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Pivotal Items—Geometry October 28, 2010 As the Virginia Department of Education revises the Standard of Learning assessments to increase
the level of rigor and include higher-level questions (Bolling, 2010), teachers will need to prepare
students to respond well to these kinds of questions. The Center for STEM Education and Outreach
at James Madison University brought together experts in the field of mathematics education to
develop sets of higher-level questions for use in Algebra I, Algebra II, and Geometry courses. These
questions are aligned with the 2009 Standards of Learning. We describe these items as “pivotal
questions” because they serve a vital and critical role in unveiling student understanding and
misconceptions in ways that knowledge-recall questions do not allow.
According to Dr. Wendy Sanchez (May 2010), an assessment expert in mathematics education,
pivotal questions have the following characteristics:
1.
2.
3.
4.
5.
Can be solved or explained in a variety of ways
Focus on conceptual aspects of mathematics
Require students to use the process standards (NCTM, 2000)
Have the potential to expose student understanding and misconceptions
Lend themselves to a scoring rubric
We encourage teachers to use these pivotal questions with their students and provide the STEM
Center feedback. We are also interested in collecting and posting student responses to share with
other teachers who are learning to implement these kinds of items in their classrooms. Please
contact Dr. LouAnn Lovin at [email protected] with feedback about the items and to learn about
how you can participate by posting student responses to these items. We have included a 6-point
scoring rubric, developed by Dr. Sanchez, on the STEM Center website
(www.jmu.edu/stem/outreach) to help in evaluating student responses to the pivotal items.
The Center for STEM Education and Outreach at James Madison University would like to thank the
following people for their work on these items:
• Dr. Wendy Sanchez, Associate Professor of Mathematics Education, Kennesaw State
University, Kennesaw, Georgia.
• Patrick Lintner, Supervisor of Mathematics, Harrisonburg City Public School Division,
Harrisonburg, VA.
• Dr. Kyle Schultz, Assistant Professor of Mathematics Education, James Madison University,
Harrisonburg, VA.
We hope these items are useful to your practice as mathematics teachers!
References:
Bolling, Michael. (2010). Virginia Department of Education Update. Fall 2010 Virginia Council for
Mathematics Supervision. Richmond, VA.
National Council of Teachers of Mathematics. (2000). Principles and Standards of School
Mathematics. NCTM: Reston, VA.
Sanchez, Wendy. (May 2010). Personal communication.
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Pivotal Items—Geometry October 28, 2010 Reasoning, Lines, and Transformations G.1
The student will construct and judge the validity of a logical argument consisting of a set of premises and a
conclusion. This will include
a) identifying the converse, inverse, and contrapositive of a conditional statement;
b) translating a short verbal argument into symbolic form;
c) using Venn diagrams to represent set relationships; and
d) using deductive reasoning.
G.1.a.1 Write a true proposition whose converse is false.
G.1.b.1 Describe propositions for each letter below so that the indicated statement is valid:
i.
p→q
ii.
~m → n
iii.
r → ~s
iv.
~t → ~w
G.1.c.1 Define the universe and sets A and B and fill in the blanks in the diagram below with
numbers or words so that the Venn Diagram appropriately models the relationships between your
sets.
A={
}
B= {
}
Universe:
G.1.d.1 Use deductive reasoning to show that the sum of two odd integers is even.
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Pivotal Items—Geometry October 28, 2010 G.2
The student will use the relationships between angles formed by two lines cut by a transversal to
a) determine whether two lines are parallel;
b) verify the parallelism, using algebraic and coordinate methods as well as deductive proofs; and
c) solve real-world problems involving angles formed when parallel lines are cut by a transversal.
G.2.a.1 List three different ways that the measures of the angles (1-8) could tell you that the two
lines are parallel.
G.2.b.1 Plot and give the coordinates of two points that determine a line that is parallel to line l.
Explain how you know your line is parallel to line l.
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Pivotal Items—Geometry October 28, 2010 G.2.c.1 Pete is building shelves for his bedroom. A triangular opening is to be cut out of a
rectangular piece of wood as shown below. Given that lines m and n are parallel, solve for p in at
least three ways.
