SM2 Curriculum Notebook

Functions–Interpreting Functions FIF9
Core Content
Cluster Title: Understand the concept of a function and use function notation.
Standard F.IF.9 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).
Concepts and Skills to Master
Conceptual
• Understand slope and intercepts for linear functions.
• Understand growth rate and intercepts for exponential functions Procedural
• Compare slopes and intercepts of two linear functions where one is represented algebraically, graphically, numerically in
• tables, or in a description and the other is modeled using a different form of representation.
• Compare slopes and intercepts of two linear functions where one is represented algebraically, graphically, numerically in
• tables, or in a description and the other is modeled using a different form of representation
Representational
• Represent linear and exponential functions algebraically, graphically, and numerically in
tables. Supports for Teachers
Critical Background Knowledge
Conceptual
• Know what slope and intercepts of linear functions are.
• Know what growth rates of exponential functions are.
Procedural
• Find slope and intercepts of linear functions.
• Find growth rates of exponential functions.
Representational
• Graph linear and exponential functions.
Academic Vocabulary and Notation
Function, Growth rate, Intercept, Interval, Rate of change, Slope
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Suggested Instructional Strategies Resources
• Compare two functions expressed Exponential functions on the web:
in different representations. Ask: Which is growing at a faster rate? Which one begins at a higher value? Why does it increase faster than the
other? How do you know?
• Match functions expressed using
different representations that have the same properties.
Sample Formative Assessment Tasks
http://facutly.gvsu.edu/goldenj/exponential.html
Geogebra
http://www.geogebra.org/cms/en/download
Graphing calculators
Skill-based Task
Which has a greater slope?
• f(x)= 3x +5
• A function representing the number of bottle caps in a shoebox where 5 are added each time
Problem Task
Create a graphic organizer to highlight your understanding of functions and their properties by comparing two functions using at least two different representations.
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Functions–Building Functions FBF1a
Core Content
Cluster Title: Build a function that models a relationship between two quantities.
Standard F.BF.1: Write a function that describes a relationship between two quantities
a. Determine an explicit expression, a recursive process, or steps for calculation from a context.
Concepts and Skills to Master
• Given a linear or exponential context, find an expression, recursive process, or steps to model a context with mathematical representations.
• Combine linear and/or exponential functions using addition, subtraction, multiplication, and division.
Supports for Teachers
Critical Background Knowledge
• Simplify expressions
Academic Vocabulary and Notation
f(x)=, Explicit expressions, Functions, Intercepts, Recursive Suggested Instructional Strategies
• Toothpick patterns
• Number of knots versus length of rope
Sample Formative Assessment Tasks
Skill-based Task
Anne is shopping and finds a $30 sweater on sale for 20% off. When she buys the sweater, she must also pay 6% sales tax. Write an expression for the final price of the sweater in such a way that the original price is still evident. (Extension: if the clerk just adds 14% will the price be correct?)
Resources
www.illuminations.NCTM.org
• Function Matching, Making it happen (NCTM)
Problem Task
Find an expression, process or calculation to determine the number of squares needed to make the next three patterns in the series.
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Functions–Building Functions FBF1b
Core Content
Cluster Title: Build a function that models a relationship between two quantities.
Standard F.BF.1: Write a function that describes a relationship between two quantities
b. Combine standard function types using arithmetic operations. For example, build a function
that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model.
Concepts and Skills to Master
• Given a linear or exponential context, find an expression, recursive process, or steps to model a context with mathematical representations.
• Combine linear and/or exponential functions using addition, subtraction, multiplication, and division.
Supports for Teachers
Critical Background Knowledge
• Simplify expressions
Academic Vocabulary and Notation
f(x)=, Explicit expressions, Functions, Intercepts, Recursive Suggested Instructional Strategies
• Give examples and use arithmetic operations to linear and exponential functions to fit the data
Sample Formative Assessment Tasks
Skill-based Task
If f(x) = x + 4 and g(x) = 3x – 5, find (f + g)(x)
www.illuminations.NCTM.org
• Function Matching, Making it happen (NCTM)
Problem Task
Find an expression, process or calculation to determine the number of squares needed to make the next three patterns in the series.
Resources
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Functions–Building Functions FBF3
Core Content
Cluster Title: Build a function that models a relationship between two quantities.
Standard F.BF.3: Identify the effect on the graph of replacing f(x) by f(x) + k for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology.
Concepts and Skills to Master
• Perform vertical translations on linear and exponential graphs
• Find the value of k given f(x) replaced by f(x) + k on a graph of linear or exponential
functions
• Relate the vertical translation of a linear function to its y-intercept
• Describe what will happen to a function when f(x) is replaced by f(x) + k for different
values of k. Supports for Teachers
Critical Background Knowledge
• Graphing linear and exponential functions Academic Vocabulary and Notation
f(x)=, Transformation, Translation, y-intercept, vertical shift Suggested Instructional Strategies
• Use graphing technology to explore translations of functions
Sample Formative Assessment Tasks
Skill-based Task
Graph the following on a single set of axes.
f(x) = 2x f(x)=2x+1 f(x)=2x+2 f(x)=2x–1 f(x)=2x–2
Resources
www.shodor.org/
Problem Task
Compare and contrast the graph of any function f(x), and the graph of f(x) + k.
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Functions–Building Functions FBF4
Core Content
Cluster Title: Build new functions from existing functions
Standard: F.BF.4: Find inverse functions
a. Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse. For example,
f(x)=2x3orf(x)=(x+1)/(x–1)forx
Concepts and skills to master
Conceptual:
• Understand that creating an inverse of a quadratic function requires a restricted domain
Procedural:
• Determine whether or not a function has an inverse • Find an inverse of a function if it exists
Representational:
• Use graphing technology to model inverse functions
Supports for Teachers
Critical Background Knowledge
Conceptual:
• Understand the definition of a function • Understand domain and range
Procedural:
• Find the domain and range of a function
Representational:
• Graph parent functions and show domains and ranges
Academic Vocabulary and Notation
f-1(x), inverse restricted domain, domain, range
Suggested Instructional Strategies
• Use technology to explore inverse functions graphically
• Compare the domains and ranges of a function and its inverse
Sample Formative Assessment Tasks
Skill-based Task:
Find the inverse of each function, if it exists:
Resources
