CHAPTER 8
2
Area Rule for Parallelograms
STUDENT BOOK PAGES 242-244
Guided Activity
Goal Develop and use a rule for calculating the area of a parallelogram.
Prerequisite Skills/Concepts
Expectations
• Measure angles with a protractor.
• Calculate the area of a rectangle.
• estimate, measure, and record quantities, using the metric measurement system
• determine the relationships among units and measurable attributes, including the area
of a parallelogram [the area of a triangle, and the volume of a triangular prism]
• estimate, measure, and record [length,] area, [mass, capacity, and volume,] using
the metric measurement system
• construct a [rectangle, a square, a triangle, and a] parallelogram, using a variety of
tools [given the area and/or perimeter]
• determine, through investigation using a variety of tools and strategies, the
relationship between the area of a rectangle and the areas of parallelograms [and
triangles,] by decomposing and composing
• develop the formulas for the area of a parallelogram [and the area of a triangle,]
using the area relationships among rectangles, parallelograms [and triangles]
• solve problems involving the estimation and calculation of [the areas of triangles
and] the areas of parallelograms
Assessment for Feedback
What You Will See Students Doing…
Students will
When Students Understand
If Students Misunderstand
• draw a perpendicular line from the base to the
opposite side of a parallelogram to find its height
• Students will use grid paper or a protractor to
draw a perpendicular line from the base to the
opposite side of a parallelogram to find its height.
• Some students may need to review the use of
protractor to draw the perpendicular line.
• calculate the area of a parallelogram using
a formula
• Students will accurately calculate the area of
a parallelogram using a formula.
• Have students who need support practice
decomposing parallelograms into rectangles by
cutting paper or by using manipulatives such as
tangram triangles or pattern blocks.
• sketch a parallelogram given its area
• Students will accurately and independently sketch
a parallelogram given its area or dimensions.
• Students may need to be explicitly shown how to
use grid paper to sketch a parallelogram given its
area or dimensions.
Preparation and Planning
Pacing
5–10 min Introduction
20–30 min Teaching and Learning
15–20 min Consolidation
Materials
•tangram sets (1/pair)
•1 cm grid transparency (1/pair)
•1 cm grid paper (several
sheets/student)
•rulers (1/student)
•calculators (1/pair)
•protractors (1/pair)
Masters
•Manipulatives Substitute:
Tangrams p. 63
•1 cm Grid Paper, Masters Booklet
p. 29
•Optional: Chapter 8 Mental Math
p. 55
Workbook
p. 72
Vocabulary/
Symbols
perpendicular, base, height
Key
Assessment
of Learning
Question
Question 4, Application of Learning
Meeting Individual Needs
Extra Challenge
• Have students select a rectangular-shaped surface on an item with which
they are familiar and calculate its area. Then have them construct several
parallelograms having the same area. For example, students could choose
to calculate the area of the cover of their math text, which measures about
26 cm by 21 cm and has an area of 546 cm2, and then construct several
parallelograms having the same area, using construction paper. Ask them
to display their results along with a photocopy or tracing of the original
item to scale, and a title such as “Different shape, same area.”
Extra Support
• Have students form a square with the two larger tangram pieces, measure its
dimensions, and find its area using a calculator. Then have them rearrange
the pieces to form a parallelogram, measure its base and height, and calculate
its area. The two areas should be almost the same (allowing some latitude
for rounding error). For example, using standard tangram pieces, the large
triangles form a 9 cm square having an area of 81 cm2. When the triangles
are rearranged as a parallelogram, its base is about 12.5 cm and its height
is about 6.5 cm. Therefore its area is about 81.25 cm2, which is very close
to 81 cm2. Alternately, students can trace the tangram pieces on grid paper
and count squares to determine the areas.
Mathematical Selecting Tools and
Processes
Computational Strategies
Copyright © 2006 by Thomson Nelson
Lesson 2: Area Rule for Parallelograms
17
1.
