Modeling in Geometry


 Amberlynn Hopkins
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1 Modeling in Geometry Overview Number of instruction days: 810 (1 day = 53 minutes) Content to Be Learned Mathematical Practices to Be Integrated Use geometric shapes and their components to represent and describe realworld objects. Use the concept of density to solve problems involving area and volume (e.g., persons per square mile, BTUs per cubic foot). Solve realworld problems by designing an object or structure that satisfies certain constraints. 2 Reason abstractly and quantitatively. Apply concepts of density and determine whether solutions are reasonable. Determine volume in modeling situations and use accurate units. 4 Model with mathematics. Solve problems that involve geometric shapes such as tree trunks or human torsos as representative of cylinders. Design an object or structure to satisfy physical constraints or minimize cost, such as a with typographic grid system based on ratios. 5 Use appropriate tools strategically. Choose tools such as graduated cylinders, measuring tapes, meter sticks, etc. strategically, depending on the nature of the problem being solved. Providence Public Schools D105
2 Modeling in Geometry (810 days) Essential Questions How would you describe realworld figures, such as a tree or a human torso, as a composite shape? When would you use geometric shapes to solve reallife problems? How would you calculate the number of threedimensional objects that will fit into another threedimensional object? For example, how many marbles will fit in a jar? Standards Common Core State Standards for Mathematical Content Modeling with Geometry GMG Apply geometric concepts in modeling situations GMG.1 GMG.2 GMG.3 Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder). Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot). Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios). Common Core State Standards for Mathematical Practice 2 Reason abstractly and quantitatively. Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the D106 Providence Public Schools
3 Modeling in Geometry (810 days) Geometry, Quarter 4, Unit 4.1 units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects. 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. 5 Use appropriate tools strategically. Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. For example, mathematically proficient high school students analyze graphs of functions and solutions generated using a graphing calculator. They detect possible errors by strategically using estimation and other mathematical knowledge. When making mathematical models, they know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of concepts. Clarifying the Standards Prior Learning In Grade 3, students used modeling to solve realworld problems involving perimeter of polygons. In Grade 4, students applied area and perimeter formulas to solve problems with realworld objects. In fifth grade, unit conversions were used to solve multistep realworld problems, which were an additional cluster. Proportional relationships were an area of indepth focus in sixth grade and were a major cluster in seventh grade. In Grades 6 and 7, students also focused on solving realworld problems Providence Public Schools D107
4 Modeling in Geometry (810 days) involving area, surface area, and volume. This work was extended in Grade 8 to include cylinders, cones, and spheres as a supporting cluster. Current Learning Modeling in high school centers on problems arising in everyday life and in the workplace. Such problems draw upon mathematical content knowledge and skills developed in previous grades and the current course. The problems encountered in this course are word problems that are richer and more open ended and task oriented than word problems students have encountered in previous grades. The problems focus on three primary areas: design, density, and use of threedimensional geometric shapes to model realworld threedimensional figures. Applying geometric concepts in modeling situations is classified as major content by the PARCC Model Frameworks for Mathematics. Future Learning In Calculus, students will find the volume of solids of rotation and volumes of solids with known cross sections. Applications of this learning are found in the electrical grid system, design, health care fields, and HVAC. This information also has applications in engineering and physics. Engineers design objects such as containers, dams, cars, etc. Additional Findings A challenge for students is that they may think that mathematics is about remembering procedures rather than questioning ideas. Now they must adapt procedures and think about them rather than just practicing them. (A Research Companion to Principles and Standards for School Mathematics p. 318). 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. D108 Providence Public Schools
