   

Welldeveloped geometry skills can provide students with spatial models
that transfer to other areas of mathematics and allow students to pursue such
everyday activities as reading maps, moving furniture and building models.
Understanding visual concepts involves identifying and using relationships
between figures. For instance, students should be able to represent geometric
figures on the coordinate plane and use these representations to examine the
properties of those figures. The visual concepts in Geometry and Spatial Sense
also involve drawing twodimensional objects and working with transformations
in a coordinate plane.
Making and justifying conjectures is another important part of Geometry
and Spatial Sense. Students should be able to use counterexamples, different
kinds of proofs, and inductive and deductive reasoning in their discussions
about geometric objects.
Even though students will not receive a score for the Mathematical Processes
standard on the OGT, it is still an important part of the curriculum. Content
and processes should be taught in tandem. To better understand Geometry and
Spatial Sense, click on the dropdown menu above and select Mathematical Processes.
The content in this Teaching Tool is based largely on the Ohio
Mathematics Content Standards and Benchmarks
and includes released
items from the Ohio Graduation Test. Additionally, these materials are aligned
with the NCTM standards
.
While there are various suggestions and activities here to use when working
with students, this Teaching Tool is designed to complement a rigorous, researchbased
curriculum, not to substitute for one.

Geometry and Spatial Sense 

1. Geometry and Spatial Sense 

Click on the following benchmarks for more information and for links to
annotated OGT items.
 
a.
 Benchmark A: Formally define geometric figures.
As students progress in their study of geometry, they are expected to identify
and use key properties of geometric figures. They should formally define terms
related to segments and formally define angles related to polygons, triangles,
and circles. They should recognize terms that are undefined.

 
b.
 Benchmark B: Describe and apply the properties of similar and congruent
figures; and justify conjectures involving similarity and congruence.
The focus of this benchmark is on using similarity and congruence in geometric
objects. Students should be able to use proportions as they relate to similar
figures. They should also be able to use what they know about two and threedimensional
figures to make and test conjectures.
Click here
for
an annotated item from the 2005 Ohio Graduation Test that addresses this benchmark.

 
c.
 Benchmark C: Recognize and apply angle relationships in situations involving
intersecting lines, perpendicular lines, and parallel lines.
The focus of this benchmark is on recognizing relationships between angles
formed by intersecting lines. The benchmark covers how the intersection of
lines affects the relationships between the angles formed. Students should
also be able to apply their knowledge of angles, segments and intersecting
lines to problems involving circles and their areas.
Click here
for
an annotated item from the 2005 Ohio Graduation Test that addresses this benchmark.

 
d.
 Benchmark D: Use coordinate geometry to represent and examine the properties
of geometric figures.
This benchmark focuses on using coordinate geometry. High school students
should be familiar with the properties of regular geometric figures. This
benchmark covers drawing or analyzing these figures on a coordinate plane.
Students should also be able to extend their knowledge of threedimensional
objects by making, justifying and testing conjectures related to properties
of these threedimensional objects.
Click here
for
an annotated item from the 2005 Ohio Graduation Test that addresses this benchmark.
Click here
for
an annotated item from the 2005 Ohio Graduation Test that addresses this benchmark.

 
e.
 Benchmark E: Draw and construct representations of two and threedimensional
geometric objects using a variety of tools, such as straightedge, compass
and technology.
The focus of this benchmark is twofold. First, students should deepen their
sense of spatial relationships by creating nets of threedimensional objects.
Second, the benchmark extends the understanding of twoand three dimensional
objects to include the variety of tools that can be used to create them. For
example, students should learn how to use a straightedge, a compass and technology
to construct, reflect and rotate twodimensional figures.
Click here
for
an annotated item from the 2005 Ohio Graduation Test that addresses this benchmark.

 
f.
 Benchmark F: Represent and model transformations in a coordinate plane
and describe the results.
The focus of this benchmark is on extending students' understanding of
transformations. Students should already understand what it means to translate,
reflect, rotate and dilate a figure. Now students should be able to perform
these transformations in a coordinate plane and determine how the coordinates
and the properties of the figures are affected by various transformations.
Click here
for
an annotated item from the 2005 Ohio Graduation Test that addresses this benchmark.
Click here
for
an annotated item from the 2005 Ohio Graduation Test that addresses this benchmark.

 
g.
 Benchmark G: Prove or disprove conjectures and solve problems involving
two and threedimensional objects represented within a coordinate system.
The focus of this benchmark is on testing conjectures and solving problems
related to two and three dimensional objects. Specifically, students are
expected to use the coordinate system to analyze twodimensional figures.

 
h.
 Benchmark H: Establish the validity of conjectures about geometric objects,
their properties and relationships by counterexample, inductive and deductive
reasoning, and critiquing arguments made by others.
In this benchmark, students are expected to determine the truth of a statement
about geometric objects. Students should be able to understand and use counterexamples,
inductive and deductive reasoning and proofs to test conjectures about geometric
figures and properties.

 
i.
 Benchmark I: Use right triangle trigonometric relationships to determine
lengths and angle measures.
In this benchmark students are introduced to the sine, cosine and tangent
ratios. They should be able to define these ratios and use them to calculate
side lengths and angle measurements of right triangles. The focus in this
benchmark is on using trigonometric ratios to set up and solve problems.
Click here
for
an annotated item from the 2005 Ohio Graduation Test that addresses this benchmark.

ABOUT THE MATH

Types of geometric transformations
A transformation is an operation that creates an image from an original
figure, or preimage. Students should be familiar with four types of transformations:

Translation  A transformation in which
an image is formed by moving every point on a figure the same distance in
the same direction.
 A translation does not change the direction that a figure is facing,
and it can be described by the number of horizontal units and the number of
vertical units of the move. Students can practice translations by tracing
an object, sliding or translating it to a new location and tracing the new
image.

