Grade 3 - Newark Public Schools · 2017. 5. 5. · Page 4 of 27 Teaching Tips Digit Name vs. Digit Value Teaching Tip 1 Stress place value in multiplication by distinguishing between

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

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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.

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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

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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.

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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.

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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.


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.

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Page 8 of 27


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





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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.




Page 10 of 27


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.



9. Area = _____ Square units 7. 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?




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Below is the floor plan for Paul’s kitchen. How many square foot tiles will he need to cover the

floor?


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?

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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:


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



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:


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



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?


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.

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Directions: Without counting, show two ways of finding the area of each object.

1.

2.

3.

4.


3in

10in

6in 2in

3in

5in

7in

6in

7in

5in 5in

5in

Area = _____ Square in







Page 17 of 27


5.

6.

7.

8.


3in 5in

6in

10in

5in

9 in

2 in

3 in

11 in

2 in

5 in








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1.

2.

3.

4.


8in

5in

7in

3in

5in 2in

3in

3in

7in

6in 7in

3in








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5.

6.

7.

8.


3in 5in

7in

6in

5in

6 in

2 in

2 in

8 in

2 in

4 in








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A storage shed is pictured below. What is the total area? How could the figure be decomposed

to help find the area?


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?

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Decompose the figure to find the total area of each figure. 1.

2.

3.

4.


4in

6in

6in

4in

5in

6in

5in

7in

4in

4in

3in

6in



world problems.

5in 3in 2in

6in 6in





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6.

7.

8.


3in

4in

2in 4in

4in

6in

4in 8in

6in

3in

6in

3in

10in

8in

5in 11in



world problems.





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Decompose the figure to find the total area of each figure.

1.

2.

3.

4.


5m

6m

4m 5m

5m

7m

6m

5m

6m

8m

2m

7m



world problems.

Area = _____ Square m




5m

10m

5m

8m

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6.

7.

8.


3m

6m

7m

5m

4m

9m

4m 8m

7m

7m

2m 4m

8m

10m

5m 9m



world problems.





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.


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

Documents

Grade 3 - Newark Public Schools · 2017. 5. 5. · Page 4 of 27 Teaching Tips Digit Name vs. Digit Value Teaching Tip 1 Stress place value in multiplication by distinguishing between