Upload
others
View
1
Download
0
Embed Size (px)
Citation preview
Grade 3
Measurement and Data
3.MD.7.a-d
2012 COMMON CORE STATE STANDARDS ALIGNED MODULES
THE NEWARK PUBLIC SCHOOLS THE OFFICE OF MATHEMATICS
Page 2 of 27
Goal: Students recognize area as an attribute of two-dimensional regions. They measure the
area of a shape by finding the total number of same size units of area required to cover the
shape without gaps or overlaps, a square with sides of unit length being the standard unit for
measuring area. Students understand that rectangular arrays can be decomposed into identical
rows or into identical columns. By decomposing rectangles into rectangular arrays of squares, students connect area to multiplication, and justify using multiplication to determine
the area of a rectangle.
MA
TH T
ASK
S TH
E N
EWA
RK
PU
BLI
C S
CH
OO
LS
Off
ice
of
Mat
hem
atic
s
Measurement and Data 3.MD.7.a-d Geometric measurement: understand concepts of area and relate area
to multiplication and addition.
Lesson 1
3. MD.7.a Using tiling to find area
Lesson 2
3. MD.7.b Using multiplication to
find area
Lesson 3
3. MD.7.c Finding areas of composite shapes
Lesson 4
3. MD.7.d Real world problem using area
Lesson 5 Golden Problem
3. MD.7.a-d
Lesson Structure: Assessment Task
Prerequisite Skills Focus Questions Guided Practice
Homework Journal Question
Embedded 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
Essential Questions:
How is area related to multiplication?
How is the area of a rectangle determined?
For what purpose do you calculate area?
Prerequisites: Simple Counting
Understanding area of square units
Description of Shapes
Partition a rectangle into rows and
columns of same-size squares and
count to find the total number
Addition
Multiplication
Division
Page 3 of 27
Multiplication Concepts
Multiplication can be defined in terms of repeated addition. For example, 3 × 6 can be viewed as 6 + 6 + 6. More generally, for any
positive integer n, n × b can be represented as n × b = b + b + … + b, where the sum on the right consists of n addends.
A rectangular array provides a visual model for multiplication. For example, the product 3 × 6 can be represented as
By displaying 18 dots as 3 rows with 6 dots in each row, this array provides a visual representation of 3 × 6 as 6 + 6 + 6. An
equivalent area model can be made in which the dots of the array are replaced by unit squares.
Besides representing 3 × 6 as an array of 18 unit squares, this model also shows that the area of a rectangle with a height of 3 units and
a base of 6 units is 3 × 6 square units, or 18 square units.
Multiplication is a binary operation that operates on a pair of numbers to produce another number. Given a pair of numbers a and b
called factors, multiplication assigns them a value a × b = c, called their product.
Multiplication has certain fundamental properties that are of great importance in arithmetic. The Commutative Property of
Multiplication states that changing the order in which two numbers are multiplied does not change the product. That is, for all
numbers a and b, a × b = b × a.
The array model can be used to make this plausible. For example, because 3 × 6 = 6 × 3, an array with 3 rows and 6 dots in each row
has the same number of dots as an array with 6 rows and 3 dots in each row.
Another important property of multiplication is the Identity Property of Multiplication. It states that the product of any number and 1
is that number. That is, for all numbers a, a × 1 = 1 × a = a.
The Zero Property of Multiplication states that when a number is multiplied by zero, the product is zero. That is, for all numbers a,
a × 0 = 0 × a = 0.
Page 4 of 27
Teaching Tips
Digit Name vs. Digit Value
Teaching Tip 1
Stress place value in multiplication by distinguishing between the name of
the digit and the value it stands for. The 2 in 24 stands for 2 × 10 = 20, not
2. Base-10 blocks and area model diagrams emphasize the value that each
digit stands for because they use expanded notation to build the answer.
