Download 16.2 Use Area Models

16.2
Use Area Models
?
Essential Question Why can you multiply to find the area of a rectangle?
Texas Essential
Knowledge and Skills
Why can you multiply
to f ind the area of a
rectangle?
Geometry and Measurement—3.6.C
Determine the area of rectangles with whole number side
lengths in problems using multiplication related to the
number of rows times the number of unit squares in each row
MATHEMATICAL PROCESSES
3.1.A Apply mathematics to problems
3.1.D Communicate mathematical ideas and reasoning
3.1.E Create and use representations
Are You Ready?
Access Prior Knowledge
Use the Are You Ready? 16.2 in the
Assessment Guide to assess students’
understanding of the prerequisite skills
for this lesson.
Vocabulary
Lesson Opener
Making Connections
Go to Multimedia eGlossary at
thinkcentral.com
Invite students to tell you what they know about painting a space.
Ask students if they have ever helped paint an area or room. For students who answer
yes, ask what they painted and about how much paint they used.
Using the Digital Lesson
It may help to draw the parking space as a rectangle for students to understand the
concepts of length and width. Label the length and width with the stated dimensions
from the problem.
Learning Task
What is the problem the students are trying to solve? Connect the story to the problem.
• How long is the parking space? (7 meters)
• How wide is the parking space? (3 meters)
• What is the problem asking you to find? (The area of the parking space)
Literacy and Mathematics
Choose one or more of the following activities.
• Point out to students that equa- in equation is similar to the word equal. Have
students explain how the words equal and equation are related.
• Have students evaluate whether or not they think the problem provides enough
information to be solved, explaining why.
Resources
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 16.2 519A
Name
16.2
?
Unlock the Problem
Geometry and
Measurement—3.6.C
MATHEMATICAL PROCESSES
3.1.A, 3.1.D, 3.1.E
Use Area Models
Essential Question
Why can you multiply to find the area of a rectangle?
Students have learned to find the area of figures by
counting unit squares. Tell them there is an easier
way, using multiplication. Relate the multiplication
strategies students have previously learned to the
area problems on this page.
You can count the number of square units to find the
area of a figure. This rectangle has an area of 6 square
units. You can also use multiplication.
One Way
Melissa has a garden that is shaped like the
rectangle below. Each unit square represents
1 square meter. What is the area of her garden?
One Way Use repeated addition.
number of unit squares are in each row.
• How can you use addition to find a total amount
for a problem involving equal groups? I can use
repeated addition to add equal groups.
6
Remember
A unit square can be different
sizes: 1 square meter, 1 square
foot, 1 square centimeter,
and so on.
6 unit squares
_
3 rows of _
6 =■
_
6 unit squares
_
6 unit squares
_
18
6 +_
6 =_
6 +_
_
18 square meters.
So, the area is _
• How is this figure like an array? The figure has rows
Another Way Use multiplication.
with the same number of unit squares in each row.
Count the number of rows. Count the
number of unit squares in each row.
• How can you use multiplication to find the total
number of unit squares in an array? I can multiply
3 rows of _
6 =■
_
the number of unit squares in each row by the number
of rows.
6 unit squares in each row
_
3 rows
_
This figure is like an array. How do you find
the total number of squares in an array?
© Houghton Mifflin Harcourt Publishing Company
Mathematical Processes
3
5
Count the number of rows.
Count the number of unit squares
in each row.
Write an addition equation.
Another Way
Math Talk
2
4
Unlock
Unlock the
the Problem
Problem
• What do you notice about the number of unit
squares in each row of Melissa’s garden? The same
It is important for students to understand that they
can use multiplication in this example because the
shape is a rectangle. Rectangles are like arrays and
involve rows with the same number of unit squares
in each row.
1
You can multiply the number in each row by
the number of rows.
3 ×_
6 =_
18
_
Math Talk
Write a multiplication equation.
18 square meters.