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Pivotal Items—Geometry October 28, 2010 G.3
The student will use pictorial representations, including computer software, constructions, and coordinate
methods, to solve problems involving symmetry and transformation. This will include
a) investigating and using formulas for finding distance, midpoint, and slope;
b) applying slope to verify and determine whether lines are parallel or perpendicular;
c) investigating symmetry and determining whether a figure is symmetric with respect to a line or a point;
and
d) determining whether a figure has been translated, reflected, rotated, or dilated, using coordinate
methods.
G.3.a.1 Given segment AB below, find a point C such that:
a. AB is a leg of right triangle ABC
b. AB is the base of isosceles triangle ABC
c. AB is the base of right isosceles triangle ABC
G.3.b.1 Determine 3 possible sets of coordinates for D that would make segments AC and BD
perpendicular.
G.3.c.1 Use the three segments provided below to design a polygon that has:
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Pivotal Items—Geometry October 28, 2010 a. One line of symmetry
b. Two lines of symmetry
c. Rotational symmetry about a point
G.3.d.1 Determine a set of transformations that maps quadrilateral A in the figure below to its
image, A’, as shown.
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Pivotal Items—Geometry October 28, 2010 G.4
The student will construct and justify the constructions of
a) a line segment congruent to a given line segment;
b) the perpendicular bisector of a line segment;
c) a perpendicular to a given line from a point not on the line;
d) a perpendicular to a given line at a given point on the line;
e) the bisector of a given angle,
f) an angle congruent to a given angle; and
g) a line parallel to a given line through a point not on the given line.
G.4.ad.1 Use a compass and straightedge to construct a segment that is √2 units long, given that
the segment below has a length of one unit.
G.4.df.1 Use a compass and straightedge to construct a right triangle with one leg congruent to the
given leg below, and the altitude to the hypotenuse congruent to the given altitude.
G.4.af.1 On the line below, construct a rhombus in which one of the angles of the rhombus is
congruent to the given angle.
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Pivotal Items—Geometry October 28, 2010 G.4.ag.1 Construct a parallelogram with the given segment as one of its sides. Prove that your
construction yields a parallelogram.
G.4.bce.1 When asked to partition the triangle below into two regions of equal area, three students
described their strategies:
Byron: Construct an altitude through point A.
Justin: Construct the bisector of angle A.
Heather: Construct the perpendicular bisector of segment BC.
Perform each student’s construction. Which method is the most effective for equally partitioning the
triangle? Would your answer to the previous question be the same for any triangle? Is there a more
effective way to partition the triangle?
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Pivotal Items—Geometry October 28, 2010 Triangles G.5
The student, given information concerning the lengths of sides and/or measures of angles in triangles, will
a) order the sides by length, given the angle measures;
b) order the angles by degree measure, given the side lengths;
c) determine whether a triangle exists; and
d) determine the range in which the length of the third side must lie.
These concepts will be considered in the context of real-world situations.
G.5.abcd.1 Triangle ABC has interior angle C measuring 105°. The segment opposite angle C has a
measure of 23 cm. Describe the range of values for the measures of the other sides and angles of
triangle ABC. Explain your reasoning.
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Pivotal Items—Geometry October 28, 2010 G.6
The student, given information in the form of a figure or statement, will prove two triangles are congruent,
using algebraic and coordinate methods as well as deductive proofs.
G.6.1 In the figure below, mark one more pair of corresponding parts of the two triangles congruent
that will ensure that the two triangles are congruent. State how you know the two triangles are
congruent.
G.6.2 Suppose you knew that the hypotenuses of two right triangles were congruent. Are the two
triangles necessarily congruent? Explain why they are or give an example to show that they are not.
G.6.3 In the figure below, can you conclude that ΔGHI ≅ ΔJKI? Explain why or why not.
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Pivotal Items—Geometry October 28, 2010 G.7
The student, given information in the form of a figure or statement, will prove two triangles are similar,
using algebraic and coordinate methods as well as deductive proofs.
G.7.1 Give reasonable measures for the least number of sides and angles possible to ensure that
ΔABC ~ ΔDEF. Explain how you determined your solution.
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Pivotal Items—Geometry October 28, 2010 G.8
The student will solve real-world problems involving right triangles by using the Pythagorean Theorem and
its converse, properties of special right triangles, and right triangle trigonometry.
G.8.1 John was experimenting with non-symmetrical designs for kite flying. Suppose he wants to
build a flying kite based on the model below. Provide possible values for segment lengths a, b, c, d,
and e in the figure below and explain your reasoning.
