Foundational Geometry - Stage 1 Desired Results

ESTABLISHED GOALS

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Standards

Students will be able to independently use their learning to…        

(G-CO.1) Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.

(G-CO.12) Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Constructions include: copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.

Meaning

UNDERSTANDINGS        

Students will understand…

1. an angle is formed by the joining of two rays, creating the vertex, and is measured in degrees.
2. to find and compare lengths of segments and angles measured.
3. a circle is equidistant around a fixed point (the center) and the center to any edge is the radius.
4. perpendicular lines form right angles.
5. parallel lines are equidistant from one another and therefore never intersect.
6. a point is a single location in space, many points for a line (which is infinite), and a portion of a line is a line segment with two endpoints.
7. distance along a line or circular arc can be measured or calculated in units of inches, centimeters, yards, etc.
8. how to create basic constructions using straightedge, protractor, and compass.

ESSENTIAL QUESTIONS        

How do you measure, calculate, and identify an angle, circle, perpendicular line, parallel line, line segment, point, line, distance along a line and circular arc?

Stage 2 - Evidence

Evaluative Criteria

Assessment Evidence

Identify and define all vocabulary terms

PERFORMANCE TASK(S):        

Standardized Test

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OTHER EVIDENCE:        

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Stage 3 – Learning Plan

Summary of Key Learning Events and Instruction

1. Create a glossary of geometric terms
2. Create a list of geometric symbols
3. Be able to provide a real-world visual representations of vocabulary terms and symbols
4. Use rulers and protractors to measure line segments and angles.

Topic   - Stage 1 Desired Results

ESTABLISHED GOALS

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Standards

Students will be able to independently use their learning to…        

Meaning

UNDERSTANDINGS        

Students will understand that…

ESSENTIAL QUESTIONS        

Acquisition

Students will independently be able to use their learning for        

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Students will be skilled at…        

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Stage 2 - Evidence

Evaluative Criteria

Assessment Evidence

what do they need to know

abilities, skills, etc.

PERFORMANCE TASK(S):        

project, test, (be specific for how each standards will be evaluated if not all standards are addressed in a summative)

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OTHER EVIDENCE:        

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Stage 3 – Learning Plan

Summary of Key Learning Events and Instruction

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Parallel and Perpendicular Lines - Stage 1 Desired Results

ESTABLISHED GOALS

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Standards

Students will be able to independently use their learning to…        

(G.CO.9) Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent, and conversely prove lines are parallel.

(G.GPE.4) Use coordinates to prove simple geometric theorems algebraically including the distance formula and its
relationship to the Pythagorean Theorem.

(G.GPE.5) Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point).

(G.GPE.6) Find the point on a directed line segment between two given points that partitions the segment in a given ratio (midpoint).

(G.GPE.7) Use coordinates to compute perimeters of polygons and areas of triangles and rectangles (e.g., using the distance formula).

Meaning

UNDERSTANDINGS        

Students will understand how:

1. to write the equation of a line that is parallel and/or perpendicular to a line given, parallel  lines have the same slope and perpendicular lines have opposite reciprocal slopes.
2. to use the distance formula, the length of a line segment on the coordinate place can be calculated and is derived from the Pythagorean Theorem.
3. parallel lines cut by a transversal create alternate interior, alternate exterior, and corresponding angles that are congruent and same-side (consecutive) interior angles that are supplementary.
4. the distance formula can be used to calculate the perimeter or area of polygons on the coordinate plane.

ESSENTIAL QUESTIONS        

1. How do you write the equation of the parallel and perpendicular line given a line in slope intercept form?
2. How is the distance formula similar to the Pythagorean theorem?
3. What is the relationship between alternate interior, alternate exterior, corresponding and same-side interior angles?

Stage 2 - Evidence

Evaluative Criteria

Assessment Evidence

Calculate slope, midpoint, endpoint, and distance.

Write the equation of a line.

Identify the slope of a parallel and perpendicular line.

Identify and calculate the angle relationships formed by parallel lines cut by a transversal.

Calculate the perimeter and area of a polygon on the coordinate plane

PERFORMANCE TASK(S):        

Open Response

Standardized Test

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OTHER EVIDENCE:        

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Stage 3 – Learning Plan

Summary of Key Learning Events and Instruction

City Map Project

                                Properties of Triangles

ESTABLISHED GOALS

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Standards

Students will be able to independently use their learning to…        

(G.CO.10) Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent, and conversely prove a triangle is isosceles; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.

