Grade 3 Measurement and Data 3.MD.5 & 6

Transcription

1 THE NEWARK PUBLIC SCHOOLS THE OFFICE OF MATHEMATICS Grade 3 Measurement and Data 3.MD.5 & COMMON CORE STATE STANDARDS ALIGNED MODULES

2 THE NEWARK PUBLIC SCHOOLS Office of Mathematics MATHTASKS Measurement and Data - 3.MD.5.a-b & 3.MD.6 Geometric measurement: understand concepts of area and relate area to multiplication and addition. Goal: Students will recognize area as an attribute of plane figures and understand concepts of area measurement. They will show understanding of the concept that a square with side length 1 unit, called a unit square, is said to have one square unit of area, and can be used to measure area. They will understand that a plane figure which can be covered without gaps or overlaps by n unit squares is said to have an area of n square units. They will measure areas by counting unit squares (square cm, square m, square in, square ft, and improvised units). Essential Questions: How do we measure area? How is the area of a rectangle determined? For what purpose do you calculate area? Prerequisites: Simple Counting Description of Shapes Partition a rectangle into rows and columns of same-size squares and count to find the total number of them 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.5.a-b & 3.MD.6 Lesson 4 3. MD.6 Real world problem using area Lesson 3 3. MD.6 Counting square units Lesson 2 3. MD.5.b Covering shapes to find area Lesson 1 3. MD.5.a Understanding Square Units Lesson Structure: Assessment Task Prerequisite Skills Focus Questions Guided Practice Homework Journal Question Page 2 of 28

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 28

4 Teaching Tips Digit Name vs. Digit Value TeachingTip1 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. TeachingTip2 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. TeachingTip3 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 28

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 c=a (b ) Area Model 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 28

6 3.MD.5a: Lesson 1 Recognize area as an attribute of plane figures and understand concepts of area measurement. a. A square with side length 1 unit, called a unit square, is said to have one square unit of area, and can be used to measure area. Below is a picture of Johan s bathroom. He wants to tile the entire floor. How many 1 unit x 1unit tiles will he need purchase in order to tile his entire floor? (Cut out the shape below and lay it out on the bathroom floor to determine how many tiles to buy.) = 1 square unit Focus Questions Question 1: What is a square unit, and what does it measure? Question 2: What does the area of a shape tell you about that shape s size? Journal Question Why is understanding how to find the area of a shape useful? Describe one way you could use area in your daily life. Page 6 of 28

7 3.MD.5a: Lesson 1 Recognize area as an attribute of plane figures and understand concepts of area measurement. a. A square with side length 1 unit, called a unit square, is said to have one square unit of area, and can be used to measure area. Create the shape that has an area equal to the one listed or determine the area of the shape for each problem. Use the key to help you determine the area of each shape. = 1 square unit (1 unit 2 ) 1. Area = 12 sq. units 2. Area = sq. units 3. Area = sq. units 4. Area = 14 sq. units 5. Area = sq. units 6. Area = 36 sq. units Page 7 of 28

8 3.MD.5a: Lesson 1 Recognize area as an attribute of plane figures and understand concepts of area measurement. a. A square with side length 1 unit, called a unit square, is said to have one square unit of area, and can be used to measure area. In each box below, create a shape that has an area equal to the one listed or determine the area of the shape for each problem. Use the key to help you determine the area of each shape. = 1 square unit (1 unit 2 ) 7. Area = sq. units 8. Area = 11 sq. units 9. Area = 20 sq. units 10. Area = sq. units 11. Area = sq. units 12. Area = 29 sq. units Page 8 of 28

9 3.MD.5a: Lesson 1 Recognize area as an attribute of plane figures and understand concepts of area measurement. a. A square with side length 1 unit, called a unit square, is said to have one square unit of area, and can be used to measure area. In each box below, create a shape that has an area equal to the one listed or determine the area of the shape for each problem. Use the key to help you determine the area of each shape. = 1 square unit (1 unit 2 ) 1. Area = 7 sq. units 2. Area = sq. units 3. Area = 24 sq. units 4. Area = sq. units 5. Area = sq. units 6. Area = 30 sq. units Page 9 of 28

10 3.MD.5a: Lesson 1 Recognize area as an attribute of plane figures and understand concepts of area measurement. a. A square with side length 1 unit, called a unit square, is said to have one square unit of area, and can be used to measure area. In each box below, create a shape that has an area equal to the one listed or determine the area of the shape for each problem. Use the key to help you determine the area of each shape. = 1 square unit (1 unit 2 ) 7. Area = sq. units 8. Area = 40 sq. units 9. Area = 45 sq. units 10. Area = sq. units 11. Area = sq. units 12. Area = 27 sq. units Page 10 of 28

