Download 3rd Grade Mathematics

 3rd Grade Mathematics Unit #3 : Relating Area to Multiplication and Division Pacing: 29 Days Unit Overview In this unit, students extend their understanding of multiplication and division (Unit 2) by applying it to real-world problems involving area. This unit is designed
to be a short, yet thorough, dive into many of the measurement concepts of the grade. Students will reason abstractly and quantitatively by connecting concepts of
area with multiplication and arrays (MP.2). The unit emphasizes modeling, as students are required to create area models and various polygons to solve area and
perimeter problems (MP.4). Students also compare rectangles with the same area but different dimensions and look for patterns in the shapes of the rectangles
(MP.7).
Prerequisite Skills
• Find the sum of an equation with 3 or more addends.
• Multiplication facts 1-12
• Partition (decompose) a rectangle into rows and
columns of same-size squares.
• Apply the distributive property to find the product of
two factors.
• Construct rectangular arrays.
• Write a multiplication equation that represents a given
array.
1 | P a g e Vocabulary
Perimeter
Area
Unit Square
Square Unit
Formula
Dimensions
Length
Width
Rectangular
Composite Figure
Decompose
Partition
Distributive property
Rectilinear Shape
Array
Column
Row
Precision
Gaps
Overlaps
Tiling
Polygon
Quadrilateral
Mathematical Practices
MP.1: Make sense of problems and persevere in solving
them
MP.2: Reason abstractly and quantitatively
MP.3: Construct viable arguments and critique the
reasoning of others
MP.4: Model with mathematics
MP.5: Use appropriate tools strategically
MP.6: Attend to precision
MP.7: Look for and make use of structure
MP.8: Look for and express regularity in repeated
reasoning
Common Core State Standards
Progression of Skills
nd
2 Grade
N/A
Additional
Standards
(10%)
3.MD.8: Solve
Problems Involving
Perimeters of Polygons
N/A
3.MD.5: Understand Concepts of
Area Measurement and Square Units
3.MD.6: Measure Area by Counting Unit Squares
3rd Grade
3.MD.5: Recognize area as
an attribute of plane figures
and understand concepts of
area measurement.
3.MD.6: Measure areas by
counting unit squares
(square cm, square m, square
in, square ft, and improvised
units).
3.MD.7: Relate Area to Multiplication and Division
3.MD.7a: Find the Area of Rectangles by Tiling
3.MD.7b: Find the Area of Rectangles by Multiplying
Major
Standards
(70%)
3.MD.7c: Area and Distributive Property
3.MD.7d: Area Addition and Decomposition
3.OA.4: Unknowns in Multiplication
and Division Equations
According to the PARCC Model Content Framework,
Standard 3.MD.7 should serve as opportunity for in-depth focus based on
the content of this unit:
“Area is a major concept within measurement, and area models must
function as a support for multiplicative reasoning in grade 3 and beyond.”
Measurement contexts for multiplication and division should serves as an
example of opportunities for connecting mathematical content and
mathematical practices:
“Students will analyze a number of situation types for multiplication and
division, including arrays and measurement contexts. Extending their
understanding of multiplication and division to these situations requires that
they make sense of problems and persevere in solving them (MP.1), look for
and make use of structure (MP.7), as they model these situations with
mathematical forms (MP.4), and attend to precision (MP.6) as they
distinguish different kinds of situations over time (MP.8).
2 | P a g e 2.G.2: Partition a rectangle
into rows and columns of
same-size squares and count
to find the total number of
them.
2.MD.1: Measure the length
of an object by selecting and
using appropriate tools such
as rulers, yardsticks, meter
sticks, and measuring tapes.
3.OA.3: Determine whether
a group of objects (up to 20)
has an odd or even number
of members, e.g., by pairing
objects or counting them by
2s; write an equation to
express an even number as a
sum of two equal addends.
3.MD.7: Relate area to the
operations of multiplication
and addition.
3.MD.8: Solve real world
and mathematical problems
involving perimeters of
polygons, including finding
the perimeter given the side
lengths, finding an unknown
side length, and exhibiting
rectangles with the same
perimeter and different areas
or with the same area and
different perimeters.
3.OA.4: Determine the
unknown whole number in a
multiplication or division
equation relating three whole
numbers. For example,
determine the unknown
number that makes the
equation true in each of the
equations 8 × ? = 48, 5 = _
÷ 3, 6 × 6 = ?
4th Grade
N/A
4.MD.3: Apply the area and
perimeter formulas for
rectangles in real world and
mathematical problems. For
example, find the width of a
rectangular room given the
area of the flooring and the
length, by viewing the area
formula as a multiplication
equation with an unknown
factor.