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Pivotal Items—Geometry October 28, 2010 G.8.2 Give reasonable side lengths for sides a, b, and c in the figures below.
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Pivotal Items—Geometry October 28, 2010 Polygons and Circles
G.9
The student will verify characteristics of quadrilaterals and use properties of quadrilaterals to solve realworld problems.
G.9.1 Suppose you were making pottery tiles and wanted to create a certain size and shape tile.
Using a compass and straightedge, construct a rhombus with the given angle and diagonal.
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Pivotal Items—Geometry October 28, 2010 G.10
The student will solve real-world problems involving angles of polygons.
G.10.1 Kyle is making a zither, a musical instrument, and plans to cut the top out of a rectangular
piece of wood. Because of the way the strings have to be angled, he will use the plan below. List
three different sets of possible values for the unknown angle measures x, y, and z.
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Pivotal Items—Geometry October 28, 2010 G.11
The student will use angles, arcs, chords, tangents, and secants to
a) investigate, verify, and apply properties of circles;
b) solve real-world problems involving properties of circles; and
c) find arc lengths and areas of sectors in circles.
G.11.ab.1 Jacob wanted to measure the distance across his room. Unfortunately, all he had was a
bike wheel to use. When he rolled from touching one wall until hitting the other, it had rolled
exactly 3 times. Explain and show how you could use this information to calculate the width of
Jacob’s room.
G.11.bc.1 Bill buys a pizza with a 16-inch diameter. He wants to cut it into equal size sector-shaped
pieces so that each slice has an area of at least 40 in.2 If he makes an initial cut along one of the
radii, how far around the circular edge should he go to make the next cut into the center to get the
most pieces possible?
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Pivotal Items—Geometry October 28, 2010 G.12
The student, given the coordinates of the center of a circle and a point on the circle, will write the equation of
the circle.
G.12.1 Bill wrote down the equations of four circles, each of which is externally tangent to two of
the other three circles. One of the circle’s equations is (x – 2)2 + (y +1)2 = 4. Give possible equations
for the other three circles.
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Pivotal Items—Geometry October 28, 2010 Three-­Dimensional Figures G.13
The student will use formulas for surface area and volume of three-dimensional objects to solve real-world
problems.
G.13.1 and G.14.c Suppose you worked for Sugary Sweet Cereal Company and the company
wanted to change the dimensions of their cereal boxes in order to give their cereal a new look.
Although the dimensions will change, Sugary Sweet wants the amount of packaging used for each
box to remain the same. Your job is to investigate possibilities for the new cereal box and how
much cereal each box design can hold. What box design would you recommend? Carefully explain
your answer.
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Pivotal Items—Geometry October 28, 2010 G.14
The student will use similar geometric objects in two- or three-dimensions to
a) compare ratios between side lengths, perimeters, areas, and volumes;
b) determine how changes in one or more dimensions of an object affect area and/or volume of the object;
c) determine how changes in area and/or volume of an object affect one or more dimensions of the object;
and
d) solve real-world problems about similar geometric objects.
G.14.ad.1 For the upcoming blockbuster movie Robots Attack!, the movie producers needed to
design and construct AL-ϕ, one of the giant robots featured in the film. A designer hired by the
producers created a scale model that was 2 m tall and had a mass of 15 kg. The design looked good
and construction of the full-sized 30 m AL-ϕ using the same materials was approved.
Unfortunately, when the full-size AL-ϕ stood up for the first time, its legs snapped under the overall
weight. The model designer was fired.
In a lawsuit, the model designer claimed that the scale model of AL-ϕ they designed was
structurally sound, so the full-size AL-ϕ should have been as well. In their view, the failure of the
full-size AL-ϕ was the movie company’s fault. The movie company argues that the original design
was flawed. As an engineer consultant, you have been hired by to settle the dispute, explain how
you would settle the argument. Provide a mathematical explanation for why the full-size AL-ϕ was
a dud.
G.14.bd.1 Chunky candy bars have always come in square-based bars. This year, they would like to
market their new Minnie Chunk—still with a square base, but half the chocolate. Your job is to
develop 3 options for the potential dimensions for the new Minnie Chunk bar.
G.14.c.1 Determine the dimensions of two rectangular prisms that have the same surface area but
different volumes.
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