Grade 3. Measurement and Data 3.MD.7.a-d 2012 COMMON CORE STATE STANDARDS ALIGNED MODULES

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1 THE NEWARK PUBLIC SCHOOLS THE OFFICE OF MATHEMATICS Grade 3 Measurement and Data 3.MD.7.a-d 2012 COMMON CORE STATE STANDARDS ALIGNED MODULES

2 THE NEWARK PUBLIC SCHOOLS Office of Mathematics MATH TASKS Measurement and Data 3.MD.7.a-d Geometric measurement: understand concepts of area and relate area to multiplication and addition. 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. 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 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 Lesson 5 Golden Problem 3. MD.7.a-d Lesson 4 3. MD.7.d Real world problem using area Lesson 3 3. MD.7.c Finding areas of composite shapes Lesson 2 3. MD.7.b Using multiplication to find area Lesson 1 3. MD.7.a Using tiling to find area Lesson Structure: Assessment Task Prerequisite Skills Focus Questions Guided Practice Homework Journal Question Page 2 of 27

3 Multiplication Concepts Multiplication can be defined in terms of repeated addition. For example, 3 6 can be viewed as 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 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 3 of 27

4 Teaching Tips Teaching Tip 1 Digit Name vs. Digit Value 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. Teaching Tip 2 Drawing Rectangles for an Area Model 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. Teaching Tip 3 Using an Area Model to Record Multiplication 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 4 of 27

5 Multiple Representations to Multiplication In the identity 3(4 + 5) = 3(4) + 3(5), the 3 is distributed over the 4 and the 5. Distributive Property a(b + c ) = ab + ac and (b + c )a = ba + ca Commutative Properties of Multiplication a b = b a 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 cm = 4 Array Model Interpret products of whole numbers 5 7 as the total number of objects in 5 groups of 7 objects each Page 5 of 27

6 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. 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 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. Focus Questions 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? Journal Question Is a square a rectangle? Why or why not? Page 6 of 27

7 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. 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. Page 7 of 27

8 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. 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 4. Area = Square units 5. Area = Square units 6. Area = Square units 7. Area = Square units 8. Area = Square units 9. Area = Square units Page 8 of 27

9 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. 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 9 of 27

10 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. 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. 1. Area = Square units 2. Area = Square units 3. Area = Square units 4. Area = Square units 5. Area = Square units 6. Area = Square units 7. Area = Square units 8. Area = Square units 9. Area = Square units Page 10 of 27

11 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. 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? Page 11 of 27

12 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. 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 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? Journal Question How would you explain how to find area to a second grader? Page 12 of 27

13 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. Name Date 1. Area = Square units 2. Area = Square cm 3. Area = Square cm 4. Area = Square ft 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: Page 13 of 27

14 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. Name Date 1. Area Area = = Square cm 2 2. Area = Square cm 4 in. 2 in. 3. ches 4. Area = Square units 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: Page 14 of 27

15 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. 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? 3ft 5ft 5ft Focus Questions 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? Journal Question What do you think distributing has to do with the distributive property? Page 15 of 27

16 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. Directions: Without counting, show two ways of finding the area of each object. 1. 7in 2. 7in 3. 3in 3in 10in 4. 2in Page 16 of 27

17 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. Directions: Without counting, show two ways of finding the area of each object in 9 in 3 in 6. 2 in 11 in 5 in 7. 3in 8. 10in Page 17 of 27

18 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. Directions: Without counting, show two ways of finding the area of each object. 1. 3in 7in 2. 7in 7in 3. 8in 3in 3in 4. 2in 3in Page 18 of 27

19 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. Directions: Without counting, show two ways of finding the area of each object in 6 in 2 in 6. 2 in 8 in 4 in 7. 7in 3in 8. Page 19 of 27

20 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. 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 15 m 6 m 6 m 5 m 5 m 10 m Focus Questions 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? Journal Question How can decomposing diagrams help you answer multiplication problems? Page 20 of 27

21 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. Decompose the figure to find the total area of each figure. 1. 2in 3in 2. 4in 4in 3. 7in 4. 4in 4in 3in Page 21 of 27

22 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. Decompose the figure to find the total area of each figure. 5. 3in 3in 6. 8in 4in 4in 7. 4in 3in 4in 2in 8. 10in 11in 8in Page 22 of 27

23 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. Decompose the figure to find the total area of each figure. 1. 5m 5m 6m 4m Area = Square m 2. 7m 5m 6m 5m Area = Square m 3. 5m 5m 8m 10m Area = Square m 4. 6m 7m 8m 2m Area = Square m Page 23 of 27

24 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. Decompose the figure to find the total area of each figure. 5. 7m 3m 6m 5m Area = Square m 6. 4m 9m 8m 4m Area = Square m 7. 7m 4m 7m 2m Area = Square m 8. 8m 9m 10m 5m Area = Square m Page 24 of 27

25 3.MD.7.a-d: Lesson 5 Relate area to the operations of multiplication and addition. 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. 1ft 1ft Focus Questions 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? Journal Question Describe one thing that you know now that you didn t know before doing these tasks. Page 25 of 27

26 LESSON 5 RUBRIC Score GOLDEN PROBLEM 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 26 of 27

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 Page 27 of 27

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