Objective
To guide the development and use of a formula for
the area of a triangle.
1
materials
Teaching the Lesson
Key Activities
Students arrange triangles to form parallelograms. They develop and use a formula for
finding the area of a triangle.
Key Concepts and Skills
• Find the areas of rectangles and parallelograms.
[Measurement and Reference Frames Goal 2]
• Develop a formula for calculating the area of a triangle.
[Measurement and Reference Frames Goal 2]
• Identify perpendicular line segments and right angles. [Geometry Goal 1]
• Describe properties of and types of triangles. [Geometry Goal 2]
• Evaluate numeric expressions containing parentheses.
[Patterns, Functions, and Algebra Goal 3]
Key Vocabulary • equilateral triangle • isosceles triangle • base • height
Math Journal 2, pp. 240–242
Study Link 8 6 (Math Masters,
p. 261)
Teaching Master (Math Masters,
p. 265)
Transparency (Math Masters,
p. 403; optional)
slate
centimeter ruler
scissors
tape
index card or other
square-corner object
Ongoing Assessment: Recognizing Student Achievement Use journal page 242.
[Measurement and Reference Frames Goal 2]
2
materials
Ongoing Learning & Practice
Students identify fractional parts of number lines, collections of objects, and regions.
Students practice and maintain skills through Math Boxes and Study Link activities.
3
materials
Differentiation Options
ENRICHMENT
Students cut apart a regular
hexagon and use the pieces
to make area comparisons.
ENRICHMENT
Students use a combination
of different area formulas
to find the area of a
nonregular hexagon.
Math Journal 2, pp. 243 and 244
Study Link Master (Math Masters,
p. 266)
EXTRA PRACTICE
Students play Rugs and
Fences to practice finding
the perimeter and area of
a polygon.
Student Reference Book,
pp. 260 and 261
Teaching Masters (Math Masters,
pp. 267 and 268)
Game Masters (Math Masters,
pp. 498–502)
scissors; centimeter ruler
See Advance Preparation
Additional Information
Advance Preparation For the optional Extra Practice activity in Part 3, consider copying the
Rugs and Fences Cards on Math Masters, pages 498–501 on cardstock.
Technology
Assessment Management System
Journal page 242, Problem 8
See the iTLG.
Lesson 8 7
693
Getting Started
Mental Math and Reflexes
Math Message
Dictate large numbers for students to write on their slates.
Suggestions:
Make a list of everything that you know
about triangles.
1,234,895
60,020,597
365,798,421
367,891
500,602
695,003
3,020,300,004
6,000,000,500
90,086,351,007
For each number, ask questions such as the following:
• Which digit is in the millions place?
• What is the value of the digit x?
• How many hundred millions are there?
Study Link 86 Follow-Up
Have small groups compare answers and explain
their strategies for finding the length of the base
when the height and the area are given.
1 Teaching the Lesson
Math Message Follow-Up
WHOLE-CLASS
ACTIVITY
As students share their responses, write them on the board.
The list might include:
A triangle is a three-sided polygon.
The sum of the measures of the angles in a triangle is 180°.
A triangle has three vertices.
A triangle is a convex polygon.
An equilateral triangle is a triangle in which all three sides
have the same measure, and all three angles have the same
measure. An equilateral triangle is a regular polygon.
Student Page
Date
LESSON
8 7
Time
Areas of Triangles
136
1. Cut out Triangles A and B from Math Masters, page 265.
DO NOT CUT OUT THE ONE BELOW. Tape the two triangles
together to form a parallelogram.
Triangle A
1 cm 2
Tape your parallelogram in the space below.
An isosceles triangle is a triangle in which two sides have
the same measure.
Tell students that in this lesson they will use the formula for
the area of a parallelogram to develop a formula for the area
of a triangle.
B
A
6
4
base height Area of triangle 6
4
base cm
height cm
12
2
cm
Links to the Future
cm
cm
Area of parallelogram 24
2
cm
2. Do the same with Triangles C and D on Math Masters, page 265.
Triangle C
Tape your parallelogram in the space below.
