Lesson 3: Area. Selected Content Standards. Translating Content Standards into Instruction


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1 Lesson 3: Area Selected Content Standards Benchmark Assessed: M.3 Estimating, computing, and applying physical measurement using suitable units (e.g., calculate perimeter and area of plane figures, surface area and volume of solids presented in realworld situations) Lesson Focus: This lesson is intended as an introductory lesson to the concept of area. Students should know by the end of this lesson the following information: The conceptual difference between area and perimeter How to use various methods to determine area of plane figures (including using a ruler, grid paper, and calculation) Use of scale in a scale drawing as it applies to area GEE 21 Connection The skills that will be addressed in this lesson include the following: Understand the concept of area Understand what a square unit is and how it is different from a linear measure Find the area of plane figures using grids Calculate the area of plane figures shown in diagrams or described in realworld terms Understand the connection between area of rectangles and triangles Understand numerical relationships among units within each system (customary and metric) Compare approximate relationships between systems in terms of intuitive reference points Translating Content Standards into Instruction A. The first goal of this lesson is to help students to understand what area is and how it is different from perimeter. 1. The first thing to do is to get students to understand what a SURFACE is something that we can cover or paint (such as floors, tabletops, covering of a ball, etc.) 2. We want students to associate the word SURFACE with the word AREA. Explain to students that The amount of surface that an object has is called its AREA. 3. Emphasize the difference between a length (show students a string and explain that length has no width to it it is onedimensional) and an area (show students a rectangular book and explain that the surface has both length and width it is twodimensional). 4. Point out that perimeter (which is a length) and area have two completely different types of units. This will be emphasized throughout the lesson. 20
2 5. Hand out Student Worksheet #1 to students, and ask them which figure they think has the most area associated with it. Show them a sticky tab (this worksheet was designed with the 2 x 3 yellow sticky notes in mind if you can, provide all the students with a set of four each) and ask students to estimate how many of the sticky notes it would take to cover each of the figures on the worksheet. Once students have made their estimates, have students actually use the sticky notes that were provided to cover the figures completely. Students may be surprised to find that all of the figures have the same area. Talk about the results with the students. 6. Have students estimate how many sheets of paper (8 ½ x 11 ) it would take to cover the blackboard in your room. Have each group come to the front of the class and write their estimates on the board. After all groups have made their estimates, talk about the strategies students came up with to get their answers. Model for students using strategies which students recommended to determine how many sheets it would take (Of course, you do not want to cover the entire board we want students to see that if we know how many rows of paper and how many in each row, we could multiply to find the answer). Compare the actual answer with estimates given by the students. B. Have students do Student Worksheet #2 in groups. During this activity, students will make a square foot and a square yard. The teacher should provide students with two large sheets of bulletin board paper in order to do so. After all students have completed the worksheet, go over the results with the class. 1. Stress the fact that area is in square units. Make sure students understand why squares are used rather than circles, etc. 2. Emphasize that we use our standard units of square centimeters, square inches, square feet, square yards, etc. when talking about units for area. Again, stress the difference between area and perimeter. 3. Make sure that students see what a square foot and square yard are and their relationship to one another. Students should see that in every square foot there are 144 square inches. They should see that it takes 9 sq. ft. to make 1 sq. yd. C. In this part of the lesson, lead students into understanding how the area of a rectangle can be found by calculation, that the area of a triangle is ½ the area of the rectangle that it comes from, and that the area of irregular figures can be found by finding the area of the individual rectangular and triangular areas that make it up (all we have to do is add the individual areas to get the total area). 1. Help students understand that unit squares do not have to be drawn on a rectangle to determine the area of the rectangle. To find the area of a rectangle, measure the length (the number of squares that will fit in each row), measure the width (the number of rows), and use this information to determine how many square units it would take to cover the rectangle. Thus, to calculate the area in square centimeters, square inches, or whatever unit is being used, multiply length by width. 2. Help students understand that measuring the length and width of a rectangle, and multiplying the two linear measures results in a new unit, called a square unit. Multiplying centimeters by centimeters results in a new unit called a 21