So, the area is _
Possible explanation: if the figure is shaped like a rectangle,
you can count the number of unit squares in all or multiply the
number of unit squares in each row by the number of rows.
Mathematical Processes
Explain when you can
use different methods to
find the same area.
Module 16 519
Use Math Talk to deepen students’ understandings
that each method gives the same result and why
they can use the methods to find the area of
rectangles.
English Language Learners
ELL Language Support
Interpersonal / Social
Small Group
Leveled Activities
ELPS
Beginning: Activity 44
1.D, 2.C.4, 4.C.1
Intermediate: Activity 3
2.D.2, 2.E.3, 3.F.2
Strategy: Draw
Advanced: Activity 46
1.F, 2.I.4, 4.G.2
Materials: 1-Centimeter Grid Paper (see eTeacher Resources)
Advanced High: Activity 35
1.B.1, 1.F, 3.C.4,
3.G.2
thinkcentral.com for the ELL Activity
Guide containing these leveled activities.
519 Module 16
ELPS 2.C.3, 2.D.1, 2.I.5
• Have students draw a rectangle on a sheet of centimeter grid paper.
What is the area of 1 unit square? 1 square centimeter
• Have students count rows. You may point to the rows in the drawing to
illustrate the word row. Ask how many rows students have. Answers will
vary.
• Have students shade unit squares in each row, one by one. Slide your
finger along a row if necessary. Ask how many unit squares there are in
each row. Answers will vary.
• Have students find the area of their rectangles. Ask how they found the
area. Answers will vary.
Share
Share and
and Show
Show
1.
Find the area of the rectangle in two ways.
Share and Show
3 rows of _
4 =■
_
The first problem connects to the learning model.
Have students use the MathBoard to explain their
thinking.
4 +_
4 +_
4 =_
12
Add. _
Math Talk
3 ×_
4 =_
12
Multiply. _
Mathematical Processes
What is the area of the rectangle?
Which method do you
prefer using? Explain.
12 square units
_
Use the checked exercises for Quick Check. Students
should show their answers for the Quick Check on the
MathBoard.
Possible explanation: I prefer using
Find the area of the rectangle.
multiplication because it is quicker than
Each unit square is 1 square foot. counting or using repeated addition.
2.
3
3.
2
Quick Check
1
a student misses the checked exercises
IF
15 square feet
14 square feet
THEN
Find the area of the rectangle.
Each unit square is 1 square meter.
4.
Differentiate Instruction with
RtI Tier 1 Lesson 76
5.
Math Talk
Mathematical Processes
Use Math Talk to continue the discussion about
which methods students prefer.
27 square meters
32 square meters
If students are having difficulty solving with one
method, suggest that they try using a different
method.
40 square meters
25 square meters
520
© Houghton Mifflin Harcourt Publishing Company
7.
6.
COMMON ERRORS
C
E
Error
Students use the incorrect unit when writing
the area.
Example Students write “square units” or “square
feet” when the measurement should be square
meters.
Springboard to Learning Tell students that they
should circle the unit given in the problem.
Enrich
Visual / Spatial
Individuals
Materials: ruler or measuring tape
• Have students estimate the area of several objects in the classroom
using two different units of measurement. Suggest the following:
computer screen: square inches, square centimeters
doorway: square feet, square meters
• Have students measure the object to the nearest whole unit using
a ruler or measuring tape. Then have students find the area using
multiplication.
• Have students repeat the activity using an object of their own.
Go to Go to thinkcentral.com for additional enrichment
activities in the Enrich Activity Guide.
Lesson 16.2 520
Name
Problem
Problem Solving
Solving
Problem Solving
8.
Problems
In Problem 9, students write their own problems.
Then, students must solve their problems using the
content of the lesson. Be sure that students explain
the method they used in their solution.
Write Math
Use Math Language
Multi-Step Compare the areas of the two
rugs at the right. Each unit square represents
1 square foot. Which rug has the greater
area? Explain.