5 Modeling in Geometry (810 days) Geometry, Quarter 4, Unit 4.1 Apply geometric shapes, concepts, measures, and properties to represent and describe realworld objects. Apply the concept of density to solve realworld situations involving area and volume models, such as persons per square mile or BTUs per cubic foot. Design an object or structure that satisfies certain constraints, such as minimizing cost or working with a grid system based on ratios. Instruction Learning Objectives Students will be able to: Describe real world objects using geometric concepts, models, and shapes. Use geometric shapes and properties to solve realworld design problems with and without technology. Apply the concept of density to solve realworld situations involving area and volume models. Review and demonstrate knowledge of important concepts and procedures related to modeling in geometry. Resources Geometry, Glencoe McGrawHill, 2010, Student/Teacher Editions Section 121 (pp ) Sections 124 through 126 (pp ) Glencoe McGrawHill Online Chapter 12 Resource Masters (pp. 6 7, 2542) Interactive Classroom CD (PowerPoint Presentations) Teacher Works CDROM CCSS Geometry Lab 1: TwoDimensional Representations of ThreeDimensional Objects (See the Supplemental Materials section of this binder) Providence Public Schools D109
6 Modeling in Geometry (810 days) CCSS Geometry Lab 14: Population Density (See the Supplemental Materials section of this binder) TINspire Teacher Software Exam View Assessment Suite Education.TI.com: Maximizing the Area of a Garden activity. See the Supplementary Unit Materials section of this binder for the student and teacher notes for this activity. Cones Launch task. See the Supplementary Unit Materials section of this binder for the student and teacher notes for this activity. Image Tool, 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 section below for specific recommendations. Materials TINspire graphing calculator Instructional Considerations Key Vocabulary constraint Planning for Effective Instructional Design and Delivery Reinforced vocabulary taught in previous grades or units: density, and ratios. The district resources in this unit are identical to the resources provided in the previous unit. In order to avoid replication of previous content, care should be taken in assigning items from the textbook sections or resource masters. The focus in this unit is application of geometric properties modeling real world problems. In this unit, students will apply modeling in the process of choosing and using the appropriate mathematics to analyze and understand geometric situations. This concept can be demonstrated using multiple methods. For example, the following link is a handson bridge building activity modeling a parabolic relationship in a reallife situation: The Mathematics Benchmarks, K12 website provided by the Dana Center the Dana Center at the University of Texas, Austin. The task Cones Launch from this resource provides the opportunity for students to connect algebra and geometry by using formulas and spatial skills to solve real world D110 Providence Public Schools
7 Modeling in Geometry (810 days) Geometry, Quarter 4, Unit 4.1 problem. This task is available in the Supplemental Unit Materials Section of this binder and on the following website: Numerous resources on the web also reinforce applications of modeling in geometry. The Illustrative Mathematics Project was developed under the guidance of members of the Common Core State Standards working group as well as other national experts in mathematics and mathematics education (PARCC Model Content Frameworks, p. 11). The project provides illustrations and example of tasks supporting the implementation of the Common Core State Standards. The following tasks can be used to support the content and are available on Tennis Balls in a Can The Lighthouse problem Toilet Roll An additional resource for geometric modeling may be found on Teachers with limited internet access in their classrooms can request a CD from Shodor.org with the website activities. The Image Tool activity from shodor.org provides students the opportunity to measure angles, distances, and areas in images. District resources for the modeling unit are also available in the Supplemental Unit Materials Section of this binder. They are also accessible online on the Glencoe McGrawHill Math Geometry Student and Teacher textbooks or on the following website: /634463/CCS_Geometry_se.pdf. To access the CCSS Supplements using the online textbooks select the CCSS icon on the homepage of the online textbook and then choose the corresponding lesson or lab to access supplementary Glencoe lessons identified in the resource section. Using technology to explore modeling in geometry is helpful for all learners. The student and teacher materials for the TI Nspire activity, Maximizing the Area of a Garden are located in the Supplementary Unit Materials Section of this binder. In this activity students solve a real world problem by determining the relationship between the width and length of a garden with a rectangular shape and a fixed amount of fencing. They derive a formula for computing the area of the garden when given the width. They also find the dimensions of the garden that has the maximum possible area. Additional TINspire resources can be found using the TINspire Teacher Software. The content tab of the TINspire desktop software contains links to the Geometry, Glencoe McGrawHill, 2010 textbook. These resources are accessible by chapter and section. As you assess students using the 5minute check transparencies, a cues, questions, and advance organizers strategy is being used, since students are answering questions about content that is important. Some of the questions help students review prior knowledge, and these should be used at Providence Public Schools D111
8 Modeling in Geometry (810 days) the beginning of a lesson; other questions, for use during and after the lesson, help students practice knowledge. Notes D112 Providence Public Schools
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