3 square centimeter. Multiplying inches by inches, we get square inches. Multiplying feet by feet, results in square feet. 3. Present the following problems to students and talk to students about each one. (a) Mrs. Peterson has a rectangular outdoor workshop that is 20 feet wide and 45 feet long. She wants to buy some carpeting to lay across the entire floor of the building. How much carpeting will she need? (b) Mrs. Peterson wants to buy some molding to put all around the edge of the building. How much molding will she need? (c) In teams of two, use your ruler to find the area of your rectangular desk (everyone should use the same size desk). Use inches as your unit of measure. What is the area? (d) Find the area and perimeter of the figure shown. Area = Perimeter= 5 meters 6 meters 2 meters 12 meters 4. Have students work in small groups on Student Worksheet #3, provided for this lesson. After students have finished the worksheet, go over all work and talk about anything which may have given students difficulties. No doubt, there will be some discrepancies in measurement for some of the items remind the students that measurement is not exact. 5. The topic of finding the area of triangles is presented on the worksheet. The teacher may have to get more in depth with the class as to the relationship between the area of a triangle and rectangle. D. Students will use a scale drawing of a basketball court and use the drawing to determine information from it. 1. Have students do the scale drawing activity, Student Worksheet #4, provided with the lesson. 2. Talk to students about finding areas of circles. This information is necessary for them to complete the activity. 3. As an enrichment activity, you may want the students to make measurements of an actual basketball court in the school gym if possible. GEE 21 Connection On the GEE 21 test, students may be required to determine the area of various geometric shapes (including areas of rectangles, triangles, circles, and composite figures). Students may also be required to apply the concept of area in a reallife applications in which the word area may never be asked for directly (such as finding 22
4 the amount of carpeting required for a room, etc.). Converting within units and from system to system may also be tested. Sources of Evidence about Student Learning A. Have students make measurements of rooms or halls around the school in order to calculate the areas. B. Have students do student worksheets in groups provided with the lesson and have students explain their work for the class. GEE 21 Connection See attachment at the end of the Lessons for sample questions related to the GEE 21. Attributes of Student Work at the GotIt level A. When students are making their measurements for the items that are in the room or around the school, make sure that all students are within an acceptable range in their measurements. Any student or group that is outside the acceptable range should be required to remeasure the item to make sure that the error being made is corrected. B. When going over student work on the provided worksheets, allow students to work alone first, then provide time to allow for students to compare their answers in groups. Let the students talk about how they got their answers and if there were any discrepancies between group members, have the members explain what they did to clear up any differences. Make sure that all students are within an acceptable range when writing down a measurement, but all students must understand that measurement is not an exact science. It is an approximation within a certain degree of accuracy, depending on the measurement tool being used. 23
5 Lesson 3: Area Student Worksheet #1 Directions: Use the figures below to do this activity. First, make estimates of how many sticky tabs it would take to cover each figure. After you make your estimates, actually take sticky tabs and try to cover the figure to see how close your estimate is. The sticky tabs cannot overlap. Figure 1: Estimate Actual: Figure 2: Estimate: Actual: Figure 3: Estimate: Actual: 24