Library of virtual manipulatives, and algebra tiles
http://nlvm.usu.edu/en/nav/vlibrary.html Geogebra software: http://www.geogebra.org/cms/
Problem Task:
Prove that the inverse of a non-horizontal linear function is also linear and that the slopes are reciprocals.
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Functions–Linear, Quadratic & Exponential Models FLE3
Core Content
Cluster Title: Construct and compare linear and exponential models and solve problems
Standard F.LE.3: Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly.
Concepts and Skills to Master
• Observe that a quantity increasing exponentially eventually exceeds a quantity increasing linearly using graphs and tables
Supports for Teachers
Critical Background Knowledge
• Identify linear and exponential functions
• Graph linear and exponential functions
• Rate of change
Academic Vocabulary and Notation
Difference, Exponential, Factor, Linear,
Suggested Instructional Strategies
• This standard should be taught in conjunction with others in this cluster
Sample Formative Assessment Tasks
Skill-based Task
Which increases faster, f(x) = 3x or g(x) = 3x ? Justify your answer.
Resources
Making it happen (NCTM)
Problem Task
What’s the better deal, earning $1000 a day for the rest of your life or earning $0.01 the first day, and doubling it every day for the rest of your life? How do you know? Do you think an 80-year-old would make the same choice? Should she?
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Functions–Trigonometric Functions FTF8
Core Content
Standard: F.TF.8: Prove the Pythagorean identity
, given
Concepts and skills to master
Conceptual:
• Know for right triangles Procedural:
and use it to find and the quadrant angle
Prove and apply trigonometric identities
Cluster Title: Prove and apply trigonometric identities
• Prove
• Given
Representational: Supports for Teachers
Critical Background Knowledge
for right triangles
for 0 < theta < 90, find
Conceptual:
• Know the Pythagorean Theorem
• Know the trigonometric ratios
Procedural:
• Apply the Pythagorean Theorem to determine unknown side lengths
• Define trigonometric ratios
Representational:
• Model right triangle relationships
Academic Vocabulary and Notation
Sin, cosine, tangent,
Suggested Instructional Strategies
• Connect trigonometric ratios to right triangle geometry • Use similar triangles to demonstrate that trigonometric
ratios are constant
Sample Formative Assessment Tasks
Skill-based Task:
Resources
“Helping students make sense of mathematics” Mathematics Teacher, Jan 2007, p. 383
Problem Task:
Show that the sin of an angel is constant regardless of the size of the triangle
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Geometry–Congruence GCO9
Core Content
Cluster Title: Prove geometric theorems
Standard: 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; points on a perpendicular bisector of a line segment are exactly equidistant from the segment’s endpoints)
Concepts and skills to master
Conceptual:
• • •
Know congruence statements for angles (e.g., vertical angles are congruent, corresponding angles of a transversal and two parallel lines are congruent)
Know that when parallel lines are cut by a transversal, supplementary angle pairs are created
Know that points on a perpendicular bisector of a line segment are equidistant from the segment’s endpoints
Procedural:
• Prove theorems about lines and angles, including angles created when a transversal cuts
two parallel lines ant the distance between a point on the perpendicular bisector to the
endpoints of the segment Representational:
• Model congruent and supplementary angles
• Model congruent lengths between a point on a perpendicular bisector and the endpoints
of the segment bisected Supports for Teachers
Critical Background Knowledge
Conceptual:
• Know the properties of supplementary, vertical, and adjacent angles • Know the properties of perpendicular bisectors
Procedural:
• Identify corresponding angles, adjacent angles, and vertical angles
Representational:
• Draw and label vertical angles, corresponding angles and adjacent angles • Draw perpendicular bisectors of a line segment
Academic Vocabulary and Notation
Proof, vertical angles, parallel lines, transversal, alternate interior angles, corresponding angles, perpendicular bisector, equidistant, the congruence symbol, congruence
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Suggested Instructional Strategies Resources
• Use multiple formats to write proofs: narrative Ladders and saws, NCTM agenda series, paragraphs, flow diagrams, two-column Measurement in the middle grades format, and diagrams without words
• Focus on the validity of the underlying reasoning while writing proofs
• Use dynamic geometry software to explore angle relationships
• Connect angle relationships to the creation of tessellation patterns
Sample Formative Assessment Tasks
Skill-based Task: Problem Task:
Prove that angle 1 is congruent to angle 8 Find as many angle relationships as possible in this pattern
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Geometry–Congruence GCO10
Core Content
Cluster Title: Prove geometric theorems
Standard: G.CO.10: Prove theorems about triangles (theorems include: measures of interior angles of a triangle sum to 180 degrees, base angles of isosceles triangles are congruent; 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)
Concepts and skills to master
Conceptual:
• Know theorems about triangles
Procedural:
• Prove theorems about triangles
Representational:
• Model the congruence of triangles
Supports for Teachers
Critical Background Knowledge
Conceptual:
• Know theorems about lines and angles
Procedural:
• Prove theorems about lines and angles
Representational:
• Model congruence from theorems about lines and angles
Academic Vocabulary and Notation
Interior angles and exterior angles of a triangle, supplementary angles, linear pairs, isosceles, base legs, base angles, vertex angles, midpoint, median of a triangle, auxiliary line
Suggested Instructional Strategies Resources
• •
•
•
Use paper folding to demonstrate Interior angles activity with Geogebra relationships in triangles
Use similar triangles or dilations to show Patty paper activities
that the mid-segment is parallel and half
the length of the third side of a triangle Use dynamic geometry software to
explore relationships in triangles Write proofs in a variety of formats
Sample Formative Assessment Tasks
Skill-based Task:
Prove that the base angles of an isosceles triangle are congruent
Problem Task:
Write a paragraph explaining why the segment joining midpoints of two sides of a triangle is parallel to the third side
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Geometry–Congruence GCO11
Core Content
Cluster Title: Prove geometric theorems
Standard: 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)
Concepts and skills to master
Conceptual:
• Know theorems about parallelograms
Procedural:
• Prove theorems about parallelograms
Representational:
• Model the congruence of parallelograms
Supports for Teachers
Critical Background Knowledge
Conceptual:
• Know the definition of a parallelogram
Procedural:
• Use the properties of parallelograms to find missing measures in geometric figures
Representational:
• Label important parts of parallelograms
Academic Vocabulary and Notation
Parallelogram, diagonal, consecutive angles, opposite angles, bisect
Suggested Instructional Strategies Resources
• •
•
Usegeometrysoftwaretoreflecta Geogebra
triangle across one of its sides Show that any two intersecting
