formula for the area of a triangle - everyday math · 2011-07-15 · students practice finding the...

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Lesson 8�7 693

Advance PreparationFor the optional Extra Practice activity in Part 3, consider copying the Rugs and Fences Cards on Math Masters, pages 498–501 on cardstock.

Teacher’s Reference Manual, Grades 4–6 pp. 180 –185, 221, 222

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.


• 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 ActivitiesStudents arrange triangles to form

parallelograms. They develop and use a

formula for finding the area of a triangle.

Ongoing Assessment: Recognizing Student Achievement Use journal page 242. [Measurement

and Reference Frames Goal 2]

Key Vocabularyequilateral triangle � isosceles triangle �

scalene triangle � right triangle � base � height

MaterialsMath Journal 2, pp. 240 – 242

Study Link 8�6

Math Masters, pp. 265 and 454A

transparency of Math Masters, p. 403

(optional) � slate � centimeter ruler �

scissors � tape � index card or other

square-corner object

Solving Fraction ProblemsMath Journal 2, p. 243

Students identify fractional parts of

number lines, collections of objects,

and regions.

Math Boxes 8�7Math Journal 2, p. 244

Students practice and maintain skills

through Math Box problems.

Study Link 8�7Math Masters, p. 266

Students practice and maintain skills

through Study Link activities.

ENRICHMENTComparing AreasMath Masters, p. 267

scissors

Students cut apart a regular hexagon and

use the pieces to make area comparisons.

ENRICHMENTFinding the Area and Perimeter of a HexagonMath Masters, p. 268

centimeter ruler

Students use a combination of different

area formulas to find the area of a

nonregular hexagon.

EXTRA PRACTICE

Playing Rugs and FencesMath Masters, pp. 498–502

Student Reference Book, pp. 260 and 261

Students practice finding the perimeter and

area of polygons.

Teaching the Lesson Ongoing Learning & Practice Differentiation Options

Formula for the Areaof a Triangle

Objective To guide the development and use of a formula

for the area of a triangle.f

��

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694 Unit 8 Perimeter and Area

Areas of TrianglesLESSON

8�7

Date Time

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 Tape your parallelogram in the space below.

base = 6 cm base = 6 cm

height = 4 cm height = 4 cm

Area of triangle = 12 cm2 Area of parallelogram = 24 cm2

2. Do the same with Triangles C and D on Math Masters, page 265.

Triangle C Tape your parallelogram in the space below.




1 cm2

A

B

C

D

219-247_EMCS_S_MJ2_G4_U08_576426.indd 240 2/1/11 1:47 PM

Math Journal 2, p. 240

Student Page

Getting Started

Math MessageMake a list of everything that you know about triangles.

Study Link 8�6 Follow-UpHave small groups compare answers and explain their strategies for finding the length of the base when the height and the area are given.

Mental Math and ReflexesDictate large numbers for students to write on their slates.Suggestions:

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?

367,891

500,602

695,003

1,234,895

60,020,597

365,798,421

3,020,300,004

6,000,000,500

90,086,351,007

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.

� An isosceles triangle is a triangle that has at least two sides with the same length and at least two angles with the same measure.

� A scalene triangle is a triangle in which there are no sides of equal length and no angles of equal measure.

� A right triangle is a triangle with one 90° angle.

� Exploring Triangle Properties WHOLE-CLASS ACTIVITY

(Math Masters, p. 454A)

Have students cut out the cards on Math Masters, page 454A. Ask students to sort the triangles into three categories: equilateral, isosceles, and scalene. equilateral: C, F; isosceles: E, G, H, N, P; scalene: A, B, D, I, J, K, L, M, O.

Discuss the angle properties of each type of triangle: equilateral triangles have three angles of equal measure, isosceles triangles have two angles of equal measure, and scalene triangles have no angles of equal measure.

NOTE Because isosceles triangles are

defined as having at least two sides of

equal length, all equilateral triangles are

isosceles. So, some students may include

the equilateral triangles in the isosceles

pile. If this occurs, you may want to have

the class decide on a definition of isosceles

triangle that would exclude the equilateral

triangles. Sample answer: An isosceles

triangle has exactly two sides that are the

same length.

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Areas of Triangles continuedLESSON

8�7

Date Time

3. Do the same with Triangles E and F.

Triangle E Tape your parallelogram in the space below.




4. Do the same with Triangles G and H.

Triangle G Tape your parallelogram in the space below.




5. Write a formula for the area of a triangle.

A = 1 _ 2 ∗ (b ∗ h), or A =

(b ∗ h)

_ 2

length of base

height

G

H

E

F

219-247_EMCS_S_MJ2_G4_U08_576426.indd 241 2/4/11 9:15 AM


Student Page

Lesson 8�7 695

height

baseThe 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.