6 Lesson 3: Area Student Worksheet #2 1. Which of the figures below would you say has the most area associated with it? Why? 2. If you were trying to cover the floor in the classroom, which figure below do you think it would be easier to use to cover the entire space without any gaps? Explain why? 3. Area is commonly found using square units. Why do you think that is? 4. How many square units,,does it take to cover the entire figure shown? 5. Use the figure below to answer the following questions. (a) What is the area of the figure? (b) What is the perimeter of the figure? (c) What is the length of the figure? width? (d) What multiplication fact does the figure model? x (e) If you multiply the length and width of the figure, what do you notice in connection with the area? 25
7 6. Use the grid at the right to draw a figure that has an area of 4 square units with a perimeter of 10 units. 7. Mr. Connor is laying out square tiles on his kitchen floor. So far he has laid out the squares shown. Determine how many squares it would take to cover the entire floor. If it took him 14 minutes to lay the tiles shown here, how long would it take to do the rest of the floor? Explain how you got your answer. 8. Shown below are two squares. Answer the following questions. Square #1 Square #2 (a) Measure the sides of square #1 with a centimeter ruler. What is the length of each side of the square? (b) Measure the sides of square #2 with an inch ruler. What is the length of each side of the square? (c) One of the squares shown is called a square centimeter and one of the square is called a square inch. Which name goes with which square? Explain why you chose your answer. (d) Using the figure shown, determine how many square centimeters it would take to cover the figure. 26
8 (e) Using the figure shown, determine how many square inches it would take to cover the below. (f) Look at the figure shown to the right do you think it would take more square centimeters or square inches to cover it? Explain why. 9. Using large sheets of bulletin board paper that you will get from your teacher, make a square foot and a square yard. Using the models you make, determine the following: (a) How many square feet does it take to make a square yard? (b) How many square inches does it take to make a square foot? (c) How many square inches will it take to cover a square yard? Show how you got your answer to this. 10. Explain in your own words what is different about perimeter and area. 27
9 Lesson 3: Area Student Worksheet #3 1. This is a square centimeter. (a) Use your centimeter ruler to find the length of each side of the square centimeter. What is the length of each side? (b) What is the perimeter of the square centimeter? (c) About how many of these square centimeters would fit inside this rectangle? What is the area in square centimeters of the rectangle? 2. This is a square inch. (a) Using your inch ruler, measure the sides of the square inch. What is the length of each side? (b) What is the perimeter of the square inch? (c) About how many of the square inches would fit inside this rectangle. What is the area of the rectangle? Measure each side of the rectangle to the nearest inch. What is the length of the rectangle? What is the width? If you multiply the length by the width, what do you get? 3. What is the area and perimeter of the rectangle shown below: Area = Perimeter = 6.25 ft ft 28
10 4. Using your ruler, measure the length and width of this rectangle. Use whole centimeters as the unit. Put dots along the length and width to mark each centimeter. Width: cm Length: cm (a) Remember this is what a square centimeter looks like. (b) How many of these square centimeters could be lined up in one row across the bottom of the rectangle? (c) How many rows of square centimeters would fit in the rectangle? (d) How many square centimeters all together will fit inside the rectangle? How did you figure this out? (e) What is the area of the rectangle in square centimeters? (f) Do you have to have all of the squares already drawn on a rectangle to figure out the area? Why or why not? 5. Using what you learned in problem 4, calculate the area of the rectangle here in square centimeters. 29
11 6. Working in teams of two or three, find the following: (a) Find the perimeter of the classroom floor in inches, feet, yards, millimeters, centimeters, and meters. The PERIMETER of the room is: inches or feet or yards millimeters or centimeters or meters (b) Find the area of the classroom floor in square centimeters, square meters, square inches, square feet, and square yards. The AREA of the room is: sq. centimeters or sq. meters sq. in. or sq. ft. or sq. yd. 7. Below is a triangle located inside a rectangle. Using what you have learned about finding areas of rectangles, answer the given questions. (a) If the area of the rectangle is 20 square feet, what would you say the area of the triangle is? Explain how you know. (b) Using your centimeter ruler, calculate the area of the rectangle in square centimeters. What do you think the area of the triangle is? Explain what you would do if you had to find the area of any triangle. 30