segments that bisect each other create the diagonals of a parallelogram
Use properties of parallelograms to find missing measures in geometric figures
http://www.geogebra.org/cms/
Sample Formative Assessment Tasks
Skill-based Task:
Write a two-column proof showing that opposite sides of a parallelogram are congruent
Problem Task:
Write a paragraph proof showing that a rectangle is a parallelogram with congruent diagonals
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Geometry–Similarity, Right Triangles, and
Trigonometry GSRT1
Core Content
Cluster Title: Understand similarity in terms of similarity transformations
Standard: G.SRT.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
Concepts and skills to master
Conceptual:
• Understand dilations
Procedural:
• Construct a dilation of an original segment given the line segment and a point not on the
line.
• Find the length of the resulting image is the length of the original segment multiplied by
the scale factor and that of the original and dilated image are parallel to each other Representational:
• Create dilations to model similar figures Supports for Teachers
Critical Background Knowledge
Conceptual:
• Understand proportions
Procedural:
• Use proportions
• Describe the effect of dilations on two-dimensional figures using coordinates
Representational:
• Represent similar figures with dilations
Academic Vocabulary and Notation
Dilation, center of dilation, scale factor, similarity, transformation
Suggested Instructional Strategies Resources
• Draw analogy of dilation to zoom-in Geogebra
and zoom-out of a camera, a document camera, an iPad, or using geometry software programs
Sample Formative Assessment Tasks
Skill-based Task:
Create a dilation of segment AB through C
with a scale factor of 2:1 to create segment following pair of triangles EF. Find the lengths of EF, AB, BC, CE and
CF
http://www.geogebra.org/cms/
Problem Task:
Locate the center of dilation and scale factor in the
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Geometry–Similarity, Right Triangles, and
Trigonometry GSRT2
Core Content
Cluster Title: Understand similarity in terms of similarity transformations
Standard: 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
Concepts and skills to master
Conceptual:
• Know the requirements for similar triangles
• Understand relationships of corresponding parts of similar triangles
Procedural:
• Decide whether two figures are similar using properties of transformations
Representational:
• Model similar triangles
Supports for Teachers
Critical Background Knowledge
Conceptual:
• Understand similarity as a sequence of transformations
Procedural:
• Identify transformations including dilations
Representational:
• Represent similar figures with dilations
Academic Vocabulary and Notation
Similarity, transformation, corresponding parts, congruence symbol, similarity symbol
Suggested Instructional Strategies Resources
•
Give students pairs of triangles, some Geogebra
of which are similar and some of which http://www.geogebra.org/cms/ are not. Have students verify or
disprove similarity using
transformations and the definition of
similarity
Sample Formative Assessment Tasks
Skill-based Task:
Are the triangles similar? How do you know?
Problem Task:
Under what conditions do two lines intersected by two transversals form similar triangles?
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Geometry–Similarity, Right Triangles, and
Trigonometry GSRT3
Core Content
Cluster Title: Understand similarity in terms of similarity transformations
Standard: G.SRT.3: Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar
Concepts and skills to master
Conceptual:
• Know the properties of similarity transformations
• Know the AA criterion for two triangles to be similar Procedural:
• Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar
Representational:
• Model the transformations needed to create similar triangles
Supports for Teachers
Critical Background Knowledge
Conceptual:
• Know the sum of the measures of the angles of a triangle is 180 degrees
• Understand that when two angles of a triangle are congruent to two angles of another
triangle then the third angle in each triangle must be congruent Procedural:
• Find the sum of the third angle if the other two angles of a triangle are given Representational:
• Model angle congruence for two similar triangles Academic Vocabulary and Notation
Similarity, transformation, AA similarity
Suggested Instructional Strategies
Resources
Geogebra
http://www.geogebra.org/cms/
Problem Task:
Write an argument to justify the AA criterion for two triangles guarantees similarity
•
Given two different-sized triangle cutouts with two corresponding angles congruent, allow the students to show that the third angle is congruent, and find a dilation that produced the two triangles
Sample Formative Assessment Tasks
Skill-based Task:
Determine whether the two triangles are congruent. Justify your answer
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Geometry–Similarity, Right Triangles, and
Trigonometry GSRT4
Core Content
Cluster Title: Prove theorems involving similarity
Standard: G.SRT.4: Prove theorems about triangles (Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity)
Concepts and skills to master
Conceptual:
• Know a line parallel to one side of a triangle divides the other two proportionally and the
converse Procedural:
• Prove theorems about triangles Representational:
• Model the theorems about triangles Supports for Teachers
Critical Background Knowledge
Conceptual:
• Understand the AA similarity criterion
Procedural:
• Use the AA similarity criterion
Representational:
• Model the AA similarity criterion
Academic Vocabulary and Notation
Parallel lines, Pythagorean Theorem, similarity, similar triangles, similar symbol
Suggested Instructional Strategies
• explore and recreate a variety of historical proofs about triangles
Sample Formative Assessment Tasks
Skill-based Task:
A triangle is intersected by a segment. Prove that the result is a proportional division of the sides
Resources
Geogebra
http://www.geogebra.org/cms/
Problem Task:
Prove the Pythagorean Theorem using similarity
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Geometry–Similarity, Right Triangles, and
Trigonometry GSRT5
Core Content
Cluster Title: Prove theorems involving similarity
Standard: G.SRT.5: Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures
Concepts and skills to master
Conceptual:
• Know congruence and similarity criteria for triangles Procedural:
• Find lengths of measures of sides and angles of congruent and similar triangles
• Solve problems in context involving sides or angles of congruent or similar triangles
• Prove conjectures about congruence or similarity in geometric figures using congruence
and similarity criteria Representational:
• Model congruence or similarity in geometric figures using congruence and similarity criteria
Supports for Teachers
Critical Background Knowledge
Conceptual:
• Know corresponding parts of triangles
Procedural:
• Use corresponding parts of triangles
Representational:
• Model corresponding parts of triangles
Academic Vocabulary and Notation
Congruence, similarity, congruent triangles, similar triangles, corresponding angles, corresponding sides, similarity symbol, congruence symbol
Suggested Instructional Strategies
Resources
Geogebra
http://www.geogebra.org/cms/
• •
measure object indirectly by using shadows formed on level ground solve puzzles involving finding the measures of missing sides or angles in artistic drawings using similar or congruent triangles
67


