Grade 3 Area and Perimeter Unit Overview

Transcription

1 Grade 3 Area and Perimeter Unit Overview Geometric measurement: Understand concepts of area and relate area to multiplication and to addition. 3.MD.C.5 Recognize area as an attribute of plane figures and understand concepts of area measurement. (a) A square with a 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. 3.MD.C.6 Measure areas by counting unit squares (square cm, square m, square in, square ft, and improvised units) 3.MD.C.7 Relate area to the operations of multiplication and addition. (a) Find the area of a rectangle with a whole-number side lengths by tiling it, and show that the area is the same as would be found by multiplying the side lengths. (b) Multiply side lengths to find the areas of rectangles with whole number side lengths in the context of solving real world and mathematical problems, and represent whole number products as a rectangular areas in mathematical reasoning. (c) 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 x b and a x c. Use area models to represent the distributive property in mathematical reasoning. (d) 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. 3.MD.C.8 Solve real world and mathematical problems involving perimeters of polygons, including finding the perimeter given the side lengths, finding an unknown side length, and exhibiting rectangles with the same perimeter and different areas or with the same area and different perimeters. Day 1: Investigations Unit 4, Session 2.5A This 5-day area unit is designed to have students explore the relationship of area and perimeter. Earlier this year students found perimeter of geometric shapes (Quarter 1) and area of rectangles (Quarter 2). This unit includes tasks and activities to provide students experience with thinking of area and perimeter simultaneously. Day 1 is Unit 4, Session 2.5A. Day 2: Solving Problems Relating Area and Perimeter Day 2 is Unit 5, Session 3.1A. This lesson focuses on the relationship between multiplication and the area of rectangles. Day 3: Finding Area of Rectilinear Figures Day 3 introduces standard units of measure for area. Students see and measure with a variety of standard units. Students are also challenged to consider which unit is most appropriate for which area (ie square inches for smaller areas like a pieces of paper, square feet for somewhat larger areas like tables or dry erase boards, and square yards or square meters for things like hallways or room floors.) Day 4-5: Robot Designer Performance Task Select 3 rd Grade Robot Designer Performance Task:

2 Day 1: Investigations Unit 4, Session 2.5A Same Area, Different Perimter; Same Perimeter, Different Area Geometric measurement: Understand concepts of area and relate area to multiplication and to addition. 3.MD.C.5 Recognize area as an attribute of plane figures and understand concepts of area measurement. (a) A square with a 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. 3.MD.C.6 Measure areas by counting unit squares (square cm, square m, square in, square ft, and improvised units) 3.MD.C.7 Relate area to the operations of multiplication and addition. (a) Find the area of a rectangle with a whole-number side lengths by tiling it, and show that the area is the same as would be found by multiplying the side lengths. (b) Multiply side lengths to find the areas of rectangles with whole number side lengths in the context of solving real world and mathematical problems, and represent whole number products as a rectangular areas in mathematical reasoning. (c) 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 x b and a x c. Use area models to represent the distributive property in mathematical reasoning. (d) 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. 3.MD.D.8 Solve real world and mathematical problems involving perimeters of polygons, including finding the perimeter given the side lengths, finding an unknown side length, and exhibiting rectangles with the same perimeter and different areas or with the same area and different perimeters. Emphasized Standards for Mathematical Practice: 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others 5. Use appropriate tools strategically. 7. Look for and make use of structure. Materials: Investigations Resource Masters C12 Rectangle Square Tiles Unit 4 SAB 33A-E Words that you should hear students using in mathematical conversations: row column square inches area perimeter length Ten Minute Math: Practicing Place Value: Say seven hundred seventy-four and ask students to write the number. Have students read, write, and say the number. Then solve mentally: ? ? ? ? ? Which places have the same digits? Which do not? Why? Before: Show students the rectangle on C13. Say: How can we use square tiles to find the area of this rectangle? (Turn and talk with a partner). Discuss describing the area with square units or square inches since each tile is one inch on each side. Today, you are going to find the area of the rectangles on SAB Unit 4, 33A-B using square tiles. During: Students find the area of the rectangles on SAB Unit 4, 33A-B using square tiles. Mid-Workshop Discussion: Focus student attention on SAB33B Say: What did you find for the area of the first rectangle? What about the second rectangle? Does everyone agree? (Make sure the class agrees that both rectangles have an area of 16 square inches.) Ask: What if we measured the perimeter of these two shapes. Do you think they are the same? Or does one have a greater perimeter than the other? (Allow students to comment). Ask, how can we check to be sure? (Work with students

3 to find perimeter. You might include a discussion of multiple ways to record the equations for the perimeter of the rectangle and square. Rectangle: Square: = 20 inches = 16 inches = 20 inches 4 x 4 = 16 inches (2 x 8) + (2 x 2) = 20 inches So, we found rectangles that look different, but have the same area. Their perimeter was not the same. Let s explore this with some other rectangles. And let s see if the opposite is true. What if the perimeter is the same, but the rectangles look different. Is the area the same? The rectangles on Unit 4, SAB 33C all have a perimeter of 12 inches. Is their area the same? Work with your partner to find out. After: Focus attention on SAB Unit 4, page 33C. Each of these rectangles had a perimeter of 12 inches. What did we find out about the area of these rectangles? How can we describe what we have learned today about the relationship of area and perimeter in rectangles? Evaluation: Unit 4, SAB33D-E (Students explore and write what they notice about rectangles with the same area, different perimeters and same perimeters, different areas.

