Formula for the Area
of a Parallelogram
Objectives To review the properties of parallelograms; and
to
t guide the development and use of a formula for the area
of a parallelogram.
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Teaching the Lesson
Key Concepts and Skills
• Find the area of a rectangle. Family
Letters
1 2
4 3
• Develop a formula for calculating the area
of a parallelogram. [Measurement and Reference Frames Goal 2]
• Calculate perimeter. [Measurement and Reference Frames Goal 2]
[Geometry Goal 1]
• Describe properties of parallelograms. [Geometry Goal 2]
Key Activities
Students construct models of parallelograms
and use them to review properties of
parallelograms.
Students cut apart and rearrange
parallelogram shapes; they develop and use
a formula for the area of a parallelogram.
Ongoing Assessment:
Informing Instruction See page 690.
Key Vocabulary
base height perpendicular
Materials
Math Journal 2, pp. 236–238
Study Link 8 5
Math Masters, p. 260
centimeter ruler straws and twist-ties scissors tape index card or other
square-cor ner object slate
Common
Core State
Standards
Curriculum
Focal Points
1 2
4 3
Playing Fraction Of
Student Reference Book, pp. 244
and 245
Fraction Of Cards (Math Masters,
pp. 477, 478, and 480)
Math Masters, p. 479
Students practice finding fractions
of collections.
Playing Angle Add-Up
Math Masters pp. 507–509
per partnership: 4 of each of number
cards 1–8 and 1 of each of number
cards 0 and 9 (from the Everything
Math Deck, if available) full-circle
protractor (transparency of Math
Masters, p. 439) dry-erase markers straightedge
Students draw angles and then use
addition and subtraction to find the
measures of unknown angles.
Interactive
Teacher’s
Lesson Guide
Differentiation Options
Ongoing Learning & Practice
[Measurement and Reference Frames Goal 2]
• Identify perpendicular line segments and
right angles. Assessment
Management
ENRICHMENT
Constructing Figures with a Compass
and Straightedge
Student Reference Book, pp. 114, 117,
and 118
compass straightedge
Students construct figures with a compass
and straightedge.
ENRICHMENT
Solving Area and Perimeter Problems
Math Masters, pp. 263, 264, and 437
scissors tape
Students explore ways of combining various
two-dimensional shapes to form new shapes.
Math Boxes 8 6
Math Journal 2, p. 239
Students practice and maintain skills
through Math Box problems.
Ongoing Assessment:
Recognizing Student Achievement
Use Math Boxes, Problem 4. [Operations and Computation Goal 5]
Study Link 8 6
Math Masters, pp. 261 and 262
Students practice and maintain skills
through Study Link activities.
Advance Preparation
For Part 1, each student needs 2 short straws, 2 long straws, and 4 twist-ties. Pairs of straws should be
the same length. Place them near the Math Message.
Teacher’s Reference Manual, Grades 4–6 pp. 180–185, 221, 222
Lesson 8 6
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Getting Started
Mental Math and Reflexes
Math Message
Pose multiplication facts and problems.
Suggestions:
Take 2 short straws, 2 long straws, and 4 twist-ties.
Use them to construct a parallelogram.
3 ∗ 7 = 21
4 ∗ 9 = 36
8 ∗ 5 = 40
9 ∗ 6 = 54
8 ∗ 52 = 416
4 ∗ 63 = 252
9 ∗ 76 = 684
88 ∗ 5 = 440
90 ∗ 8 = 720
10 ∗ 90 = 900
60 ∗ 70 = 4,200
80 ∗ 30 = 2,400
Study Link 8 5 Follow-Up
Have partners compare answers and discuss
how they found the missing side measure in
Problems 5 and 6.
1 Teaching the Lesson
Math Message Follow-Up
WHOLE-CLASS
ACTIVITY
Ask students to tell what they know about parallelograms, using
their straw constructions as models, while you list the properties
they name on the board. The list should include:
A parallelogram is a four-sided polygon called a quadrangle or
quadrilateral.
Opposite sides of a parallelogram are parallel.