Sample Formative Assessment Tasks
Skill-based Task:
Prove that the base angles of an isosceles triangle are congruent
Problem Task:
The length of George Washington’s face at Mt Rushmore is 60 feet. Describe a method for determining the length of his nose using similar triangles. Justify your reasoning
68


Geometry–Similarity, Right Triangles, and
Trigonometry GSRT6
Core Content
Cluster Title: Define trigonometric ratios and solve problems involving right triangles
Standard: G.SRT.6: Understand that by similarity, side ratios of right triangles are properties of the angles in the triangle leading to definitions of trigonometric ratios for acute angles Concepts and skills to master
Conceptual:
• Understand that the ratio of two sides in one triangle is equal to the ratio of the corresponding two sides of all other similar triangles
Procedural:
• Define sine, cosine, and tangent as the ratio of sides in a right triangle
Representational:
• Model right triangle trigonometric ratios
Supports for Teachers
Critical Background Knowledge
Conceptual:
• Understand that corresponding angles of similar triangles are congruent and ratios of
corresponding sides are equal Procedural:
• Use corresponding angles of similar triangles are congruent and ratios of corresponding sides are equal
Representational:
• Model corresponding angles of similar triangles are congruent and ratios of
corresponding sides are equal
Academic Vocabulary and Notation
similar triangles, ratio, right triangle, sin, cos, tan, side opposite, side adjacent, hypotenuse
Suggested Instructional Strategies
Resources
Geogebra
http://www.geogebra.org/cms/
• •
Use special triangles to develop the concept of trigonometric ratios
Use interactive geometry software to produce tables demonstrating the equivalence of ratios formed by the measures of corresponding sides of similar triangles
69


Sample Formative Assessment Tasks
Skill-based Task: Problem Task:
Find sin, cos and tan of x Explain why sin x is the same regardless of which triangle is used to find it in the figure below
70


Geometry–Similarity, Right Triangles, and
Trigonometry GSRT7
Core Content
Cluster Title: Define trigonometric ratios and solve problems involving right triangles
Standard: G.SRT.7: Explain and use the relationship between the sine and cosine of complementary angles
Concepts and skills to master
Conceptual:
• Understand the relationship between the sine and cosine of complementary angles Procedural:
• Explain relationship between the sine and cosine of complementary angles
• Use the relationship between the sine and cosine of complementary angles Representational:
• Model the relationship between the sine and cosine of complementary angles Supports for Teachers
Critical Background Knowledge
Conceptual:
• Understand that the acute angles of a right triangle are complementary
• Know the right triangle definitions of sine and cosine
Procedural:
• Use the right triangle definitions of sine and cosine
Representational:
• Model the right triangle definitions of sine and cosine
Academic Vocabulary and Notation
Complementary angles, sine, cosine, sin, cos, side opposite, side adjacent, hypotenuse
Suggested Instructional Strategies
• Use a properly labeled right triangle to demonstrate that acute angles are complementary and by definitions of their ratios, the
Sample Formative Assessment Tasks
Skill-based Task:
Find the second acute angle of a right triangle given thatthefirstacuteanglehasameasureof39degrees
Complete the following statement: If sin 30 degrees = 1⁄2 the the cos ________ = 1⁄2
Resources
Geogebra
http://www.geogebra.org/cms/
Problem Task:
Find: sinA,sinB,cosA,cosB
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Geometry–Similarity, Right Triangles, and
Trigonometry GSRT8
Core Content
Cluster Title: Define trigonometric ratios and solve problems involving right triangles
Standard: G.SRT.8: Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems
Concepts and skills to master
Conceptual:
• Know the trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems
Procedural:
• Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied
problems Representational:
• Model trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems
Supports for Teachers
Critical Background Knowledge
Conceptual:
• Understand the role of the Pythagorean Theorem in real-world and mathematical
situations Procedural:
• Apply the Pythagorean Theorem in real-world and mathematical situations
• Apply right triangle trigonometric ratios to solve right triangles Representational:
• Model right triangle trigonometric ratios to solve right triangles
• Model the Pythagorean Theorem in real-world and mathematical situations
Academic Vocabulary and Notation
Pythagorean Theorem, sine, cosine, tangent, sin, cos, tan, side opposite, side adjacent, hypotenuse, angle of elevation, angle of depression
Suggested Instructional Strategies
Resources
Geogebra
http://www.geogebra.org/cms/
• •
Demonstrate angles of elevation and depression with a laser pointer
Use a hypsometer to measure the height of a building or other tall object
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Sample Formative Assessment Tasks
Skill-based Task:
A teenager whose eyes are 5 feet above
ground level is looking into a mirror on the
ground and can see the top a building that is the peak is 3.5 degrees. After driving 14 miles
30 feet away from the teenager. The angle ofcloser to the mountain, the angle of elevation is 9 elevation from the center of the mirror to the degrees 24’ 36”. Explain how you would set up the top of the building is 75 degrees. How tall is problem, and find the approximate height of the the building? How far away from the mountain
teenager’s feet is the mirror?
Problem Task:
While traveling across flat land, you see a mountain directly in front of you. The angle of elevation to
73


Geometry–Circles GC1
Core Content
Cluster Title: Understand and apply theorems about circles
Standard: G.C.1: Prove that all circles are similar
Concepts and skills to master
Conceptual:
• Know the definition of a circle
Procedural:
• Define a circle as the set of points equidistant to a given center point
• Prove all circles are similar
Representational:
Model the proof that all circles are similar Supports for Teachers
Critical Background Knowledge
Conceptual:
• Understand dilations and scale factors
Procedural:
• Apply dilations and scale factors
Representational:
• Model dilations and scale factors
Academic Vocabulary and Notation
Circle, radius, diameter, dilation, center, on the circle, circumference
Suggested Instructional Strategies Resources
• Employ multiple strategies to prove Geogebra
all circles are similar, dilating the http://www.geogebra.org/cms/ radius, construct similar triangles in
two different circles in which one
vertex is at the center of the circle
and the other two vertices are on
the circle
Sample Formative Assessment Tasks
Skill-based Task: Problem Task:
Given a circle of a radius of 3 and another Prove the two circles are similar circle with a radius of 5, compare the ratios
of the two radii, the two diameters and the
two circumferences
74