N/A
4.MD.3: Apply the area and
perimeter formulas for
rectangles in real world and
mathematical problems. For
example, find the width of a
rectangular room given the
area of the flooring and the
length, by viewing the area
formula as a multiplication
equation with an unknown
factor.
4.OA.2: Multiply or divide
to solve word problems
involving multiplicative
comparison, e.g., by using
drawings and equations with
a symbol for the unknown
number to represent the
problem, distinguishing
multiplicative comparison
from additive comparison.
Big Ideas
Students Will…
Know/Understand…
• Perimeter is additive: the sum of all side
lengths equals the perimeter.
• Area is measured in square units.
• What is a square unit? Why do we use square • A square unit is a rectangle with equal side
units to find and report the area of twolengths.
dimensional shapes?
• The size of the square unit affects the area.
• A figure that can be covered by n unit squares
• Rectangles are composed of arrays (rows and
with no gaps or overlaps has an area of n
columns). Therefore, we can use tiling and
square units.
multiplication to find the area of a rectangle
• The dimensions of a rectangle are called its
by multiplying the number of columns that
length and width.
compose its length by the number of rows that
• A composite figure is made up of two or more
compose its height
figures.
• Rectangles may have the same area, but
• Given a perimeter, you can create shapes with
different perimeters and vice versa.
various dimensions and therefore different
areas. Given an area, you can create a
rectangle with various dimensions and
therefore various perimeter.
• Perimeter is a measurement of the distance
around a figure. Area is a measurement of the
amount of flat space a figure takes up.
• The Distributive Property can be used as a
tool to find the area of rectangles with greater
side lengths.
• You can find the area of complex or
composite shapes by decomposing them into
familiar (rectangular) shapes
3 | P a g e Be Able To…
• Add side lengths to determine the perimeter of
a figure.
• Solve for missing side lengths given the
perimeter of a figure.
• Tile a rectangle and count square units to
determine the area.
• Find the area and perimeter of a rectangle on a
grid.
• Multiply side lengths of a figure to determine
the area.
• Use the Distributive Property to find the area
of rectangles with side lengths greater than
12.
• Find the area of a composite figure by
decomposing into or composing rectangles
Unit Sequence SWBAT…
1
Measure the perimeter
of a figure in
centimeters and inches
using string and rulers
Key Points/
Teaching Tips
•
•
•
•
2
Use grid paper, rulers,
and the perimeter
formula to find the
perimeter of a figure.
•
•
Exit Ticket
This should be a hands-on, kinesthetic,
conceptual lesson about perimeter. In
addition to using string and rulers/meter
sticks, students may explore perimeter
through geoboards and rubber bands, by
walking around the perimeter of the
room, by tracing the boundaries of
shapes, or by searching the school for
examples of perimeter (i.e. bulletin board
boarders, door frames, etc.).
Students will understand perimeter as the
distance around the outside of a figure
Students do not need to be adding side
lengths until the next lesson.
Students should measure the perimeter of
objects to the nearest whole inch and
centimeter in order to support standard
3.MD.2 and their understanding of the
relationship between the size of a unit
and the number of units needed.
1. Trace the perimeter of the shape below with a
crayon:
According to the unpacked standards
guide, students should work through
determining perimeter in the following
progression:
1. Side lengths are marked off so that
students can count unit lengths.
2. Side lengths are labeled with
numerals.
3. Students mark off unit lengths with a
ruler and label the length of each
side.
This lesson should give students an
opportunity to practice all three.
1. Find the perimeter of the shape below:
Instructional
Resources1
My Math
Chapter 13, Lesson 1
EngageNY
Module 7, Lesson 10
Homework
(Appendix C)
a. Explain in writing how you know you
outlined the perimeter of the shape.
2. Use string and your rulers to find the perimeter of
the shape above to the nearest inch and the nearest
centimeter. Be sure to label your answers with the
correct units.
3. Compare the perimeters of the objects in
centimeters to the perimeters in inches. What do
you notice?
My Math
Chapter 13, Lesson 2
EngageNY
Module 7
Lessons 12 & 13
(Appendix C)
2. Find the perimeter of Shape B below. Write an
equation to support your answer.
1 Teachers should be aware that the EngageNY curriculum covers geometry before perimeter; therefore, resources must be adapted accordingly. 4 | P a g e •
When using grid paper, students must
attend to precision when counting around
the corners (i.e. a common misconception
is that students count square units instead
of unit lengths). Students may benefit
from being required to trace each square
unit as they count.