(G.SRT.5) Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.

Meaning

UNDERSTANDINGS        

Students will understand…

1. Various classifications of triangles (scalene, isosceles, equilateral, equiangular, acute, obtuse, right triangles).
2. Isosceles Triangle Theorem and its converse
3. Triangle Angle Sum Theorem
4. Triangle Midsegment Theorem
5. How to use congruence theorems (SSS, SAS, ASA, AAS, HL) to prove two triangles congruent
6. Lines inside a Triangle-intersection of the medians is the centroid (orthocenter, circumcenter, incenter??) A triangle’s three medians are always concurrent (time permitting)

ESSENTIAL QUESTIONS        

1. How do you classify triangles?
2. How do you show that two triangles are congruent? (SSS, SAS, ASA, AAS, HL)
3. How do you identify corresponding parts of congruent triangles?
4. How do you use coordinate geometry to find relationships within triangles?
5. How do you solve problems that involve measurements of triangles?
6. What are some special lines inside a triangle? (midsegment, centroid)

Stage 2 - Evidence

Evaluative Criteria

Assessment Evidence

Identify and calculate the side lengths and angles of a scalene, isosceles, and equilateral triangle.

Determine if two triangles are congruent using one of the congruence theorems.

Understand and determine the sequence of geometric definitions, postulates and theorems that prove two triangles congruent.

PERFORMANCE TASK(S):        

Open Response

Standardized Test

Geometric Proofs

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OTHER EVIDENCE:        

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Stage 3 – Learning Plan

Summary of Key Learning Events and Instruction

Properties of Quadrilaterals   - Stage 1 Desired Results

ESTABLISHED GOALS

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Standards

Students will be able to independently use their learning to…        

(G.CO.11) Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals.

Meaning

UNDERSTANDINGS        

Students will understand…

1. a rhombus, rectangles, and squares all have the properties of a parallelogram, in addition to properties that make them unique from one another.
2. A square is a rectangle, but a rectangle is not a square.
3. use the relationships among sides and angles and parallelograms.
4. how to determine whether a quadrilateral is a parallelogram
5. how to use the properties of diagonals of all parallelograms.

ESSENTIAL QUESTIONS

1. How can you classify quadrilaterals?
2. What are the properties of a parallelogram?
3. What is the difference between a rhombus, a rectangle, and a square?

Stage 2 - Evidence

Evaluative Criteria

Assessment Evidence

Identify, using properties of a figure, the difference between a quadrilateral, parallelogram, rhombus, square, and rectangle.

Calculate the side lengths and angle measures of parallelograms based on the properties of each.

PERFORMANCE TASK(S):        

Open Response

Standardized Test

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OTHER EVIDENCE:        

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Stage 3 – Learning Plan

Summary of Key Learning Events and Instruction

Categorize quadrilaterals through measurement of angles and side lengths

Similar Figures  - Stage 1 Desired Results

ESTABLISHED GOALS

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Standards

Students will be able to independently use their learning to…        

(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.

(G.SRT.3) Use the properties of similarity transformations to establish the Angle-Angle (AA) criterion for two triangles to be similar.

Meaning

UNDERSTANDINGS        

Students will understand how:

1. by definition, two figures are similar if they have corresponding congruent angles.
2. if two figures are similar, their corresponding sides are proportional.
3. using the AA theorem, two figures can be proven similar.

ESSENTIAL QUESTIONS        

1. What is the difference between congruent and similar figures?
2. How do you calculate the side lengths of similar figures?
3. How do you prove two figures similar?

Stage 2 - Evidence

Evaluative Criteria

Assessment Evidence

Identify if two figures are similar.

Use proportions to solve for missing side lengths.

Use AA to prove figures similar.

PERFORMANCE TASK(S):        

Open Response

Standardized Test

Geometric Proof

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OTHER EVIDENCE:        

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Stage 3 – Learning Plan

Summary of Key Learning Events and Instruction

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Right Triangles and Trigonometry   - Stage 1 Desired Results

ESTABLISHED GOALS

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Standards

Students will be able to independently use their learning to…        

(G.SRT.6) Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.

(G.STR.7) Explain and use the relationship between the sine and cosine of complementary angles.

(G.STR.8) Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.

Meaning

UNDERSTANDINGS        

Students will understand that…

1. using the properties of similar figures, a specific relationship creates the basic trig functions of sine, cosine and tangent.
2. sine and cosine of complementary angles of a right triangle are reverse of one another.
3. Pythagorean theorem can only be used to solve for a missing side of a right triangle, where the basic trig functions can calculate both.

