16.4
Relate Figures, Fractions, and Area
?
INVESTIGATE
Essential Question How can you divide figures into parts with equal areas and
write the area as a unit fraction of the whole?
Texas Essential
Knowledge and Skills
How can you divide f igures
into par ts with equal areas
and write the area as a unit
fraction of the whole?
Geometry and Measurement—3.6.E
Decompose two congruent two-dimensional figures into
parts with equal areas and express the area of each part as a
unit fraction of the whole and recognize that equal shares of
identical wholes need not have the same shape
MATHEMATICAL PROCESSES
3.1.C Select tools, technology, and techniques
3.1.E Create and use representations
Are You Ready?
Access Prior Knowledge
Use the Are You Ready? 16.4 in the
Assessment Guide to assess students’
understanding of the prerequisite skills
for this lesson.
Vocabulary
Lesson Opener
Making Connections
Invite students to tell you what they know about kites.
Have you ever flown a kite? What shape was your kite? Where did you fly it?
Go to Multimedia eGlossary at
thinkcentral.com
Materials
Pattern blocks, color pencils, ruler
Using the Digital Lesson
Model the shape of the kite with different colored pattern blocks, as in the problem.
Have students make observations about the different shapes and parts of the kite
shape.
Learning Task
What is the problem the students are trying to solve? Connect the story to the problem.
• What are you being asked to find? (How much of the kite is green)
Resources
• What figures can you see in the kite? (Triangles and parallelograms)
• What does Doc want to use to describe the figures in the kite? (Fractions)
Literacy and Mathematics
Choose one or more of the following activities.
• Have students design a kite and write three reasons why their design is creative,
original, and interesting.
• Have students work in pairs to clarify information by asking and answering
questions about the problem.
For the student
For the teacher
Interactive
Student Edition
provides students
with an interactive learning
environment!
Digital Management
Center organizes program
resources by TEKS!
Math on the Spot
Video Tutor
eTeacher
Edition
Online Assessment
System
iTools Virtual
Manipulatives
Soar to Success Math
Online Intervention
Lesson 16.4 531A
Name
16.4
?
Investigate
Be sure students have partitioned the hexagon
tracing into equal parts.
Essential Question
How can you divide figures into parts with equal areas
and write the area as a unit fraction of the whole?
Hands
On
Investigate
Investigate
Point out that since the denominator of the fraction
should show the total number of equal parts, and
there are 2 parts in all, the fraction named in Part D
should be 2_2 .
Materials ■ pattern blocks ■ color pencils ■ ruler
Connect Use what you know about combining and
separating pattern blocks to explore the relationship
between fractions and area.
• How do you know the two figures have the same
area? Possible answer: I can cover the area with two
A. Trace a hexagon pattern block.
trapezoid pattern blocks. The blocks are the same size and
shape.
B. Divide your hexagon into two parts with equal area.
What new figures have you drawn?
trapezoids
or pentagons
________
• What is another way you can be sure the figures
have the same area? Possible answer: cut out the figure
Math Idea
C. Write the fraction that names each part of the
and fold it in half; if the sides match, they are the same
size.
Equal parts of a whole
have equal area.
1
_
2
whole you divided. _
Each part is 1_2 of the whole figure’s area.
Go Deeper
2
_
2
D. Write the fraction that names the whole area. _
To extend students’ thinking, ask what would
happen if they divided the hexagon into three
figures with equal area. Have them tell what fraction
names the area of each part of the divided hexagon
and what fraction names the whole area.
Make
Make Connections
Connections
The rectangle at the right is divided into four parts with equal area.
1
_
4
• Write the unit fraction that names each part of the divided whole. _
5 square units
• What is the area of each part? ___
Make Connections
4 Math Talk
• How many equal shares does it take to make one whole? _
© Houghton Mifflin Harcourt Publishing Company
Remind students that a unit fraction will always have
a numerator of 1.
• What unit fraction names 1 part of the divided
whole? 1_4
Guide students to see how counting the squares in
each part has the same effect as covering each part
with pattern blocks to determine whether the areas
are equal, even though the shapes are different.
English Language Learners
Mathematical Processes
No
• Does each equal share of the whole have the same shape? _
• Is the area of each equal share the same? Explain how you know.