11 3.MD.5b: Lesson 2 Recognize area as an attribute of plane figures and understand concepts of area measurement. b. A plane figure which can be covered without gaps or overlaps by n unit squares is said to have an area of n square units. Paul is trying to figure out the area of his back yard. Below is a drawing of his yard. He also wants to put a vegetable garden along one of the sides of his yard. His wife wants to plant a lot of vegetables, so the garden needs to be 30 square feet in area. Determine the entire area of Paul s backyard Draw in where you think his garden should go. (Be sure it measures 30 sq. ft.) After he builds his garden, how much area will Paul still have left in his yard? Paul s Yard (Each = 1 sq. 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 of a rectangle? Journal Question In your own words, describe one strategy for finding the area of a rectangle. Page 11 of 28

12 3.MD.5b: Lesson 2 Recognize area as an attribute of plane figures and understand concepts of area measurement. b. A plane figure which can be covered without gaps or overlaps by n unit squares is said to have an area of n square units. 1. Determine the area of the shaded shape in square units. 2. Area = 13 square units 3. Area = 5 square units 4. Create a shape with 25 square units on the grid. 5. Area = 8 square units 6. Area = 12 square units Page 12 of 28

13 3.MD.5b: Lesson 2 Recognize area as an attribute of plane figures and understand concepts of area measurement. b. A plane figure which can be covered without gaps or overlaps by n unit squares is said to have an area of n square units. 7. Create a shape with 42 square units on the grid. 8. Area = 9 square units 9. Area = 15 square units 10. Determine the area of the shaded shape in square units. 11. Area = 18 square units 12. Area = 21 square units Page 13 of 28

14 3.MD.5b: Lesson 2 Recognize area as an attribute of plane figures and understand concepts of area measurement. b. A plane figure which can be covered without gaps or overlaps by n unit squares is said to have an area of n square units. 1. Determine the area of the shaded shape in square units. 2. Area = 3 square units 3. Area = 5 square units 4. Create a shape with the 25 square units on the grid. 5. Area = 14 square units 6. Area = 6 square units Page 14 of 28

15 3.MD.5b: Lesson 2 Recognize area as an attribute of plane figures and understand concepts of area measurement. b. A plane figure which can be covered without gaps or overlaps by n unit squares is said to have an area of n square units. 7. Create a shape with the 50 square units on the grid. 8. Area = 4 square units 9. Area = 10 square units 10. Determine the area of the shaded shape in square units. 11. Determine the area of the shaded shape in square units. 12. Area = 6 square units Page 15 of 28

16 3.MD.6: Lesson 3 Measure areas by counting unit squares (square cm, square m, square in, square ft, and improvised units). Sarah s little sister Hanna asked her to help decorate a wall in her bedroom with picture frames. Sarah decided to use rectangular shapes. Help Hanna determine the area of each shape by counting the number of square inches. = 1 in 2 A F E B D C Area of Shape A = Area of Shape B = Area of Shape C = Area of Shape D = Area of Shape E = Area of Shape F = Focus Questions Question 1: What do you notice about the size of a square inch when it is compared to a square foot? Question 2: How many square centimeters are in a square meter? Journal Question How many square inches are in 1 square foot? Explain how you know using mathematical reasoning. Page 16 of 28

17 3.MD.6: Lesson 3 Measure areas by counting unit squares (square cm, square m, square in, square ft, and improvised units). 1. Determine how many square inches are contained in the shaded figure. = 1 in 2 2. Determine how many square inches are contained in the shaded figure. = 1 in 2 3. Determine how many square inches are contained in the shaded figure. = 1 in 2 4. Determine how many square inches are contained in the shaded figure. = 1 in 2 5. Determine how many square inches are contained in 6. Determine how many square inches are contained in the shaded figure. = 1 in 2 the shaded figure. = 1 in 2 Page 17 of 28

18 3.MD.6: Lesson 3 Measure areas by counting unit squares (square cm, square m, square in, square ft, and improvised units). 7. Determine how many square inches are contained in the shaded figure. = 1 in 2 8. Determine how many square inches are contained in the shaded figure. = 1 in 2 9. Determine how many square inches are contained in the shaded figure. = 1 m Determine how many square inches are contained in the shaded figure. = 1 m Determine how many square inches are contained 12. Determine how many square inches are contained in the shaded figure. = 1 cm 2 in the shaded figure. = 1 cm 2 Page 18 of 28

19 3.MD.6: Lesson 3 Measure areas by counting unit squares (square cm, square m, square in, square ft, and improvised units). 1. Determine how many square inches are contained in the shaded figure. = 1 in 2 2. Determine how many square inches are contained in the shaded figure. = 1 in 2 3. Determine how many square inches are contained in the shaded figure. = 1 m 2 4. Determine how many square inches are contained in the shaded figure. = 1 m 2 5. Determine how many square inches are contained in 6. Determine how many square inches are contained in the shaded figure. = 1 cm 2 the shaded figure. = 1 cm 2 Page 19 of 28