Opposite sides of a parallelogram are the same length.
Rectangles and squares are special kinds of parallelograms.
Have students form a rectangle with their straw constructions,
and then ask them to pull gently on the opposite corners. They
should get a parallelogram that is not a rectangle. Ask the
following questions:
height
he
base
igh
t
ba
se
NOTE Height is the distance perpendicular
to the base of a figure. Any side of a
parallelogram can be the base. The choice
of the base determines the height.
●
Does the perimeter remain the same? yes
●
Does the area remain the same? No, because although the
length of the base stays the same, the height decreases, so
the area decreases.
Draw a parallelogram on the board. Choose one of the sides, for
example, the side on which the parallelogram “sits,” and label
it the base. Explain that base is also used to mean the length
of the base.
The shortest distance between the base and the side opposite the
base is called the height of the parallelogram. Draw and label
a dashed line to show the height. Include a right-angle symbol.
Point out that the dashed line can be drawn anywhere between
the two sides as long as it is perpendicular to (forms a right
angle with) the base.
Remind students that rectangles are parallelograms whose sides
form right angles. If you think of one side of a rectangle as its
base, then the length of an adjacent side is its height.
688
Unit 8 Perimeter and Area
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Student Page
Tell students that in this lesson they will use the formula for
the area of a rectangle to develop a formula for the area of a
parallelogram.
Date
Time
LESSON
Areas of Parallelograms
86
1.
Parallelogram A
Sample answer:
The use of a formula to calculate the area of a parallelogram is a Grade 5 Goal.
base =
height =
6
2
WHOLE-CLASS
ACTIVITY
2.
6 cm
2 cm
12 cm
Area of rectangle =
length of base =
cm
width (height) =
cm
Area of parallelogram =
the Area of a Parallelogram
1 cm2
Tape your rectangle in the space below.
Links to the Future
Developing a Formula for
135
Cut out Parallelogram A on Math Masters, page 260.
DO NOT CUT OUT THE ONE BELOW. Cut it into
2 pieces so that it can be made into a rectangle.
12
cm2
2
Do the same with Parallelogram B on Math Masters, page 260.
Parallelogram B
Tape your rectangle in the space below.
Sample answer:
PROBLEM
PRO
P
RO
R
OB
BLE
BL
L
LE
LEM
EM
SO
S
SOLVING
OL
O
LV
L
VIN
V
IIN
NG
(Math Journal 2, pp. 236 and 237;
Math Masters, p. 260)
Point out that Parallelogram A on journal page 236 is the same as
Parallelogram A on Math Masters, page 260.
4
4
base =
height =
1. Cut out Parallelogram A from the master.
width (height) =
cm
Area of parallelogram =
Guide students through the following activity:
4 cm
4 cm
16 cm
Area of rectangle =
length of base =
cm
16
cm2
2
Math Journal 2, p. 236
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2. Cut the parallelogram into two pieces along one of the
vertical grid lines.
3. Tape the pieces together to form a rectangle.
cut
4. Tape this rectangle in the space next to the parallelogram in
the journal.
Discuss the relationship between the parallelogram and the
rectangle formed from the parallelogram.
●
Why must the parallelogram and the rectangle both have
the same area? The rectangle was constructed from the
parallelogram. Nothing was lost or added.
Student Page
Date
Areas of Parallelograms
86
3.
continued
Do the same with Parallelogram C.
Parallelogram C
Tape your rectangle in the space below.
Sample answer:
5. Record the dimensions and area of the parallelogram and the
rectangle. Length of base of parallelogram and length of base
of rectangle = 6 cm; height of parallelogram and width
(height) of rectangle = 2 cm; area of each figure = 12 cm2
Have students repeat these steps with Parallelograms B, C, and
D, working on their own or with a partner.
Time
LESSON
base =
height =
4
3
4.
width (height) =
cm
Area of parallelogram =
4 cm
3 cm
12 cm
Area of rectangle =
length of base =
cm
12
cm2
2
Do the same with Parallelogram D.
Parallelogram D
Tape your rectangle in the space below.