ESSENTIAL QUESTIONS        

1. What are the sine, cosine and tangent functions and what can you solve when using them?
2. What is the relationship between the sine and cosine values between the complementary angles of a right triangle?
3. What problems would you use the pythagorean theorem? basic trig functions?

Stage 2 - Evidence

Evaluative Criteria

Assessment Evidence

Identify what problems use the pythagorean theorem and which use trigonometry.

Identify which trig functions are needed to solve for missing angles/side lengths of a right triangle.

Correctly identify the hypotenuse when using the pythagorean theorem to solve for missing side length.

PERFORMANCE TASK(S):        

Open Response

Standardized Test

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OTHER EVIDENCE:        

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Stage 3 – Learning Plan

Summary of Key Learning Events and Instruction

Baseball Project

Transformations   - Stage 1 Desired Results

ESTABLISHED GOALS

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Standards

Students will be able to independently use their learning to…        

(G.CO.2) Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs.

(G.CO.3) Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.

(G.CO.4) Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.

(G.CO.5) Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another.

(G.STR.1) Verify experimentally the properties of dilations given by a center and a scale factor:
         a. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged.
         b. The dilation of a line segment is longer or shorter in the ratio given by the scale factor.

Meaning

UNDERSTANDINGS        

Students will understand that…

1. translations, rotations and reflections maintain congruence between the image and preimage.
2. a dilation is a specified ratio making the image larger or smaller to the preimage.

ESSENTIAL QUESTIONS

1. How can you change a figures position without changing its size and shape?
2. How can you change a figures size without changing its shape?
3. How can you represent a transformation in the coordinate plane?
4. How do you recognize congruence and similarity in figures?

Stage 2 - Evidence

Evaluative Criteria

Assessment Evidence

Identify and complete a translation, rotation, reflection and dilation to a figure in the coordinate plane.

Determine the series of transformations completed to the preimage that created the final image.

PERFORMANCE TASK(S):        

Open Response

Standardized Test

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OTHER EVIDENCE:        

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Stage 3 – Learning Plan

Summary of Key Learning Events and Instruction

Logo Project

Surface Area and Volume    - Stage 1 Desired Results

ESTABLISHED GOALS

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Standards

Students will be able to independently use their learning to…        

(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.

(G.GMD.3) Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.

Meaning

UNDERSTANDINGS        

Students will understand that…

1. through the use of nets, the basis for the formulas for area, surface area, and volume of the 3D figures are derived.
2. the B in the volume formulas represents the area of the base of the 3D figure

ESSENTIAL QUESTIONS        

1. How are the formulas for area, surface area, and volume related?

Stage 2 - Evidence

Evaluative Criteria

Assessment Evidence

Identify the correct formula for each 3D solid to calculate the area, surface area, and volume.

Use the formula of two figures to determine the remaining space between them.

When given the total area, surface area, or volume solve for a missing measurement within the formula.

PERFORMANCE TASK(S):        

Open Response

Standardized Test

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OTHER EVIDENCE:        

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Stage 3 – Learning Plan

Summary of Key Learning Events and Instruction

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Circles   - Stage 1 Desired Results

ESTABLISHED GOALS

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Standards

Students will be able to independently use their learning to…        

(G.C.1) Prove that all circles are similar.

(G.C.2) Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle.

(G.C.3) Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral and other polygons inscribed in a circle.

(G.C.5) Derive, using similarity, the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector.

(G.GPE.1) Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation.

Meaning

UNDERSTANDINGS        

Students will understand that…

1. a central angle is formed by 2 radii of the circle and the measure is the same as the arc it creates, in degrees.
2. inscribed angles are formed by chords within the circle whose measure is half of the arc it creates, in degrees.
3. tangent lines are perpendicular to the any radius, forming a right angle.
4. the arc length and area of a sector of a circle is proportional to the angle of the circle and the perimeter and area respectively.
5. the equation of a circle on the coordinate plane can be derived from the pythagorean theorem, to determine the length of the radius and the center of the circle.

ESSENTIAL QUESTIONS        

1. What is the relationship between radii and central angles, and chords and inscribed angles of a circle?
2. What is the relationship between the perimeter and area of the circle and the arc length and area of a sector respectively?
3. How can the pythagorean theorem be used to write the equation of a circle in the coordinate plane?
4. Why are tangent lines always perpendicular to the radius of a circle?
5. How do you determine the angle measure of inscribed figures within a circle?

