21st Century Lessons
Area of Parallelogram Lesson
Primary Lesson Designers:
Sarah Cook
Nicola Larcombe
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This project is funded by the
American Federation of Teachers.
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21st Century Lessons – Teacher Preparation
Please do the following as you prepare to deliver this lesson:
•
Spend AT LEAST 30 minutes studying the
Lesson Overview, Teacher Notes on each
slide, and accompanying worksheets.
•
Set up your projector and test this PowerPoint file to make
sure all animations, media, etc. work properly.
•
Feel free to customize this file to match the language and
routines in your classroom.
*1st Time Users of 21st Century Lesson:
Click HERE for a detailed description of our project.
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Lesson Overview (1 of 3)
Lesson Objective
Lesson Objective: SWBAT demonstrate that any parallelogram can
be decomposed and recomposed into a rectangle, and as such to
calculate the area of a parallelogram multiply the base times the
height (and not slant height/side length).
Student- Friendly Objective: SWBAT use an efficient method to find
the area of any parallelogram and explain why it makes sense.
Lesson Description
The overarching design of this lesson is to launch students on a brief
and somewhat directed explore time on how to decompose and
compose a parallelogram in order to find an efficient way to find area.
Following that exploration, a succinct summary will make the key
connections between a parallelogram and a rectangle, revealing that
the same area formula can be used for parallelograms as is used for
rectangles.
The remainder of the lesson involves interleaving practice for
students to identify base and height on parallelograms and calculate
area using the formula. An exit ticket will be used to assess both
student understanding of the concept and accuracy at calculating
area of a parallelogram.
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Lesson Overview (2 of 3)
Lesson
Vocabulary
Base – A side of a figure that a height can be drawn from.
Height –The distance of a line, perpendicular to the base, measured from the base to the opposite side or vertex.
Area – The number of square units that cover a closed figure
Square Unit – Units used to measure area (in2, cm2, ft2, etc.)
Parallelogram – Four-sided figure with opposite sides equal and parallel
Compose – Combining shapes to construct new ones.
Decompose – Breaking shapes apart into familiar pieces.
Perimeter – Distance around the outside of a figure.
Materials
Parallelogram Lab Sheet, Scissors, Area of Parallelogram Class Work handout, Lesson 1
Homework
Scaffolding
Throughout the Explore, Summary, and Practice portions, a handout will be used by students to
organize their notes.
Scaffolding buttons throughout the lesson provide additional supports and hints to help students
make important connections.
Enrichment
Advanced Objective: SWBAT prove two other methods for decomposing and composing
parallelograms that help find the area.
To support students in doing this, give students multiple copies of the lab sheet during the
Explore time, and challenge them to come up with additional methods of decomposing
and composing.
Online
Resources
for Absent
Students
http://www.mathexpression.com/area-of-a-parallelogram.html
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Lesson Overview (3 of 3)
Common Core
State Standard
Common Core State Standard: 6.G.1: Find the area of right triangles, other triangles, special quadrilaterals,
and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these
techniques in the context of solving real-world and mathematical problem
Before and
After
Before: Many of the concepts in this lesson build on learning that has taken place over the past 8 years of the
students’ schooling.
v Introduction to Concept of Area – PK.MD.MA1 (pre-k)
v Composing Shapes – 1.G.2 (first grade)
v Square units – 3.MD.5 (third grade)
v Perpendicular and Parallel Lines – 4.G.1 (fourth grade)
After: The objectives of this lesson build the foundation for students’ future learning in middle and high
school
v Solving real-life math problems with area, surface area, and volume – 7.G.4
v Shape orientation (transformations) – 8.G.4
v Solving equations with one variable – 8.EE.7
Topic
Background
Ancient Egyptian mathematicians have had a significant influence on the development of geometric
concepts. Sources such as the Rhind Papyrus and the Moscow Papyrus demonstrate that the Ancient
Egyptians knew how to compute areas of several geometric shapes (triangles, rectangles, circles, etc.) and
the volumes of cylinders and pyramids – the pyramids being one of the world’s wonders for which they are
famous.
Greek mathematicians were also fundamental contributors to the development of geometric concepts.
Euclid, often referred to as the “Father of Geometry,” produced “Elements,” a series of books that covered
various geometry concepts (including area) as well as much of what is now known as algebra, trigonometry,
and advanced arithmetic.
“The Pythagorean Theorem,” one of the most famous geometric concepts, has been attributed to the Greek
philosopher and mathematician Pythagoras. While the Theorem is not directly related to this lesson or unit,
a strong understanding of triangles and special quadrilaterals will lay the foundation for later learning.
This lesson incorporates the research-based practice of interleaving repeated teacher-guided problems with
individual student practice.