Sample answer:
Bring students together to develop a formula for the area of a
parallelogram. These are three possible lines of reasoning:
base =
height =
3
4
The area of the rectangle is equal to the length of its base times
its width (also called the height).
5.
width (height) =
cm
Area of parallelogram =
3 cm
4 cm
12 cm
Area of rectangle =
length of base =
cm
12
cm2
Write a formula for the area of a parallelogram.
A=b∗h
2
height
The area of each parallelogram is the same as the area of the
rectangle that was made from it.
base
Math Journal 2, p. 237
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Lesson 8 6
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abo
ut 2
2 cm
.2 c
m
The length of the base of the parallelogram is equal to the
length of the base of the rectangle. The height of that
parallelogram is equal to the width (height) of that rectangle.
Therefore, the area of the parallelogram is equal to the length
of its base times its height. Using variables:
6 cm
A=b∗h
Perimeter = about 16.4 cm
where b is the length of the base and h is the height.
Area = 12 cm2
2 cm
2 cm
Have students record the formula at the bottom of journal
page 237.
Ongoing Assessment: Informing Instruction
Watch for students who think that the perimeter of each parallelogram and
rectangle pair is also the same. Point out that although the height and base are
the same measure, the height of a parallelogram is only used in computing
its perimeter when the parallelogram is a rectangle or square. (See margin.)
6 cm
Perimeter = 16 cm
Area = 12 cm2
Solving Area Problems
PARTNER
ACTIVITY
(Math Journal 2, p. 238)
Algebraic Thinking Work with the whole class on Problem 6,
journal page 238. Students can place an index card (or other
square-corner object) on top of the shape, align the bottom edge
of the card with the base, and then use the edge of the card to
draw a line for the height. They will need a centimeter ruler to
measure the length of the base and the height.
index card
Time
LESSON
Areas of Parallelograms
8 6
䉬
6.
continued
Draw a line segment to show the height of Parallelogram DORA.
D
Use your ruler to measure the base and height.
Then find the area.
base ⫽
height ⫽
Area ⫽
7.
height
Student Page
Date
5
4
20
b.
c.
cm
2
cm
O
R
A rectangle whose area is 12 square centimeters
A parallelogram, not a rectangle, whose area is 12 square centimeters
A different parallelogram whose area is also 12 square centimeters
Sample answers:
a.
b.
c.
8.
What is the area of:
a. Parallelogram ABCD?
24
b.
Trapezoid EBCD?
18
cm2
A
E
c.
Problem 7 illustrates the fact that shapes that do not look the
same can have the same area.
Problem 8b lends itself to a variety of solution strategies. Some
students may have partitioned the trapezoid into a rectangle
flanked by two triangles. The rectangle covers 12 grid squares.
If one triangle were cut apart and placed next to the other
triangle to form a rectangle, the pair would cover 6 squares.
The rectangle and two triangles cover 12 + 6 = 18 cm2.
E
Triangle ABE ?
6
cm2
D
cm2
D
B
C
Math Journal 2, p. 238
690
Drawing the height of a parallelogram
Have partnerships complete Problems 7 and 8.
cm
Draw the following shapes on the grid below:
a.
A
B
C
Problem 8b
Unit 8 Perimeter and Area
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Student Page
Problem 8c can be solved without using a formula for the area
of a triangle. The parallelogram area minus the trapezoid area
is the triangle area. 24 - 18 = 6 cm2
Date
Time
LESSON
Math Boxes
86
1.
Dimensions for actual rectangles are
given. Make scale drawings of each
rectangle described below.
a.
Scale: 1 cm represents 20 meters.
2 Ongoing Learning & Practice
a.
Length of rectangle: 80 meters
Width of rectangle: 30 meters
b.
Length of rectangle: 90 meters
Width of rectangle: 50 meters
b.
145
2.
What is the area of the parallelogram?
3.
8 blue blocks,
7"
Playing Fraction Of
A jar contains
4 red blocks,
3"
9 orange blocks, and
PARTNER
ACTIVITY
4 green blocks.