Stage 2 - Evidence

Evaluative Criteria

Assessment Evidence

Calculate the measure of a central and inscribed angles, arc length, and area of a sector.

Write the equation of a circle on a coordinate plane, by determining the radius length and center of the circle.

Determine the angle measure of an inscribed polygon within a circle.

PERFORMANCE TASK(S):        

Open Response

Standardized Test

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OTHER EVIDENCE:        

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Stage 3 – Learning Plan

Summary of Key Learning Events and Instruction

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 Probability   - Stage 1 Desired Results

ESTABLISHED GOALS

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Standards

Students will be able to independently use their learning to…        

(S.CP.1) Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events (“or,” “and,” “not”).

(S.CP.2) Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities, and use this characterization to determine if they are independent.

(S.CP.3) Understand the conditional probability of A given B as P(A and B)∕P(B), and interpret independence of A and B as saying that the conditional probability of A given B is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B.

(S.CP.4) Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities.

(S.CP.5) Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations.

(S.CP.6) Find the conditional probability of A given B as the fraction of B’s outcomes that also belong to A, and interpret the answer in terms of the model.

(S.CP.7) Apply the Addition Rule, P(A or B) = P(A) + P(B) – P(A and B), and interpret the answer in terms of the model.

Meaning

UNDERSTANDINGS        

Students will understand…

1. basic probability of any event can be represented by a fraction (part to whole or number to total).
2. the difference between independent and dependent events.
3. the difference between the probability of “either” two events occurring and “both” events occurring.

ESSENTIAL QUESTIONS        

1. How does the probability of an independent event relate to what we know about fractions?
2. How does probability vary between independent and dependent events?

Stage 2 - Evidence

Evaluative Criteria

Assessment Evidence

PERFORMANCE TASK(S):        

Open Response

Standardized Test

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OTHER EVIDENCE:        

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Stage 3 – Learning Plan

Summary of Key Learning Events and Instruction

Calculate probability of real-world events.

Statistical Analysis   - Stage 1 Desired Results

ESTABLISHED GOALS

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Standards

Students will be able to independently use their learning to…        

(AI.S.ID.1) Represent data with plots on the real number line (dot plots, histograms, and box plots).

(AI.S.ID.2) Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets.

(AI.S.ID.3) Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers).

(AI.S.ID.5) Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data.

(AI.S.ID.6) Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.

1. Fit a linear function to the data and use the fitted function to solve problems in the context of the data. Use functions fitted to data or choose a function suggested by the context (emphasize linear and exponential models).
2. Informally assess the fit of a function by plotting and analyzing residuals.
3. Fit a linear function for a scatter plot that suggests a linear association.

(AI.S.ID.7) Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.

(AI.S.ID.8) Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.

(AI.S.ID.9) Distinguish between correlation and causation.

Meaning

UNDERSTANDINGS        

Students will understand how to…

1. create and interpret a frequency table, scatter plot, histogram, and box and whisker plot for a set of data.
2. calculate the mean, median, interquartile range, and standard deviation for a set of data.
3. identify the function type that best fits the given a set(s) of data to predict and solve problems.
4. explain the meaning of the slope and y-intercept of a linear function for a given function.
5. determine the difference between correlation and causation.

ESSENTIAL QUESTIONS

1. What is the difference between a frequency table, scatter plot, histogram, and box and whisker plot?
2. Given a real-world example, determine which of the following statistical representations is best to represent the data given?
3. How do you calculate the mean, median, interquartile range and standard deviation for a set of data?
4. What qualities in a data set determine a linear and/or exponential function?
5. What do the rate of change and y-intercept represent in a real world example linear?
6. What is the difference between correlation and causation?

Stage 2 - Evidence

Evaluative Criteria

Assessment Evidence

Calculate percents, ratios, averages, range, slope and y-intercept for a linear function.

Determine the best statistical representation for a set of data.

Draw and interpret a histogram, scatter, line and box and whisker plots.

Describe the meaning of “rate of change” and “y-intercept” for a given real world example.

Determine the change in the mean, median, and range given an outlier in the data set.

Identify the difference between correlation and causation.

PERFORMANCE TASK(S):        

Open Response

Standardized Test

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OTHER EVIDENCE:        

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Stage 3 – Learning Plan

Summary of Key Learning Events and Instruction

NFL Stats