Geometry–Circles GC2
Core Content
Cluster Title: Understand and apply theorems about circles
Standard: G.C.2: Identify and describe relationships among inscribed angles, radii, and chords. (Including 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)
Concepts and skills to master
Conceptual:
• Know relationships for central, inscribed, and circumscribed angles of a circle
Procedural:
• • •
Use circle relationships to find measures of central, inscribed, and circumscribed angles of a circle
Use circle relationships to show that the measure of the inscribed angle on a diameter is a right angle
Use circle relationships to show that the radius of a circle is perpendicular to a line tangent to the endpoint of the radius
Representational:
• Model circle relationships to find measures of central, inscribed, and circumscribed angles
of a circle Supports for Teachers
Critical Background Knowledge
Conceptual:
• Understand all circles are similar
Procedural:
• Define circle, radius, diameter, and circumference
Representational:
• Model circles and their significant parts
Academic Vocabulary and Notation
Circle, radius, diameter, dilation, center, on the circle, circumference, inscribed angle, circumscribed angle, perpendicular, tangent line, central angle,
Suggested Instructional Strategies
• Use geometry software to explore relationships
Sample Formative Assessment Tasks
Skill-based Task:
Given the measures of a central angle of a
circle is 100 degrees, find the measure of anpoints equal regardless of where the vertex is on the inscribed angle that intersects the circle at circle?
the same points as the central angle
Resources
Geogebra
http://www.geogebra.org/cms/
Problem Task:
Why are all inscribed angles that intersect the same
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Geometry–Circles GC3
Core Content
Cluster Title: Understand and apply theorems about circles
Standard: G.C.3: Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle
Concepts and skills to master
Conceptual:
• Understand that opposite angles in a quadrilateral inscribed in a circle are supplementary
• Understand the meaning of inscribed and circumscribed Procedural:
Inscribe a circle in a triangle Circumscribe a circle about a triangle
• • •
• Model opposite angles in a quadrilateral inscribed in a circle are supplementary Supports for Teachers
Critical Background Knowledge
Conceptual:
• Know the relationship between an inscribed angle and its intercepted arc
Procedural:
• Use a variety of construction methods
Representational:
• Construct various models to show the relationship between an inscribed angle and its
intercepted arc
Academic Vocabulary and Notation
inscribed angle, circumscribed angle, quadrilateral
Suggested Instructional Strategies Resources
• Use geometry software to measure the Geogebra
Prove that opposite angles in a quadrilateral inscribed in a circle are supplementary Representational:
angles of several quadrilaterals inscribed in circles in order to find any relationships between the angles
Sample Formative Assessment Tasks
Skill-based Task:
Find the other two angle measures
http://www.geogebra.org/cms/
Problem Task:
Find the unique relationships between the angles of a quadrilateral inscribed within a circle if the quadrilateral is:
• A square
• A rectangle
• An isosceles trapezoid
76


Geometry–Expressing Geometric Properties with Equations GGPE1
Core Content
Cluster Title: Translate between geometric description and the equation for a conic section
Standard: 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
Concepts and skills to master
Conceptual:
• Know the Pythagorean Theorem and its relation to the equation of a circle Procedural:
• Find the center of a circle given its equation
• Use the Pythagorean Theorem to derive the equation of a circle Representational:
• Model the relationship between the Pythagorean Theorem and the equation of a circle Supports for Teachers
Critical Background Knowledge
Conceptual:
• Understand completing the square
• Understand the Pythagorean Theorem
Procedural:
• Use the Pythagorean Theorem to find the distance between two points
• Use the method of completing the square to transform equations into desired forms
Representational:
• Model the Pythagorean Theorem
Academic Vocabulary and Notation
Circle, center of circle, radius, completing the square
77


Suggested Instructional Strategies
Resources
Geogebra
http://www.geogebra.org/cms/
http://www.learnzillion.com/lessons/282-change- equations-into-the-standard-form-of-a-circle
http://www.learnzillion.com/lessons/283-prove-the- pythagorean-theorem-using-similar-triangles
http://www.learnzillion.com/lessons/284-prove-the- pythagorean-theorem-using-the-areas-of-squares-and- triangles
http://www.learnzillion.com/lessons/280-derive-the- equation-of-a-circle-using-the-pythagorean-theorem
http://www.learnzillion.com/lessons/281-find-the- center-and-radius-of-a-circle
Problem Task:
A circle is inscribed in an equilateral triangle. The equilateral triangle lies in the first quadrant with one vertex at the origin and a second vertex at (4√3, 0) Find the equation of the circle.
•
•
Import images of circle crop fields from Google Earth into a coordinate grid system and find their equations
Use geometry software
Sample Formative Assessment Tasks
Skill-based Task:
A circle is tangent to the x-axis and y-axis in the first quadrant. A point of tangency has coordinates (4, 0). Find the equation of the circle
78


Geometry–Expressing Geometric Properties with Equations GGPE2
Core Content
Cluster Title: Translate between geometric description and the equation for a conic section
Standard: G.GPE.2: Derive the equation of parabola given a focus and directrix
Concepts and skills to master
Conceptual:
• Know the distance formula and its relation to the equation of a parabola
• Know the geometric definition of a parabola, including a focus and directrix
Procedural:
• Develop the geometric definition of a parabola, including a focus and directrix
• Use the distance formula and its relation to the equation of a parabola
Representational:
• Model the relationship between the distance formula and the equation of a parabola
Supports for Teachers
Critical Background Knowledge
Conceptual:
• Know the distance formula
• Understand the definition of a midpoint
Procedural:
• Find the distance between two points
• Find the midpoint of a segment
Representational:
• Use graphical representations for finding the distance between two points and a
midpoint of a segment
Academic Vocabulary and Notation
Focus, directrix, midpoint, focal point, vertex
Suggested Instructional Strategies
Resources
Geogebra
http://www.geogebra.org/cms/
Problem Task:
A parabola has focus (-2, 1) and directrix y = -
3. Determine whether or not the point (2, 1) is part of the parabola. Justify your response
•
•
•
Begin with a parabola with vertex (0, 0)
Define the focus and directrix in terms of distance from the vertex Use geometry software
Sample Formative Assessment Tasks
Skill-based Task:
Write the equation of a parabola with focus (3, 5) and directrix x = - 1
79


