Summer 2013: TTE 596 1
Understanding by Design Final Project by Lawrence Schneider
Stage 1 – Desired Results
Established Goals
What content standards and
program goals will the unit
address?
Program Goals
Content Standards
(HS.G-MG.3) Apply geometric methods
to solve design problems (e.g. designing
an object or structure to satisfy physical
constraints or minimizing cost; working
with typographic grid systems based on
ratios)
(HS.N-Q.2) Define appropriate quantities
for the purposes of descriptive modeling.
(HS.N-Q.3) Choose a level of accuracy
appropriate to limitations on measurement
when reporting quantities.
(HS.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).
(HS.MP.1) Make sense of problems and
persevere in solving them.
(HS.G-SRT.2) Given two figures, use the
definition of similarity in terms of
similarity transformations to decide if they
are similar; explain using similarity
transformations the meaning of similarity
for triangles as the equality of all
corresponding pairs of angles and the
proportionality of all corresponding pairs
of sides.
(HS.G-GMD.1) Give an informal
argument for the formulas for the
circumference of a circle, area of a circle,
volume of a cylinder, pyramid, and cone.
Use dissection arguments, Cavalieri’s
principle, and informal limit arguments.
(HS.G-GMD.3) Use volume formulas for
cylinders, pyramids, cones, and spheres to
solve problems.
Transfer
-
Measuring items in theory (by definition) to actual objects (physics)
Identifying measured values
Communication between a designer of an object and the producer of the object.
Understandings
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Meaning
Essential Questions
The unit of measurement chosen to
measure an object has a direct role in
how accurate your measurements will
be.
The use of
properties/definitions/characteristics
of geometric shapes on actual objects
can influence how one measures said
objects.
Because we use properties of
geometric shapes to measure actual
objects, how we measure the objects
may be generalized in some way.
Students will know …

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If polygons can be constructed using
triangles, what polyhedra can be
constructed by another polyhedron?
What are the pros and cons to using
metric units of measurement over
standard units of measurement and
vice-versa?
What does one have to consider when
using a described model or drawing to
construct a physical object?
What does one have to consider when
using geometric shapes and their
properties when constructing physical
objects?
Acquisition
Students will be skilled at
Pre-Skills
Pre-Knowledge
 That negative exponents in scientific
notation affects the order of
magnitude in a value’s representation
 That there is a specific relationship
between vertices, edges, and faces of
polyhedron, Euler’s Formula.
 The different formulas (and how they
are related) for polyhedron (i.e., how
the surface area of pyramid is related
to the surface area of a cone).
Post-Knowledge
 That any prism can be deconstructed
into pyramids with no gaps between
them at all.
 That to construct a physical model of
a described object requires
consideration of the relationships
between defined geometric properties
and practical realities (i.e. error in
measurement, approximation of
irrational numbers).
 Provide a mathematically rigorous
description of a physical model


Use measuring devices in metric and
standard forms
Convert fractions to decimals for
measuring purposes
Post-Skills
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Generalize the properties of geometric
shapes in order to analyze given
objects
Identify the proportion between a
physical object and its scaled
drawing/model
Convert between general
measurements and specific
measurements
Convert between different magnitudes
of measurement in terms of decimal
values (i.e. 100 thousandths versus
1/10th)
Recognize constraints in their
measuring tools and the constraints’
implications on accuracy
Summer 2013: TTE 596 2
Understanding by Design Final Project by Lawrence Schneider
Stage 2 - Assessments
Performance Task (in GRASPS format)
Goal:
- You have been asked by the American Society of Mechanical Engineers to develop a set of equations that will work given any CNC
machine that uses Cartesian Coordinate inputs.
- Your job is to help both the engineering community and the manufacturing community to recognize the relationships that exist
between the theoretical and practical aspects of mathematics.
- Your task is to algebraically generalize lengths of each edge of the three rectangular pyramids that fit together to form a rectangular
prism.
- The goal is to be able to construct the three rectangular pyramids that form any rectangular prism.
- The challenge you will face will be to take a specific rectangular prism, identify the three rectangular pyramids that form the prism
and generalize the lengths of each edge in relation to the given prism so that no matter what values exist for a rectangular prism, you
can identify the twelve (12) edge lengths.
- You will have to identify the obstacles you will face inaccurately measuring the lengths of a rectangular prism, and then taking those
measurements and physically construct the three triangular prisms. You will have to consider the relationship between definitions and
reality in respect to geometric concepts.
Role:
- You are an apprentice engineer. As such, you are developing a relationship with the machinists who will be using the generalizations
you develop to program their CNC machines. In essence, not only are you an apprentice engineer, you are also an apprentice
machinist. You will be gaining a perspective that provides a point of view from the engineering world (design) and the manufacturing
world (production).
Audience:
- Your client is the American Society of Mechanical Engineers, ASME. Each year the ASME makes refinements to the numerous set
of Standards that they have been developing since 1945.
- The target audience are two board member of ASME who are looking for the clearest set of generalizations that will make sense to
an engineer and a machinist.
- You will need to convince the board members that your set of equations are true given any circumstance, and that the set of
equations will make sense to both an engineer and to a machinist.
Situation:
- The context you find yourself in is this. Engineers use math on a daily basis, and so do machinists. The difference is that engineers
use theoretical/pure mathematics while machinists use practical/applied mathematics. Miscommunications arise because theoretical
mathematics and equations do not LOOK like the types of inputs a machinist will input into a CNC machine. There needs to be a way
to bridge the communication gap between engineers and machinists.
- The challenge involves taking an abstract math concept and developing a “Rosetta Stone” for the machinist, a set of equations that a
machinist can view as a template for the machines that they use on a daily basis.
Product, Performance and Purpose:
- You will create a matrix that determines which equation to use given a specific scenario in order that the engineer knows what
equation to give a machinist when the machinist is required to create a desired piece, and that the machinist knows what equation to
ask the engineer for when they are creating a desired piece.
- You need to develop this matrix so that the level of communication between the engineering community and the machinist
community is strong and transparent. This matrix may not solve every problem that exists, but will provide a path towards consistent
communication.
Standards and Criteria for Success:
- Your performance needs to be professional in nature. You will develop a Power Point presentation, and a written report. The
matrices and systems of equations you are developwill be presented to two ASME board members. You will demonstrate how your
systems of equations work, given the dimensions of any rectangular prism.
- Your work will be judged by the board members of ASME. Because it may actually become a part of the standards the ASME
produces and refines, your work will be clear, concise, and accessible (meaning that both engineers and machinists will be able to
understand it).
- Your product must meet the following requirements:
 General equations for lines in relation to one another in such a way that when folded construct pyramids.
 The equations will be algebraically sound, having been peer reviewed by your classmates and reviewed by the board
members of ASME.
 The equations will be general in nature, applicable to a variety of scenarios and only needing a minimum amount of
information to be used.
Summer 2013: TTE 596 3
Understanding by Design Final Project by Lawrence Schneider

Your product will give clear and thoughtful suggestions to the members of the American Society of Mechanical Engineers.
These suggestions will be centered on your work and the generalized equations you produce.
Other Evidence: (quizzes, tests, prompts, work samples, labs, etc.)
Work Samples
 2-dimensional constructions of the nets of 3-dimensional objects, showing measurements in specific form and generalized
 A system of equations in standard form that represent the edge lengths of the 2-dimensional constructions laid on a Cartesian
Coordinate Plane
Labs
 Constructing a system of equations to map a given net on a Cartesian Coordinate System.
 Taking a 2-dimensional figure, generalizing the characteristics of that figure based on a datum (reference characteristic).
Quizzes
 The quizzes will be physical constructions or systems of equations that model the Work Samples and Labs
Student Self-Assessment and Reflection
Reflections
 As part of the group project, the students will do a behind-the-scenes video of their project. They will detail the areas they
struggled in, the areas they were successful in, and show the class the process of their constructions.\
 The students will participate in weekly online reflections where they can share their frustrations. They will be prompted to
reflect on specific aspects of their understanding.
Self-Assessment
 The students will participate in Likert Scale surveys to measure their understanding of the differences between a novice
engineers and expert engineers.
Stage 3 – Learning Plan
1.
Entry Question - There is no such thing as a straight line. There is no such thing as a
circle. There is no such thing as flat surface. Are these statements true? (H)
2. Essential Questions and discussion of the culminating unit performance tasks- Why
are there differences between theory and definition and practical uses of mathematics?
Why do these differences exist? What are some of those differences? ( H, R)
3. Presentation of Concept–The students will be given the scenario of being an engineer
who is developing generalized systems of equations that a machinist can use to construct
pyramids that are part of the whole of a rectangular prism. (W)
4. Group and Role Assignment – The students are put into groups of 4. (O, T, E)
a. Person responsible for the keeping of records, documentation,
b. Person responsible for organizing the work materials
c. Person responsible for the communication between the team and their Supervisor
(Mr. Schneider)
d. Person responsible for the final presentation.
5. I will give a timeline for the project, share with them the concepts that will be covered and
necessary for them to master in order to be successful in constructing their models and
developing their systems of equations. The students will understand that this unit will take
a few weeks to complete, that it is not just a day or two day, or even one week project.
(W, O, Eq)
6. The students will be given a pre-assessment to gage their current level of understanding.
7. The first lesson will focus on systems of equations and piecewise functions that represent
an unfolded pyramid. (Eq.)
8. The students will reflect on previous knowledge and identify connections to what they
have learned in the past (E)
9. The second lesson will focus on properties of triangles, including the Pythagorean
Theorem and its proof. (Eq) This lesson will also begin to describe the relationship
between measurements as definitions and as actual constructions
10. The students will draw right triangles that are Pythagorean Triples and that also have
irrational hypotenuses. (R, E)
11. The students will write a reflection on what they learned concerning the relationship
between measurements and actual constructions (R).
Pre-Assessments
The attached Pre-Assessment will be
used at the beginning of this lesson unit.
The same assessment will be given after
as a Post-Assessment to measure
growth.
Progress Monitoring will take place in
two ways: Through me and summative
and formative assessments, and through
the students through journaling and
Likert-type surveys online (see
attached). As the students gage their
own learning, progress will be
monitored. I will also be posting class
grades and trends so that students have a
prompt to reflect on.
• What are potential rough spots and
student misunderstandings?
In mathematics, if you miss one day,
much of the concepts may be
difficult to catch up on. This is my
biggest concern.
• How will students get the feedback
they need
The feedback should be pretty
instantaneous in regards to the
Likert-scale surveys my students
will be taking weekly. The students
will see right away their own
progress in terms of their behaviors
and perceptions. In terms of actual
work, the students will have a rubric
in which to grade themselves and
their group members.
Summer 2013: TTE 596 4
Understanding by Design Final Project by Lawrence Schneider
12. The third lesson will introduce the students to prisms. The properties of polyhedron will
be introduced. Students will discuss the relationship between geometric properties and
definitions and objects that actually exist. (R, H)
13. The fourth lesson will introduce students to the fact that prisms can be constructed using
pyramids, specifically that rectangular prisms can be constructed using three pyramids.
The students will learn how to draw the two calculate the explicit lengths of edges of the
three pyramids given the length, width, and height of the prism (Eq).
14. The fifth lesson will focus on generalizing the edge lengths given variable edge lengths for
a rectangular prism (Eq).
15. The sixth lesson will have the students take the three-dimensional objects and construct
two-dimensional nets. This will be the first lesson where students can physically see the
relationship between definitions of measurement and physical constructions. The students
will discuss and reflect on what they see as best practices for measurement and
construction. Is the metric system better than the standard system? (H, Eq, R, E)
16. The seventh lesson will have the students construct rectangular prisms that have specific
dimensions using three pyramids. The students will discuss and reflect on the best
practices to construct the pyramids and prisms. The students will discuss and reflect on the
considerations they have to make in constructing the pyramids and prisms. (Eq)
17. The seventh lesson will have the students transfer their ability to draw the twodimensional nets to the Cartesian Coordinate Plane. The students will take these drawings
and identify the system of piecewise equations that would result in the figures. (Eq)
18. The eighth lesson will have students develop a system of piecewise functions that would
satisfy the required conditions for a pyramid with specific dimensions. (Eq)
19. The ninth lesson will have students develop three systems of piecewise functions that
would satisfy the required conditions for a prism, constructed using three pyramids. (Eq)
20. The students will being developing their presentations. They will be required to identify
the techniques necessary to produce three pyramids that construct a prism. They will be
required to identify the techniques they used to accurately construct said pyramids. They
will be required to identify the techniques used to identify the three systems of piecewise
functions that would draw on a two-dimensional plane the net of three pyramids that
construct a prism. They will also construct a prism using three pyramids and present how
they did it, what problems they faced, considerations that were knew to them, etc. (R, E,
T)
Self Check:
- Are all three types of goals (acquisition, meaning, and transfer) addressed in the learning plan?
- Does the learning plan reflect principles of learning and best practices?
- Is there tight alignment with Stages 1 and 2?
- Is the plan likely to be engaging and effective for all students?
Summer 2013: TTE 596 5
Understanding by Design Final Project by Lawrence Schneider
Appendices
Relationship to Internship Experience
Measurement is a practical application of geometric properties and concepts. Measurement takes place
every day. In the grocery store, prices of bulk items are often based on weight. The amount of money one pays
for a tank of gas depends on the volume of gas purchased. The ratio between time and energy used by a home
determines the monthly utility bill. These ubiquitous experiences are dependent on the accuracy of said
measurements, yet the general public usually takes these measurements for granted. When was the last time
anyone checked the scales at the grocery store, or checked to see if the gallons of gas they are purchasing are
actually gallons?
The goal of this UbD lesson is to have students recognized the nuances of measurement. Geometric
properties define many of the lengths people deal with on a daily basis. In the design of a modular system, it is
often the case that one shape contains smaller shapes (i.e. solar panels on a satellite, or the folds in an airbag).
At Raytheon, modular systems are the norm and the words accuracy and precision are part of everyday
conversation. In the metrology department, measurement is the sole focus. Metrology means the study of
measurement. Every day, metrology engineers and technicians check the accuracy of measurement devices and
measurement practices.
The students will face many of the issues that metrology engineers face. In particular, they will have to
recognize the relationship between standards and definitions of measurement that are derived through geometric
properties and measurements of physical objects. Geometry is a set of abstract ideas, definitions, properties, and
theorems that industry applies to physical objects, objects that cannot satisfy said properties. This unit of
instruction will give students the opportunities to see these relationships firsthand.
In the metrology department at Raytheon, the tools of measurement used by engineers and technicians
are pretty advanced technologically speaking. Although these tools will not be available to a general high
school classroom, the basic ruler will allow the students to participate in, and experience, the same standards of
measurement used by engineers and technicians. The systems of equations that the students are developing can
Summer 2013: TTE 596 6
Understanding by Design Final Project by Lawrence Schneider
be used by CNC technicians and engineers to cut the two-dimensional shapes. Through my research during the
past two years at Raytheon, there are many programs that CNC machines use to cut material, both in twodimensions and three-dimensions. Cartesian systems of equations are not the only types of systems that exist,
but they are the type my students will be familiar with and that can be further developed in the scope of the
geometry class. The students will be given the opportunity to construct their pyramid and prism objects using
materials of their choice, however paper and poster board will be the most accessible. I am looking forward to
the types of materials my students use.
Summer 2013: TTE 596 7
Understanding by Design Final Project by Lawrence Schneider
Assessment Tool
Name:
Period:
1. Consider the shape below. Describe how you would identify the lengths of every unlabeled side. Assume
that the triangles are right triangles. Your description should be in written form. I am not asking you, yet, to
actually identify the side lengths.
4
4
2
3
Answer:
Summer 2013: TTE 596 8
Understanding by Design Final Project by Lawrence Schneider
2. Using the techniques you described in answering the first question, identify the side lengths of the nonlabeled sides. Your lengths should be in exact form.
Answer:
3. Consider the shape below. Describe how you would identify the side lengths that are not labeled. Assume
the triangles are right triangles.
z
z
y
x
Answer:
Summer 2013: TTE 596 9
Understanding by Design Final Project by Lawrence Schneider
4. Use the techniques you described in answering Question 3 to identify the side lengths that are not labeled.
Your values need to be in terms of x, y, and z.
Answer:
5. Compare the techniques you described in answering Question 1 and Question 3. What similarities do you
see? What differences do you see?
Answer:
6. What do these similarities and differences mean in terms of the concept of identifying side lengths?
Answer:
Summer 2013: TTE 596 10
Understanding by Design Final Project by Lawrence Schneider
7. Consider the figure below. No side lengths are labeled. Describe the minimum amount of information
necessary to be able to identify all of the side lengths in a general form.
Answer:
Summer 2013: TTE 596 11
Understanding by Design Final Project by Lawrence Schneider
Rubric
Level
Understanding
Novice
There is no solution, or the solution has no
relationship to the task. Inappropriate concepts
are applied and/or procedures are used. The
solution addresses none of the mathematical
components presented in the task.
Apprentice
The solution is not complete, indicating that
parts of the problem are not understood. The
solution addresses some, but not all, of the
mathematical components presented in the task.
Practitioner
The solution shows that the student has a broad
understanding of the problem and the major
concepts necessary for its solution. The
solution addresses all of the components
presented in the task.
Expert
The solution shows a deep understanding of the
problem, including the ability to identify the
appropriate mathematical concepts and the
information necessary for its solution. The
solution completely addresses all mathematical
components presented in the task. The solution
puts to use the underlying mathematical
concepts upon which the task was designed.
Strategies, Reasoning,
Procedures
No evidence of a strategy or
procedure, or uses a strategy
that does not help solve the
provelm. No evidence of
mathematical reasoning. There
were so many errors in
mathematical procedures that
the problem could not be
solved.
Uses a strategy that is partially
useful, leading some way
towards a solution, but not to a
full solution of the problem.
Some evidence of
mathematical reasoning. Could
not completely carry out
mathematical procedures.
Some parts may be correct, but
a correct answer is not
achieved.
Uses a strategy that leads to a
solution of the problem. Uses
effective mathematical
reasoning. Mathematical
procedures used. All parts are
correct and a correct answer is
achieved.
Uses a very efficient and
sophisticated strategy leading
directly to a solution. Employs
refined and complex reasoning.
Applies procedures accurately
to correctly solve the problem
and verify the results. Verifies
solutions and/or evaluates the
reasonableness of the solution.
Makes mathematically relevant
observations and/or
connections.
Communication
There is no explanation of the
solution, the explanation
cannot be understood or it is
unrelated to the problem. There
is no use or inappropriate use
of mathematical
representations (e.g. figures,
diagrams, graphs, tables, etc.)
There is no use, or mostly
inappropriate use, of
mathematical terminology and
notation.
There is an incomplete
explanation, it may not be
clearly represented. There is
some use of appropriate
mathematical representation.
There is some use of
mathematical terminology and
notation appropriate of the
problem.
There is a clear explanation.
There is appropriate use of
accurate mathematical
representation. There is
effective use of mathematical
terminology and notation.
There is a clear, effective
explanation detailing how the
problem is solved. All of the
steps are included so that the
reader does not need to infer
how and why decisions were
made. Mathematical
representation is actively used
as a means of communicating
ideas related to the solution of
the problem. There is precise
and appropriate use of
mathematical terminology and
notation.