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1 THE STATE EDUCATION DEPARTMENT / THE UNIVERSITY OF THE STATE OF NEW YORK / ALBANY, NY 12234 Curriculum, Instruction, and Instructional Technology Team - Room 320 EB www.emsc.nysed.gov/ciai email: emscnysmath@mail.nysed.gov Geometry Sample Tasks for Int..

     THE STATE EDUCATION DEPARTMENT / THE UNIVERSITY OF THE STATE OF NEW YORK / ALBANY, NY 12234 Curriculum, Instruction, and Instructional Technology Team - Room 320 EB

     www.emsc.nysed.gov/ciai email: emscnysmath@mail.nysed.gov

    Geometry

    Sample Tasks for Integrated Algebra, developed by New York State teachers, are clarifications, further explaining the language and intent of the associated Performance Indicators. These tasks are not test items, nor are they meant for students' use.

    Note: There are no Sample Tasks for the Number Sense and Operations, Measurement, and Statistics and Probability Strands. Although there are no Performance Indicators for these strands in this section of the core curriculum, these strands are still part of instruction within the other strands as an ongoing continuum and building process of mathematical knowledge for all students.

     Strands

    Process Content

     Problem Solving Number Sense and Operations

     Reasoning and Proof Algebra

     Communication Geometry

     Connections Measurement

     Representation Statistics and Probability

    Problem Solving Strand

Students willbuild new mathematical knowledge through problem solving.

    G.PS.1 Use a variety of problem solving strategies to understand new mathematical content

G.PS.1a

    Obtain several different size cylinders made of metal or cardboard. Using stiff paper, construct a cone with the same base and height as each cylinder. Fill the cone with rice, then pour the rice into the cylinder. Repeat until the cylinder is filled. Record your data.

    What is the relationship between the volume of the cylinder and the volume of the corresponding

    cone?

    Collect the class data for this experiment.

    Use the data to write a formula for the volume of a cone with radius r and height h.

G.PS.1b

    Use a compass or dynamic geometry software to construct a regular dodecagon (a regular12-sided polygon).

    What is the measure of each central angle in the regular dodecagon?

    Find the measure of each angle of the regular dodecagon.

    Extend one of the sides of the regular dodecagon.

    What is the measure of the exterior angle that is formed when one of the sides is extended? 1

    Students will solve problems that arise in mathematics and in other contexts.

    G.PS.2 Observe and explain patterns to formulate generalizations and conjectures

G.PS.2a

    Examine the diagram of a right triangular prism below.

     Describe how a plane and the prism could intersect so that the intersection is:

    a line parallel to one of the triangular bases

    a line perpendicular to the triangular bases

    a triangle

    a rectangle

    a trapezoid

G.PS.2b

    ABCUse a compass or computer software to draw a circle with center. Draw a chord .

    ABChoose and label four points on the circle and on the same side of chord.

    Draw and measure the four angles formed by the endpoints of the chord and each of the four points.

    What do you observe about the measures of these angles?

    ~ACBMeasure the central angle,. Is there any relationship between the measure of an inscribed angle formed using the endpoints of the chord and another point on the circle and the central angle formed using the endpoints of the chord?

    Suppose the four points chosen on the circle were on the other side of the chord.

    How are the inscribed angles formed using these points and the endpoints of the chord related to the

    inscribed angles formed in the first question?

G.PS.2c

    Consider the following conjecture: The intersection of two distinct planes can be a point. Find a “real world” example that supports the conjecture or provides a counterexample to the conjecture. Share your example with a partner and use your knowledge of geometry in three dimensional space to justify it.

G.PS.2d

    Using dynamic geometry software, locate the circumcenter, incenter, orthocenter, and centroid of a given triangle. Use your sketch to answer the following questions:

    Do any of the four centers always remain inside the circle?

    If a center is moved outside of the triangle, under what circumstances will it happen?

    Are the four centers ever collinear? If so, under what circumstances?

    Describe what happens to the centers if the triangle is a right triangle.

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G.PS.2e

    RR(,)(,)xyxy:?The equation for a reflection over the y-axis, , is . x0x0

    Find a pattern for reflecting a point over another vertical line such as x = 4.

    Write an equation for reflecting a point over any vertical line y = k

G.PS.2f

    RR(,)(,)xyxy:?The equation for a reflection over the x-axis, , is . y0y0

    Find a pattern for reflecting a point over another horizontal line such as y = 3.

    Write an equation for reflecting a point over any horizontal line y = h

    G.PS.3 Use multiple representations to represent and explain problem situations (e.g., spatial,

    geometric, verbal, numeric, algebraic, and graphical representations)

G.PS.3a

    Consider the following conjecture: The intersection of two distinct planes can be a point. Find a “real

    world” example that supports the conjecture or provides a counterexample to the conjecture. Share your

    example with a partner and use your knowledge of geometry in three dimensional space to justify it.