80


Geometry–Expressing Geometric Properties with Equations GGPE4
Core Content
Cluster Title: Use coordinates to prove simple geometric theorems algebraically
Standard: G.GPE.4: Use coordinates to prove simple theorems algebraically (for example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove
or disprove that the point lies on the circle centered at the origin and containing the point (0,2))
Concepts and skills to master
Conceptual:
• Understand geometric theorems algebraically Procedural:
• Use coordinates to prove simple geometric theorems algebraically Representational:
• Model simple geometric theorems Supports for Teachers
Critical Background Knowledge
Conceptual:
Understand slope of a line
Understand the relationship between the slopes of parallel lines and perpendicular lines
• • •
• Calculate distances using the distance formula
• Calculate slope of lines Representational:
• Model distances using the distance formula
Academic Vocabulary and Notation
Radius, center, diameter, inscribed, altitude, diagonal, perpendicular, bisector, median, parallel, midpoint, Pythagorean Theorem, coordinates, slope
Understand the basic properties of polygons and circles Procedural:
Suggested Instructional Strategies
Resources
Geogebra
http://www.geogebra.org/cms/
http://www.learnzillion.com/lessons/285-prove- whether-a-figure-is-a-rectangle-in-the-coordinate- plane
http://www.learnzillion.com/lessons/286-prove- whether-a-point-is-on-a-circle
• •
Generalize coordinates of geometric figures using variables for one or more of the vertices
Use geometry software
81


Sample Formative Assessment Tasks
Skill-based Task:
Given a circle with center (-2, 3) determine whether or not the points (-4, -1) and (3, 5) are on the same circle. Justify your response
Problem Task:
Prove that a triangle with vertices at (4, 3) , (8, 6) and (8, 3) is a right triangle
82


Geometry–Expressing Geometric Properties with Equations GGPE6
Core Content
Cluster Title: Use coordinates to prove simple geometric theorems algebraically
Standard: G.GPE.6: Find the point on a directed line segment between two given points that partitions the segment in a given ratio
Concepts and skills to master
Conceptual:
• Know the properties of line segments on a coordinate system Procedural:
• Use coordinate geometry to divide a segment into a given ratio Representational:
• Model partitioning of a segment on a coordinate system
Supports for Teachers
Critical Background Knowledge
Conceptual:
• Understand number lines, both vertical and horizontal
Procedural:
• Find distances on number lines
Representational:
• Represent distances on numberlines
Academic Vocabulary and Notation
Coordinates, a:b, a/b, ratio, directed line segment
Suggested Instructional Strategies Resources
• Usegeometrysoftwaretodividea Geogebra
segment into a given ratio in a coordinate system
Sample Formative Assessment Tasks
Skill-based Task:
A segment with endpoints A(3, 2) and B(6, 11) is partitioned by a point C such that AC andCBforma2:1ratio. FindC
http://www.geogebra.org/cms/
Problem Task:
A point B(4, 2)on a segment with endpoints A(2, -1) and C(x, y) partitions the segment in a 3:1 ratio. Find xandy
83


Geometry–Geometric Measurement and Dimension GGMD1
Core Content
Cluster Title: Understand and apply theorems about circles
Standard: G.CMD.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)
Concepts and skills to master
Conceptual:
• Know the formulas for the circumference of a circle, area of a circle, volume of a cylinder,
pyramid, and cone Procedural:
• Use dissection arguments, Cavalieri’s principle, and informal limit arguments to explain the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone
Representational:
• Model the arguments explaining the formulas for the circumference of a circle, area of a
circle, volume of a cylinder, pyramid, and cone Supports for Teachers
Critical Background Knowledge
Conceptual:
• Know mathematical language needed to express ideas of mathematical arguments • Understand the logical progression of ideas to present an argument
Procedural:
• Use mathematical language and a logical progression of ideas to present an argument
Representational:
• Model the logical progression of ideas to present an argument
Academic Vocabulary and Notation
Cylinder, right prism, pyramid, cone, dissection argument, Cavalieri’s principle, limit argument
Suggested Instructional Strategies
•
•
•
Geogebra
Resources
Use Archimedes’ argument of
fitting n-gons of increasing
number of sides into a circle to
approximatethecircumference Estimatingcircumferenceofacircle
or area of a circle
Dissect a circle into pizza slices, then rearrange the slices into a shape that approximates a parallelogram; find the area Use geometry software
http://www.cut-the- knot.org/curriculum/calculus/goodlimit
pi filling, Archimedes style:
http://illuminations.nctm.org
Circle: wolfgram math world:
http://mathworld.wolfgram.com/circle.html
Problem Task:
Find the volume of the Great Pyramid of Giza
http://www.geogebra.org/cms/
Sample Formative Assessment Tasks
Skill-based Task:
Explain why the volume of a cylinder is
84


Geometry–Geometric Measurement and Dimension GGMD3
Core Content
Cluster Title: Understand and apply theorems about circles
Standard: G.CMD.3: Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems
Concepts and skills to master
Conceptual:
• Know the formulas for the volume of a cylinder, pyramid, cone and sphere Procedural:
• Find the volume of a cylinder, pyramid, cone and sphere Representational:
• Model the volume of a cylinder, pyramid, cone and sphere Supports for Teachers
Critical Background Knowledge
Conceptual:
• Understand volume
Procedural:
• Calculate values with formulas
Representational:
• Model multiple area models
Academic Vocabulary and Notation
Cylinder, pyramid, cone, sphere, volume, width, length, height, base, radius, pi
Suggested Instructional Strategies Resources
•
•
Have students bring household objects Geogebra
with the given characteristics. Provide http://www.geogebra.org/cms/ opportunities for students to measure
with rulers or tape measures to gather
needed information. Compute the
volume of the objects
Make connections between metric measurements (for example, using rice to fill a cylinder, compare liquid volume in liters and the geometric volume in cubic meters
Sample Formative Assessment Tasks
Skill-based Task:
Find the volume of a cylindrical oatmeal container
Problem Task:
Given a three-dimensional object, compute the effect on volume of doubling or tripling one or more dimensions (for example, how is the volume of a cone affected when the height is doubled?)
85