7 ∗ 3 = 21
Number model:
(Student Reference Book, pp. 244 and 245; Math Masters, pp. 477–480)
21
Area =
You put your hand in the jar and without
looking pull out a block. About what
fraction of the time would you expect to
get a blue block?
2
in
8
_
4.
Add or subtract.
a.
3 +_
7 =
_
16
16
2 +_
1 =
b. _
Playing Angle Add-Up
25
135
Students play Fraction Of to practice finding fractions of
collections. See Lesson 7-3 for additional information.
3
PARTNER
ACTIVITY
c.
d.
Multiply. Use a paper-and-pencil algorithm.
6,142
= 83 ∗ 74
6
6
6
_
_3
10 , or 5
_3
9 -_
3
=_
10
10
= _3 - _3
8
(Math Masters, pp. 439 and 507–509)
5.
10
_
_5
16 , or 8
_5
45
4
8
18 19
55 57
To further explore the idea that angle measures are additive, have
students play Angle Add-Up. See Lesson 7-9 for more information.
Math Boxes 8 6
Math Journal 2, p. 239
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INDEPENDENT
ACTIVITY
(Math Journal 2, p. 239)
Mixed Practice Math Boxes in this lesson are paired
with Math Boxes in Lesson 8-8. The skill in Problem 5
previews Unit 9 content.
Ongoing Assessment:
Recognizing Student Achievement
Math Boxes
Problem 4
Study Link Master
Use Math Boxes, Problem 4 to assess students’ ability to solve fraction addition
and subtraction problems. Students are making adequate progress if they are
able to solve Problems 4a and 4c, which involve fractions with like denominators.
Some students may be able to solve Problems 4b and 4d by using equivalent
fractions with like denominators, using manipulatives, or drawing pictures.
Name
Date
STUDY LINK
86
1.
2.
9'
3 cm
8 cm
4'
INDEPENDENT
ACTIVITY
4 ∗ 9 = 36
36
Area =
square feet
3.
4 ft
(Math Masters, pp. 261 and 262)
8 ∗ 3 = 24
24 square centimeters
Number model:
Area =
4.
65 cm
72 cm
Number model:
135
Find the area of each parallelogram.
[Operations and Computation Goal 5]
Study Link 8 6
Time
Areas of Parallelograms
6 ft
Home Connection Students calculate the areas of
parallelograms on Math Masters, page 261.
Number model:
6 ∗ 4 = 24
24
Area =
square feet
Number model: 65
Area =
∗ 72 = 4,680
4,680 square centimeters
Try This
5.
6.
59 m
NOTE Math Masters, page 262 should be completed before Lesson 9-1,
in which students share and discuss examples of percents they have collected.
The area of each parallelogram is given. Find the length of the base.
2 in.
?
?
Area = 26 square inches
base =
13
inches
Area = 5,015 square meters
base =
85
meters
Math Masters, p. 261
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Teaching Master
LESSON
Perimeter and Area
86
䉬
150
3 Differentiation Options
INDEPENDENT
ACTIVITY
ENRICHMENT
Constructing Figures with
30+ Min
a Compass and Straightedge
(Student Reference Book, pp. 114, 117, and 118)
To apply students’ understanding of the properties of
parallelograms, have them construct parallelograms and
perpendicular line segments as described on pages 114,
117, and 118 of the Student Reference Book.
PARTNER
ACTIVITY
ENRICHMENT
263
Math Masters, p. 263
Solving Area and
30+ Min
Perimeter Problems
(Math Masters, pp. 263, 264, and 437)
To apply students’ understanding of area and perimeter, have
them explore different ways of combining various 2-dimensional
shapes to form new shapes.
Possible solutions to Problem 6:
Teaching Master
Name
Date
Teaching Aid Master
Time
Name
LESSON
Perimeter and Area
86
䉬
continued
131
133–136
1.
Make a square out of 4 of the shapes. Draw the square on the centimeter dot
grid on Math Masters, page 437. Your picture should show how you put the
square together.
2.
Make a triangle out of 3 of the shapes. One of the shapes should be the
shape you did not use to make the square in Problem 1. Draw the triangle
on Math Masters, page 437.
