# Number Models for Area

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1 Number Models for Area Objectives To guide children as they develop the concept of area by measuring with identical squares; and to demonstrate how to calculate the area of rectangles using number models. epresentations etoolkit Algorithms Practice EM Facts Workshop Game Family Letters Assessment Management Common Core State Standards Curriculum Focal Points Interactive Teacher s Lesson Guide Teaching the Lesson Ongoing Learning & Practice Differentiation Options Key Concepts and Skills Use multiplication facts to find the areas [Operations and Computation Goal 3] Draw and use arrays to find the areas [Operations and Computation Goal 6] Count unit squares to determine the areas [Measurement and Reference Frames Goal 2] Use arrays to write number models to find the areas [Patterns, Functions, and Algebra Goal 2] Key Activities Children discuss ways to find the area of a room that is to be carpeted. They write number models to show how to find areas Ongoing Assessment: Recognizing Student Achievement Use journal page 74. [Measurement and Reference Frames Goal 2] Materials Math Journal 1, p. 74 Student Reference Book, pp. 156, 156A, and 156B Home Link 3 7 transparency of Math Masters, p. 417 (optional) 1-yard squares of paper tape (optional) yardstick (optional) Simulating a Shopping Trip Math Masters, p. 75 Student Reference Book, pp. 214, 216, and 217 play money Children simulate a shopping trip to practice calculating money. Math Boxes 3 Math Journal 1, p. 75 Children practice and maintain skills through Math Box problems. Home Link 3 Math Masters, p. 74 Children practice and maintain skills through Home Link activities. READINESS Modeling Area with a Geoboard geoboard rubber bands pennies Children use geoboards to find areas ENRICHMENT Finding the Areas of Squares and Triangles Math Masters, pp. 76 and 41 square pattern blocks ruler scissors Children find and compare the areas of two triangles and a square. EXTRA PRACTICE Investigating Perimeter and Area Math Masters, p. 416 Children draw different rectangles with areas of 12 square centimeters and find the perimeter of each. Advance Preparation For Part 1, you will need enough 1-yard squares to lay along the length of one classroom wall. Make a transparency of Math Masters, page 417 or draw a 10-by-10 grid on the board. Teacher s Reference Manual, Grades 1 3 pp Unit 3 Linear Measures and Area

2 Getting Started Mathematical Practices SMP1, SMP2, SMP4, SMP5, SMP7, SMP Content Standards 3.MD.5, 3.MD.5a, 3.MD.5b, 3.MD.6, 3.MD.7a, 3.MD.7b, 3.MD. Mental Math and Reflexes Pose fact extensions like the following: = = = = = = = = = = = = 105 Math Message Suppose you want to order carpet to cover the whole classroom floor. How would you find out how many square yards of carpet to buy? Be ready to talk about it. Home Link 3 7 Follow-Up Children share their answers with a partner. 1 Teaching the Lesson Math Message Follow-Up DISCUSSION To find out how many square yards of carpet to order, it is necessary to find the area of the floor. Discuss ways to do this. Someone may suggest tiling the entire floor with 1-yard squares. This might be difficult since furniture would get in the way. Links to the Future Using 1-Yard Squares to Estimate Area The activities in this lesson lay the foundation for the formula for the area of a rectangle: Area = length of base height. Using the formula to calculate the area of a rectangle is a Grade 5 Goal. To find the area of the room with very little moving of furniture, try the following activity with the class: Rules 1. Place 1-yard squares in a row along one wall of the room. Ask: How many squares are in the row? 2. Ask children to imagine a second row of 1-yard squares next to the first row. Ask: How many squares would be in the second row? How many squares are there in all in the 2 rows? 3. Point out that to find the area of the whole floor, you need to find out how many rows of 1-yard squares would cover the floor. Ask: How would you find the number of rows? Sample answers: You could place 1-yard squares next to each other along the adjacent wall: Each square represents 1 row. Or you could simply measure the adjacent wall with a yardstick: Each yard represents 1 row. 4. After children have figured out the number of rows, have them estimate the area of the floor. rows with square yards in each row gives a total of square yards. Area = length of base height rows in each row = square yards square yards Adjusting the Activity Have children tape the 1-yard squares together and roll them into a scroll. The scroll can be useful if children want to find the areas of other large surfaces, such as the hall or various parts of the playground. AUDITORY KINESTHETIC TACTILE VISUAL Lesson 3 213