Links to the Future

Draw a right triangle on the board:

Ask students to tell you the definition of a right triangle. A right triangle is a triangle with one 90° angle.

Ask: Do you see any right triangles in any of the piles? Yes. Triangles A, H, I, and P are right triangles. Can an equilateral triangle be a right triangle? Explain. No. All the angles in an equilateral triangle have the same measure. Can an isosceles triangle be a right triangle? Explain. Yes. Triangles H and P are isosceles and right. Can a scalene triangle be a right triangle? Explain. Yes. Triangles A and I are scalene and right.

Draw the chart shown in the margin on the board. Ask students to help you fill in the chart with “yes” and “no” to show which types of triangles can be right triangles.

Tell students that in this lesson they will develop a formula for the area of a triangle using the formula for the area of a parallelogram.

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.

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.)

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.

PROBLEMBBBBBBBBBBBOOOOOOOOOOBBBBBBBBBBBBBBBBBBBBBBBBBBB MMMMEEEEMLBLELBLEBLELLLBLEBLEBLEBLEBLEBLEBLEEEEMMMMMMMMMMMMMOOOOOOOOOOOOBBBBBLBLBLBBLBLLBLLLLLPROPROPROPROPROPROPROPROPROPROPROPPRPROPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPROROROROROOROOPPPPPPP MMMMMMMMMMMMMMMMMMMMEEEEEEEEEEEEELELELEEEEEEEEEELLLLLLLLLLLLLLLLLLLLLLRRRRRRRRRRRRRRRRPROBLEMSOLVING

BBBBBBBBBBBBBBBBBBBB EEEMMMMMMMOOOOOOOOOBBBBBBBOOOROROROROROROROROROROO LELELELEEEEEELEMMMMMMMMMMMMLEMLLLLLLLLLLLLLLLLLLLLLLRRRRRRRRRRRGGGLLLLLLLLLLLLVINVINVINNNNVINVINVINNVINVINVINVINVINVV GGGGGGGGGGGOLOOOLOOLOLOLOO VINVINVVINLLLLLLLLLVINVINVINVINVINVINVINVINVINVINVINVINNGGGGGGGGGGGOLOOLOLOLOLOLOLOOO VVVLLLLLLLLLLVVVVVVVVOSOSOOSOSOSOSOSOSOSOOSOSOSOSOSOOOOOSOSOSOSOSOSSOOSOSOSOSOSOSOSOSOSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSS VVVVVVVVVVVVVVVVVVVVVLLLLLLVVVVVVVVLVVVVVVVVLLLLLLLLVVVVVLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLSSSSSSSSSSSSSSSSSSSSSSS GGGGGGGGGGGGGGGGGGOOOOOOOOOOOOOOOOOOO GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGNNNNNNNNNNNNNNNNNNNNNNNNNNIIIIIIIIIIIIIIIIIIISOLVING

Right

Equilateral no

Isosceles yes

Scalene yes



1. Circle 1

_ 6 of the triangles. Mark Xs on 2. a. Shade 2

_ 5 of the pentagon.2

_ 3 of the triangles.

b. Shade 3

_ 5 of the pentagon.

3. There are 56 musicians in the school band:1

_ 4 of the musicians play the flute and 1

_ 8 play

the trombone.

a. How many musicians

play the flute?

b. How many musicians

play the trombone?

5. Complete.

a. 3

_ 4 of 120 is 90. b.

of 27 is 18.

c. 5

_ 6 of 120 is 100 . d. 3

_ 10 of 50 is 15.

e.

of 72 is 24. f. 5

_ 4 of 16 is 20 .

6. Fill in the missing fractions on the number line.

2

_ 3

Fractions of Sets and WholesLESSON

8�7

Date Time

59

4. Wei had 48 bean-bag animals in her

collection. She sold 18 of them to another

collector. What fraction of her collection

did she sell?

3

_ 8

0 116

26

36

46

56

1

_ 3

14

7



Student Page

Adjusting the Activity

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.

base =

cm

height =

cm

Area =

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.

8. See the shapes below. Which has the larger area—the star or the square? Explain your answer.

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.

5

2

2

Areas of Triangles continuedLESSON

8�7

Date Time

136

A M

S

Sample answers:

�



Student PageDiscuss 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

parallelogram = 24 cm2; area of triangle = 1 _ 2 the area of parallelogram = 12 cm2.

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.

Since the base and the height of a triangle are the same as the base and the height of the corresponding parallelogram, then:

Area of the triangle = 1 _ 2 the Area of the parallelogram, or

Area of the triangle = 1 _ 2 of (base ∗ height)

Using variables: A = 1 _ 2 of (b ∗ h), or A = 1 _ 2 ∗ (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 241.

