formula for the area of a parallelogram - everyday math - … · 2012-05-04 · • describe...
TRANSCRIPT
www.everydaymathonline.com
Common Core State Standards
Interactive Teacher’s
Lesson Guide
CurriculumFocal Points
AssessmentManagement
Family Letters
EM FactsWorkshop Game™
Algorithms Practice
eToolkitePresentations
Lesson 8�6 687
Advance PreparationFor 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
Key Concepts and Skills• Find the area of a rectangle.
[Measurement and Reference Frames Goal 2]
• Develop a formula for calculating the area
of a parallelogram.
[Measurement and Reference Frames Goal 2]
• Calculate perimeter.
[Measurement and Reference Frames Goal 2]
• Identify perpendicular line segments and
right angles.
[Geometry Goal 1]
• Describe properties of parallelograms.
[Geometry Goal 2]
Key ActivitiesStudents 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 Vocabularybase � height � perpendicular
MaterialsMath 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
Playing Fraction OfStudent 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.
Math Boxes 8�6Math 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�6Math Masters, pp. 261 and 262
Students practice and maintain skills
through Study Link activities.
ENRICHMENTConstructing Figures with a Compass and StraightedgeStudent Reference Book, pp. 114, 117,
and 118
compass � straightedge
Students construct figures with a compass
and straightedge.
ENRICHMENTSolving Area and Perimeter ProblemsMath Masters, pp. 263, 264, and 437
scissors � tape
Students explore ways of combining various
two-dimensional shapes to form new shapes.
Teaching the Lesson Ongoing Learning & Practice
132
4
132
4
Differentiation Options
Formula for the Areaof a Parallelogram
Objectives To review the properties of parallelograms; and
to guide the development and use of a formula for the area
of a parallelogram.
t
��������
687_EMCS_T_TLG1_U08_L06_576906.indd 687687_EMCS_T_TLG1_U08_L06_576906.indd 687 2/3/11 11:53 AM2/3/11 11:53 AM
688 Unit 8 Perimeter and Area
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.
heightheight
base
base
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:
● 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.
Getting Started
Math MessageTake 2 short straws, 2 long straws, and 4 twist-ties. Use them to construct a parallelogram.
Study Link 8�5 Follow-Up Have partners compare answers and discuss how they found the missing side measure in Problems 5 and 6.
Mental Math and ReflexesPose multiplication facts and problems. Suggestions:
3 ∗ 7 = 21
4 ∗ 9 = 36
8 ∗ 5 = 40
9 ∗ 6 = 54
90 ∗ 8 = 720
10 ∗ 90 = 900
60 ∗ 70 = 4,200
80 ∗ 30 = 2,400
8 ∗ 52 = 416
4 ∗ 63 = 252
9 ∗ 76 = 684
88 ∗ 5 = 440
688-692_EMCS_T_TLG1_U08_L06_576906.indd 688688-692_EMCS_T_TLG1_U08_L06_576906.indd 688 2/2/11 10:28 AM2/2/11 10:28 AM
Areas of ParallelogramsLESSON
8�6
Date Time
135
1. 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.
Parallelogram A Tape your rectangle in the space below.
base = 6 cm length of base = 6 cm
height = 2 cm width (height) = 2 cm
Area of parallelogram = 12 cm2 Area of rectangle = 12 cm2
2. Do the same with Parallelogram B on Math Masters, page 260.
Parallelogram B Tape your rectangle in the space below.
base = 4 cm length of base = 4 cm
height = 4 cm width (height) = 4 cm
Area of parallelogram = 16 cm2 Area of rectangle = 16 cm2
1 cm2
Sample answer:
Sample answer:
219-247_EMCS_S_MJ2_G4_U08_576426.indd 236 2/1/11 1:47 PM
Math Journal 2, p. 236
Student Page
3. Do the same with Parallelogram C.
Parallelogram C Tape your rectangle in the space below.
base = 4 cm length of base = 4 cm
height = 3 cm width (height) = 3 cm
Area of parallelogram = 12 cm2 Area of rectangle = 12 cm2
4. Do the same with Parallelogram D.
Parallelogram D Tape your rectangle in the space below.
base = 3 cm length of base = 3 cm
height = 4 cm width (height) = 4 cm
Area of parallelogram = 12 cm2 Area of rectangle = 12 cm2
5. Write a formula for the area of a parallelogram.
A = b ∗ h
Areas of Parallelograms continuedLESSON
8�6
Date Time
Sample answer:
Sample answer:
base
he
igh
t
219-247_EMCS_S_MJ2_G4_U08_576426.indd 237 2/1/11 1:47 PM
Math Journal 2, p. 237
Student Page
Lesson 8�6 689
Links to the Future
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.
The use of a formula to calculate the area of a parallelogram is a Grade 5 Goal.
� Developing a Formula for WHOLE-CLASS ACTIVITY
the Area of a Parallelogram(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.
Guide students through the following activity:
1. Cut out Parallelogram A from the master.
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.
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.
Bring students together to develop a formula for the area of a parallelogram. These are three possible lines of reasoning:
� The area of each parallelogram is the same as the area of the rectangle that was made from it.