3. Measure and label the side lengths of the shape
below in centimeters. Then find the perimeter.
3
Given the perimeter of
a figure, students will
find a missing side
length.
•
•
Students should be able to name squares
and rectangles and pay special attention
to patterns in the side lengths of each.
They should be able to use these patterns
and what they know about the nature of
squares and rectangles to reason about
missing side lengths for these shapes.
For other polygons, students should
practice writing equations with letters or
symbols standing for the unknown
(missing side length).
1. The perimeter of the triangle shown below is 48
inches. Find the unknown side length.
16 in.
2. The perimeter of the figure below is 26 feet.
10 ft.
3
Sample PARCC EOY assessment question:
10 ft.
5 | P a g e EngageNY
Module 7, Lesson 14
(Appendix C)
20 in.
?
?
My Math
Chapter 13, Lesson 2
a. Write an equation to represent the
perimeter of the figure. Find the
unknown.
b. Use your knowledge about this figure to
explain another way to find the unknown.
3. A garden has eight equal sides and has a perimeter
of 56 meters. Circle the equation that gives the
length, in meters, of each side. Use pictures,
words, and your knowledge of mathematical
operations to defend your answer.
a. 56 + 8 = 65
b. 56 – 8 = 48
c. 56 ÷ 8 = 7
4
Given side lengths in a
real-world context,
students will solve
word problems to
determine perimeter.
•
When solving real-world problems about
perimeter, a common misconception is
for students to only add the side lengths
given instead of finding the distance all
the way around the shape (i.e. 7 + 6 for a
rectangle with dimensions 7 by 6, instead
of 7 + 6 + 7 + 6). This should be an
explicit teaching point.
Sample PARCC EOY assessment
question:
Lavina wants to place a fence around a
rectangular play area for her rabbits. The
play area will be 7-feet long and 4-feet wide.
What is the total length of the fence, in feet,
Lavina needs to place around the play area?
•
5
6 | P a g e 1. Marvin draws a border around a letter that is 9
inches wide and 8 inches long. How many inches
of border does Marvin draw? Draw a picture to
support your answer.
2. Marlene ropes off a square section of her yard
where she plants grass. One side of the square
measures 9 yards. What is the total length of rope
Marlene uses?
3. Write your own word problem for a classmate that
would require him or her to find the perimeter of a
figure. Create an answer key and use words and
pictures to explain why your classmate would need
to find the perimeter.
Flex Day (Instruction Based on Data)
Recommended Resources:
“Engage NY Module 7 Lesson 23” (Appendix C)
“Finding and Measuring Perimeter” (Appendix C)
“Perimeter with Color Tiles” (Appendix C)
EngageNY
Module 7, Lesson 15
(Appendix C)
6
Explore the area of
rectangles by using
tiles; attend to
precision when tiling
by ensuring there are
no gaps or overlaps.
•
•
•
According to the unpacked standards
guide, students should have “ample
experiences filling a region with square
tiles before transitioning to pictorial
representations on graph paper.”
Students should understand area as the
amount of flat space a shape takes up
Students should be able to articulate why
we use square units to measure area.
“Tiling a Tabletop” provides a good
foundation for an inquiry lesson on this
topic. In addition to using notebook
paper and index cards, students might
also try describing the amount of flat
space a shape takes up using circle
counters so that they discover the
importance of square units in being able
to cover a figure.
Error analysis:
7 | P a g e 1.
Use square unit tiles to find the area of the shape
below. Make sure to label your answer correctly.
(Include picture of a rectangle that can be measured
exactly using square unit tiles.)
2. Does the picture below accurately show the area
of Rectangle A? Why or why not?
My Math Chapter 13
Foldable: 1 square
unit
LearnZillion: “Find
the Area of a Shape”
(Appendix C)
Tiling a Tabletop
(Appendix C)
3. Why do we measure area in square units?
7
Use grid paper to find
the area of a figure
and to create figures
with a given area.
•
Students may benefit from exploring this
objective with geoboards before working
with grid paper.
•
Students should understand that the size
of the square unit affects the area of the
figure.
•
Students may notice that multiple
different shapes can have the same area.
This will be revisited later in the unit.
1. Each
is 1 square unit. What is the area of each
of the following rectangles?
My Math
Chapter 13
Lessons 3 & 4
EngageNY
Module 4, Lesson 3
(Appendix C)
2. Find the area of the figure below. Label your
answer in square units.
8
Use rulers or labeled
side lengths to
separate rectangles
into square units.