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Warm Up
OBJECTIVE: SWBAT use an efficient method to find the area of any
parallelogram, and explain why it makes sense.
1) What is the area of this
shape?
2) What is the area of this rectangle?
A = 24 sq units
A = 6 sq units
3) Which of the following shapes are parallelograms? Explain
how you know. Shape A, B and D are all parallelograms!
A.
B.
C.
D.
Agenda
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Agenda:
OBJECTIVE: SWBAT use an efficient method to find the area of any
parallelogram, and explain why it makes sense.
1) Warm Up
2) Launch – Building Blocks A, B & C
3) Explore – Partners: Area of Parallelogram
4) Summary – Formula for Area of Parallelogram
5) Practice – Interleaving
6) Assessment – Exit Ticket
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Launch A
Vocabulary
What is the definition of a parallelogram?
A parallelogram is a quadrilateral that has 2 pairs of parallel
sides. Opposite sides have the same length and opposite
angles have equal measurements.
Agenda
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Launch B
(Wait time: 30 seconds)
Can you quickly find the area of this parallelogram by
counting the unit squares?
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1 2 3 4 5
6 7 8 9 10
12
Area = 12 square units
Agenda
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Launch B
Can you quickly find the area of this parallelogram by
counting the unit squares?
Agenda
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Launch B
“I don’t have all day. Counting takes way too long!”
Your challenge:
Develop a more efficient method to determine the area of the
parallelogram.
Agenda
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Launch C
What does it mean to find an efficient method?
Remember the warm up problems?
A = 24 sq units
A = 6 sq units
Did you have to count squares to find the area of the rectangle?
NO!
Multiplying length x width is a more efficient method for finding
the area of a rectangle than counting squares.
Agenda
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Launch C
Let’s consider the first shape in the Warm Up.
How could decomposing (cutting) and composing (putting back
together) into another shape help you find the area of this
shape?
Agenda
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Explore
Click on the timer!
Part 1 - (10 Min)
Work with your PARTNER to find an
efficient method of finding the area
of a parallelogram.
1-Partners
2-Share Out
3-Worksheet
You will get a parallelogram and a
pair of scissors. You can:
-Write on the shape
-Draw on it
-Use scissors on it
In 10 minutes you will be asked to stop and think about it!
HINT
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Agenda
Explore – Student Share Out
Part 2 - (3 Min)
Students share out work.
Classwork Questions
Agenda
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Explore – Complete top half of worksheet
Part 3 - (5 Min)
Fill out the top half of your worksheet.
Agenda
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Summary – Sharing Questions #1-5
#1) Explain what you did to find a quicker way to find the area of the
parallelogram.
Agenda
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Summary – Sharing Questions #1-5
#2) Draw the shapes you decomposed (cut apart) your parallelogram into.
Do you know the names of these shapes?
#3) Did you create any new shape or shapes by composing (putting back
together in a different way)?
Agenda
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Summary – Sharing Questions #1-5
rectangle
original parallelogram
8 cm
12 cm
8 cm
12 cm
#4) What dimensions does your new shape (rectangle) have?
The base is 12 cm. The height is 8 cm.
#5) Can you identify those dimensions on the original parallelogram?
The base is ...?
also 12 cm. The height is ...?
also 8 cm.
Agenda
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Summary – Interactive Worksheet
We are going to complete the rest of the worksheet together. You
will fill in the boxes at the bottom of the first side as we go.
Agenda
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Summary
original parallelogram
8 cm
rectangle
8 cm
12 cm
12 cm
The base and the height in the rectangle match the base and height
in the parallelogram!
sq cm = 12 cm x 8 cm
#6) 96Area
= base x height
A= bxh
Agenda
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Summary
Okay, so that worked with one parallelogram. But can any
parallelogram be decomposed and composed into a rectangle
with the same base and height?
Let’s look at the example from earlier today…
2 cm
2 cm
6 cm
6 cm
A= bxh
= 6 x 2 = 12 cm2
Agenda
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Summary
So…. could we find the area of this rectangle without
cutting and changing it to a rectangle?
Agenda
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Summary
A=bxh
Height
#7) Now that you know this is the formula for area of a
parallelogram, what dimensions must you always know in
order to find area?
 base and height
#8) If we don’t rearrange the shape into a rectangle, could we
still find the height?
 Yes, the height is the
perpendicular distance from
the top to the base.
Base
Agenda
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Summary
slant
height?
8 cm
#9) Can you tell what the
length of the other side
(the slant height) of the
parallelogram is?
 No, not exactly
12 cm
#10) Do you need to know this length in order to find the area of
the parallelogram?
 No, you only need the base and height.