Introduction (Pairs/Whole Class)
➧ 5–10 min
Distribute tangram sets to pairs of students. Ask the pairs
to work with the two small triangles to find out what other
tangram shapes they can make by putting the two triangles
together in different ways. As a class, discuss what must be
true about the area of all of the shapes formed. Next, have the
pairs measure a side of the square formed from the triangles
and use their calculators to determine the square’s area.
Sample Discourse
“Using the two small tangram triangles, what other
tangram shapes can you form?”
• We made the square.
• We made the parallelogram and medium-sized triangle.
“What must be true about the area of each of the shapes
that you formed? Explain.”
• They were all made from the same two triangles, so they
must all have the same area.
“How is the square different from the other shapes you
could make?”
• It has square corners that form right angles.
“How could we check that the areas of all of these shapes
are the same?”
• We could trace them onto a grid and count the squares covered.
“Another way to say that corners form right angles is to say
that their sides are perpendicular to each other. We can check
this by using a protractor to find out if the sides form an
angle of 90°.” Ask how a protractor can be used to prove that
something is perpendicular. Review its use as you field answers.
Tell students that, in this lesson, they will develop a rule
for calculating the area of a parallelogram.
2.
Teaching and Learning (Whole Class/Pairs/Individual) ➧ 20–30 min
Distribute protractors, centimetre grid transparencies, and
1 cm grid paper to students and have them turn to Student
Book page 242. Together read about Denise’s plans for a
stained glass pane, the central question, and Denise’s solution.
Next, direct student attention to the definitions in the
right margin. Ask students to identify several examples
of perpendicular lines in the classroom. Then draw a
parallelogram on the chalkboard and its height, and
explain how the height of a parallelogram is measured.
Now carefully draw a perpendicular grid over the
parallelogram, making sure that one of the grid lines is
in line with the base of the parallelogram. Have students
identify that the grid lines are perpendicular. Discuss how
a figure can be traced onto grid paper and why its corners
and sides should align with grid lines where possible.
Sample Discourse
“Why are we going to trace shapes onto grid paper?
• The lines are at right angles or perpendicular to each other.
18
Chapter 8: Area
• If we align as many sides of the parallelogram as possible on
the 1 cm grid lines its area will be easy to measure.
Have pairs work through Prompt A. Ensure that they
draw the height correctly and accurately. Show examples of
student work to ensure that students understand that the
parallelogram can be drawn in two different ways and that
the perpendicular can be drawn anywhere along the base.
Allow several pairs to share their findings in C, D, E,
and F. Have students complete Prompts G and H
individually. Remind students to use their protractors to
determine the height of the other two parallelograms.
Reflecting
Here, students should be thinking about how they could
make a rectangle out of a parallelogram and why the rule
for finding the area of a parallelogram is therefore similar.
They should realize that the rule can be used regardless of
how a parallelogram is situated in space when they first
encounter it.
Copyright © 2006 by Thomson Nelson
D. The base measures 3 cm, which is the same as the length
of the rectangle.
E. The height is 2 cm, which is the same as the width of
the rectangle.
F. I could multiply its base times its height, just like I
multiply a rectangle’s length times its width.
G. The area of the second parallelogram is 4 cm × 2 cm = 8 cm2.
The area of the third parallelogram is 3 cm × 3 cm = 9 cm2.
2
3
(Lesson 2 Answers continued on p. 74)
3.
Consolidation ➧ 15–20 min
Checking (Pairs)
Key Assessment of Learning Question (See chart on next page.)
Answers
A. Student drawings should be similar to the following. The
height can be drawn in the position shown or anywhere to
the right along the base of the parallelogram.
height
B. Observe student work to be sure it follows the steps shown.
The height, and thus, the cut, can be in the position shown
or anywhere to its right along the base of the parallelogram.
For intervention strategies, refer to Meeting Individual
Needs or the Assessment for Feedback chart.
3. a) Have students work together in pairs but suggest
that each partner draw a different parallelogram for
the given dimensions.