Statistics–Conditional Probability and the Rules
of Probability SCP1
Core Content
Standard: 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”)
Concepts and skills to master
Conceptual:
• Identify an event as a subset of a set of outcomes (sample space)
Procedural:
• Use correct set notation, with appropriate symbols and words, to identify sets and
subsets within a sample space Representational:
• Represent relationships (unions, intersections, or complements) between sets within a sample space with Venn diagrams
Supports for Teachers
Critical Background Knowledge
Conceptual:
• Understand sample spaces
Procedural:
• Collect and organize data
Representational:
• Represent sample spaces
Academic Vocabulary and Notation
Sample space, subset, outcome, union, intersection, complement, (Various notations for
complement)
Suggested Instructional Strategies
Cluster Title: Understand independence and conditional probability and use them to interpret data
• •
Create and use Venn diagrams to illustrate relationships between sample spaces and events
Perform chance experiments, such as rolling dice or tossing coins, to generate sample spaces and identify events within the sample spaces
Resources
http://www.shodor.org
interactive Venn Diagram Shape Sorter
86


Sample Formative Assessment Tasks
Skill-based Task:
Describe the event that the summing of two rolled dice is larger than 7 and even, and contrast it with the event that the sum is larger than 7 or even
Problem Task:
Create a Venn diagram to display the information in the table. Describe the set of students who have a curfew but don’t do chores as a subset of the group
Chores: Yes13
Chores: No 12
Total 25 8
Curfew: Yes
Curfew: No
Total
5 18
3
15
87


Statistics–Conditional Probability and the Rules
of Probability SCP2
Core Content
Standard: 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
Concepts and skills to master
Conceptual:
• Understand that independent events satisfy the relationship P(A) x P(B) with their joint
probability (P(AB)) Procedural:
• Use appropriate probability notation for individual events as well as their intersection (joint probability)
• Calculate probabilities for events, including joint probabilities, using various methods (e.g., Venn diagram, frequency table)
• Compare the product of probabilities for individual events P(A) x P(B) = (P(AB)) Representational:
• Represent relationships (unions, intersections, or complements) between sets within a sample space with Venn diagrams
Supports for Teachers
Critical Background Knowledge
Conceptual:
• Understand basic properties of probability
• Know that (P(AB)) is the equivalent of the probability of A and event B occurring together
Procedural:
• Approximate probabilities of chance events through experiment
Representational:
• Use Venn diagrams and two-way frequency tables to represent relationships
Academic Vocabulary and Notation
Joint probability, intersection, event, independent events, P(A), P(AB), P(A and B)
Suggested Instructional Strategies Resources
Cluster Title: Understand independence and conditional probability and use them to interpret data
• •
•
Convert frequencies from a Venn diagram Scheaffer, Richard, “Streaky Behavior” in
or a two-way frequency table into probabilities with correct notation Generate a two-way frequency table to describe characteristics of your class (e.g., gender and eye color) and use the table to determine if eye color and gender are independent
Compare experimental results to theoretical (long run) probabilities
Activity-based statistics, student guide, 2nd edition
88


Sample Formative Assessment Tasks
Skill-based Task: Problem Task:
When rolling two six-sided dice: Roll a pair of six-sided dice 100 times and keep 1. what is the probability of rolling a sum thattrack of the outcomes. Find pairs of events
is greater than7?
2. What is the probability of rolling a sum
that is odd?
3. What is the probability of rolling a sum
that are independent and pairs that are
not. Justify your conclusions. (For example, the probability of rolling doubles and the probability of rolling 7 vs the probability of rolling doubles and the probability of rolling a
that is greater than 7 and is odd?
4. Are the events rolling a sum greater than 7sum that is less than 4)
and rolling a sum that is odd independent? Justify your answer.
89


Statistics–Conditional Probability and the Rules
of Probability SCP3
Core Content
Standard: 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
Concepts and skills to master
Conceptual:
• Understand conditional probability and how it applies to real-life events
• Understand that events A and B are independent if and only if they satisfy P(A|B) = P(A)
or satisfy P(B|A) = P(B) Procedural:
• Use to calculate conditional probabilities
• Apply the definition of independence to a variety of chance events Representational:
• Model P(A|B) Supports for Teachers
Critical Background Knowledge
Conceptual:
• Understand independent events
Procedural:
• Use basic probability notation, particularly
Representational:
• Model probability of independent events
Academic Vocabulary and Notation
Conditional, independence, conditional probability, P(A|B)
Suggested Instructional Strategies Resources
• Use Venn diagrams to exploreCut the knot – conditional probability and independent
Cluster Title: Understand independence and conditional probability and use them to interpret data
and compute conditional probabilities
events:
http://www.cut-the- knot.org/curriculumprobability/conditionalProbability.shtml Texas A&M – conditional probaibility applet: http://www.stat.tamu.edu/~west/applets/venn1.html
90


Sample Formative Assessment Tasks
Skill-based Task:
Given the following Venn diagram, determine whether events A and B are independent
Problem Task:
Roll a pair of six-sided dice 100 times and keep track of the
outcomes. Find pairs of events that are independent and pairs that are not. Justify your conclusions. (For example, the probability of rolling doubles and the probability of rolling 7 vs the probability of rolling doubles and the probability of rolling a sum that is less than 4)
91


Statistics–Conditional Probability and the Rules
of Probability SCP4
Core Content
Standard: 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. (For example, collect data from a random sample of students in your school on their favorite subject among math, science, and English. Estimate the probability that a randomly selected student from your school will favor science given that the student is in the tenth grade. Do the same for other subjects and compare the results)
Concepts and skills to master
Conceptual:
• Understand how to read two-way frequency tables
Procedural:
• Use to calculate conditional probabilities from a two-way frequency table
• Apply the definition of independence to a variety of chance events as represented by a
two-way frequency table Representational:
• Model real-life data using two-way frequency tables Supports for Teachers
Critical Background Knowledge
Conceptual:
• Understand what it means for two events to be independent
Procedural:
• Summarize categorical data in a variety of ways
Representational:
• Represent categorical data in a variety of ways
Academic Vocabulary and Notation
Conditional, independence, joint probability P(AB), Conditional probability P(A|B), marginal probability
Suggested Instructional Strategies Resources
Cluster Title: Understand independence and conditional probability and use them to interpret data
•
•
Constructtwo-wayfrequencytableson Dataandstorylibrary(DASL):
data from news media and investigate http://lib.stat.cmu.edu/cgi-
independence by computing conditional bin/dasl.cgi?query=contingency+table&submit= probabilities search!&metaname=method&sort=swishrank
Analyze two-way tables to determine independence and conditional probability
92