#11) When would you need to know this length?
 You would need to know the slant height to measure perimeter.
Agenda
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(2 minutes)
Summary
#12) Oops! Your sleepy friend
slept through the last 20 minutes
of class! Can you help her out?
• In the space for #12, write her a
note explaining what you learned
so far today.
•Use complete sentences.
Scaffolding
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Agenda
Practice – Interleaving Worksheet
Many kids learn better when the alternate solving problems with
their teacher. Watch me solve one, and then you’ll do one,
then I’ll do one…
Agenda
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Practice
#1)
10 cm
#2)
8 cm
17 in
14 in
16 cm
base = _____
16 cm
height = 8
____
cm
20 in
in
base = 20
_____
height = ______
14 in
Agenda
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Practice
#3)
18 ft
#4)
14 ft
32 m
20 ft
base = _____
20 ft
height = 14
____
ft
27 m
48 m
m
base = 48
_____
height = ______
27 m
Agenda
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Practice
#5)
28 in
#6)
22 in
55 m
16 in
45 m
30 m
16 in
base = _____
m
base = 30
_____
22 in
height = ____
height = ______
45 m
Agenda
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Practice
#7)
#8)
12 ft
25 cm
8 ft
10 ft
12 ft
base = _____
8 ft
height = ____
18 cm
25 cm
base = _____
20 cm
18 cm
height = ______
Agenda
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Practice: Which rectangle has the same area as the green parallelogram?
#9)
#10)
9m
7m
40 in
13 m
9m
A.
30 in
35 in
B.
13 m
A.
30 in
13 m
7m
35 in
40 in
B.
30 in
Agenda
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Practice: Which rectangle has the same area as the blue parallelogram?
#11)
19 ft
#12)
16 ft
20 cm
16 cm
23 ft
32 cm
16 ft
A.
23 ft
19 ft
20 cm
B.
23 ft
16 cm
A
. cm
32
B
32
. cm
Agenda
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Practice: What is the area of the parallelogram?
#14)
#13)
4 in
9 ft
8 in
A=bxh
A = 8 in x 4 in
A = 32 in2
7 ft
A=bxh
A = 7 in x 9 in
A = 63 ft2
Agenda
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Practice: What is the area of the parallelogram?
#16)
#15)
6.7 m
6m
5.4 cm
12 m
A=bxh
A = 12 m x 6 m
A = 72 sq. m
3 cm
15 cm
A=bxh
A = 15 cm x 3 cm
A = 45 sq. cm
Agenda
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Practice: What is the area of the parallelogram?
#17)
#18)
15 in
13 in
7 cm
5 cm
14 cm
9 in
A=bxh
A = 9 in x 13 in
A=bxh
A = 14 cm x 5 cm
A = 70 cm2
A = 117 in2
Agenda
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Practice: What is the area of the parallelogram?
#19)
#20)
4.5 m
6 ft
9m
3.2 ft
6 ft
6.1 m
5.8 ft
A=bxh
A = 6 ft x 3.2 ft
A = 19.2
ft2
A=bxh
A = 9 m x 4.5 m
A = 40.5 m2
Agenda
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Assessment – Exit Ticket!
Complete and hand in the Exit Ticket before you leave!
Agenda
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21st Century Lessons
The goal…
The goal of 21st Century Lessons is simple: We want to assist teachers, particularly in
urban and turnaround schools, by bringing together teams of exemplary educators
to develop units of high-quality, model lessons. These lessons are intended to:
• Support an increase in student achievement;
• Engage teachers and students;
• Align to the National Common Core Standards and the Massachusetts curriculum
frameworks;
• Embed best teaching practices, such as differentiated instruction;
• Incorporate high-quality multi-media and design (e.g., PowerPoint);
• Be delivered by exemplary teachers for videotaping to be used for professional
development and other teacher training activities;
• Be available, along with videos and supporting materials, to teachers free of charge via the
Internet.
• Serve as the basis of high-quality, teacher-led professional development, including mentoring
between experienced and novice teachers.
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21st Century Lessons
The people…
Directors:
Kathy Aldred - Co-Chair of the Boston Teachers Union Professional Issues Committee
Ted Chambers - Co-director of 21st Century Lessons
Tracy Young - Staffing Director of 21st Century Lessons
Leslie Ryan Miller - Director of the Boston Public Schools Office of
Teacher Development and Advancement
Kevin Qazilbash - Co-director of 21st Century Lessons
Lesson Designers:
Nicola Larcombe
Sarah Cook
Meghan McGoldrick
Brian Connor
Tracy Young
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Technology Coordinator:
Shane Ulrich
PowerPoint Designers:
Alex Robinson
LaQueena Williams