Drawing Rectangles for an Area Model
Teaching Tip 2
The area model is an alternative and efficient way to multiply. Encourage
students to draw rectangles, even though the rectangles may not be drawn
to scale. If students need to use base-10 blocks as a transitional step,
change the numbers in the problems to match the quantity of blocks that
are available.
Using an Area Model to Record Multiplication
Teaching Tip 3
Is it okay to permit students to use the area model as a recording
method for multiplication? Yes. An area model not only helps to explain
why the standard algorithm commonly taught in the United States for
multiplication works, it is an efficient recording alternative. Some students
(especially visual learners and those who have difficulty keeping numbers
lined up in multiplication problems) may prefer it. Furthermore, this
method has certain benefits. It illuminates important mathematical
concepts (such as the distributive property), allows for computational
flexibility (expanded notations allow students to use derived facts), and
reinforces the concept of area. Finally, when students take algebra, they
are likely to see the area model when they learn to multiply and factor
polynomials.
Page 5 of 27
Multiple Representations to Multiplication
Distributive Property
a(b + c ) = ab + ac and
(b + c )a = ba + ca
Commutative Properties of Multiplication
a • b = b • a
Array Model
In the identity 3(4 + 5)
= 3(4) + 3(5), the 3 is
“distributed” over the
4 and the 5.
3 • 4 = 4 • 3
(3 • 4) • 5 = 12 • 5 = 60
or 3 • (4 • 5) = 3 • 20 = 60
Associative Properties of
Multiplication (a • b) • c = a • (b • c )
Area Model
4 cm · 3 cm = 12 cm2
2 · 2 = 4
Interpret products of
whole numbers
5 × 7 as the total number of objects in 5 groups of 7
objects each
Page 6 of 27
Imagine that each square in the picture measures one centimeter on each side. What is the area of each shape?
Try to work it out without counting each square individually.
1.
2. Decompose the object below in to rectangles to find the area of the entire object.
3. Decompose the object below in to rectangles to find the area of the entire object.
Introductory Task Guided Practice Collaborative Homework Assessment
Focus Questions
Journal Question
Is a square a rectangle?
Why or why not?
Question 1: Show how you divided each object to find the area.
Question 2: How do the squares covering a rectangle compare
to an array?
3.MD.7.a: Lesson 1 Find the area of a rectangle with whole-number side lengths by tiling it, and show that the area is the
same as would be found by multiplying the side lengths.
Page 7 of 27
4. Decompose the object below in to rectangles to find the area of the entire object.
5. Decompose the object below in to rectangles to find the area of the entire object.
Introductory Task Guided Practice Collaborative Homework Assessment
3.MD.7.a: Lesson 1 Find the area of a rectangle with whole-number side lengths by tiling it, and show that the area is the
same as would be found by multiplying the side lengths.
Page 8 of 27
Introductory Task Guided Practice Collaborative Homework Assessment
Name_______________________________ Date_____________________
Finding Area Using Square Units
Find the area of each figure. A quick hint is to rearrange the composition of each figure to make
a shape you can work with.
1. Area = _____ Square units 2. Area = _____ Square units 3. Area = _____ Square units
6. Area = _____ Square units 5. Area = _____ Square units 4. Area = _____ Square units
9. Area = _____ Square units 8. Area = _____ Square units 7. Area = _____ Square units
3.MD.7.a: Lesson 1 Find the area of a rectangle with whole-number side lengths by tiling it, and show that the area is the
same as would be found by multiplying the side lengths.
Page 9 of 27
10. Tanya built this rectangular model using 39 tiles.
a. List two number sentences this model represents.
b. Tanya found one more tile. Draw a new rectangular model using all of Tanya’s tiles.
c. List two multiplication number sentences this new model represents.
Introductory Task Guided Practice Collaborative Homework Assessment
3.MD.7.a: Lesson 1 Find the area of a rectangle with whole-number side lengths by tiling it, and show that the area is the
same as would be found by multiplying the side lengths.