G.PS.3b

    Draw a line on a piece of cardboard. Use additional pieces of cardboard to construct two planes that are perpendicular to the line that you drew. Make a conjecture regarding those two planes and share your example with a partner and use your knowledge of geometry in three dimensional space to justify your conjecture.

G.PS.3c

    Determine the point(s) in the plane that are equidistant from the points A(2,6), B(4,4), and C(8,6).

G.PS.3d

    BSIn figure 1 a circle is drawn that passes through the point (-1,0). is perpendicular to the y-axis at B the

    SCpoint where the circle crosses the y-axis. is perpendicular to the x-axis at the point where C crosses

    the x-axis. As point S is dragged, the coordinates of point S are collected and stored in L1 and L2 as shown in figure 2. A scatter plot of the data is shown in figure 3 with figure 4 showing the window settings for the graph. Finally a power regression is performed on this data with the resulting function displayed in figure 5 with its equation given in figure 6.

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In groups of three or four discuss the results that you see in this activity. Answer the following questions in

    your group:

    Is the function reasonable for this data?

    Did you recognize a pattern in the lists of data?

    DCSCExplain why and are related.

    What is the significance of A being located at the point (-1,0)?

    State the theorem that you have studied that justifies these results.

    Students will apply and adapt a variety of appropriate strategies to solve problems.

    G.PS.4 Construct various types of reasoning, arguments, justifications and methods of proof

    for problems

G.PS.4a

    Consider a cylinder, a cone, and a sphere that have the same radius and the same height.

    Sketch and label each figure.

    What is the relationship between the volume of the cylinder and the volume of the cone?

    What is the relationship between the volume of the cone and the volume of the sphere?

    What is the relationship between the volume of the cylinder and the volume of the sphere?

    Use the formulas for the volume of a cylinder, a cone, and a sphere to justify mathematically that

    the relationships in the previous parts are correct.

G.PS.4b

    ~ABCUse a straightedge to draw an angle and label it . Then construct the bisector of ~ABC by

    following the procedure outlined below:

    BABCStep 1: With the compass point at B, draw an arc that intersects and . Label the

    intersection points D and E respectively.

    Step 2: With the compass point at D and then at E, draw two arcs with the same radius that

    intersect in the interior of ~ABC. Label the intersection point F.

    Step 3: Draw ray BF.

    Write a proof that ray BF bisects ~ABC.

G.PS.4c

    ABUse a straightedge to draw a segment and label it . Then construct the perpendicular bisector of

    ABsegment by following the procedure outlined below:

    Step 1: With the compass point at A, draw a large arc with a radius greater than ?AB but less than

    ABthe length of AB so that the arc intersects .

    Step 2: With the compass point at B, draw a large arc with the same radius as in step 1 so that the

    ABABarc intersects the arc drawn in step 1 twice, once above and once below . Label the

    intersections of the two arcs C and D.

    CDStep 3: Draw segment .

    ABCDWrite a proof that segment is the perpendicular bisector of segment .

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G.PS.4d

    Prove: The bisector of the vertex angle of an isosceles triangle is perpendicular to the base.

    G.PS.5 Choose an effective approach to solve a problem from a variety of strategies (numeric,

    graphic, algebraic)

G.PS.5a

    Students in one mathematics class noticed that a local movie theater sold popcorn in different shapes of containers, for different prices. They wondered which of the choices was the best buy. Analyze the three popcorn containers below. Which is the best buy? Explain.

G.PS.5b

    10Find the number of sides of a regular n-gon that has an exterior angle whose measure is

G.PS.5c

    The equations of two lines are 2x + 5y = 3 and 5x = 2y 7. Determine whether these lines are parallel,

    perpendicular, or neither and explain how you determined your answer.

G.PS.5d

    (2)180nJeanette invented the rule to find the measure of A of one angle in a regular n-gon. Do An

    you think that Jeannette’s rule is correct? Justify your reasoning. Use the rule to predict the measure of

    one angle of a regular 20-gon. As the number of sides of a regular polygon increases, how does the measure of one of its angles change? When will the measure of each angle of a regular polygon be a whole number?

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    G.PS.6 Use a variety of strategies to extend solution methods to other problems

G.PS.6a

    10Find the number of sides of a regular n-gon that has an exterior angle whose measure is

G.PS.6b

    (2)180nJeanette invented the rule to find the measure of A of one angle in a An

    regular n-gon. Do you think that Jeannette’s rule is correct? Justify your reasoning.

    Use the rule to predict the measure of one angle of a regular 20-gon. As the number of sides of a regular polygon increases, how does the measure of one of its angles change? When will the measure of each angle of a regular polygon be a whole number?

    G.PS.7 Work in collaboration with others to propose, critique, evaluate, and value alternative

    approaches to problem solving

G.PS.7a

    As a group, examine the four figures below:

    A plane that intersects a three dimensional figure such that one half is the reflected image of the other half is called a symmetry plane. Each figure has new many symmetry planes?