D
C
base height 4
4
Area of triangle base cm
height cm
8
cm2
4
4
cm
cm
Area of parallelogram 240
Math Journal 2, p. 240
694
Unit 8 Perimeter and Area
16
cm2
The use of a formula to calculate the area of a triangle is a Grade 5 Goal.
Developing a Formula for
WHOLE-CLASS
ACTIVITY
the Area of a Triangle
(Math Journal 2, pp. 240 and 241; Math Masters, p. 265)
Draw a triangle on the board. Choose one of the sides—the side on
which the triangle “sits,” for example—and call it the base. Label
the base in your drawing. Explain that base is also used to mean
the length of the base.
height
The shortest distance from the vertex above the base to the base is
called the height of the triangle. Draw a dashed line to show the
height and label it. Include a right-angle symbol. (See margin.)
base
The height of a triangle is measured
along a line segment perpendicular
to the base. As with parallelograms,
any side of a triangle can be the base.
The choice of the base determines
the height.
Ask the class to turn to journal page 240 while you distribute
copies of Math Masters, page 265. Point out that Triangles A and
B on the master are the same as Triangle A on the journal page.
Guide students through the following activity:
1. Cut out Triangles A and B from the master. Make sure students
realize that the triangles have the same area and are congruent.
2. Tape the triangles together at the shaded corners to form a
parallelogram.
3. Tape the parallelogram in the space next to Triangle A in
the journal.
Discuss the relationship between the area of the triangle and the
area of the parallelogram. Triangles A and B have the same area.
Therefore, the area of either triangle is half the area of the
parallelogram.
4. Record the dimensions and areas of the triangle and the
parallelogram. Base of triangle and parallelogram 6 cm;
height of triangle and parallelogram 4 cm; area of
1
parallelogram 24 cm2; area of triangle 2 the area of
2
parallelogram 12 cm .
Have students repeat these steps with Triangles C and D, E and
F, and G and H. Then bring the class together to state a rule and
write a formula for the area of a triangle.
Student Page
Date
LESSON
8 7
Time
Areas of Triangles
continued
3. Do the same with Triangles E and F.
Triangle E
Tape your parallelogram in the space below.
Since the base and the height of a triangle are the same as the
base and the height of the corresponding parallelogram, then:
1
Area of the triangle 2 of (base height)
Using variables:
1
1
A 2 of (b h), or A 2 (b h)
F
1
Area of the triangle 2 the Area of the parallelogram, or
E
7
2
base height base cm
height cm
7
Area of triangle cm2
cm
cm
Area of parallelogram 14
cm2
4. Do the same with Triangles G and H.
Triangle G
Tape your parallelogram in the space below.
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 241.
7
2
G
4
3
base height Area of triangle base cm
height cm
6
cm2
5. Write a formula for the area of a triangle.
A
1
2
(b h), or A 4
3
cm
cm
Area of parallelogram (b h)
2
12
cm2
height
length of base
241
Math Journal 2, p. 241
Lesson 8 7
695
Student Page
Date
Solving Area Problems
Time
LESSON
Areas of Triangles
8 7
continued
136
(Math Journal 2, p. 242; Math Masters, p. 403)
6. Draw a line segment to show the height of Triangle SAM.
Use your ruler to measure the base and height of the
triangle. Then find the area.
5
2
2
base height Area S
cm
A
cm
PARTNER
ACTIVITY
Work with the class on Problem 6. Students can place an index
card (or other square-corner object) on top of the triangle, align the
bottom edge of the card with the base (making sure that one edge
of the card passes through point S), and then draw a line for the
height. Students will need a centimeter ruler to measure the base
and the height.
M
cm2
7. Draw three different triangles on the grid below. Each triangle must have
an area of 3 square centimeters. One triangle should have a right angle.
Sample answers:
index card
S
8. See the shapes below. Which has the larger area—the star or the square? Explain your answer.
height
Neither. Both have an area of 16 sq units. Area of
square 4 4 16; area of star area of square in
center (4 area of a triangle) 4 (4 3) 16.