Page 10 of 27
Introductory Task Guided Practice Collaborative Homework Assessment
8. Area = _____ Square units
Name_______________________________ Date_____________________
Area of Unusual Shapes with Square Units
Find the area of each figure. A quick hint is to rearrange the composition of each figure to make
a shape you can work with.
3.MD.7.a: Lesson 1 Find the area of a rectangle with whole-number side lengths by tiling it, and show that the area is the
same as would be found by multiplying the side lengths.
9. Area = _____ Square units 7. Area = _____ Square units
6. Area = _____ Square units 5. Area = _____ Square units 4. Area = _____ Square units
3. Area = _____ Square units 2. Area = _____ Square units 1. Area = _____ Square units
Page 11 of 27
10. Amanda wants to cover the top of her doll’s table with colored paper. The top of the table is
shown below.
How many square centimeters of paper does Amanda need if each square equals 1 square centimeters?
Introductory Task Guided Practice Collaborative Homework Assessment
3.MD.7.a: Lesson 1 Find the area of a rectangle with whole-number side lengths by tiling it, and show that the area is the
same as would be found by multiplying the side lengths.
Page 12 of 27
Below is the floor plan for Paul’s kitchen. How many square foot tiles will he need to cover the
floor?
Introductory Task Guided Practice Collaborative Homework Assessment
10 ft
5 ft
3.MD.7.b: Lesson 2
Multiply side lengths to find areas of rectangles with whole number side lengths in the context of
solving real world and mathematical problems, and represent whole-number products as rectangular
areas in mathematical reasoning.
Journal Question
How would you explain how to
find area to a second grader?
Focus Questions
Question 1: What strategies can be used to find the area of a
shape?
Question 2: How is multiplication related to finding area?
Page 13 of 27
Name _______________________ Date __________________
5. What is the area of a rectangle with side length of 5 inches and a side width of 8 inches?
Number sentence:
6. What is the area of a rectangle with the side length of 7 feet and a side width of 3 feet?
Number sentence:
Introductory Task Guided Practice Collaborative Homework Assessment
1. Area = _____ Square units 2. Area = _____ Square cm
3. Area = _____ Square cm 4. Area = _____ Square ft
3.MD.7.b: Lesson 2 Multiply side lengths to find areas of rectangles with whole number side lengths in the context of
solving real world and mathematical problems, and represent whole-number products as rectangular
areas in mathematical reasoning.
Page 14 of 27
Name _______________________ Date __________________
Area = ______cm2
5. What is the area of a rectangle with side length of 6 meters and a side width of 7 meters?
Number sentence:
6. What is the area of a rectangle with the side length of 5 feet and a side width of 2 feet?
Number sentence:
Introductory Task Guided Practice Collaborative Homework Assessment
4 in.
2 in.
1. Area = _____ Square cm 2. Area = _____ Square cm
3. Area = _____ Square inches 4. Area = _____ Square units
3.MD.7.b: Lesson 2 Multiply side lengths to find areas of rectangles with whole number side lengths in the context of
solving real world and mathematical problems, and represent whole-number products as rectangular
areas in mathematical reasoning.
Page 15 of 27
Joe and John are installing windows in their new home. The first window is 5’ by 3’ and the
second window is 5’ by 5’. They are placing the windows in the wall side-by-side so that there
was no space between them. How much area will the two windows cover?
Introductory Task Guided Practice Collaborative Homework Assessment
5ft
3ft 5ft
Focus Questions
Journal Question
What do you think distributing
has to do with the distributive
property?
Question 1: Can you write an equation for the situation above?
Question 2: Is there a simpler way to find the area that the two
windows will cover?
Question 3: Can you write an equation for Question 2?
3.MD.7.c: Lesson 3 Use tiling to show in a concrete case that the area of a rectangle with whole-number side lengths a and
b + c is the sum of a × b and a × c. Use area models to represent the distributive property in
mathematical reasoning.