� Solving Area Problems PARTNER ACTIVITY

(Math Journal 2, p. 242; Math Masters, p. 403)

Algebraic Thinking 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.

index card

height

MA

S

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:

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

ELL


Math Boxes LESSON

8�7

Date Time

4. If you throw a die 420 times, about how

many times would you expect to

come up? Circle the best answer.

A 70 times

B 100 times

C 50 times

D 210 times

5. Complete.

a. 43 is half as much as 86.

b. 48 is twice as much as 24 .

c. 150 is 3 times as much as 50.

d. 40 is 1

_ 5 of 200 .

e. 135 is 5 times as much as 27.

6. Divide with a paper-and-pencil algorithm.

7,653 � 6 =

1. Write three equivalent fractions for

each fraction.

a. 4

_ 9 ,

,

b. 3

_ 8 ,

,

c. 2

_ 5 ,

,

d. 7

_ 10 ,

,

70

_ 100 35

_ 50

14

_ 20

20

_ 50

10

_ 25 4

_ 10

30

_ 80

15

_ 40 6

_ 16

40

_ 90

20

_ 45 8

_ 18

2. Measure the sides of the figure to the

nearest centimeter to find its perimeter.

Perimeter = 17 cm

3. Complete the “What’s My Rule?” table,

and state the rule.

Rule: -4.6

Sample answers:

131

81

22 23179

162–166

49–51

cm

cm

cm

cm

cm2

2

2

6

5

in out

8.69 4.09

11.03 6.43

19.94 15.34

26.05

1,275 3 _ 6 , or 1,275 1 _ 2

21.45



Student Page

Lesson 8�7 697

Name Date Time

The area of each triangle is given. Find the length of the base.

5. 6.

STUDY LINK

8�7 Areas of Triangles 136

Find the area of each triangle.

1. 2.

Number model: Number model:

Area = square feet Area = square cm

3. 4.

Number model: Number model:

Area = square in. Area = square cm

Try This

4'

8'

5 cm

12 cm

75 cm

34 c

m2 in.

10 in.

12 in.

?

Area = 18 in2

base = in.

5 m

?

Area = 15 m2

base = m

Practice

7. 18, , , 45, , 63, 8. , 16, , 32, , , 56484024872543627

1

_ 2 ∗ (8 ∗ 4) = 16

1

_ 2 ∗ (10 ∗ 2) = 10

1

_ 2 ∗ (12 ∗ 5) = 30

1

_ 2 ∗ (34 ∗ 75) = 1,275

16

10

3

30

1,275

6

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Math Masters, p. 266

Study Link Master

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.

� 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. (See margin.)

Ongoing Assessment: Journal

page 242

Problem 8 �

Recognizing Student Achievement

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.


2 Ongoing Learning & Practice

� Solving Fraction Problems INDEPENDENTACTIVITY

(Math Journal 2, p. 243)

Students identify fractional parts of number lines, collections of objects, and regions.

� Math Boxes 8�7 INDEPENDENTACTIVITY

(Math Journal 2, p. 244)

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.

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 INDEPENDENTACTIVITY

(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.



1 cm

10 cm24 cm2

24 cm2

6 cm2 48 cm2

24 cm2

10 cm

12 cm5 cm

8 cm

3 cm

LESSON

8�7

Name Date Time

Area and Perimeter

131134–136

1. 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.

Total area of hexagon � cm2

2. Find the perimeter of the hexagon. Use a centimeter ruler.

Perimeter � cm

Sample answer:

48

126


Teaching Master

LESSON

8�7

Name Date Time

Comparing Areas

1. 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.

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?

3. 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 trianglescover the equilateral triangle. Six of thesmaller triangles cover the entire hexagon.

There are 4 triangles in the hexagon.

� The large triangle is called an equilateraltriangle. All 3 sides are the same length.

� The smaller triangles are called isoscelestriangles. Each of these triangles has

2 sides that are the same length.

yes

yes


Teaching Master

3 Differentiation Options

ENRICHMENT PARTNERACTIVITY

� Comparing Areas 5–15 Min

(Math Masters, p. 267)

To apply students’ understanding of area, have them compare the areas of a rhombus and a hexagon.

ENRICHMENT INDEPENDENTACTIVITY

� Finding the Area and 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.

One strategy is to partition the polygon as shown below. Another strategy can be found on the reduction of Math Masters, page 268.

B

C

A

EXTRA PRACTICE PARTNER ACTIVITY

� Playing Rugs and Fences 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.

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Copyright

© W

right

Gro

up/M

cG

raw

-Hill

454A

Name Date Time

Exploring Triangle Properties

A

BC

D

E

F G H

I

J K L

M NO

P

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formula for the area of a triangle - everyday math · 2011-07-15 · students practice finding the...

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