20 3.MD.6: Lesson 3 Measure areas by counting unit squares (square cm, square m, square in, square ft, and improvised units). 7. Determine how many square inches are contained in the shaded figure. = 1 in 2 8. Determine how many square inches are contained in the shaded figure. = 1 in 2 9. Determine how many square inches are contained in the shaded figure. = 1 m Determine how many square inches are contained in the shaded figure. = 1 m Determine how many square inches are contained 12. Determine how many square inches are contained in the shaded figure. = 1 cm 2 in the shaded figure. = 1 cm 2 Page 20 of 28

21 3.MD.6: Lesson 4 Measure areas by counting unit squares (square cm, square m, square in, square ft, and improvised units). Given the measurements of a junior peewee soccer field below, determine the area of the parts of the field listed below. 4 ft 2 ft 3 ft 4ft 8 ft 5 ft 10 ft 20 ft 1. Area of the entire field = 2. Area of one half of the field = 3. Area of the goal box = 4. Area of the penalty box = 5. Area of the goal = 6. Area of half the field = Focus Questions Question 1: Which is larger, an inch or a centimeter? Question 2: Compare one square foot to one square meter, which one is larger? Journal Question If you had to choose between using the Metric System or the American Standard System of Measurement only, which would you choose and why? Page 21 of 28

22 3.MD.6: Lesson 4 Measure areas by counting unit squares (square cm, square m, square in, square ft, and improvised units). Area in square centimeters (Determine the area of each shape in cm 2.) A D E C B F Area of Shape A = Area of Shape B = Area of Shape C = Area of Shape D = Area of Shape E = Area of Shape F = Page 22 of 28

23 3.MD.6: Lesson 4 Measure areas by counting unit squares (square cm, square m, square in, square ft, and improvised units). Create your own shapes with the following areas. Be sure to label each shape with the correct letter. Area of Shape A = 8 cm 2 Area of Shape B = 10 cm 2 Area of Shape C = 3 cm 2 Area of Shape D = 12 cm 2 Area of Shape E = 15 cm 2 Area of Shape F = 9 cm 2 Page 23 of 28

24 3.MD.6: Lesson 4 Measure areas by counting unit squares (square cm, square m, square in, square ft, and improvised units). Name Date Determine how many square feet or square meters are contained in each figure. = 1 Ft Page 24 of 28

25 3.MD.6: Lesson 4 Measure areas by counting unit squares (square cm, square m, square in, square ft, and improvised units). = 1 m Page 25 of 28

26 3.MD.5 & 6: Lesson 5 5. Recognize area as an attribute of plane figures and understand concepts of area measurement. a. A square with side length 1 unit, called a unit square, is said to have one square unit of area, and can be used to measure area. b. A plane figure which can be covered without gaps or overlaps by n unit squares is said to have an area of n square units. 6. Measure areas by counting unit squares (square cm, square m, square in, square ft, and improvised units). Patricia wants to buy a new set of bedroom furniture. She knows that her room is 100 square feet in area, and she has created a drawing of it below. Her problem is that she doesen t know if the new furniture that she wants to buy will fit in her bedroom and still allow her space to move around. Below is a list of items that Patricia wants to buy for her bedroom. Place each item in the room to determine if she has enough space to fit it all. Then determine if she will have enough space to move around the bedroom once the furniture is placed in it. List of Furniture Items Item Dimensions Bed 6ft. x 5 ft. Night Stand 2 ft. x 2ft. Dresser 6 ft. x 3 ft. Chest of Drawers 2 ft. x 3 ft. Desk 4 ft. x 2 ft. 1ft 1ft Focus Questions Question 1: What strategies can be used to find the area of a shape? Question 2: Can you describe another method for determining area that does not require you to count all of the squares inside a shape? Journal Question Describe one thing that you know now that you didn t know before doing these tasks. Page 26 of 28

27 LESSON 5 RUBRIC Score GOLDEN PROBLEM Description 3 Student has an understanding of what area is. Student correctly placed all of the items in the room and was able to determine the correct area for each item. In addition, the student was then able to determine the total area taken up by all of the items combined. Finally, the student was able to subtract that area from the original area of the room to correctly determine how much space is left once the furniture is placed into the room. 2 Student has an understanding of what area is. Student correctly placed all of the items in the room and was able to determine the correct area for each item, however he/she either does not come up with the total amount of area taken up by the items, does not determine the total area of the room, shows a lack of understanding of how to determine the remaining area, or simply makes mathematical mistakes in computation. 1 Student places the items in the room but is unable to determine the remaining area, or fails to place all of the items in the room. 0 Does not address task, unresponsive, unrelated or inappropriate. Page 27 of 28

28 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 Two Plus Two Digit Addition Version 4 Answer Key 2. Addition Worksheet Three Plus Two Digit Addition Version 4 Answer Key 3. Multiplication All Two Minute Test Version 1 Answer Key 4. Multiplication All Two Minute Test Version 2 Answer Key Page 28 of 28