign_num_list

2. Assign_num_list

5. Assign_num_list

Assign_para

6. Assign_num_list

Assign_para

Assign_para

Assign_para


4. Calculate the area of the triangle.3. Identify a base and corresponding height

for the given triangle. Determine the

area of the triangle.

2. Calculate the area of the parallelogram.

20 mm

11 mm

PracticeAnswer each question for the given figure.

1. Identify a base and corresponding height for the

given parallelogram. Determine the area of the

parallelogram.

14 yd

32 yd

Assignment

WriteDefine each term in your own

words.

1. height of a parallelogram

2. height of a triangle

RememberThe area of a parallelogram, a triangle, or a trapezoid can be

determined by composing or decomposing it into one or more

shapes with an equal total area.

Area of a parallelogram 5 bh

Area of a triangle 5 1–2 bh

Area of a trapezoid 5 1–2 (b1 1 b2)h


Assignment

Write1. Assign_num_list

2. Assign_num_list

PracticeAnswer each question for the given figures.

1. Assign_num_list

RememberAssign_para

Assign_mid

2. Assign_num_list

3. Identify a base and corresponding height

Assign_para

Assign_para

4. Assign_num_list


Stretch1. What is the area of a parallelogram that has a base of 43–

4 ft and a height of 11–

3 ft?

2. Calculate the area of the triangle.

5. Identify the two bases and the height

for the given trapezoid. Determine the

area of the trapezoid.

6. Yvonne cut a picture into the shape of a

trapezoid to place into her scrapbook.

The picture is shown. What is the area of

the picture?

10 m 10.54 m

8.66 m

11 m

5 in.

4 in.

7 in.

ReviewUse the Distributive Property to write an equivalent addition expression for each.

1. 6(9 1 1)

2. (14 1 3)7

3. 1–2 (7 1 10)

Decompose each rectangle into two or three smaller rectangles to demonstrate the Distributive Property.

Then write each in the form a(b 1 c) 5 ab 1 ac.

4.

5.

192

8

512

4



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