3.
Find the area of the following:
the small triangle
b.
the square
c.
the parallelogram
4. a.
b.
5.
8
16
16
What is the perimeter of the large square
you made in Problem 1?
What is the area of that square?
What is the area of the large triangle you
made in Problem 2?
32
64
32
Time
Dot Paper
Cut out and use only the shapes in the top half of Math Masters,
page 263 to complete Problems 1–5.
a.
Date
cm2
cm2
cm2
cm
cm2
cm2
a square
b.
a rectangle
c.
a trapezoid
d.
any shape you choose
Tape your favorite shape onto the back of this sheet. Next to the shape, write
its perimeter and area.
Math Masters, p. 264
692
g
Answers vary.
a.
py g
Cut out the 5 shapes in the bottom half of Math Masters, page 263 and add
them to the other shapes. Use at least 6 pieces each to make the following
shapes.
p
Try This
6.
Math Masters, p. 437
Unit 8 Perimeter and Area
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Name
Date
Time
Angle Add-Up
Materials
1 2
4 3
□ number cards 1–8 (4 of each)
□ number cards 0 and 9 (1 of each)
□ dry-erase marker
□ straightedge
□ full-circle protractor (transparency of Math Masters, p. 439)
□ Angle Add-Up Record Sheet (Math Masters, p. 509)
Players
2
Skills
Drawing angles of a given measure
Recognizing angle measures as additive
Solving addition and subtraction problems to find the measures
of unknown angles
Objective
To score the most points in 3 rounds.
Copyright © Wright Group/McGraw-Hill
Directions
1.
Shuffle the cards and place the deck number-side down on the table.
2.
In each round, each player draws the number of cards indicated
on the Record Sheet.
3.
Each player uses the number cards to fill in the blanks and form
angle measures so the unknown angle measure is as large
as possible.
4.
Players add or subtract to find the measure of the unknown angle
and record it in the circle on the Record Sheet. The measure of the
unknown angle is the player’s score for the round.
5.
Each player uses a full-circle protractor, straightedge, and marker
to show that the angle measure of the whole is the sum of the angle
measures of the parts.
6.
Players play 3 rounds for a game. The player with the largest total
number of points at the end of the 3 rounds wins the game.
507
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Name
Date
Time
1 2
4 3
Angle Add-Up Example
Example:
In Round 1, Suma draws a 2, 7, 1, and 5. She creates the angle
measures 51° and 72° and records them on her record sheet.
°
Round 1:
Draw 4 cards.
5
1 °+ 7
m∠ABD
2 °=
m∠DBC
m∠ABC
Using addition, Suma finds the sum of the measures of angles ABD and DBC.
She records the measure of angle ABC on her record sheet and scores
123 points for the round.
Round 1:
Draw 4 cards.
5
123
2 °=
1 °+ 7
m∠ABD
m∠DBC
°
m∠ABC
Suma uses her full-circle protractor to show that m∠ABD + m∠DBC = m∠ABC.
A
degrees
11 12 1
10
2
9
3
B
8
7
4
6
5
Copyright © Wright Group/McGraw-Hill
D
C
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Name
Date
Time
1 2
4 3
Angle Add-Up Record Sheet
Game 1
°
Round 1:
Draw 4 cards.
°+
m∠ABD
°=
m∠DBC
m∠ABC
°
Round 2:
Draw 2 cards.
°+
m∠ABD
= 90°
m∠DBC
m∠ABC
°
Round 3:
Draw 2 cards.
° = 180°
+
m∠ABD
m∠DBC
m∠ABC
Total Points =
Game 2
°
Round 1:
Draw 4 cards.
°+
Copyright © Wright Group/McGraw-Hill
m∠ABD
°=
m∠DBC
m∠ABC
°
Round 2:
Draw 2 cards.
°+
m∠ABD
= 90°
m∠DBC
m∠ABC
°
Round 3:
Draw 2 cards.
° = 180°
+
m∠ABD
m∠DBC
m∠ABC
Total Points =
509
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