Page 16 of 27
Directions: Without counting, show two ways of finding the area of each object.
1.
2.
3.
4.
Introductory Task Guided Practice Collaborative Homework Assessment
3in
10in
6in 2in
3in
5in
7in
6in
7in
5in 5in
5in
Area = _____ Square in
Area = _____ Square in
Area = _____ Square in
Area = _____ Square in
3.MD.7.c: Lesson 3 Use tiling to show in a concrete case that the area of a rectangle with whole-number side lengths a and
b + c is the sum of a × b and a × c. Use area models to represent the distributive property in
mathematical reasoning.
Page 17 of 27
Directions: Without counting, show two ways of finding the area of each object.
5.
6.
7.
8.
Introductory Task Guided Practice Collaborative Homework Assessment
3in 5in
6in
10in
5in
9 in
2 in
3 in
11 in
2 in
5 in
Area = _____ Square in
3.MD.7.c: Lesson 3 Use tiling to show in a concrete case that the area of a rectangle with whole-number side lengths a and
b + c is the sum of a × b and a × c. Use area models to represent the distributive property in
mathematical reasoning.
Area = _____ Square in
Area = _____ Square in
Area = _____ Square in
Page 18 of 27
Directions: Without counting, show two ways of finding the area of each object.
1.
2.
3.
4.
Introductory Task Guided Practice Collaborative Homework Assessment
8in
5in
7in
3in
5in 2in
3in
3in
7in
6in 7in
3in
3.MD.7.c: Lesson 3 Use tiling to show in a concrete case that the area of a rectangle with whole-number side lengths a and
b + c is the sum of a × b and a × c. Use area models to represent the distributive property in
mathematical reasoning.
Area = _____ Square in
Area = _____ Square in
Area = _____ Square in
Area = _____ Square in
Page 19 of 27
Directions: Without counting, show two ways of finding the area of each object.
5.
6.
7.
8.
Introductory Task Guided Practice Collaborative Homework Assessment
3in 5in
7in
6in
5in
6 in
2 in
2 in
8 in
2 in
4 in
3.MD.7.c: Lesson 3 Use tiling to show in a concrete case that the area of a rectangle with whole-number side lengths a and
b + c is the sum of a × b and a × c. Use area models to represent the distributive property in
mathematical reasoning.
Area = _____ Square in
Area = _____ Square in
Area = _____ Square in
Area = _____ Square in
Page 20 of 27
A storage shed is pictured below. What is the total area? How could the figure be decomposed
to help find the area?
Introductory Task Guided Practice Collaborative Homework Assessment
10 m
5 m
10 m
5 m
5 m 15 m
6 m
6 m
3.MD.7.d: Lesson 4 Recognize area as additive. Find areas of rectilinear figures by decomposing them into non-overlapping
rectangles and adding the areas of the non-overlapping parts, applying this technique to solve real
world problems.
Focus Questions
Journal Question
How can decomposing diagrams
help you answer multiplication
problems?
Question 1: How can decomposing a figure into smaller
figures help solve complex math problems? Question 2: How do multiplication equations help solve area
problems?
Page 21 of 27
Decompose the figure to find the total area of each figure. 1.
2.
3.
4.
Introductory Task Guided Practice Collaborative Homework Assessment
4in
6in
6in
4in
5in
6in
5in
7in
4in
4in
3in
6in
3.MD.7.d: Lesson 4 Recognize area as additive. Find areas of rectilinear figures by decomposing them into non-overlapping
rectangles and adding the areas of the non-overlapping parts, applying this technique to solve real
world problems.
5in 3in 2in
6in 6in
Area = _____ Square in
Area = _____ Square in
Area = _____ Square in
Area = _____ Square in
Page 22 of 27
Decompose the figure to find the total area of each figure. 5.
6.
7.
8.
