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MG-A: Apply geometric concepts in modeling situations

**Standards**:

- MG-A.1. Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).
- MG-A.2. Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot).
- MG-A.3. 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).

**Technology-enhanced investigations:**

MIT BLOSSOMS: Video lessons with additional teacher and learner resources. Descriptions are from the video summaries.

- Building Cryptosystems: "This video module presents an introduction to cryptography – the method of sending messages in such a way that only the intended recipients can understand them. In this very interactive lesson, students will build three different devices for cryptography and will learn how to encrypt and decrytp messages." Aligns with MG-A.3.
- Engineering Innovation and Entrepreneurship: "This video lesson introduces students to the worlds of engineering innovation and entrepreneurship. ... The video will familiarize students with all the steps in the innovation process: from conception to launch. By the end of this lesson, students will be prepared for an optional long-term innovation project." Aligns with MG-A.3.
- Using Geometry to Design Simple Machines: "This video is meant to be a fun, hands-on session that gets students to think hard about how machines work. It teaches them the connection between the geometry that they study and the kinematics that engineers use -- explaining that kinematics is simply geometry in motion. In this lesson, geometry will be used in a way that students are not used to." Aligns with MG-A.3.

National Math and Science Initiative: Lesson: Using Linear Equations to Define Geometric Solids: Use this lesson for geometry within a unit on volume applications. Students write the equations of lines and graph linear equations that define bounded regions, calculate the areas and perimeters of the regions, revolve the planar regions about horizontal and vertical lines to create solids, and calculate the volumes of the resulting solids; they use technology (virtual manipulatives) to create and revolve planar figures about horizontal and vertical lines and model the conceptual understanding using a real world situation. Targeted standards include GMD-A.3, GMD-B.4, GPE-B.7 and MG-A.1. Common Core Math Practice standards 1-7 are also addressed.

NCTM Illuminations: Geometric Solids: Use this virtual manipulative to "learn about various geometric solids and their properties. You can manipulate and color each shape to explore the number of faces, edges, and vertices." View the solid or its net. You can also use this tool to help determine the relationship between the number faces, vertices, and edges.

Wolfram Demonstrations Project: Download the free Wolfram CDF player to interact with the following manipulatives:

- US Common Core State Standard Math HSG-MG.A.1: 57 manipulatives--all are worth viewing. Among those: Interlocking disks, wooden bucket, throw a hammer, telescopic tower, ball in a box, collapsible cup, collapsible umbrella, dome frames made of arches, Jefferson National Expansion Memorial, deploying and steering a parabolic antenna, modeling a simple roller coaster, controlling airplane flight, perspective drawing of billiards table, basic views in technical drawing, potter's wheel, Hello Block world, motion of planet around a star, and more.
- US Common Core State Standard Math HSG-MG.A.2: 2 manipulatives: Truck packing (aligns with MG-A.2, MG-A.3) and Floating sphere.
- US Common Core State Standard Math HSG-MG.A.3:
28 manipulatives. Among those:
- Maximizing the Volume of a Cup Made from a Square Sheet of Paper III: Square is 12 cm. by 12 cm. Aligns with MG-A.3.
- Circles packed in a circle: Aligns with MG-A.3.
- Sphere packing in a box: Aligns with MG-A.3.
- A Multiobjective Optimization of a Room Configuration: Aligns with MG-A.3.

**Multiple Choice:**

Khan Academy: Practice questions with videos:

- Solid geometry word problems: Aligns with MG-A.1 and GMD-A.3.
- Density word problems: Aligns with MG-A.2 and GMD-A.3.

**Constructed-response:**

OpenEd:

- Geometric modelling of everyday objects: 4 questions focus on free response. Includes related review resources. Aligns with GM-A.1.
- Density based on area and volume: 5 questions focus on free response. Includes related review resources. Aligns with GM-A.2.
- Modelling with geometry: 3 questions focus on free response. Includes related review resources. Aligns with GM-A.3.

**Performance tasks:**

Illustrative Mathematics: Geometry:

- Eratosthenes and the Circumference of the Earth: Aligns with MG-A.
- Indiana Jones and the Golden Statue: Aligns with MG-A.2.

Mathematics Assessment Project: Standards: High School: Geometry: https://www.map.mathshell.org/stds.php?standardid=1367

- Task 428: Modeling: Rolling Cups
- Task 432: Solving Quadratic Equations: Cutting Corners

Mathematics Vision Project, Secondary 3 Student Edition:

- Module 5: Modeling with Geometry: This module contains 8 classroom tasks. Multiple standards are addressed. Standards MG-A.1 is found in task 3; MG-A.1, MG-A.2, MG-A.3 are found in task 4.

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