Areas are the same. Possible explanation: the green rug has 4 rows of 3 or 4 × 3 = 12
square feet; the purple rug has 3 rows of 4 or 3 × 4 = 12 square feet.
Problem 10 requires students to understand the
concept of area in order to draw three different
rectangles with the same area, 24 square units. The
problem requires students to start with the area to
find appropriate side lengths. Since students have not
learned multiplication facts for 12 or 24, encourage
them to write an addition equation if they draw a
rectangle that is 2 units by 12 units or 1 unit by
24 units.
9.
Pose a Problem
A tile company tiled a wall
using square tiles. A mural is
painted in the center. The
drawing shows the design.
The area of each tile used is
1 square foot.
Write a problem that can be solved by using
the drawing. Then solve your problem.
Possible problem: A wall is tiled using yellow tiles and hand-painted tiles. How many more
square feet of yellow tiles were used than square feet of hand-painted tiles?; 2 more square feet
M
Math
on the Spot
Video Tutor
V
Math on the Spot videos are in the
Interactive Student Edition and at
thinkcentral.com.
© Houghton Mifflin Harcourt Publishing Company
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.
10.
Draw and shade three different rectangles
with an area of 24 square units. Then write an addition or
multiplication equation for each area.
Possible equations: 12 + 12 = 24 square units; 3 × 8 = 24 square units;
4 × 6 = 24 square units
Module 16 • Lesson 2 521
3
RtI Tier 1 Lesson 76
2
1
Enrich 75
Name
Name
LESSON
76
Enrich 75
1
Use Area Models
3.6.C
OBJECTIVE Relate area to addition and multiplication by using area models.
Area Riddles
Use the clues to solve the riddle.
You may use grid paper to draw the figure.
Use multiplication to find the area of the rectangle.
Each unit square is 1 square meter.
1.
1
1
2
3
4
5
6
7
2.
I am a square. One of my sides is
9 feet long. What is my area?
4.
I am a rectangle. Two of my sides
are each 7 inches long. My area is
28 square inches. What is the length
of each of my other two sides?
6.
I am a square. My area is 64 square
feet. What is the length of one of my
sides?
8.
Possible answer: I found
Stretch Your Thinking Suppose
you know that a figure is a rectangle
and its area is 8 square meters.
What are all the different wholenumber side lengths the rectangle
could have?
what number, when
1 and 8 meters or 2 and
multiplied by 7, equals 28:
4 meters
81 square feet
3 meters
Step 2 Count the number of unit
squares in each row.
There are 10 unit squares.
Step 1 Count the number of rows.
There are 6 rows.
My sides are all the same length.
My area is 9 square meters. What is
the length of one of my sides?
3.
I am a rectangle. One of my sides is
8 centimeters long. Another side is
6 centimeters long. What is my area?
5.
I am a rectangle. Each of my shorter
sides measure 5 meters. My area is
45 square meters. What is the length
of each of my longer sides?
8 9 10
2
3
4
5
4 inches
48 square centimeters
6
Step 3 Multiply the number in each row by the
number of rows to find the area.
number of rows × number in each row = area
6
×
= 60
10
8 feet
9 meters
© Houghton Mifflin Harcourt Publishing Company
So, the area of the rectangle is 60 square meters.
Find the area of the rectangle.
Each unit square is 1 square meter.
1.
2.
27 square meters
521 Module 16
7.
Geometry and Measurement
How did you find the
answer in Exercise 4?
4
35 square meters
151
Enrich
© Houghton Mifflin Harcourt Publishing Company
E75
Mathematical Processes
Model ¥ Reason ¥ Communicate
Daily
Daily Assessment
Assessment Task
Task
3
Daily Assessment Task
Fill in the bubble for the correct answer choice.
11.
12.
13.
Brianna is coloring a bookmark. Each unit square
is 1 square inch. Which multiplication equation
can be used to find the area of the bookmark?