Introductory Task Guided Practice Collaborative Homework Assessment
3in
4in
2in 4in
4in
6in
4in 8in
6in
3in
6in
3in
10in
8in
5in 11in
3.MD.7.d: Lesson 4 Recognize area as additive. Find areas of rectilinear figures by decomposing them into non-overlapping
rectangles and adding the areas of the non-overlapping parts, applying this technique to solve real
world problems.
Area = _____ Square in
Area = _____ Square in
Area = _____ Square in
Area = _____ Square in
Page 23 of 27
Decompose the figure to find the total area of each figure.
1.
2.
3.
4.
Introductory Task Guided Practice Collaborative Homework Assessment
5m
6m
4m 5m
5m
7m
6m
5m
6m
8m
2m
7m
3.MD.7.d: Lesson 4 Recognize area as additive. Find areas of rectilinear figures by decomposing them into non-overlapping
rectangles and adding the areas of the non-overlapping parts, applying this technique to solve real
world problems.
Area = _____ Square m
Area = _____ Square m
Area = _____ Square m
Area = _____ Square m
5m
10m
5m
8m
Page 24 of 27
Decompose the figure to find the total area of each figure. 5.
6.
7.
8.
Introductory Task Guided Practice Collaborative Homework Assessment
3m
6m
7m
5m
4m
9m
4m 8m
7m
7m
2m 4m
8m
10m
5m 9m
3.MD.7.d: Lesson 4 Recognize area as additive. Find areas of rectilinear figures by decomposing them into non-overlapping
rectangles and adding the areas of the non-overlapping parts, applying this technique to solve real
world problems.
Area = _____ Square m
Area = _____ Square m
Area = _____ Square m
Area = _____ Square m
Page 25 of 27
A New Jersey farmer is thinking of growing crops. The farm has an area of 100 ft². The farmer
has multiple pieces of land attached to each other: one growing corn, one growing potatoes and
one growing carrots. Create a diagram that meets the farmer’s needs. Label the side lengths of
all the pieces of land and write a multiplication equation for each piece of land.
Introductory Task Guided Practice Collaborative Homework Assessment
3.MD.7.a-d: Lesson 5
Relate area to the operations of multiplication and addition.
Focus Questions
Journal Question
Describe one thing that you know
now that you didn’t know before
doing these tasks.
Question 1: What strategies can be used to find the area of a shape?
Question 2: How can decomposing a figure into easier figures help
solve complex math problems with multiplication equations?
1ft
1ft
Page 26 of 27
LESSON 5 RUBRIC
GOLDEN PROBLEM
Score Description 3 Student has an understanding of what area is. Student used
length and width to develop an equation for an area of 100
square feet of land. The student is able to create a diagram that
meets the farmer’s needs of 100ft². The student illustrates
multiplication equation for each piece of land with correct
labeling of length and width throughout the problem. Student
has solved the problem using accurate computation throughout
the problem arriving at 100 sq. feet.
2 Student has an understanding of what area is and has used
length and width to develop equation for an area of 100 square
feet, however does not come up with an equation. The diagram
has multiple pieces of land attached with the dimensions of the
land labeled. The student illustrates multiplication equation for
each piece of land. Student comes up with 100 sq. feet,
however does provide an equation to arrive at the answer.
1 Student uses length and width to develop an area of 100 square
feet but does not come up with an equation for an area of 100
square feet. Student does not use multiple pieces of land.
0 Does not address task, unresponsive, unrelated or
inappropriate.
Page 27 of 27
Third Grade CCSSM Fluencies
Skills
Multiply/divide within 100 (By end of year, know from
memory all products of two one‐digit numbers)
Add/subtract within 1000
Skill builders for the above fluencies.
1. Addition Worksheet #9
Answer Key
2. Addition Worksheet #10
Answer Key
3. Multiplication Worksheet 1
Answer Key
4. Multiplication Worksheet 2
Answer Key