Practising (Individual)
4. b) & c) Observe students to see if they use their
protractors correctly to find the perpendicular heights.
6. You may need to remind students that the appliqué
will be used 12 times.
Related Question to Ask
Ask
Possible Responses
About Question 5:
• If the base of one parallelogram
was 6 cm, what height would
give you an area if 18 cm2?
• It would have to be 3 cm,
because 3 cm × 6 cm = 18 cm2.
Closing (Whole Class)
C. The area of the rectangle is 2 × 3 = 6 cm2. The area of
the parallelogram is also 6 cm2 because I made the rectangle
from the parallelogram and I used all of it to do so.
Copyright © 2006 by Thomson Nelson
Have students draw a rectangle on grid paper having a width
of 5 cm and a length of 8 cm, find its area, and label it
accordingly. Then ask them to decompose the rectangle into
a parallelogram and explain how the areas of the two shapes
are related by responding to the following prompt: “The rule
for calculating the area of a parallelogram is to multiply its
base times its height. This makes sense because …. ”
Lesson 2: Area Rule for Parallelograms
19
Assessment of Learning—What to Look for in Student Work…
Assessment Strategy: Written Answer
Application of Learning
Key Assessment Question 4
• Calculate the area of each parallelogram. Use a ruler and protractor. Show your work.
1
2
• demonstrates limited ability to apply
mathematical knowledge and skills
in familiar contexts (e.g., has difficulty
using a rule [i.e., base × height] to
calculate the area of each
parallelogram)
• demonstrates some ability to apply
mathematical knowledge and skills
in familiar contexts (e.g., demonstrates
some ability to use a rule [i.e., base
× height] to calculate the area of
each parallelogram)
3
• demonstrates considerable ability
to apply mathematical knowledge
and skills in familiar contexts (e.g.,
uses a rule [i.e., base × height] to
calculate the area of each
parallelogram)
4
• demonstrates sophisticated ability
to apply mathematical knowledge
and skills in familiar contexts (e.g.,
demonstrates sophisticated ability
to use a rule [i.e., base × height] to
calculate the area of each
parallelogram)
Extra Practice and Extension
At Home
• You might assign any of the questions related to this lesson,
which are cross-referenced in the chart below.
• Ask students to find one or more rectangular-shaped items
at home such as a newspaper page, magazine cover, or paper
towel, and then use the item(s) to demonstrate the area rule
for parallelograms to other household members. They should
start by showing their audiences how to measure the dimensions
of the first item and find its area. Then they should draw
a straight line from a top vertex to the base, cut out this
triangle, and add it to the other end of the shape to form
a parallelogram. At this point they should explain to their
audience that the new shape has the same area as the rectangle,
and point out its base and height. They should then state
the area rule for parallelograms and reconstruct the rectangle
before continuing their demonstrations with other items.
Mid-Chapter Review
Student Book p. 249, Questions 3 & 4
Skills Bank
Student Book p. 255, Questions 2, 3, & 4
Problem Bank
Student Book p. 257, Question 3
Chapter Review
Student Book p. 260, Questions 3, 4, & 5
Workbook
p. 72, all questions
Nelson Web Site
Visit www.mathK8.nelson.com and follow
the links to Nelson Mathematics 6, Chapter 8.
Math Background
Students should be very familiar with the area rule for
rectangles. To help them discover the rule for the area of a
parallelogram, ask them to consider how a parallelogram
can be changed into a rectangle, that is, by cutting off a
right triangle at one end and attaching it to the other. A
parallelogram can always be transformed into a rectangle
with the same base, the same height, and the same area.
The rectangle is, in fact, a special kind of parallelogram,
which is why the area rule for parallelograms can also be
applied to rectangles.
Manipulatives Substitute:
Tangrams p. 63
1 cm Grid Paper, Masters
Booklet p. 29
Optional: Chapter 8
Mental Math p. 55
20
Chapter 8: Area
Copyright © 2006 by Thomson Nelson