A
7 + 7 = 14
C
2 × 7 = 14
B
7 × 7 = 49
D
2×2=4
A
9 square feet
C
16 square feet
B
18 square feet
D
20 square feet
YES
15 square feet
B
40 square feet
D
25 square feet
•
Soar to Success Math
Warm-Up 48.28
•
•
Enrich 76
Homework and Practice
Lesson 16.2
Multi-Step Tom and Bill are painting a fence.
Tom paints the yellow area. Bill paints the
white area. Each unit square is 1 square foot.
How many more square feet does Bill paint
than Tom?
C
THEN
IF
NO
10 square feet
1
Can students understand why they multiply to
find the area of a rectangle?
Maria is painting a tile design for her porch.
Each square tile is 1 square foot. What is the area
of the design?
A
2
TEXAS Test Prep Coach
Test Prep Coach helps teachers to identify common
errors that students can make.
In the Test Prep exercise, if students selected:
B They multiplied the number in each row by itself.
TEXAS Test Prep
C They used an addition equation and counted only
4 rows.
Heather drew this rectangle. Which equation
can be used to find the area of the rectangle?
A
5 × 8 = 40
B
8 × 8 = 64
C
8 + 8 + 8 + 8 = 32
D
5 × 5 = 25
D They multiplied the number of rows by itself.
© Houghton Mifflin Harcourt Publishing Company
14.
522
?
Essential Question
Write
Math
Why can you multiply to find the area of a rectangle?
A rectangle is like an array involving rows with an equal
number of unit squares, so I can use multiplication to find
the area of a rectangle just like I can use an array to solve a
multiplication problem.
Differentiated Centers Kit
Activities
Jump to 9
Students complete blue Activity
Card 6 by measuring and then
adding lengths.
Activities
Hurray for Arrays!
Students complete blue Activity
Card 15 by using arrays to model
multiplication facts.
Lesson 16.2 522
5
Ho mewo rk
and Practice
16.2
Geometry and Measurement—3.6.C
MATHEMATICAL PROCESSES 3.1.A, 3.1.D, 3.1.E
Name
Fill in the bubble completely to show your answer.
6.
Use Area Models
2.
21 square meters
What is the area of
the puzzle?
A
5 + 8 = 40
A
27 square feet
B
5 × 5 = 25
B
28 square feet
C
8 × 5 = 40
C
32 square feet
D
8 × 8 = 64
D
25 square feet
4.
36 square meters
35 square meters
Problem
Problem Solving
Solving
© Houghton Mifflin Harcourt Publishing Company
Marisol puts together a floor puzzle.
Each puzzle piece is 1 square foot.
24 square meters
8.
5.
7.
Shoshauna is using one-inch tiles to cover
a table. She wants to put a green border
around the edge of the table. She wants
to make the center of the table yellow.
How many more yellow tiles does she need
than green tiles? Explain.
Multi-Step Lu Chen draws this
diagram of his patio. Each square is
1 square meter. He wants to add 12
square meters to the patio. What will
be the size of the new patio?
A
40 square meters
B
36 square meters
C
42 square meters
D
48 square meters
8 more yellow tiles; Possible explanation: The area
Multi-Step Aliki and Deke are covering a
wall with panels. Aliki covers the yellow
area. Deke covers the white area. Each
panel is 1 square foot. How many more
square feet does Aliki cover than Deke?
of the center of the table is 36 square inches; the area of the border is 28 square inches.
A
6 square feet
C
9 square feet
36 − 28 = 8
B
8 square feet
D
7 square feet
9.
Module 16 • Lesson 2
Homework and Practice
Use the Homework and Practice pages to provide
students with more practice on the concepts and
skills of this lesson.
523-524
Module 16
523
524
© Houghton Mifflin Harcourt Publishing Company
3.
Nolan draws this diagram for a rug.
Which equation
can be used to
find the area?
Find the area of the rectangle.
Each unit square is 1 square meter.
1.
TEXAS Test Prep
Lesson
Lesson Check
Check