4 Day 2: Solving Problems Relating Area and Perimeter Geometric measurement: Understand concepts of area and relate area to multiplication and to addition. 3.MD.C.5 Recognize area as an attribute of plane figures and understand concepts of area measurement. (a) A square with a 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. 3.MD.C.6 Measure areas by counting unit squares (square cm, square m, square in, square ft, and improvised units) 3.MD.C.7 Relate area to the operations of multiplication and addition. (a) Find the area of a rectangle with a whole-number side lengths by tiling it, and show that the area is the same as would be found by multiplying the side lengths. (b) Multiply side lengths to find the areas of rectangles with whole number side lengths in the context of solving real world and mathematical problems, and represent whole number products as a rectangular areas in mathematical reasoning. (c) 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 x b and a x c. Use area models to represent the distributive property in mathematical reasoning. (d) 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. 3.MD.D.8 Solve real world and mathematical problems involving perimeters of polygons, including finding the perimeter given the side lengths, finding an unknown side length, and exhibiting rectangles with the same perimeter and different areas or with the same area and different perimeters. Emphasized Standards for Mathematical Practice: 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others 5. Use appropriate tools strategically. 7. Look for and make use of structure. Materials: Relating Area and Perimeter of Rectangles (below) Area and Perimeter Exit ticket (below) Words that you should hear students using in mathematical conversations: row column square units area perimeter length Ten Minute Math: Breaking Arrays Apart: Have students build an array to represent 8 x 7. Ask how can we break this array apart into two smaller arrays to make the product easier to find? How can we use an equation to represent how we broke the array apart? Before: Ask: Yesterday, we learned something special about the relationship between area and perimeter of rectangles. Who can remind us of two claims that we made about area and perimeter? Today, we are going to solve some problems in which we need to think about how to use what we know about area and perimeter to find a solution. Introduce the problems below. Discuss what is happening in the problems or any unfamiliar vocabulary (ie pen as a dog pen) During: Students work in pairs to solve the problems below. After: Focus student attention on the first problem. Ask for different sizes of dog pens students found. Ask students if they found all of the possible solutions, and how they know that they did. You might consider recording them (or rerecording them in numeric order by length (11 x 1, 10 x 2, 9 x 3, 8 x 4, 7 x 5, 6 x 6). What pattern do you notice in the shape of the rectangle as the area increases? Which pen would give Austin s puppy the most area?

5 Evaluation: Area and Perimeter exit ticket (below)

6 Relating Area and Perimeter of Rectangles Name: 1. Austin s grandpa bought him a new puppy and 24 yards of fence to build a pen for the puppy. Austin wanted to build the biggest area he could for his new puppy. How many different sizes does Austin have to choose from if his dad said the pen must be a rectangle? Which pen would be the largest? What are the dimensions of the rectangles you made? Which rectangular fence would give Austin s puppy the most area?

7 2. Ms. Rusk is planning a special dinner for friends and family. She has 12 square tables that she can push together to make one big rectangular table for her guests. There is enough space for 1 person on each side of each table. When she puts more tables together, more people can sit at the tables. So when she uses two square tables, six people can sit at the table. Ms. Rusk wants to use all 12 square tables. How many different ways can she arrange the 12 square tables into a rectangle? Since the perimeter around the table changes for each way, how many guests can she invite for each way?

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9 Day 3: Find Area of Rectilinear Figures Geometric measurement: Understand concepts of area and relate area to multiplication and to addition. 3.MD.C.5 Recognize area as an attribute of plane figures and understand concepts of area measurement. (a) A square with a 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. 3.MD.C.6 Measure areas by counting unit squares (square cm, square m, square in, square ft, and improvised units) 3.MD.C.7 Relate area to the operations of multiplication and addition. (a) Find the area of a rectangle with a whole-number side lengths by tiling it, and show that the area is the same as would be found by multiplying the side lengths. (b) Multiply side lengths to find the areas of rectangles with whole number side lengths in the context of solving real world and mathematical problems, and represent whole number products as a rectangular areas in mathematical reasoning. (c) 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 x b and a x c. Use area models to represent the distributive property in mathematical reasoning. (d) 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. 3.MD.D.8 Solve real world and mathematical problems involving perimeters of polygons, including finding the perimeter given the side lengths, finding an unknown side length, and exhibiting rectangles with the same perimeter and different areas or with the same area and different perimeters. Emphasized Standards for Mathematical Practice: 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others 5. Use appropriate tools strategically. 7. Look for and make use of structure. Materials: Area of Rectilinear Figures (below) Square tiles Words that you should hear students using in mathematical conversations: square units area sum length Ten Minute Math: Breaking Arrays Apart: Have students build an array to represent 9 x 6. Ask how can we break this array apart into two smaller arrays to make the product easier to find? How can we use an equation to represent how we broke the array apart? Before: Say: We have been finding area of lots of rectangles. Sometime we need to find the area of other shapes that are made of rectangles. Create two small arrays using tiles or grid paper (3 x 7 and 2 x 5). Ask for the area of each of the arrays. Push the two arrays together to make one shape. How could we find the area of this whole shape? (Have students turn and talk.) Discuss as a class how the area is of the new shape is the sum of first two rectangles. During: Students work in pairs to solve the problems below. After: Focus student attention on the last page. How did you find the area of the first figure? How did you decide how to decompose the shape into rectangles? How could we use an equation to represent you found the area? Evaluation: Area of Rectilinear Figures Exit Ticket (below)

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13 Area of Rectilinear Figures Exit Ticket What is the area of this figure? Name: Write an equation to show how you found the area of this figure: Area of Rectilinear Figures Exit Ticket What is the area of this figure? Name: Write an equation to show how you found the area of this figure: