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STAGE 1:
IDENTIFY DESIRED RESULTS
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Content Standard(s)
Generalizations about what students should know and be able to do
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Established Goals:
State Standards:
Model and describe patterns and functional
relationships.
Model and analyze quantitative data.
Using algebraic symbols to represent and
interpret data and physical phenomena.
National (NCTM)
Standards
Understand patterns, relations, and functions.
Represent and analyze mathematical situations
and structures using algebraic symbols.
Use mathematical models to represent and
understand quantitative relationships.
Analyze change in various contexts.
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Enduring Understandings
Insights earned from exploring generalizations via the essential
questions (Students will understand THAT…)
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Essential Questions
Inquiry used to explore generalizations
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- Algebraic
models help to solve problems and allow us to reach conclusions or make
predictions.
- Many
different kinds of problems can be solved using the same technique.
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- How
do you know for sure that your solution is correct?
- How
can you convince others that your algebraic solution is correct by also
using pictures, charts, graphs, and/or simple word explanations?
- What
is algebra?
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Knowledge and Skills
What students are expected to know and be able to do
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Students will know that…
- Algebraic
equations may be used as problem solving tools.
- A
constant rate of change between two variables (slope) will yield a
straight line graph (linear), but if the rate of change varies, then the
graph is not a line (nonlinear).
- Some
relationships are continuous, others are not continuous (discrete) and
the graphs of data points should reflect this.
- The
values of slope and of intercepts (the points where a graph intersects
each axis) facilitate writing equations and graphing linear
relationships.
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Students will be able to…
- Solve
problems involving simple linear equations using concrete verbal,
graphical and tabular representations.
- Recognize
and generate equivalent forms for simple algebraic expressions and
equations.
- Use
graphs, tables and equations to represent and analyze changes in linear
and nonlinear relationships.
- Identify
the independent and dependent variables in a given situation.
- Recognize
that the constant rate of change of a function is the slope.
- Identify
how change in the values of data patterns in tables affects changes in
the corresponding equations and graphs of linear functions.
- Identify
points on the graph where the x or y values are equal to zero as
representing the x- or y-intercepts, respectively.
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STAGE 2:
DETERMINE ACCEPTABLE EVIDENCE
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Performance Task(s)
Authentic application in
new context to evaluate student achievement of desired results designed
according to GRASPS (Goal, Role, Audience, Setting Performance, Standards)
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Other Evidence
Application that is functional in a classroom context only to
evaluate student achievement of desired results
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Goal:
The goal is to collect and record data in a
table using correct labels and scale to recognize what varies, and name the
independent and dependent variables in the experiment.
Role:
You are a member of a personal training team.
Audience:
The audience is your fitness client.
Situation
(Setting):
The setting is the classroom.
You are a team to help your client determine
the number of jumping jacks he/she can perform in a given amount of
time. One member is the timer, one the
recorder and another is the counter as the client performs the jumping jacks.
Product Performance
and Purpose:
You need to create a table from the data
generated by the client doing the jumping jacks. From this table, you must make a graph that
shows the jumping jack rate as time passes.
Compare the table to the graph for an appropriate interpretation of the data.
Standard and
Criteria for Success:
Your report should include:
a table of jumping jack data.
a graph of jumping of jumping jack data.
a summary of the results for the client.
Scoring Rubric:
Score of 3: Student shows
a correct and/or appropriate answer and shows work and/or an explanation that
demonstrates full and complete understanding.
Score of 2: Student has
minor flaws in the answer, but the work and/or explanation is acceptable and
the reasoning is appropriate.
Score of 1: Student does
not have a reasonable explanation or show sufficient work, resulting in a
demonstration of only limited understanding.
Score of 0: Student shows no understanding of the
problem or how to arrive at a solution.
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Daily Homework,
Observation of
Classroom Activities
Quizzes
District
Assessments
State Assessments
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RESOURCES
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Connected Mathematics: Algebra, Variables and Patterns,
Dale Seymour Publications, 2002
“Navigating through Algebra in Grades 6-8,” National
Council of Teachers of Mathematics, 2001
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STAGE 1:
IDENTIFY DESIRED RESULTS
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Content Standard(s)
Generalizations about what students should know and be able to do
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Established Goals:
State Standards:
Use spatial reasoning, location and geometric
relationships to solve problems.
Use attributes of two- and three-dimensional
shapes and geometric theorems to describe relationships, communicate ideas,
and solve problems.
Develop and apply units, systems, formulas and
appropriate tools to estimate and measure.
National (NCTM)
Standards:
Analyze characteristics and properties of two-
and three-dimensional geometric shapes and develop mathematical arguments
about geometric relationships.
Specify locations and describe spatial
relationships using coordinate geometry and other representational systems.
Apply transformations and use symmetry to
analyze mathematical situations.
Use visualization, spatial reasoning, and
geometric modeling to solve problems.
Understand measurable attributes of objects
and the units, systems, and processes of measurement.
Apply appropriate techniques, tools, and
formulas to determine measurements.
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Enduring Understandings
Insights learned from exploring generalizations via the essential
questions (Students will understand THAT…)
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Essential Questions
Inquiry used to explore generalizations
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1.
Geometry enables us to describe, analyze, and
understand our physical world.
2.
Geometry introduces an experience of
mathematics that complements and supports the study of other aspects of
mathematics such as number and measurement.
3.
Geometry offers powerful tools for
representing and solving problems in all areas of mathematics, in other
school subjects, and in everyday applications.
4.
Both metric and customary systems of
measurement are used in everyday life.
5.
There are relationships among units within the
same system.
6.
There are appropriate units for measuring angles, perimeter, area,
surface area, and volume.
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- How
would our classrooms, schools, and homes be different without geometric
shapes/angles?
- What’s
your position in the world?
- How
are size and distance related?
- How
would our lives be different if we couldn’t measure?
- How
does what we measure change how we measure?
- When
is it better to use an exact measurement rather than an estimate?
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Knowledge and Skills
What students are expected to know and be able to do
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Students will know that…
- Subdividing polygons and solids into simpler
shapes and prisms can be used to solve geometric and measurement
problems.
- The properties of polygons influence the number
of flips and turns needed to return a shape to its original orientation.
- Base plans (footprints), orthogonal views (from
the front, side and top) and isometric drawings (on a triangle-based
grid) are ways to represent three-dimensional objects in two-dimensional
diagrams.
- Problems involving
measurement can be solved through the use of appropriate tools,
techniques and strategies.
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Students will be able to…
- Estimate
and develop formulas to find the surface area and volume of prisms and
cylinders.
- Explore
the relationships among angles, sides, perimeters and areas of congruent
and similar polygons using models and diagrams on the rectangular
coordinate plane.
- Explore
constructing various angles and polygons using a compass and
straightedge.
- Examine
and describe the effect of transformations on polygons with line and/or
rotational symmetry.
- Draw
and interpret nets, cross-sections and front, side, top views of various
solids.
- Develop
and use estimation and measurement strategies to solve problems
involving the areas of irregular polygons and volumes of irregular
solids.
- Understand
the relationship of a cubic centimeter and a milliliter. Explore ways to determine the volume
of irregular solids using the relationship.
- Choose
appropriate units and use standard and nonstandard referents as
benchmarks when estimating length, area, volume, weight, mass, time,
temperature, and angle.
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STAGE 2:
DETERMINE ACCEPTABLE EVIDENCE
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Performance Task(s)
Authentic application in
new context to evaluate student achievement of desired results designed
according to GRASPS (Goal, Role, Audience, Setting Performance, Standards)
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Other Evidence
Application that is functional in a classroom context only to
evaluate student achievement of desired results
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Task 1
Goal:
The goal is to design a pattern of geometric
shapes on a square template for a floor tile to cover a specific area.
Role:
You are a custom product designer for a tile
company.
Audience:
The audience is the homeowner who has
contracted your tile company for a custom floor design.
Situation(Setting):
The setting is your design studio at the tile
company.
You are designing patterns of geometric shapes
in a square template for floor tiles.
Product Performance
and Purpose:
You are to produce a tile design that will
best fit the area of the specific floor.
Standards and
Criteria for Success:
Your
square template should:
include at least two different geometric
shapes within it.
be between 8 and 12 inches on a side.
include at least two different colors in the
design.
include at least one line of symmetry.
Your proposal for the specific floor should:
include the
of dimensions of the floor
include the area of the floor.
include the number of tiles needed to cover
the floor.
Scoring rubric:
Score of 3: Student shows
a correct and/or appropriate answer and shows work and/or an explanation that
demonstrates full and complete understanding.
Score of 2: Student has
minor flaws in the answer, but the work and/or explanation is acceptable and
the reasoning is appropriate.
Score of 1: Student does
not have a reasonable explanation or show sufficient work, resulting in a
demonstration of only limited understanding.
Score of 0: Student shows no understanding of the
problem or how to arrive at a solution.
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Daily Homework,
Observation of
Classroom Activities
Quizzes
District
Assessments
State Assessments
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Task 2
Goal:
The goal is to draw an enlarged cartoon that
would produce an enlarged image.
Role:
You are graphic designer.
Audience:
The audience is the consuming public.
Situation (Setting):
You are to create an advertising poster to
sell a product using an enlarged similar image of a cartoon. It’s your job to supply the similar image
for the poster.
Product Performance
and Performance:
Your final advertising poster project must be
presented in a format that can be displayed for others to view.
Standards and
Criteria for Success:
Your poster should…
show the original picture and image.
show the scale factor.
include the text for advertisement.
Your report should…
describe the method you used to create your
image.
describe the changes in the lengths, angles,
and area between the original picture and your image.
include a paragraph or more about other
details you think would be interesting or would help the reader understand
what they are seeing, such as a description of any problems or surprises you
encountered while completing the project and the decisions you made as a
result.
Scoring rubric:
Score
of 3: Student shows a correct
and/or appropriate answer and shows work and/or an explanation that
demonstrates full and complete understanding.
Score of 2: Student has
minor flaws in the answer, but the work and/or explanation is acceptable and
the reasoning is appropriate.
Score of 1: Student does
not have a reasonable explanation or show sufficient work, resulting in a
demonstration of only limited understanding.
Score of 0: Student shows no understanding of the
problem or how to arrive at a solution.
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RESOURCES
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Connected Mathematics, Filling and Wrapping, Dale
Seymour, 2002
Connected Mathematics, Geometry, Stretching and
Shrinking, Dale Seymour Publications, 2002
Navigating Through Geometry in Grades 6-8, National
Council of Teachers of Mathematics, 2002
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STAGE 1:
IDENTIFY DESIRED RESULTS
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Content Standard(s)
Generalizations about what students should know and be able to do
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Established Goals:
State Standards:
Use a variety of numerical representations in
the base ten system to describe quantitative relationships
Use numbers and their properties to compute
flexibly and fluently, and to reasonably estimate measures and quantities.
National Standard:
Compute fluently and make reasonable
estimates.
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Enduring Understandings
Insights learned from exploring generalizations via the essential
questions (Students will understand THAT…)
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Essential Questions
Inquiry used to explore generalizations
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1.
Estimation serves as an important companion to
computation.
2.
Estimation provides an effective tool for
judging the reasonableness of calculator, mental, and paper-and-pencil
computations.
3.
There are appropriate times to using
estimation rather than computing an exact answer.
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- Why
is estimation important?
- When
does estimating make more sense than finding an exact answer?
- How
will estimation make solving a problem easier?
- What
is the difference between rounding and estimating?
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Knowledge and Skills
What students are expected to know and be able to do
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Students will know that…
- Computation with
positive and negative numbers may be modeled in the context of
increasing and decreasing value or changes in measurements.
- Very large and very
small numbers may be written using scientific notation, which is based
on powers of ten.
- Fractions, decimals
and percents are equivalent ways to represent real-world situations and
the choice of which symbolic form to use may make it easier to describe
a relationship or solve a problem.
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Students will be able to…
1.
Use number theory concepts (primes, factors, multiples, divisibility) to
estimate and solve problems.
2.
Use and explain estimation strategies that involve powers of ten and
scientific notation.
3.
Develop, describe and use a variety of methods
to estimate and calculate with very large numbers.
4.
Estimate and use the calculator to do
computations involving fractions, decimals, mixed numbers, improper
fractions, ratios, proportions and percents.
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