� The area of the rectangle is equal to the length of its base times its width (also called the height).
PROBLEMBBBBBBBBBBBOOOOOOOOOOOBBBBBBBBBBBBBBBBBBBBBBBBB MMMMMEEEEMMMLEBLELBLEBLELLLBLEBLEBLEBLEBLEBLEBLEEEEMMMMMMMMMMMMMOOOOOOOOOOOOBBBBBLBLBLBLBLBLBLLLLPROPROPROPROPROPROPROPROPROPROPROPPRPROPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPROROROROROOROOPPPPPPP MMMMMMMMMMMMMMMMMMMEEEEEEEEEEEELLELEEEEEEEEEELLLLLLLLLLLLLLLLLLLLLRRRRRRRRRRRRRRRRPROBLEMSOLVING
BBBBBBBBBBBBBBBBBBBB ELEELEEMMMMMMMMMOOOOOOOOOBBBLBLBLBBLBBBLOOORORORORORORORORORORORO LELELELEEEEEELEMMMMMMMMMMMLEMLLLLLLLLLLLLLLLLLLLLLRRRRRRRRRRGGGGLLLLLLLLLLLLLVINVINVINVINNNNVINVINVINNVINVINVINVINVV GGGGGGGGGGGOLOOOLOOLOLOLOO VVINVINLLLLLLLLLVINVINVINVINVINNVINVINVINVINVINVINNGGGGGGGGGGOOOLOLOLOLOLOLLOO VVVLLLLLLLLLLVVVVVVVVOSOSOSOOSOSOSOSOSOSOOSOSOSOSOOOSOOOSOSOSOSOSOSOSOSOOSOSOSOSOSOSOSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSS VVVVVVVVVVVVVVVVVVVVLLLLLLLLVVVVVVVVVLVVVVVVVVLLLLLLLLVVVVVLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLSSSSSSSSSSSSSSSSSSSSS GGGGGGGGGGGGGGGGGGOOOOOOOOOOOOOOOOOO GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGNNNNNNNNNNNNNNNNNNNNNNNNNIIIIIIIIIIIIIIIIIIIISOLVING
688-692_EMCS_T_TLG1_U08_L06_576906.indd 689688-692_EMCS_T_TLG1_U08_L06_576906.indd 689 2/2/11 11:36 AM2/2/11 11:36 AM
690 Unit 8 Perimeter and Area
Areas of Parallelograms continuedLESSON
8 � 6
Date Time
6. Draw a line segment to show the height of Parallelogram DORA.
Use your ruler to measure the base and height.
Then find the area.
base � cm
height � cm
Area � cm2
7. Draw the following shapes on the grid below:
a. A rectangle whose area is 12 square centimeters
b. A parallelogram, not a rectangle, whose area is 12 square centimeters
c. A different parallelogram whose area is also 12 square centimeters
8. What is the area of:
a. Parallelogram ABCD? b. Trapezoid EBCD? c. Triangle ABE ?
cm2 cm2 cm261824
2045
D A
O R
Sample answers:a.
b.
c.
A
B C
E D
Math Journal 2, p. 238
Student Page
2 cm
6 cm
abou
t 2.2
cm
Perimeter = about 16.4 cm
Area = 12 cm2
2 cm
6 cm
2 cm
Perimeter = 16 cm
Area = 12 cm2
� 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:
A = b ∗ h
where b is the length of the base and h is the height.
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.)
� 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
heig
ht
Drawing the height of a parallelogram
Have partnerships complete Problems 7 and 8.
� 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 D
CBProblem 8b
688-692_EMCS_T_TLG1_U08_L06_576906.indd 690688-692_EMCS_T_TLG1_U08_L06_576906.indd 690 2/2/11 10:28 AM2/2/11 10:28 AM
Math Boxes LESSON
8�6
Date Time
4. Add or subtract.
a. 3 _ 16
+ 7 _ 16
=
b. 2 _ 3 + 1 _
6 =
c. = 9 _ 10
- 3 _ 10
d. = 3 _ 4 - 3 _
8
3
_ 8
6
_ 10 , or 3
_ 5
5
_ 6
10
_ 16 , or
5
_ 8
1. Dimensions for actual rectangles are
given. Make scale drawings of each
rectangle described below.
Scale: 1 cm represents 20 meters.
a. Length of rectangle: 80 meters
Width of rectangle: 30 meters
b. Length of rectangle: 90 meters
Width of rectangle: 50 meters
2. What is the area of the parallelogram?
Number model: 7 ∗ 3 = 21
Area = 21 in2
3. A jar contains
8 blue blocks,
4 red blocks,
9 orange blocks, and
4 green blocks.
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?
8
_ 25
5. Multiply. Use a paper-and-pencil algorithm.
6,142 = 83 ∗ 74
145
135 45
55 57 18 19
a.
b.
3"
7"
�
219-247_EMCS_S_MJ2_G4_U08_576426.indd 239 2/1/11 1:47 PM
Math Journal 2, p. 239
Student Page
Name Date Time
STUDY LINK
8�6 Areas of Parallelograms
Find the area of each parallelogram.