8 | P a g e •
A common misconception is for students
to fill a rectangle with square units
without attending to the given side
lengths. (For example, a student may
make 4 columns even though the side
length is labeled as 3 units.) This should
be an explicit teaching point, and students
should be required to attend to precision
by checking their drawings to ensure that
it matches the labeled side length.
3. Use the appropriate resources to draw a rectangle
with an area of 8 square centimeters and a
rectangle with an area of 8 square inches. What do
you notice about these two rectangles? Explain.
1. Use a ruler to measure the side lengths of the
rectangle in centimeters. Use your measurements
to tile the rectangle with square centimeters. Find
the area.
My Math
Chapter 13, Lesson 5
EngageNY
Module 4, Lesson 4
(Appendix C)
•
Students should learn the word
“dimensions” and recognize that this
refers to the side lengths of a shape.
Area = _____ square centimeters
2. Use the given side lengths to tile the rectangle with
square units. Find the area.
3 units
6 units
3. Why is perimeter measured in units and area
measured in square units?
9
Relate area to arrays
in order to apply skipcounting and
multiplication
strategies to determine
the area of a rectangle.
9 | P a g e •
Students should draw on their foundation
with arrays from Unit 2. They should be
able to describe area as repeated groups
of equal size of square units.
1. Darren has a total of 28 square-centimeter tiles.
He arranges them into 7 equal rows. Draw
Darren’s rectangle and label the side lengths.
a. Write an addition equation to find the total
area.
b. Write a multiplication equation to find the
total area.
2. How is the area of a rectangle related to
multiplication?
My Math
Chapter 13, Lesson 5
EngageNY
Module 4, Lesson 5
(Appendix C)
10
Complete rows or
columns to complete
arrays and form
rectangles
•
According to the unpacked standards
guide, “many activities that involve
seeing and making arrays of squares to
form a rectangle might be needed to build
robust conceptions of a rectangular area
structured into squares.” This lesson is
included to provide students with
additional exposure to arrays.
1. Each
represents a 1-cm square. Draw to find
the number of rows and columns in each array.
Then, fill in the blanks to make a true equation to
represent the array’s area
___ x ___ = ____ square centimeters
2. Label the side lengths of Rectangle A on the grid
below. Use a straight edge to draw a grid of equal
size squares within Rectangle A. Find the total
area of Rectangle A.
3. Mary skip-counts by 6s and Julio skip-counts by
7s to find the total number of square units in the
array. Who is correct? Use pictures, numbers, and
words to explain your answer.
11
Flex Day (Instruction Based on Data)
Recommended Resources:
“Doubling, Halving, Tripling” (Appendix C)
“Find the Area” (Appendix C)
“Area Compare” (Appendix C)
“EngageNY Module 7, Lesson 10” (Appendix C)
“How Big is a Desk?” (Appendix C)
10 | P a g e EngageNY
Module 4
Lessons 6 & 7
12
Use equations and fact
families to find the
missing side length of
a rectangle.
EngageNY Module 4, Lesson 8 Exit Ticket
1. Write a multiplication sentence to find the
area of the rectangle below:
2. Write a multiplication sentence and a
division sentence to find the unknown side
length for the rectangle below:
3. Ria draws a rectangle that has a side length
of 4 inches and an area of 28 square inches.
What is the other side length? Use words,
pictures, and equations to support your
answer.
11 | P a g e EngageNY
Module 4, Lesson 8
(Appendix C)
My Math
Chapter 13, Lesson 6
13
Apply the distributive
property to find the
area of rectangles.
Pacing: 2 days
Students should practice first with rectangles
with grids, then with rectangles where only the
side lengths are labeled.
• Inquiry-based question from Smarter Balanced:
In the picture below, Rectangle A has a width of 3
ft., Rectangle B has a width of 2 ft., and both of
them have a length of 4 ft.
Rectangle C was formed by sticking Rectangle A
and Rectangle B together along one of their
lengths. This is shown with the dotted line.
•
•
1. Label the side lengths of the shaded and
unshaded rectangles. Then find the total area
of the large rectangle by adding the areas of
the 2 smaller rectangles.
2. Find the area of the rectangle below by using
the Distributive Property to decompose the
longer side into a sum.
15 ft.
12 | P a g e EngageNY
Module 4
Lessons 9 & 10
(Appendix C)
“Breaking Apart
Arrays”
(Appendix C)
14
a. What is the area of Rectangle A?
b. What is the area of Rectangle B?
c. What is the sum of the areas of Rectangles
A and B?
d. What is the width of Rectangle C? How do
you know?
e. What is the area of Rectangle C?
f. Compare your answers in (c) and (e). Are
they the same? Why or why not?
