Describing and Solving for Area and Perimeter


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1 Grade 3 Mathematics, Quarter 2,Unit 2.2 Describing and Solving for Area and Perimeter Overview Number of instruction days: 810 (1 day = 90 minutes) Content to Be Learned Distinguish between linear and area measurements. Solve real world problems involving area (rectangular figures) and perimeter (regular polygons). Use tiling to show the area of a rectangle. Find area by decomposing rectilinear figures into nonoverlapping rectangles and adding those areas. Find perimeter given side lengths. Find perimeter when one side length is not given. Identify rectangles with the same perimeter and different areas. Identify rectangles with the same area and different perimeters. Mathematical Practices to Be Integrated 1 Make sense of problems and persevere in solving them. Identify the relationship between area and multiplication with given measurements. Understand the approaches of others and identify correspondences between different approaches. 4 Model with mathematics Identify and use appropriate math strategies (diagrams, tables, equations) to solve problems. Exhibit rectangles given area and perimeter measurement parameters. Essential Questions How can you describe area as a form of measurement? How can you describe perimeter as a form of measurement? How can you solve real world problems involving area and perimeter? How can you build and record all the possible rectangles with the same perimeter (fixed perimeter) and different areas? How can you record all the possible rectangles with the same area (fixed area) and different perimeters? Providence Public Schools D41
2 Grade 3 Mathematics, Quarter 2, Unit 2.2 Describing and Solving for Area and Perimeter (810 days) Standards Common Core State Standards for Mathematical Content Measurement and Data 3.MD Geometric measurement: understand concepts of area and relate area to multiplication and to addition. 3.MD.7 Relate area to the operations of multiplication and addition. b. Multiply side lengths to find areas of rectangles with wholenumber side lengths in the context of solving real world and mathematical problems, and represent wholenumber products as rectangular areas in mathematical reasoning. c. Use tiling to show in a concrete case that the area of a rectangle with wholenumber side lengths a and b + c is the sum of a b and a 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 nonoverlapping rectangles and adding the areas of the nonoverlapping parts, applying this technique to solve real world problems. Geometric measurement: recognize perimeter as an attribute of plane figures and distinguish between linear and area measures. 3.MD.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. Common Core State Standards for Mathematical Practice 1 Make sense of problems and persevere in solving them. Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and D42 Providence Public Schools
3 Describing and Solving for Area (810 days) Grade 3 Mathematics, Quarter 2, Unit 2.2 relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, Does this make sense? They can understand the approaches of others to solving complex problems and identify correspondences between different approaches. 4 Model with mathematics. Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, twoway tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose. Clarifying the Standards Prior Learning In Grade 2, students became fluent with addition and subtraction within 100. They developed a foundation for understanding area through building and drawing shapes, and they partitioned a rectangle into rows and columns of samesize squares and counted to find the total. Students measured and estimated lengths in standard units and solved addition and subtraction problems in the context of length. Current Learning Recognizing area as an attribute of twodimensional regions is a critical area of study in Grade 3 and is closely linked with the expectation of fluency in multiplication and division. Students show the relationship between tiling (drawing unit squares) and multiplication through their work. They recognize area as an attribute of plane figures and understand concepts of area measurement. They also relate area to the operations of multiplication and addition. Students also solve real world math problems involving perimeter of polygons. Students find the area of a polygon with all side lengths given. They also find the perimeter of a polygon with an unknown side length. Grade 3 students build and record all possible rectangles with the same perimeter and different areas. They also build and record all possible rectangles with the same area and different perimeters. Providence Public Schools D43