Yes; possible explanation: I can count the unit squares. There are 5 unit squares in
each 1_4 part, so the area of each 1_4 part is the same.
ELL Language Support
Leveled Activities
ELPS
Beginning: Activity 20
1.A.1, 3.G.2, 4.C.3
Intermediate: Activity 3
2.D.2, 2.E.3, 3.F.2
Strategy: Model Language
Advanced: Activity 58
2.C.2, 4.C.3, 4.F.9
Materials: pattern blocks
Advanced High: Activity 59
1.F, 3.E, 3.H.3, 4.C.3
thinkcentral.com for the ELL Activity
Guide containing these leveled activities.
531 Module 16
Geometry and
Measurement—3.6.E
MATHEMATICAL PROCESSES
3.1.C, 3.1.E
Relate Figures, Fractions,
and Area
Module 16 531
Visual
Small Group
ELPS 1.B.1, 2.E.3, 3.F.2
•
•
•
•
Teachers model language to teach pronunciation.
Show each pattern block and tell students its proper name.
Have students repeat the name for each block.
Trace a pattern block and divide it into equal parts. Have students
name the shape of each part.
Share
Share and
and Show
Show
Divide the trapezoid into 3 parts with equal area. Write
the names of the new shapes. Then write the fraction
that names the area of each part of the whole.
1.
Share and Show
The first problem connects to the learning model.
Have students use the MathBoard to explain their
thinking.
Math Talk
3 triangles; _13
Mathematical Processes
Explain how you know
the areas of all the parts
are equal.
Possible explanation: I can cover the area with 3 triangle
pattern blocks. The blocks are the same size and shape,
so the areas of the parts are equal.
• How do you know that your figures in Exercise 1
have the same area? Possible answer: I can cover the
Draw lines to divide the figure into equal parts
that show the fraction given. Possible drawings are shown.
2.
3.
area with 3 triangle pattern blocks.
4.
1
8
1
2
1
6
Math Talk
Mathematical Processes
Use Math Talk to focus on students’ understanding
of partitioning a figure into parts with equal areas.
Draw lines to divide the figure into parts with equal area.
Write the area of each part as a unit fraction. Possible drawings are shown.
5.
6.
Use the checked exercises for Quick Check. Students
should show their answers for the Quick Check on the
MathBoard.
7.
3
2
1
_
3
3 equal parts
6 equal parts
1
_
6
4 equal parts
Quick Check
1
1
_
4
IF
a student misses the checked exercises
Problem
Problem Solving
Solving
THEN
Write Math
If the area of three
is equal
to the area of one , the area of how many
equals
four ? Explain your answer.
© Houghton Mifflin Harcourt Publishing Company
8.
12 blue rhombuses; Possible explanation: I can multiply 3 × 4 = 12.
9.
Apply Show how you can divide the hexagon
into four shapes with equal area. Check students’ drawings.
1
_
4
of the whole shape’s area.
Each part is _
532
Enrich
Differentiate Instruction with
RtI Tier 1 Lesson 81
Problem Solving
Problems
Problem 8 requires students to use proportional
reasoning using given relationships between
objects. Students can use drawings to represent the
relationship. For each yellow hexagon, students
should draw three blue rhombuses.
Problem 9 requires students to partition the hexagon
into four equal parts. Emphasize that all parts should
be equal when they draw the lines to divide the
figure.
Visual / Kinesthetic
Partners
Materials: pattern blocks, paper, scissors
• Have students trace a pattern block and cut it into equal parts. Then
have them label each piece with the unit fraction that names each part
of the divided whole.
• Ask students to trade their pieces with a
1
partner. Partners should use the pieces
6
1
1
to reassemble the whole and name the
6
6
original figure.
• Challenge students to divide the figures
1
1
into as many equal parts as possible
6
6
1
before trading.
6
Go to Go to thinkcentral.com for additional enrichment
activities in the Enrich Activity Guide.
Hexagon
Lesson 16.4 532
Name
Multi-Step Sense or Nonsense?
10.
Problem
Check students’ drawings.
For Problem 10, ask students to compare the figures
that result from dividing the hexagon into 6 equal
parts with the figures that result from dividing the
trapezoid into 3 equal parts.
Divide the hexagon into six
equal parts.
Divide the trapezoid into three
equal parts.