3 10 9 rectangle Finding the Areas of DISCUSSION Rectangles (Student Reference Book, p. 156; Math Masters, p. 417, optional) Show children how to draw a 10-by-9 rectangle (10 squares down, 9 squares across) on the 10-by-10 grid you prepared for the lesson. Tell the class that this rectangle represents the floor of a room for which you want to order carpet. Shade a row of squares along the shorter side of the rectangle. Say that each square represents 1 square yard. How many squares are there in this row? 9 squares 10 rows 9 square yards in each row 10 9 = 90 Answer: The area is 90 square yards. NOTE Some children might be curious about the symbols for the paper units they are using or for other commonly used units of area. square foot sq ft ft 2 square yard sq yd yd 2 square inch sq in. in. 2 square centimeter sq cm cm 2 square meter sq m m 2 How many equal rows of squares are there inside the rectangle? 10 equal rows How many squares are there in all? 90 squares Write 10 9 = 90, and tell the class that you can use this number model to represent the area of the rectangle. The number model uses the multiplication symbol. Children who used Second Grade Everyday Mathematics have had experience with arrays, multiplication, and the multiplication symbol. Remind children that is read as times or multiplied by. How much carpet should you order? At least 90 square yards Repeat this routine with other rectangles. In discussing solutions, use such phrases as an x-by-y rectangle and x rows with y squares in each row, for a total of z squares. Summarize the activity by reviewing Student Reference Book, page 156 with the class. Measurement Some surfaces are too large to cover with squares. It would take too long to count a large number of squares. To find the area of a rectangle, you do not need to count all of the squares that cover it. The example below shows a shortcut for finding the area. Find the area of this rectangle. Each square is 1 square foot. There are 4 rows of squares. Each row has 10 squares. So there are 4 10 squares, or 40 squares in all. The area is 40 square feet. Student Page Summary To find the area of a rectangle: 1. Count the number of rows. 2. Count the number of squares in each row. 3. Multiply: (number of rows) (number of squares in each row) Finding the Area of Rectilinear Figures (Student Reference Book, pp. 156A and 156B; Math Masters, p. 417) DISCUSSION Draw a rectilinear figure on the 10-by-10 grid, as shown below on the left. Explain to children that this figure represents an outdoor reading area that you want to cover with rubber tiles. Say that each tile measures 1 square yard and that each square on the grid represents 1 square yard. A B Find the area of each rectangle Which area is larger, 1 square yard or 1 square meter? Check your answers on page 341. Student Reference Book, p _16_EMCS_S_G3_SRB_MeA_ indd Unit 3 Linear Measures and Area 2/16/11 1:42 PM

6 Example: 5-by-4 rectangle (See margin.) How many squares are there in a row? 4 squares How many equal rows of squares are there inside the rectangle? 5 equal rows How many squares are there in all? 20 squares Discuss strategies for finding the total number of squares. Ask children to describe their rectangles without showing them to the class. For example, My rectangle has 4 squares in a row. There are 5 equal rows. The area of my rectangle is 20 squares. After children have described a rectangle, ask the other children to make the same rectangle on their geoboards. ENRICHMENT Finding the Areas of Squares and Triangles (Math Masters, pp. 76 and 41) PARTNER Min 3 Finding and Comparing Areas 1. Cut out the square. 2. Use square pattern blocks to find the area of the square. 3. Cut the square into 2 equal triangles. 4. Find the area of each triangle. Names: Area of square: sq. in. 2. Area of first triangle: about sq. in. 3. Area of second triangle: about sq. in. 4. How do the areas of the two triangles compare? They are about the same. 5. How does the area of one triangle compare to the area of the original square? It is about half the area of the square. Math Masters, p. 76 Teaching Master To further explore area concepts, have children find and compare the areas of triangles and squares. Partners cut a 4-inch square from Math Masters, page 76. Using square pattern blocks, inch grid paper (Math Masters, page 41), and rulers, partners work together to find the area of the square. Then they cut the square into 2 equal triangles and find the area of each triangle. They answer the questions on Math Masters, page 76. EXTRA PRACTICE Investigating Perimeter and Area (Math Masters, p. 416) SMALL-GROUP Min To provide additional experience with finding the areas and perimeters of rectangles, children draw as many different rectangles as they can with areas of 12 square centimeters. Distribute several copies of Centimeter Grid Paper (Math Masters, page 416) to each small group for children to draw on. Remind them to label their rectangles with the lengths of each side. Then have them find and record the perimeter of each rectangle. After children complete this task, bring the class together to go over their answers. There are three possible rectangles with an area of 12 square centimeters whose sides have whole-number lengths: 1 cm 12 cm, 3 cm 4 cm, and 2 cm 6 cm. The perimeters of these rectangles are 26 cm, 14 cm, and 16 cm. Some children may suggest other rectangles, such as 1 1_ cm cm. The perimeter of 2 this rectangle is 19 cm. Over the next few days, encourage children who show interest in this activity to draw as many rectangles as they can for other specified areas. You may want to make a bulletin board of the results and discuss patterns that children observe. One-Inch Grid 41 Teaching Aid Master Math Masters, p = 20 Lesson 3 217

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