A
M
ELL
Adjusting the Activity
On the board, draw a triangle that has an obtuse angle as one of its
base angles. Have students draw the height of the triangle. Demonstrate by
extending the base along the side of the obtuse angle and drawing a
perpendicular line from the opposite vertex to the extended base. For example:
242
Math Journal 2, p. 242
A U D I T O R Y
K I N E S T H E T I C
T A C T I L E
V I S U A L
Have students complete Problems 7 and 8.
There are many possibilities for Problem 7. You can use a
transparency of a 1-cm grid (Math Masters, page 403) on the
overhead projector to display a number of them.
Student Page
Date
It may surprise some students that the star and the square in
Problem 8 have the same area. One way to find the area of the
star is to think of it as a square with a triangle attached to
each of its sides.
Time
LESSON
Fractions of Sets and Wholes
8 7
1
6
1. Circle of the triangles. Mark Xs on
2
of the triangles.
3
2. a. Shade
2
5
59
of the pentagon.
3
b. Shade of the pentagon.
5
4. Wei had 48 bean-bag animals in her
3. There are 56 musicians in the school band:
1
1
of the musicians play the flute and play
4
8
collection. She sold 18 of them to another
collector. What fraction of her collection
did she sell?
the trombone.
a. How many musicians
3
8
14
play the flute?
b. How many musicians
7
play the trombone?
5. Complete.
3
a. of
4
120
5
c. of 120 is
6
1
3
e.
is 90.
100
2
3
b.
3
d. of
10
.
50
5
f. of 16 is
4
of 72 is 24.
Ongoing Assessment:
Recognizing Student Achievement
of 27 is 18.
is 15.
20
1
6
2
6
3
6
4
6
5
6
Use journal page 242, Problem 8 to assess students’ ability to describe a
strategy for finding and comparing the areas of a square and a polygon.
Students are making adequate progress if they are able to count unit squares
and partial squares to find the areas of these two shapes. Some students may
describe the use of a formula to calculate the areas of the triangles.
.
6. Fill in the missing fractions on the number line.
0
Journal
page 242
Problem 8
1
[Measurement and Reference Frames Goal 2]
243
Math Journal 2, p. 243
696
Unit 8 Perimeter and Area
Student Page
Date
Math Boxes
8 7
1. Write three equivalent fractions for
4
a. 9
Solving Fraction Problems
INDEPENDENT
ACTIVITY
3
b. 8
2
c. 5
(Math Journal 2, p. 243)
7
d. 10
Students identify fractional parts of number lines, collections of
objects, and regions.
Math Boxes 8 7
Sample answers:
,
,
,
,
20
45
15
40
10
25
35
50
,
,
,
,
40
90
30
80
20
50
70
100
Mixed Practice Math Boxes in this lesson are paired
with Math Boxes in Lesson 8-5. The skill in Problem 6
previews Unit 9 content.
out
A
70 times
4.09
B
100 times
11.03
6.43
C
50 times
19.94
15.34
D
210 times
26.05
21.45
1
d. 40 is of
5
e.
Writing/Reasoning Have students write a response to the
following: For Problem 4, write two probability questions
for which the correct answer would be D—210 times.
Sample answers: If you throw a die 420 times, about how many
times would you expect an even number to come up? If you throw a
die 1,260 times, how many times would you expect a 6 to come up?
Study Link 8 7
150
135
17
cm
cm
131
4. If you throw a die 420 times, about how
in
162–166
81
6. Divide with a paper-and-pencil algorithm.
3
7,653 / 6 is half as much as 86.
b. 48 is twice as much as
c.
Perimeter 8.69
43
cm
many times would you expect
to
come up? Circle the best answer.
4.6
5. Complete.
a.
5
cm
cm
6
and state the rule.
(Math Journal 2, p. 244)
nearest centimeter to find its perimeter.
2
3. Complete the “What’s My Rule?” table,
INDEPENDENT
ACTIVITY
2. Measure the sides of the figure to the
2
49–51
Rule:
8
18
6
16
4
10
14
20
cm
each fraction.