Sample Formative Assessment Tasks
Skill-based Task:
Problem Task:
1. 2.
3. 4.
Find the probability that a randomly selected student attends summer school Find the probability that a student is a boy given that they attend summer school
Find the probability that a randomly selected student is a boy who attends summer school
Are the events “attending summer school” and “boy” independent? Justify your answer
Select two categorical variables and conduct research to answer various probability questions and determine independence. Write a “newsworthy” article for the school newspaper that interprets the interesting relationships between events
Gender
Summer school
Summer job
total
Girls 25 20 Boys 35 20 total
93


Statistics–Conditional Probability and the Rules
of Probability SCP5
Core Content
Standard: S.CP.5: Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations. (For example, compare the chance of having lung cancer if you are a smoker with the chance of being a smoker if you have lung cancer)
Concepts and skills to master
Conceptual:
• Recognize the concepts of conditional probability and independence in every day language and everyday situations
Procedural:
• Explain the concepts of conditional probability and independence in every day language
and everyday situations
• Interpret conditional probabilities and independence in context
Representational:
• Model real-life conditional probabilities
Supports for Teachers
Critical Background Knowledge
Conceptual:
• Understand independence in probability
• Understand conditional probabilities
Procedural:
• Calculate conditional probabilities
• Find probabilities of events using tree diagrams
• Summarize categorical data in a variety of ways
Representational:
• Represent categorical data in a variety of ways
Cluster Title: Understand independence and conditional probability and use them to interpret data
Academic Vocabulary and Notation
Conditional probability P(A|B), independence
Suggested Instructional Strategies
Resources
Stat trek:
http://stattrek.com/ap-statistics- 1/association.aspx
• • •
Practice representing conditional probabilities using tree diagrams
Find the probability that a randomly selected athlete is an honors student Have students generate questions similar to the example in the standard and pursue the answers
94


Sample Formative Assessment Tasks
Skill-based Task:
Is owning a smart phone independent from grade level?
10th grade 204 170 11th grade 192 160 12th grade 198 165 total
Problem Task:
Have students find and interpret probability statements in media
Own smart phone
Do not own smart phone
total
95


Statistics–Conditional Probability and the Rules
of Probability SCP6
Core Content
Standard: 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
Concepts and skills to master
Conceptual:
• Understand the difference between compound and conditional probabilities
• Interpret conditional probabilities using a two-way frequency table, Venn diagram, or tree
diagram Procedural:
• Find conditional probabilities using a two-way frequency table, Venn diagram, or tree diagram
Representational:
• Model real-life conditional probabilities
Supports for Teachers
Critical Background Knowledge
Conceptual:
• Understand compound probabilities
• Understand conditional probabilities
Procedural:
• Find probabilities of compound events
• Find conditional probabilities
• Summarize categorical data in two-way frequency tables
Representational:
• Represent categorical data with two-way frequency tables
Academic Vocabulary and Notation
Random variable, probability model
Cluster Title: Use the rules of probability to compute probabilities of compound events in a uniform probability model
96


Suggested Instructional Strategies
Resources
Make a “human Venn diagram” where the sample space is all the students in the
class. Use lengths of rope to create three overlapping circles. Assign an event to each of the three circles such as: ate breakfast, brought a cell phone to school, and got at least 7 hours of sleep. Have students place themselves in the appropriate locations. Using correct probability notation, identify each of the spaces in the Venn diagram (don’t forget to include the space outside the circles). Analyze, explore and record the results in terms of conditional probabilities
Connect probability models from other
standards
Sample Formative Assessment Tasks
Skill-based Task:
From the table, determine the probability of getting theflu,andcomparethattotheprobabilityofgetting the flue given that the individual takes high doses of vitamin C
•
•
Cold
No cold
total
placebo Vitamin C total
31 109 17 122
Problem Task:
Life is like a box of chocolates. Suppose yourboxof36chocolateshavesomedark and some milk chocolate, divided into cream or nutty centers. Out of the dark chocolate, 8 have nutty centers. Out of the milk chocolates, 6 have nutty
centers. One-third of the chocolates are dark chocolate. What is the probability that you randomly select a chocolate with a nutty center? Given that it has a nutty center, what is the probability you chose a dark chocolate? Show how you determined your answers
97


Statistics–Conditional Probability and the Rules
of Probability SCP7
Core Content
Standard: 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
Concepts and skills to master
Conceptual:
• Understand the formula P(A or B) = P(a) + P(B) – P(A and B) Procedural:
• UsetheformulaP(AorB)=P(a)+P(B)–P(AandB)
• Define the probability of event (A or B) as the probability of their union Representational:
• Model the probability of event (A or B) as the probability of their union Supports for Teachers
Critical Background Knowledge
Conceptual:
• Understand probabilities of compound events
Procedural:
• Find probabilities of compound events
Representational:
• Model probabilities of compound events
Cluster Title: Use the rules of probability to compute probabilities of compound events in a uniform probability model
Academic Vocabulary and Notation
Or, and, P(A), ∪, ∩
Suggested Instructional Strategies
• · Make a connection between the formula for the addition rule and a probability model
Resources
98


Sample Formative Assessment Tasks
Skill-based Task:
Given the following table, which includes data regarding boating preferences of boys and girls, use the addition rule to fine P(L ∪ G)
Girls (G) 21 29 Boys(B) 32 18 total
Problem Task:
Sally shaded the following Venn diagram to illustrate the addition rule. What was wrong with her
reasoning? How could you represent the addition rule pictorially?
Lake (L)
River (R)
total
99


Course Pacing Guide
The order and timeline of the instruction of standards, objectives, indicators, and Essentials over the span of a course (semester or year).
Course
Core Standards
Text
Term 1
Term 2
Term 3
Term 4
100