4 Grade 3 Mathematics, Quarter 2, Unit 2.2 Describing and Solving for Area and Perimeter (810 days) Future Learning In Grade 4, students will apply their understanding of models to multiplication. They will apply formulas for area and perimeter to solve realworld and mathematical problems involving rectangles. Students will deepen their understanding of twodimensional objects by using them to solve problems. Grade 4 students will understand the concept of angle and how to measure angles. Additional Findings The basic idea of measuring area is that of covering a region by units that just fit (an idea that is sometimes called tiling). This is important to note due to the emphasis in standard 3.MD.5b on accurate representation of the square units of a figure. (Adding It Up, p. 283) Assessment When constructing an endofunit assessment, be aware that the assessment should measure your students understanding of the big ideas indicated within the standards. The CCSS for Mathematical Content and the CCSS for Mathematical Practice should be considered when designing assessments. Standardsbased mathematics assessment items should vary in difficulty, content, and type. The assessment should comprise a mix of items, which could include multiple choice items, short and extended response items, and performancebased tasks. When creating your assessment, you should be mindful when an item could be differentiated to address the needs of students in your class. The mathematical concepts below are not a prioritized list of assessment items, and your assessment is not limited to these concepts. However, care should be given to assess the skills the students have developed within this unit. The assessment should provide you with credible evidence as to your students attainment of the mathematics within the unit. Distinguish between linear and area measurements Solve real world problems involving area (rectangular figures) and perimeter (regular polygons) Use tiling to show the area of a rectangle. Find area by decomposing rectilinear figures into nonoverlapping rectangles and adding those areas. Find the perimeter of a polygon, given the side lengths. Find the perimeter of a polygon when one side length is not given. Identify rectangles with the same perimeter and different areas. Identify rectangles with the same area and different perimeters. D44 Providence Public Schools
5 Describing and Solving for Area (810 days) Grade 3 Mathematics, Quarter 2, Unit 2.2 Instruction Learning Objectives Students will be able to: Compare attributes of area and perimeter measurements. Solve real world problems involving area (rectangular figures) and perimeter (regular polygons) Use tiling to show the area of a rectangle. Find area by decomposing rectilinear figures into nonoverlapping rectangles and adding those areas. Find the perimeter of a polygon, given the side lengths. Find the perimeter of a polygon when one side length is not given. Identify rectangles with the same perimeter and different areas. Identify rectangles with the same area and different perimeters. Demonstrate understanding of the concepts and skills learned in this unit. Resources envision Math Grade 3, Pearson Education, Inc., 2009 Topic 16 Perimeter, Area, and Volume Teacher Edition Resource Masters Student Pages Investigations in Numbers, Data, and Space Grade 3, Pearson Education, Inc., 2008 Implementing Investigations in Grade 3 Implementation Guide Unit 4 Teacher Edition, Perimeter, Angles, and Area Unit 5 Teacher Edition Equal Groups Teacher Resources Binder Also see Section I, Supplemental Materials EnVision: Lesson 162A Tools and Units for Perimeter Providence Public Schools D45
6 Grade 3 Mathematics, Quarter 2, Unit 2.2 Describing and Solving for Area and Perimeter (810 days) Lesson 166a: Covering Regions Lesson 166b: Areas and Units Lesson 167A Area of Squares and Rectangles Lesson 167B Area and the Distributive Property Lesson 167C Area of Irregular Shapes Investigations Unit 4 Session 2.5A Same Area, Different Perimeter; Same Perimeter, Different Area Investigations Unit 5 Session 3.1A: What s the Area? Session 3.5A: Using What you Know Session 3.5B: Learning Multiplication Combinations Pearson Success Net, Investigations (TERC), Exam View Assessment Suite Note: The district resources may contain content that goes beyond the standards addressed in this unit. See the Planning for Effective Instructional Design and Delivery and Assessment for ample resources to refer to when planning your unit and individual lessons. Materials grid paper, color tiles, twocolor tiles, array cards, nonstretchy string Instructional Considerations Key Vocabulary area perimeter linear measurement area measurement square units row column dimensions array factor D46 Providence Public Schools
7 Describing and Solving for Area (810 days) Grade 3 Mathematics, Quarter 2, Unit 2.2 factor pair Planning for Effective Instructional Design and Delivery A critical area of study in grade 3 is to develop an understanding of the structure of rectangular arrays and of area. The concept of perimeter and understanding the relationship between perimeter and area is also addressed in this unit. It is important that students build an understanding of area and perimeter in relationship to each other. Through various hands on learning opportunities, students grow to understand that the perimeter of a shape is the total length of its boundary, is a onedimensional, linear measurement, and is measured in units of length. The area of a shape is how much surface it takes up, and is measured in two dimensions. Area is a major concept within measurement. Area models support students multiplicative reasoning ability. Area models for products are also an important part of developing fluency with multiplication and division. In this