2
2 Ongoing Learning & Practice
Time
LESSON
24
1
1,275 6, or 1,275 2
.
is 3 times as much as 50.
200
.
is 5 times as much as 27.
22 23
179
244
Math Journal 2, p. 244
INDEPENDENT
ACTIVITY
(Math Masters, p. 266)
Home Connection Students calculate the areas of
triangles. They continue to work on Math Masters,
page 262, which should be completed before Lesson 9-1.
Study Link Master
Name
Date
STUDY LINK
Time
Areas of Triangles
8 7
136
Find the area of each triangle.
2.
5 cm
1.
4'
8'
12 cm
1
2
Number model:
16
Area (8 4) 16
1
Number model: 2
square feet
Area (12 5) 30
square cm
4.
2 in.
75 cm
34 cm
3.
30
10 in.
1
Number model: 2
10
Area (10 2) 10
1
Number model: 2
Area square in.
(34 75) 1,275
1,275 square cm
Try This
The area of each triangle is given. Find the length of the base.
Area 18 in2
base 3
Area 15 m2
6.
in.
base 5m
5.
12 in.
6
m
?
?
Practice
7.
18,
27 , 36 , 45, 54 , 63, 72
8.
8
, 16,
24 , 32, 40 , 48 , 56
Math Masters, p. 266
Lesson 8 7
697
Teaching Master
Name
Date
LESSON
87
Time
Cut out the hexagon below. Then cut out the large equilateral triangle.
You should end up with one large triangle and three smaller triangles.
2.
Use the large triangle and the three smaller triangles to form a rhombus.
3.
3 Differentiation Options
Comparing Areas
1.
PARTNER
ACTIVITY
ENRICHMENT
a.
Sketch the rhombus in the
space to the right.
b.
Is the area of the rhombus the
same as the area of the hexagon?
c.
Is it possible for two different
shapes to have the same area?
Comparing Areas
yes
5–15 Min
(Math Masters, p. 267)
yes
Put all the pieces back together to form a hexagon with an equilateral
triangle inside.
How can you show that the area of the hexagon is twice the area of the
large triangle?
Sample answer: The three smaller triangles
cover the equilateral triangle. Six of the
smaller triangles cover the entire hexagon.
To apply students’ understanding of area, have them compare the
areas of a rhombus and a hexagon.
INDEPENDENT
ACTIVITY
ENRICHMENT
There are 4 triangles in the hexagon.
Finding the Area and
The large triangle is called an equilateral
triangle. All 3 sides are the same length.
The smaller triangles are called isosceles
triangles. Each of these triangles has
2 sides that are the same length.
5–15 Min
Perimeter of a Hexagon
(Math Masters, p. 268)
To apply students’ understanding of area formulas, have
them find the area and perimeter of a nonregular
hexagon. Counting squares to find the area is not
permitted; students are encouraged to divide the hexagon into
figures and then use a formula to calculate the area of each figure.
Math Masters, p. 267
One strategy is to partition the polygon as shown below.
Another strategy can be found on the reduction of Math Masters,
page 268.
Teaching Master
Name
LESSON
8 7
1.
Date
Find the area of the hexagon below without counting squares.
Hint: Divide the hexagon into figures for which you can calculate
the areas: rectangles, parallelograms, and triangles. Use a formula
to find the area of each of the figures. Record your work.
126
131
134–136
C
cm2
Find the perimeter of the hexagon. Use a centimeter ruler.
Perimeter 48
cm
Sample answer:
10 cm
24 cm2
EXTRA PRACTICE
12 cm
5 cm
48 cm2
6 cm2
2
8 cm 24 cm
24 cm2
10 cm
1 cm
3 cm
Math Masters, p. 268
698
B
Area and Perimeter
Total area of hexagon 2.
A
Time
Unit 8 Perimeter and Area
Playing Rugs and Fences
PARTNER
ACTIVITY
15–30 Min
(Math Masters, pp. 498–502; Student Reference Book,
pp. 260 and 261)
To practice calculating the area and perimeter of a polygon, have
students play Rugs and Fences. See Lesson 9-2 for additional
information.