Number Models for Area


 Annabelle Atkinson
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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) 1yard 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 1yard squares to lay along the length of one classroom wall. Make a transparency of Math Masters, page 417 or draw a 10by10 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 FollowUp Children share their answers with a partner. 1 Teaching the Lesson Math Message FollowUp 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 1yard squares. This might be difficult since furniture would get in the way. Links to the Future Using 1Yard 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 1yard 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 1yard 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 1yard squares would cover the floor. Ask: How would you find the number of rows? Sample answers: You could place 1yard 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 1yard 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 10by9 rectangle (10 squares down, 9 squares across) on the 10by10 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 xbyy 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 10by10 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
4 Ask children to think about how they might find the number of rubber tiles needed to cover an outdoor reading area of this shape. Expect that some children may view the rectilinear figure as having the same area as the by9 rectangle surrounding it, and may suggest multiplying times 9 to get the area. Point out that this strategy will not work because the shape is not a rectangle; it is not made up entirely of equal rows of squares. Explain that the figure can be partitioned into 2 nonoverlapping rectangles and model how to do it as shown. You may want to label one rectangle A and the other B or outline each rectangle with a different color. For Rectangle A, ask: How many square yards are there in one row? 9 square yards How many equal rows of square yards are there inside the rectangle? 4 equal rows Write 9 4 = 36 and remind children that this number model can be used to represent the area of Rectangle A. In the same manner, determine the area of Rectangle B. 24 square yards Next, ask children to calculate the area for the whole figure. 60 square yards. Have them share their strategies with the class. Sample answer: I added the area of Rectangle A and the area of Rectangle B together to find the area of the whole figure. Some children may suggest finding the area of the 9by rectangle and then subtracting the cutout area. 9 = 72; 3 4 = 12; = 60. Praise this strategy, but keep the focus of the activity on partitioning the figure into rectangles. How many squareyard rubber tiles are needed to cover the entire outdoor reading area? 60 tiles Summarize the activity by reading Student Reference Book, pages 156A and 156B with the class. Practicing Finding the Areas of Rectangles (Math Journal 1, p. 74) INDEPENDENT PROBLEM SOLVING Date 3 1. Draw a 5by7 rectangle. More Areas of Rectangles Number of rows: Squares in a row: Area = square units Number model: = Time Draw the rectangle on the grid. Then fill in the blanks Draw an by rectangle. Number of rows: Squares in a row: Area = square units Number model: 3. Draw a 3by9 rectangle. Area = square units Number model: = = Follow these steps to find the area of this shape. Sample answer: 1. Divide the shape into 2 rectangles. Area of shape = _EMCS_S_SMJ_G3_U03_ indd Find the area of each rectangle. 3. Add the areas to find the total area of the shape. square units Math Journal 1, p. 74 Student Page 64 Sample answers: 3 A Shopping Trip Teaching Master 1. List the items you are buying in the space below. You must buy at least four items. Answers vary. If you buy the same item 2 times, list it 2 times. Item Sale Price 27 1/26/11 9:07 AM Algebraic Thinking Children write number models and find the areas Ongoing Assessment: Recognizing Student Achievement Journal page 74 Problems 1 3 Use journal page 74, Problems 1 through 3 to assess children s progress with using strategies to calculate the area of rectangular shapes. Children are making adequate progress if they are able to count the square units to determine the area. Some children may use multiplication to determine the area. [Measurement and Reference Frames Goal 2] 2. Estimate how many dollar bills you will need to pay for your items. 3. Give the clerk the dollar bills. 4. The clerk calculates the total cost. You owe $. 5. The clerk calculates the change you should be getting. $ 6. Record your change. Use Î, Â, Í,. Try This Use the Stationery Store Poster in your Student Reference Book, page Linda wants to buy a box of pens and a box of pencils. How much will she save by buying them on sale? Regular price Sale price Difference pens $ $ Regular total $ pencils $ 1.9 $ 1.49 Sale total $ Total cost $ 6.3 $ 4.4 Amount saved $ 1.90 Math Masters, p. 75 Lesson 3 215
5 Date 3 Math Boxes 1. Circle the best unit of measurement. distance to Spain miles centimeters inches width of a crayon miles centimeters feet length of your journal miles yards inches 3. When I left home, I had $4.00. I spent $0.73 at the fruit stand and $2.59 at the grocery store. How much did I spend in all? $3.32 How much do I have left? $ Describe 2 events that you are certain will not happen today. Answers vary Math Journal 1, p _EMCS_S_SMJ_G3_U03_ indd 75 Student Page 92 Time 2. Find the perimeter. 2 cm 2 cm 5 cm 3 cm Fill in the circle for the best answer. A. 13 cm B. 12 cm C. cm D. 4 cm 4. John s dad gave him 107 pennies. Now John has 392 pennies. With how many pennies did John start? Change Start End? Fill in the empty frames. Rule Number model: =? or? = 392 Answer: 25 pennies /26/11 9:07 AM 2 Ongoing Learning & Practice Simulating a Shopping Trip PARTNER (Math Masters, p. 75; Student Reference Book, pp. 214, 216, and 217) This activity was introduced in Lesson To buy items on the shopping trip, children can use any of the following sources: the ad display suggested by Home Link 110 the Stationery Store Poster on Student Reference Book, page 214 either of the two StockUp Sale Posters in the Student Reference Book, pages 216 and 217 Working in partnerships, children first select the items they want to buy; record the information on Math Masters, page 75; and estimate the number of bills to give the shopkeeper. Then they take turns acting out their shopping trip. After each child completes the shopping trip, partners work together to complete the Try This problem. Math Boxes 3 (Math Journal 1, p. 75) INDEPENDENT Mixed Practice Math Boxes in this lesson are paired with Math Boxes in Lesson 36. The skill in Problem 6 previews Unit 4 content. HOME LINK 3 Family Note Area Today we discussed area as an array, or diagram. An array is a rectangular arrangement of objects in rows and columns. Help your child draw an array of the tomato plants in Problem 3. Use that diagram to find the total number of plants Please return this Home Link to school tomorrow. Mr. Li tiled his kitchen floor. This is what the tiled floor looks like. 1. How many tiles did he use? tiles 2. Each tile cost $2. How much did all the tiles cost? $ Mrs. Li planted tomato plants. She planted Sample answer: 5 rows with 6 plants in each row. Draw a diagram of her tomato plants. Hint: You can show each plant with a large dot or an X. 4. How many tomato plants are there in all? 30 plants Practice Write these problems on the back of this page. Fill in a unit box. Write a number model for your ballpark estimate. Use any method you wish to solve each problem. Show your work Home Link Master 0 Math Masters, p Unit 216 Unit 3 Linear Measures and Area Home Link 3 (Math Masters, p. 74) INDEPENDENT Home Connection Children solve number stories involving equal rows of things. They also solve addition and subtraction problems by estimating and then finding an exact answer. 3 Differentiation Options READINESS Modeling Area with a Geoboard SMALLGROUP 5 15 Min To provide concrete experience with area, have children use geoboards to model and calculate the area of a rectangle. Have them use a rubber band to make various rectangles on a geoboard. When counting the number of squares in a row or rows in the rectangle, have children place a penny in each of the squares.
6 Example: 5by4 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 4inch 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) SMALLGROUP 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 wholenumber 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. OneInch Grid 41 Teaching Aid Master Math Masters, p = 20 Lesson 3 217
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