• Are any of the triangles larger than the others? no
• Why can the triangle areas be described with
different fractions if they are the same size?
Possible answer: the fractions compare the area of one part
to the area of the whole. The original shapes of the wholes
were different, so the denominators will be different.
Which pattern block represents
1_ of the whole area?
COMMON ERRORS
C
Which pattern block represents
6
1 of the whole area?
_
3
triangle
____
triangle
____
E
Error
Students may draw lines of division that do
not result in equal parts.
Alexis said the area of 1_3 of the trapezoid is greater than the
area of 1_6 of the hexagon because 1_3 > 1_6 . Does her statement
make sense? Explain your answer.
Example
No; possible explanation: Alexis just looked at the fractional parts _16 and _13 . All the triangle
parts have the same area, but since the wholes are not the same size, you cannot
compare the fractions.
Write a statement that makes sense.
Possible answer: the area of _13 of the trapezoid is equal to the area of _16 of the hexagon.
11.
© Houghton Mifflin Harcourt Publishing Company
Springboard to Learning Use pattern blocks to
show that the hexagon cannot be covered using
the figure created by the partition. Remind
students that unless the parts are equal, you cannot
write a unit fraction to represent the area of a part.
What if you divide the hexagon into 3 equal parts?
Use Math Language to write a sentence that compares
the area of each equal part of the hexagon to each equal
part of the trapezoid.
Possible answer: the area of 1_3 of the hexagon is greater than the area
of 1_3 of the trapezoid.
Module 16 • Lesson 4 533
M
Math
on the Spot
Video Tutor
V
Through the Math on the Spot Video Tutor,
students will be guided through an interactive
solving of this type of H.O.T. problem. Use this
video to also help students solve the H.O.T.
problem in the Interactive Student Edition. With
these videos and the H.O.T. problems, students
will build skills needed in the TEXAS assessment.
3
RtI Tier 1 Lesson 81
2
1
Enrich 77
Name
Name
LESSON
81
3.6.E
Secret Message
Draw lines to divide the figure into the given number of
parts of equal area. Write the fraction that names each part
of the whole.
Possible drawings are shown.
You can separate a plane figure into equal parts to explore
the relationship between fractions and area.
A 10 equal parts
Divide the rectangle into 6 parts with equal area. Write
the fraction that names the area of each part of the whole.
F 4 equal parts
H 8 equal parts
Step 1 Draw lines to divide the rectangle into 6 parts
with equal area. Use the grid to help you.
Step 2 Write the fraction that names each part of the
divided whole.
Math on the Spot videos are in the
Interactive Student Edition and at
thinkcentral.com.
Think: Each part is 1 part out of 6 equal parts.
__ of the whole figure’s area.
Each part is 1
6
Step 3 Write the fraction that names the whole area.
1
__
1
_
10
4
I 6 equal parts
1
_
8
M 12 equal parts
N 2 equal parts
Think: There are 6 equal parts.
__.
The fraction that names the whole area is 6
6
Possible
drawings are shown.
© Houghton Mifflin Harcourt Publishing Company
Draw lines to divide the figure into parts with equal area.
Write the area of each part as a unit fraction.
1
_
1
__
6
12
S 5 equal parts
1
_
2
T 9 equal parts
U 3 equal parts
2. 8 equal parts
1. 4 equal parts
1
_
5
1
__
4
of
Each part is
the whole figure’s area.
533 Module 16
Enrich 77
1
Relate Figures, Fractions, and Area
OBJECTIVE Partition figures into parts with equal areas and express the area
as a unit fraction of the whole.
Geometry and Measurement
1
__
8
1
_
1
_
9
3
Write the fractions in order from least to greatest.
Use the letter for each fraction to write the secret message.
Each part is
of
the whole figure’s area.
1
__
1
__
12 10
M A
161
1
_
9
T
Enrich
© Houghton Mifflin Harcourt Publishing Company
1
_
8
H
1
_
6
I
E77
1
_
5
S
1
_
4
F
1
_
3
U
1
_
2
N
Mathematical Processes
Model ¥ Reason ¥ Communicate
Daily
Daily Assessment
Assessment Task
Task
3
Daily Assessment Task
Fill in the bubble for the correct answer choice.
12.
Inez made a stained glass design with
the rectangles shown at the right. Which
rectangles have the same shape and the
same area?
