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Approximate Time Frame: 1 – 2 Weeks
Connections to Previous Learning:
Throughout Math I and II, students have been provided with numerous opportunities to model problem situations using algebraic equations and geometric
figures. In particular, in Math II, students applied properties of congruent and of similar triangles to solve problems that could be modeled with triangles. Many
of the geometric relationships previously investigated and proven by students will play a role in determining the type of model for a given physical phenomena
and how exactly to solve the problem being modeled. In Math I and II, students worked with rotations as part of their study of triangle congruence and triangle
similarity. Back in Grade 8, students solve real-world and mathematical problems involving cones, cylinders, and spheres.
Focus of this Unit:
Key to this unit is providing students with opportunities to see mathematics as a tool to model real-world situations. Students will draw on their growing
knowledge of the relationships between geometric objects to use geometric shapes to describe a physical objects. Important to this process is determining
exactly what type of model to use in a given situation and physical object (e.g., is it best to model a coin as a three-dimensional cylinder or as a two-dimensional
disk in the context of a given problem?). Using construction tools, physical, computational, and virtually (e.g., dynamic geometry environments) helps students
to create geometric representations to model problem situation and can lead them to conjecture about and solve these problems.
Connections to Subsequent Learning:
Modeling physical phenomena and problem situations with geometric representations will prove to be a useful tool in subsequent units in Math III where
students will be asked to model physical phenomena and problem situations using a variety of functions (quadratic, exponential, trigonometric). This work may
also have applications to students' work in other subject area (such as science or engineering).
Desired Outcomes
Standard(s):
Apply geometric concepts in modeling situations.
 G.MG.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.
 G.MG.2 Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot)
 G.MG.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.
Visualize relationships between two-dimensional and three-dimensional objects.
 G.GMD.4 Identify the shapes of two-dimensional cross sections of three-dimensional objects, and identify three-dimensional objects generated by rotations
of two-dimensional objects.
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WIDA Standard: (English Language Learners)
English language learners communicate information, ideas and concepts necessary for academic success in the content area of Mathematics.
English language learners benefit from:
 explicit attention to units of measurement and geometry vocabulary.
 guidance understanding the context of modeling situations, with specific attention to vocabulary and visualization.
 tactile and virtual tools for designing and analyzing geometric models.
Understandings: Students will understand…
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Geometric objects may be used to model various physical phenomena.
Different geometric objects can be used to model the same physical phenomena and the object chosen to model the phenomena will be dependent upon
how the model is to be used.
The concept of density and how it may be applied in modeling problems involving area or volume.
The relationship between a three-dimensional object and two-dimensional cross sections of that object.
The relationship between a rotated two-dimensional figure and its resulting three-dimensional object.
Essential Questions:
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How can geometric figures be used to model physical phenomena or problem situations?
How can geometric properties and relationships be applied to solve problems that are modeled by geometric objects?
What is density as it relates to area or volume?
If a two-dimensional cross section of a three-dimensional object is taken, what geometric figure is formed by this cross section?
If a two-dimensional geometric figure is rotated 180° about a line, what three-dimensional object is formed?
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Mathematical Practices: (Practices to be explicitly emphasized are indicated with an *.)
*1. Make sense of problems and persevere in solving them. Students will apply the information provided about physical phenomena to create geometric
models of the phenomena and then use the geometric properties of the model to solve problems related to the phenomena.
*2. Reason abstractly and quantitatively. Students will be able to use figures and information pertaining to a specific geometric figure as an aid in reasoning
about objects being modeled by that geometric figure.
3. Construct viable arguments and critique the reasoning of others.
*4. Model with mathematics. Students will use two- and three-dimensional geometric objects to model physical phenomena and then use the geometric
properties of these figures to solve problems related to these physical phenomena.
*5. Use appropriate tools strategically. Students will be able to use drawings and dynamic geometry environments to form geometric models of physical
phenomena and then use those models to form conjectures about and solve problems related to the physical phenomena being modeled.
6. Attend to precision. Students will attend to appropriate units of measure in density models.
*7. Look for and make use of structure. Students will be able to use the structure of geometric objects to gain insights into, make conjectures about, and solve
problems pertaining to these objects.
8. Look for and express regularity in repeated reasoning.
Prerequisite Skills/Concepts:
Students should already be able to:
Advanced Skills/Concepts:
Some students may be ready to:
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Identify, describe and use geometric objects such a point, line,
angle, triangle, parallelogram, circle, etc.
Identify, describe and analyze three-dimensional geometric objects
such cylinders, cones, and spheres.
Sketch two- and three dimensional geometric figures that model
given physical phenomena or problem situations.
Recall various geometric relationships proven in previous courses.
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Use less common three-dimensional shapes and find their volume.
Use less common geometric shapes (such as frustums, trapezoidal prisms), their
measures, and their properties to describe objects.
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Knowledge: Students will know…
Skills: Students will be able to …
All standards in this unit go beyond the knowledge level.
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Sketch a figure or object that represents specific given information.
Use geometric shapes (such as cylinders, spheres, cones, pyramids, prisms), their
measures, and their properties to describe objects (e.g., modeling a tree trunk or a
human torso as a cylinder.)
Use Geometric properties and relationships to solve problems that are modeled by
geometric objects.
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.)
Determine the three-dimensional object generated when a two-dimensional
geometric figure is rotated 180° about a line.
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Academic Vocabulary:
Critical Terms:
Supplemental Terms:
Modeling
Sphere
Cylinder
Cone
Cross section
Rotation
Density
Frustum
Trapezoidal prism
Platonic solids
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