12 8. Julie wants to build an exercise pen for her pet rabbit. She has 36 feet of fencing and 4 metal posts to build a rectangular enclosure. She wants to carefully plan her project, measuring in units of whole feet. Find all the possible ways that Julie could build her pen and have a perimeter of 36 feet and whole units for length and width. Fill in the table below and find the area of each case. Length (ft.) Width (ft.) Perimeter (ft.) Area (sq. ft.) (a) What dimensions (length and width) for the pen should Julie build for her rabbit if she wants the largest possible area? (b) What would be a benefit of building the pen in one of the other ways (the pen without the most area)? 9. Carlos bought some land. The scale drawing below shows the layout of his land. What is the total area (approximately) in square feet, for the land that he bought? (Scale: 1 cm = 30 feet) 31
13 Lesson 3: Area Student Worksheet #4 Directions: Below is a scale drawing of a basketball court. Use the scale drawing to answer the given questions. Jump Area Free Throw Lane ft 19 ft 6 ft radius Circular Outer Jump Area 2 ft inner radius The Arc What is the total area of the basketball court? 2. What is the area of the free throw lane? 3. What is the area of the Jump Area? 4. What is the area of the Circular Outer Jump Area? 5. What is the area of The Arc? 6. What is the perimeter of the basketball court? 7. How many times would you have to run around the basketball court if you wanted to run a mile? Show or explain how you got your answer. 32
14 Lesson 3 Area Answer Sheet Student Worksheet #1 *The answers here depend on the size of the sticky tabs used. Use answers based upon their own particular classes. However, it should be noted that all of the figures have the EXACT SAME AREA. Student Worksheet #2 1. The rectangle has the most area associated with it (it has the most surface). 2. It would be easier to cover the floor with the squares because there would be no empty spaces in the flooring. 3. Area is found in square units because other units would leave spaces, and would not entirely cover the object square units 5. (a) 24 square units (b) 20 units (c) l = 6 units, w = 4 units (or vice versa) (d) 4 x 6 or 6 x 4 (d) You get the area of the figure 6. Answers may vary. 7. It would take him 80 squares to cover the entire floor; every 14 minutes he can lay 26 tiles. In another 14 minutes he would lay an additional 26 tiles (52 so far). In another 14 minutes he would lay an additional 26 tiles (78 so far) So he would need to lay another two tiles at that point (each tile takes about thirty seconds to lay so this is an additional minute of time). Altogether, it would take an about an additional 29 minutes, for a total time to lay the floor of about 43 minutes. (This is of course one way of doing this problem through proportional reasoning). 8. (a) 1 cm (b) 1 inch (c) Square # 1 is the square centimeter; Square #2 is square the square inch (d) 8 square centimeters (e) 4 square inches (f) It would take more square centimeters to cover it because they are smaller than the square inches. 9. All of these answers should be found by actually making the square foot and square yard. DO NOT just show students the answers on a chart. (a) 9 square feet = 1 square yard (b) 144 square inches = 1 square foot (c) 9x144 sq in = 1296 square inches 10. Perimeter is a linear measurement (1 dimension) and is the total distance around a figure. Area is a measure of the amount of surface an object has. Its units are totally different than that of perimeter called square units (2 dimensions). 33
15 Lesson 3: Area Answer Sheet Student Worksheet #3 1. (a) 1 cm (b) 4 cm (c) Since it would take 18 square centimeters to fit inside the figure, this is also the area associated with the figure. 2. (a) 1 in (b) 4 in (c) 9 sq in; 9 sq in; length = 3 in, width = 3 in; 9 sq in 3. Area is about 147 sq ft; Perimeter = 59.5 ft 4. length = 8 cm; width = 6 cm (b) 8 (c) 6 (d) 48 square centimeters (e) 48 sq cm (f) You do not have to have all of the squares drawn because you can just measure each side of the rectangle and multiply the two numbers to find the area sq cm 6. Answers will vary from class to class. 7. (a) The area of the triangle is ½ the area of the rectangle, so area of triangle is 10 square feet. (b) The area of the rectangle is 15 sq cm, so the area of the triangle is ½ this amount, or 7.5 sq cm 8. Length (ft) Width (ft) Perimeter (ft) Area (sq ft) 1 ft 17 ft 36 ft 17 sq ft 2 ft 16 ft 36 ft 32 sq ft 3 ft 15 ft 36 ft 45 sq ft 4 ft 14 ft 36 ft 56 sq ft 5 ft 13 ft 36 ft 65 sq ft 6 ft 12 ft 36 ft 72 sq ft 7 ft 11 f. 36 ft 77 sq ft 8 ft 10 ft 36 ft 80 sq ft 9 ft 9 ft 36 ft 81 sq ft 10 ft 8 ft 36 ft 80 sq ft Starts to repeat (a) The dimensions which result in the largest possible area are 9ft by 9 ft. (b) Maybe you wanted a longer pen for the rabbit to be able to run a longer distance without having to turn around, or maybe the space which the yard has won t permit the length or width associated with one of the other measures. 8. The approximate area is 28,800 sq ft. Student Worksheet # sq ft sq ft 3. About 12 sq ft 4. About 96 sq ft 5. About 54 sq ft ft is about 18 times around the gym floor to make a mile (It s actually a little more than that.) 34
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