unit, students solve real world math problems involving perimeters of polygons. Students find the area of a polygon with all side lengths given and they also find the perimeter of a polygon with an unknown side length. When a measurement length is missing, students use what they know about the properties of the figure, to determine the missing side length. (e.g. opposite sides of a rectangle are equal). Students also build and record all possible rectangles with the same perimeter and different areas (fixed perimeter). Consider providing a nonstretchy string for students to tie into a loop measuring 12 inches. Students form different sized rectangles using the loop with the fixed perimeter. On grid paper record the area and side lengths for each new rectangle. They also build and record all possible rectangles with the same area (fixed area) and different perimeters. Consider giving students 24 color tiles to find all of rectangles that can be made with an area of 24 square units. On grid paper record the perimeter and side lengths for each new rectangle. Developing understanding of the structure of rectangular arrays and of area is a critical area in grade 3. Area is a major concept within measurement, and area models support students multiplicative reasoning ability. Area models for products are also an important part of developing fluency with multiplication and division. When tiling on grid paper, be sure that the tile size and the grid size are congruent, ensuring accurate area measurement without gaps or overlaps. The area of a rectangle is the number of unit squares it takes to cover the region. Area can be found by tiling or counting unit squares on grid paper. The area of a rectangular array can also be found by multiplying the measures of the length and width. The resulting product is the area in square units. Meeting the grade 3 fluency requirement (multiplication and division within 100) requires time, support, and practice throughout the year. Students explicitly connect area measurement to multiplication and division and practice in the context of area measurement. Encourage students to share their strategies Providence Public Schools D47
8 Grade 3 Mathematics, Quarter 2, Unit 2.2 Describing and Solving for Area and Perimeter (810 days) (based on the properties of multiplication for finding products and quotients through modeling and explanation. The Distributive Property, the inverse relationship between multiplication and division, the family of facts, and knowledge of factor pairs can be easily illustrated using the array model. Each multiplication and division situation involves three quantities, each of which can be unknown. An array model shows the different equations in the fact family, supporting students conceptual understanding of the relationship between multiplication and division. Using arrays, array cards and rectangles, students can use known multiplication combinations to determine the product of more difficult combinations, Students can apply their understanding of area and perimeter to real world situations. For example, Students can find the area of their neighborhood, a park, or compare the area of Rhode Island to other states. Students can trace a map on grid paper, and use the grid paper to determine the area of the identified place. Another example of a real world situation could be determining how much paint is needed to repaint a bedroom. To do this, determine the surface area of the walls in square units. This information will help to decide how much paint is needed to paint a room. This information, coupled with the approximate coverage per gallon of paint (as indicated on the paint label) will indicate the number of gallons of paint needed to complete the project. The following Content Standard is applied in this unit. They are not specifically called out in the Learning Objectives or Assessment sections, but should be embedded in instruction and practice. Operations and Algebraic Thinking 3.OA Multiply and divide within OA.7 Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division (e.g., knowing that 8 5 = 40, one knows 40 5 = 8) or properties of operations. By the end of Grade 3, know from memory all products of two onedigit numbers. Incorporate Ten Minute Math Activities, the Problem of the Day, Daily Spiral Review and Quick Checks that are aligned to The Common Core State Standards for Mathematics. EnVision Center Activities and Investigations Activities offer additional practice for student learning and support small group differentiated instruction. Use teacher created common tasks as formative assessments to monitor student progress and understanding of critical content and essential questions. Use data from formal and informal assessments to guide your instruction and planning. D48 Providence Public Schools
9 Describing and Solving for Area (810 days) Grade 3 Mathematics, Quarter 2, Unit 2.2 For planning considerations, read through the teacher editions for suggestions about scaffolding techniques, using additional examples, and differentiated instruction as suggested by the envision and Investigations resources (particularly the Algebra Connections and Teacher Notes sections) Notes Providence Public Schools D49
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