My Math
Chapter 13, Lesson 7
4 ft.
3. How are the operations of addition and
multiplication used when finding area using
the Distributive Property?
15
Apply the distributive
property to solve word
problems about the
area of rectangles.
Students should have practice matching
expressions with real world situations. See the
example below from the NY State Assessment:
The garden below was divided into regions—one
for carrots and one for peas.
Which expression represents the area, in square
units, of the whole garden?
a. (5 + 10) + (5 + 6)
b. (5 x 10) + (5 x 6)
c. (5 x 10) + (5 x 6)
d. (5 + 10) x (5 + 6)
Sample question from Arizona Common Core:
Joe and John made a poster that was 4’ by 5’.
Mary and Amir made a poster that was 4’ by 3’.
They placed their posters on a wall side-by-side so
that there was no space between them. How much
area will the two posters cover? Draw a picture to
help explain your answer.
Sample PARCC assessment question:
1. There is a large mural made of colored tiles
at the entrance to Rena’s school. A part of
the mural was damaged in a heavy storm.
The part of the mural that was NOT damaged
is 5 tiles long and 4 tiles high.
My Math
Chapter 13, Lesson 7
Rena wants to know how many tiles need to be
replaced.
a. Choose an expression from the
options box to write in each of the
two blanks to make a true statement.
“Designing a Flower
Bed”
(Appendix C)
b. How many tiles need to be replaced
in the mural? Explain how you
found your answer.
Sample Smarter Balanced:
2. Jasper used the expression 5 x (10 + 3) to
find the area of a rectangular closet floor, in
square feet.
a. On a grid, draw a rectangle that
could be the one Jasper measured.
b. What is the area of the closet floor in
square feet?
3. Jasper has 200 square feet of tile. He will use
some of the tile to cover the closet floor. He will
only use whole tiles.
a. How many square feet of tile will Jasper
have left after covering the closet floor with
13 | P a g e EngageNY
Module 4 Lesson 12
(Appendix C)
tile?
b. Jasper wants to use some of the remaining
tile to cover the floor of a kitchen. The
kitchen is 12 feet long and 12 feet wide.
Does Jasper have enough tiles to cover the
kitchen floor? Show how you got your
answer using drawings, mathematical
expressions or equations, and words.
16
Use grid paper to find
the area of composite
figures.
•
•
•
•
14 | P a g e Provide an opportunity for students to do the
heavy lifting by presenting them with a
complex shape with given side lengths – first
ask them to calculate the perimeter, then give
them “think time” to brainstorm how they
could find the area (allow them to discuss in
small groups). You may provide tiles for them
to recreate the shape and test their theories.
Students should begin their work with
composite figures by using grid paper to:
• Decompose rectilinear figures into
rectangles (2 or more)
• Compose rectangles in order to subtract a
known area
EngageNY Module 4, Lesson 13 Concept
Development Problems 1 & 2 (with grid
paper), Problem Set #1, Homework #1-2
Students may benefit from coloring the
rectangles they have created with different
colors. They should be required to label the
dimensions of these new shapes.
1. Find the total area of the figure below by
decomposing it into 2 rectangles.
EngageNY
Module 4, Lesson 13
(Appendix C)
“Finding the Area of
Complex Figures:
Lessons 13 & 14”
(Appendix C)
2. Find the total area of the figure above by
composing a rectangle in order to subtract a
known area.
3. For either #1 or #2, explain how you found
the area. Include the following words in
your answer: compose/decompose,
add/subtract, area, and composite figure.
17
Find the area of
composite figures in
which all side lengths
are labeled.
•
Students should continue their work with
composite figures by working without grid
paper, but solving problems in which necessary
side lengths are labeled. They must attend to
precision when choosing which dimensions to
use to find the area of the decomposed
rectangles.
1.
Find the area of the figure below three ways.
Show two ways to decompose the figure and
one way to compose a rectangle around it
My Math
Chapter 13, Lesson 8
“Finding the Area of
Complex Figures:
Lesson 15”
(Appendix C)
EngageNY Module 4, Lesson 13 Problem Set #2
EngageNY Module 4, Lesson 14 Problem Set #3-4
My Math Chapter 13, Lesson 8 Example 2,
Independent Practice #2-3, 5
2. Max found the area of the composite figure
below:
(10 x 4) + (8 x 3) = 40 + 24 = 64 square
inches
Is Max correct? Use pictures, words, and numbers to
explain your reasoning. If you believe he is
incorrect, find the correct answer.
15 | P a g e EngageNY
Module 4
Lessons 13 & 14
(Appendix C)