MathGeomSample.htm

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

Number Sense and Operations

Algebra

Geometry

Measurement

Statistics and Probability

Problem Solving Strand

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

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?

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

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

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

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

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

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

Measure 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?

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.

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

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

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

G.PS.3 Use multiple representations to represent and explain problem situations (e.g., spatial, geometric, verbal, numeric, algebraic, and graphical representations)

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.

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

In figure 1 a circle is drawn that passes through the point (-1,0).

is perpendicular to the y-axis at B the point 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.

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

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

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

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.

Use a straightedge to draw an angle and label it ÐABC . Then construct the bisector of ÐABC by following the procedure outlined below:

Step 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 ÐABC. Label the intersection point F.

Use a straightedge to draw a segment and label it

. Then construct the perpendicular bisector of segment

by following the procedure outlined below:

Step 1: With the compass point at A, draw a large arc with a radius greater than ½AB but less than the 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 arc intersects the arc drawn in step 1 twice, once above

and once below

. Label the intersections of the two arcs C and D.

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)

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.

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

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.

Jeanette invented the rule

to find the measure of A of one angle in a 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.6 Use a variety of strategies to extend solution methods to other problems

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

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

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.

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.

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

to the symmetry plane.

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

to the symmetry plane.

Within your group use a straightedge to draw an angle and label it ÐABC. Then construct the bisector of ÐABC by following the procedure outlined below:

Step 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 ÐABC. Label the intersection point F.

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

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.

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

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?

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.

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

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

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.

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.

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

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

G.RP.2 Recognize and verify, where appropriate, geometric relationships of perpendicularity, parallelism, congruence, and similarity, using algebraic strategies

A symmetry plane is a plane that intersects a three dimensional figure so that one half is the reflected image of the other half. On a copy of the figure sketch a symmetry plane.

On a copy of the figure sketch another symmetry plane. Then write a description that uses the word perpendicular.

What changes the volume of a cylinder more, doubling the diameter or doubling the height? Provide evidence for your conjecture. Then write a mathematical argument for why your conjecture is true.

G.RP.3 Investigate and evaluate conjectures in mathematical terms, using mathematical strategies to reach a conclusion

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

A rectangular gift box with whole number dimensions has a volume of 36 cubic inches.

Find the dimensions of all possible boxes. Determine the box that would require the least amount of wrapping paper.

Find the dimensions of all possible boxes if the volume is 30 cubic inches. Determine the box that would require the least amount of wrapping paper.

Write a conjecture about the dimensions of a rectangular box with any fixed volume that would require the least amount of wrapping paper. Write a mathematical argument for why your conjecture is true.

What changes the volume of a cylinder more, doubling the diameter or doubling the height? Provide evidence for your conjecture. Then write a mathematical argument for why your conjecture is true.

G.RP.4 Provide correct mathematical arguments in response to other students’ conjectures, reasoning, and arguments

Given acute triangles

and

with

, and

. Norman claims that he can prove

using Side-Side-Angle congruence. Is Norman correct? Explain your conclusion to Norman.

Use a straightedge to draw a segment and label it

. Then construct the perpendicular bisector of segment

by following the procedure outlined below:

Step 1: With the compass point at A, draw a large arc with a radius greater than ½AB but less than the 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 arc intersects the arc drawn in step 1 twice, once above

and once below

. Label the intersections of the two arcs C and D.

Justify the fact that if one edge of a triangular prism is perpendicular to its base then the prism is a right triangular prism.

Prove that if a radius of a circle passes through the midpoint of a chord, then it is perpendicular to that chord. Discuss your proof with a partner.

Using dynamic geometry, draw a circle and its diameter. Through an arbitrary point on the diameter (not the center of the circle) construct a chord perpendicular to the diameter. Drag the point to different locations on the diameter and make a conjecture. Discuss your conjecture with a partner.

A rectangular gift box with whole number dimensions has a volume of 36 cubic inches.

Find the dimensions of all possible boxes. Determine the box that would require the least amount of wrapping paper.

Find the dimensions of all possible boxes if the volume is 30 cubic inches. Determine the box that would require the least amount of wrapping paper.

Write a conjecture about the dimensions of a rectangular box with any fixed volume that would require the least amount of wrapping paper.

Compare your arguments with a partner and discuss the validity of each argument.

Prove that a quadrilateral whose diagonals bisect each other must be a parallelogram.

Compare your arguments with a partner and discuss the validity of each argument.

Prove that if a radius of a circle passes through the midpoint of a chord, then it is perpendicular to that chord. Discuss your proof with a partner.

Compare your arguments with a partner and discuss the validity of each argument.

Prove that a quadrilateral whose diagonals are perpendicular bisectors of each other must be a rhombus.

Compare your arguments with a partner and discuss the validity of each argument.

Students will select and use various types of reasoning and methods of proof.

G.RP.7 Construct a proof using a variety of methods (e.g., deductive, analytic, transformational)

Use a straightedge to draw a segment and label it

. Then construct the perpendicular bisector of segment

by following the procedure outlined below:

Step 1: With the compass point at A, draw a large arc with a radius greater than ½AB but less than the 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 arc intersects the arc drawn in step 1 twice, once above

and once below

. Label the intersections of the two arcs C and D.

Consider the theorem below. Write three separate proofs for the theorem, one using synthetic techniques, one using analytical techniques, and one using transformational techniques. Discuss the strengths and weakness of each of the different approaches.

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

G.RP.8 Devise ways to verify results or use counterexamples to refute incorrect statements

Examine the diagonals of each type of quadrilateral (parallelogram, rhombus, square, rectangle, kite, trapezoid, and isosceles trapezoid).

For the quadrilaterals whose diagonals are lines of symmetry, identify other properties that are a direct result of the symmetry.

G.RP.9 Apply inductive reasoning in making and supporting mathematical conjectures

Explain how the pencil illustrates the fact that if two lines are perpendicular to the same line, then they must be parallel.

Explain how the pencil illustrates the fact that if two lines are parallel, then they must be perpendicular to the same line.

Analyze the following changes in dimensions of three-dimensional figures to predict the change in the corresponding volumes.

One box of pasta has dimensions that are three times the dimensions of a smaller box.

The dimensions of one triangular prism are x times the dimensions of another triangular prism.

Communications

Students will organize and consolidate their mathematical thinking through communication.

G.CM.1 Communicate verbally and in writing a correct, complete, coherent, and clear design (outline) and explanation for the steps used in solving a problem

Jim is a carpenter and would like to install a flagpole in his front yard. Carpenters use a tool called a level, shown in the figure below, to determine if objects are level (horizontal) or plumb (vertical). Describe how Jim can use a level to ensure that the flagpole appears vertical from any direction.

In the accompanying diagram, figure

is a parallelogram and

and

are diagonals that intersect at point

. Working with a partner determine at least two pairs of triangles that are congruent and discuss which properties of a parallelogram are necessary to prove that the triangles are congruent.

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

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

In the accompanying diagram figure

is an isosceles trapezoid and

and

are diagonals that intersect at point

. Working with a partner, determine a pair of triangles that are congruent and state which properties of an isosceles trapezoid are necessary to prove that the triangles are congruent.

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

G.CM.2 Use mathematical representations to communicate with appropriate accuracy, including numerical tables, formulas, functions, equations, charts, graphs, and diagrams

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

Jeanette invented the rule

to find the measure of A of one angle in a regular n-gon. Do you think that Jeannette’s rule is correct? Justify your reasoning.

In figure 1 a circle is drawn that passes through the point (-1,0).

is perpendicular to the