A
B
13.
C
A and D
B and D
D
B and C
B
C
B
1
_
3
1_
6
C
D
D
A
B
C
D
THEN
IF
NO
•
Soar to Success Math
Warm-Up 5.10
1
_
8
1
_
2
YES
•
•
Multi-Step Anastasia is painting one wall of her room.
She divides the wall into 8 squares with the same area.
She paints half of the squares light green and half of the
squares dark green. What fraction of the wall is one dark
green square?
1
_
2
1_
8
1
Can students divide figures into parts with equal
areas and write the area as a unit fraction of the
whole?
Paulo buys a mirror. The frame is made of 6 same-size
trapezoids. What unit fraction of the frame is each
trapezoid?
A
14.
A and C
A
2
Enrich 77
Homework and Practice
Lesson 16.4
TEXAS Test Prep Coach
1
_
4
1
_
1
Test Prep Coach helps teachers to identify common
errors that students can make.
In the Test Prep exercise, if students selected:
A They counted the number of equal parts.
TEXAS Test Prep
B They multiplied the number of equal parts by the
number of unit squares in each part.
Rav divided the rectangle at the right into six parts with
equal area. What is the area of each part?
A
6 square units
B
24 square units
C
4 square units
D
10 square units
534
© Houghton Mifflin Harcourt Publishing Company
15.
D They added the number of equal parts and the
number of unit squares in each part.
?
Essential Question
Write
Math
How can you divide figures into parts with equal
areas and write the area as a unit fraction of the
whole? Possible answer: I can trace pattern blocks and
then draw lines that divide the figure into equal parts. Then
I can write the area of each part as a fraction by using 1
as the numerator and the number of equal parts as the
denominator.
Differentiated Centers Kit
Activities
Fish for Fractions
Students complete
orange Activity
Card 11 by playing
a Go Fish game in
which they match
fraction symbols,
words, and pictures.
Literature
The Whole Picture
Students read the book
and model fractional
parts.
Activities
Classification Act
Students complete
orange Activity
Card 18 by classifying
two-dimensional
shapes based on their
attributes.
Lesson 16.4 534
5
Geometry and Measurement—3.6.E
MATHEMATICAL PROCESSES 3.1.C, 3.1.E
16.4
Name
Fill in the bubble completely to show your answer.
7.
Relate Figures, Fractions, and Area
Draw lines to divide the figure into equal parts that
show the fraction given. Possible drawings are shown.
1.
2.
Georgie has a rectangular window
that is divided into 6 same-size
squares. What fraction of the
window is 1 square?
A
3.
B
C
1
_
2
D
1
_
6
1
_
4
Draw lines to divide the figure into parts with equal
area. Write the area of each part as a unit fraction. Possible drawings are shown.
4.
9.
5.
4 equal parts
3 equal parts
10.
Problem
Problem Solving
Solving
© Houghton Mifflin Harcourt Publishing Company
Sanchez puts a window in his house
made of 2 same-size triangles. What
unit fraction of the window is
each triangle?
A
4 square units
B
5 square units
C
7 square units
D
6 square units
Multi-Step Perry makes a quilt
with same-size squares. He makes
2 red squares, 2 blue squares,
2 yellow squares, and 2 green
squares. What fraction of the quilt
are the red squares?
B
1
_
2
C
D
Module 16 • Lesson 4
Use the Homework and Practice pages to provide
students with more practice on the concepts and
skills of this lesson.
Module 16
1
_
2
1_
4
1_
3
1_
6
Marcy has a square window that
is divided into 8 same-size squares.
What fraction of the window is
1 square?
A
B
C
D
1
_
3
1_
4
1_
8
1_
2
Rory divides the rectangle into
7 parts with equal area. What is the
area of each part?
A
Homework and Practice
535-536
8.
1
_
3
1
_
4
6.
TEXAS Test Prep
Lesson
Lesson Check
Check
535
536
1
_
4
1_
2
1_
8
1_
3
11.
Multi-Step Doreen makes a
painting with 12 same-size squares.
She has painted one-half of the
squares with a design. How many
squares has Doreen painted?
A
1
B
4
C
3
D
6
© Houghton Mifflin Harcourt Publishing Company
Ho mewo rk
and Practice