1. 2.
Number model: Number model:
Area = square feet Area = square centimeters
3. 4.
Number model: Number model:
Area = square feet Area = square centimeters
6 ft
4 ft 65 cm
72 c
m
Try This
The area of each parallelogram is given. Find the length of the base.
5. 6.
Area = 26 square inches Area = 5,015 square meters
base = inches base = meters
8 cm
3 cm
2 in.
?
59 m
?
135
9'
4'
8 ∗ 3 = 24
65 ∗ 72 = 4,680
4 ∗ 9 = 36
6 ∗ 4 = 24
24
4,680
85
36
24
13
247-277_EMCS_B_MM_G4_U08_576965.indd 261 2/1/11 2:17 PM
Math Masters, p. 261
Study Link Master
Lesson 8�6 691
� 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
2 Ongoing Learning & Practice
� Playing Fraction Of PARTNER ACTIVITY
(Student Reference Book, pp. 244 and 245; Math Masters, pp. 477–480)
Students play Fraction Of to practice finding fractions of collections. See Lesson 7-3 for additional information.
� Playing Angle Add-Up PARTNER ACTIVITY
(Math Masters, pp. 439 and 507–509)
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 INDEPENDENTACTIVITY
(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: Math Boxes
Problem 4 �Recognizing Student Achievement
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.
[Operations and Computation Goal 5]
� Study Link 8�6 INDEPENDENTACTIVITY
(Math Masters, pp. 261 and 262)
Home Connection Students calculate the areas of parallelograms on Math Masters, page 261.
NOTE Math Masters, page 262 should be completed before Lesson 9-1,
in which students share and discuss examples of percents they have collected.
688-692_EMCS_T_TLG1_U08_L06_576906.indd 691688-692_EMCS_T_TLG1_U08_L06_576906.indd 691 2/3/11 12:35 PM2/3/11 12:35 PM
692 Unit 8 Perimeter and Area
LESSON
8�6 Perimeter and Area
263
150
Math Masters, p. 263
Teaching Master
LESSON
8�6
Name Date Time
Perimeter and Area continued
131133–136
Cut out and use only the shapes in the top half of Math Masters,
page 263 to complete Problems 1–5.
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:
a. the small triangle cm2
b. the square cm2
c. the parallelogram cm2
4. a. What is the perimeter of the large square
you made in Problem 1? cm
b. What is the area of that square? cm2
5. What is the area of the large triangle you
made in Problem 2? cm2
6. 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.
a. 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.
32
64
32
16
16
8
Try This
Answers vary.
Math Masters, p. 264
Teaching Master
3 Differentiation Options
ENRICHMENT INDEPENDENTACTIVITY
� 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.
ENRICHMENT PARTNER ACTIVITY
� 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:
Name Date Time
Dot Paper
pyg
gp
Math Masters, p. 437
Teaching Aid Master
688-692_EMCS_T_TLG1_U08_L06_576906.indd 692688-692_EMCS_T_TLG1_U08_L06_576906.indd 692 2/2/11 10:28 AM2/2/11 10:28 AM
Name Date Time
507
Copyright
© W
right
Gro
up/M
cG
raw
-Hill
132
4Angle Add-Up
Materials □ 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.
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.
Unaffected Converted436-440_457_471_EMCS_B_MM_G4_PROJ_576965.indd 507436-440_457_471_EMCS_B_MM_G4_PROJ_576965.indd 507 3/10/11 2:55 PM3/10/11 2:55 PM
Name Date Time
508
Copyrig
ht ©
Wrig
ht G
roup/M
cG
raw
-Hill
132
4Angle 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 2 °
=
°
m∠ABD 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 1 ° +
7 2 °
=
°
m∠ABD m∠DBC m∠ABC
Suma uses her full-circle protractor to show that m∠ABD + m∠DBC = m∠ABC.
123
12
6
11
5
10
4
1
7
2
8
39
degrees
A
B
D
C
Unaffected Converted436-440_457_471_EMCS_B_MM_G4_PROJ_576965.indd 508436-440_457_471_EMCS_B_MM_G4_PROJ_576965.indd 508 3/10/11 2:55 PM3/10/11 2:55 PM
Name Date Time
509
Copyright
© W
right
Gro
up/M
cG
raw
-Hill
132
4Angle Add-Up Record Sheet
Game 1
Round 1:
Draw 4 cards. ° +
°
=
°
m∠ABD m∠DBC m∠ABC
Round 2:
Draw 2 cards. ° +
°
=
90°
m∠ABD 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. ° +
°
=
°
m∠ABD m∠DBC m∠ABC
Round 2:
Draw 2 cards. ° +
°
=
90°
m∠ABD m∠DBC m∠ABC
Round 3:
Draw 2 cards.
°
+
°
=
180°
m∠ABD m∠DBC m∠ABC
Total Points =
Unaffected Converted436-440_457_471_EMCS_B_MM_G4_PROJ_576965.indd 509436-440_457_471_EMCS_B_MM_G4_PROJ_576965.indd 509 3/10/11 2:55 PM3/10/11 2:55 PM