    Describe the location of all the symmetry planes for each figure within your group. Record your

    answers.

G.PS.7b

    Consider the following conjecture: The intersection of two distinct planes can be a point. Find a “real world” example that supports the conjecture or provides a counterexample to the conjecture. Share your

    example with a partner and use your knowledge of geometry in three dimensional space to justify it.

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G.PS.7c

    A symmetry plane is a plane that intersects a three-dimensional figure so that one half is the reflected image of the other half. The figure below shows a right hexagonal prism and one of its symmetry planes.

     Discuss the following questions:

    ADHow is the segment related to the symmetry plane?

    Describe any other segments connecting points on the prism that have the same relationship as

    ADsegment to the symmetry plane.

    BFHow is segment related to the symmetry plane?

    Describe any other segments connecting points on the prism that have the same relationship as

    BFsegment to the symmetry plane.

    ADBFHow are segments and related?

G.PS.7d

    ~ABCWithin your group use a straightedge to draw an angle and label it . Then construct the bisector of

    ~ABC by following the procedure outlined below:

    BABCStep 1: With the compass point at B, draw an arc that intersects and . Label the

    intersection points D and E respectively.

    Step 2: With the compass point at D and then at E, draw two arcs with the same radius that

    intersect in the interior of ~ABC. Label the intersection point F.

    BFStep 3: Draw ray .

    As a group write a proof that ray BF bisects ~ABC.

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Students will monitor and reflect on the process of mathematical problem solving.

    G.PS.8 Determine information required to solve a problem, choose methods for obtaining the

    information, and define parameters for acceptable solutions

    G.PS.8a

    The Great Pyramid of Giza is a right pyramid with a square base. The measurements of the Great Pyramid include a base b equal to approximately 230 meters and a slant height s equal to approximately 464 meters.

    Calculate the current height of the Great Pyramid to the nearest meter.

    Calculate the area of the base of the Great Pyramid.

    Calculate the volume of the Great Pyramid.

G.PS.8b

    A swimming pool in the shape of a rectangular prism has dimensions 26 feet long, 16 feet wide, and 6 feet deep.

    How much water is needed to fill the pool to 6 inches from the top?

    How many gallons of paint are needed to paint the inside of the pool if one gallon of paint covers

    approximately 60 square feet?

    How much material is needed to make a pool cover that extends 1.5 feet beyond the pool on all

    sides?

    How many 6 inch square tiles are needed to tile the top of the inside faces of the pool?

G.PS.8c

    Students in one mathematics class noticed that a local movie theater sold popcorn in different shapes of containers, for different prices. They wondered which of the choices was the best buy. Analyze the three popcorn containers below. Which is the best buy? Explain.

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     G.PS.9 Interpret solutions within the given constraints of a problem

G.PS.9a

    A manufacturing company is charged with designing a can that is to be constructed in the shape of a right circular cylinder. The only requirements are that the can must be airtight, hold at least 23 cubic inches and should require as little material as possible to construct. Each of the following cans was submitted for consideration by the engineering department.

    Which can would you choose to produce?

    Justify your choice.

    Proposal #1

Proposal #2

Proposal #3

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G.PS.9b

    A swimming pool in the shape of a rectangular prism has dimensions 26 feet long, 16 feet wide, and 6 feet deep.

    How much water is needed to fill the pool to 6 inches from the top?

    How many gallons of paint are needed to paint the inside of the pool if one gallon of paint covers

    approximately 60 square feet?

    How much material is needed to make a pool cover that extends 1.5 feet beyond the pool on all

    sides?

    How many 6 inch square tiles are needed to tile the top of the inside faces of the pool?

G.PS.9c

    Use the information given in the diagram to determine the measure of ~ACB.

    B

    4x+3

    222x+3x-2+1x

    DAC

    G.PS.10 Evaluate the relative efficiency of different representations and solution methods of a

    problem

G.PS.10a

    The equations of two lines are 2x + 5y = 3 and 5x = 2y 7. Determine whether these lines are parallel,

    perpendicular, or neither and explain how you determined your answer.

    Compare your answer with others. As a class discuss the relative efficiency of the different

    representations and solution methods.

G.PS.10b

    Consider the following theorem: The diagonals of a parallelogram bisect each other. Write three separate proofs for the theorem, one using synthetic techniques, one using analytical techniques, and one using transformational techniques. Discuss with the class the relative strengths and weakness of each of the different approaches.

     Back to top

    Reasoning and Proof

    Students will recognize reasoning and proof as fundamental aspects of mathematics.

    G.RP.1 Recognize that mathematical ideas can be supported by a variety of strategies

G.RP.1a

    Investigate the two drawings using dynamic geometry software. Write as many conjectures as you can for each drawing.

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