Reflection  A transformation that results
in a mirror image of the original shape.
 Students can practice reflecting physical objects by reversing their
orientation (flipping them) across a given line.

Rotation  A rotation is a transformation
about a fixed point such that every point in the object turns through the
same angle relative to that fixed point.
 Students will best understand the idea by rotating physical objects
without lifting them off the table.

Dilation  A transformation that preserves
the shape of a figure, but allows the size to change.
 Enlarging photographs or reducing images on a copy machine are examples
of dilations.

Spatial reasoning
Developing a student's capacity for spatial reasoning at an early age is
very important. Some students will naturally have more talent in this area;
however, given enough time and the right kind of practice all students will
be able to develop their skills. Give students plenty of opportunities to
look at different views of the same object, to fold and unfold threedimensional
figures, and to match threedimensional objects to their corresponding twodimensional
nets.

Types of reasoning
A conjecture is a hypothesis based on limited information. Students are
encouraged to form conjectures by looking at a few examples of similar problems.
For example, after looking at several different quadrilaterals, students may
form the conjecture that rectangles are a special kind of parallelogram. Conjectures
are not necessarily correct and should be proven before they are used as valid
mathematical conclusions.
Students are likely to develop initial conjectures that are wrong. That
is acceptable and expected. As they work through the process of testing their
conjectures against a variety of problems, they will continue to revise their
thinking until they arrive at a "correct" conjecture. Developing
students' reasoning abilities can be time consuming. However, when students
take an active role in constructing their own learning, they are more likely
to remember and understand new skills.

Inductive vs. deductive reasoning
Inductive reasoning and conjecture are related. A conjecture is a hypothesis
based on limited information while inductive reasoning uses logic to make
generalizations based on observation of specific cases and consideration of
patterns. For example, students may look at isosceles trapezoids and use inductive
reasoning to form the conjecture that all isosceles trapezoids have exactly
one pair of parallel sides and two legs equal in length.
Deductive reasoning, on the other hand, uses logic to arrive at a conclusion
from a given premise. For example, you know that perpendicular lines form
right angles. When you find a pair of perpendicular lines, you then use deductive
reasoning to conclude that the angles formed are right angles.

Right triangle trigonometry
Students should be able to find the side lengths of a right triangle given
one acute angle measure and the length of one side. Or, they should be able
to find acute angle measures given two side lengths. The equations for the
trigonometric functions are the following:
 Sine: ,
where the sine of an acute angle can be found by dividing the length of the
side opposite the angle by the length of the hypotenuse. One way to remember
the formula is the term SOH: Sine = Opposite
divided by Hypotenuse.
 Cosine: ,
where the cosine of an acute angle can be found by dividing the length of
the side adjacent to (next to) the angle by the length of the hypotenuse.
One way to remember the formula is the term CAH: Cosine
= Adjacent divided by Hypotenuse.
 Tangent: ,
where the tangent of an acute angle can be found by dividing the length of
the side opposite the angle by the length of the side adjacent to the angle.
One way to remember the formula is the term TOA: Tangent
= Opposite divided by Adjacent.


Strategies 

Help With Fundamentals
Listed here are a few of the difficulties students might have with this
topic, along with suggestions for addressing these difficulties. Pay close
attention to your students' work to identify other problem areas. Once you
have identified a particular difficulty, find a strategy to help students
overcome it.

Additional Instruction and Practice
If your students need additional instruction and practice, you can supplement
your instruction with activities. Talk to your colleagues and use your curriculum
to come up with activities for addressing these topics. Here are a couple
of activities you might want to try.


Activity 1
Have students use appropriate tools to construct the centroid,
incenter, orthocenter, and circumcenter of acute, right and obtuse triangles.
Have students label each point of concurrency and compare their location.
Discuss such questions as, "Is each point inside or outside the triangle?
Are any always inside or always outside the triangle? How can we tell the
points apart?"


Activity 2
Have students define various quadrilaterals in terms of other
quadrilaterals. For example, a square is a rhombus with four right angles.
Have students defend their definitions or critique each others arguments using
counterexample, inductive reasoning or deductive reasoning.


Activity 3
Have students draw a picture on the coordinate plane using
only straight line segments. To emphasize coordinate geometry, have them calculate
the slope of each line segment, and require them to include at least one pair
of perpendicular and/or parallel line segments to reinforce the relationships
of their slopes.

Advanced Work
Properties of Circles
The definition of a circle is the set of points in a plane that are equidistant
from the center point. There are many terms and properties associated with
circles, such as arcs, chords, and tangents. Challenge your students with
questions that combine using these terms with their other geometry skills.


Problem 1


The radius of circle C is 5 inches. If the
length of arc AB is inches,
find the length of line segment CD and show mathematical support for your
solution.


Method


This problem seems to gives us too little information, but we can
use deductive reasoning to determine the length of CD. First, since we are
given the arc length of AB, it makes sense to calculate the circumference
of our circle to determine angle ACB. The circumference of this circle is inches.
If we take the ratio of arc length to the circumference, ,
we see that the arc is actually one quarter of our circle. Therefore, the
measure of central angle ACB is onefourth of 360 degrees, or 90 degrees.
Since this angle is a right angle, we know that triangle ACB is a right triangle.
Line segments AC and CB are radii of the circle, so their lengths are each
5 inches.
Now there are two ways we can finish solving the problem.
First, using the Pythagorean theorem gives us: ,
so the length of AB is .
Since triangle ACB is isosceles, line segment CD is the perpendicular bisector
of AB, so we know that line segment DB equals .
Using the Pythagorean theorem again with the lengths of line segments DB and
BC in place, .
We can simplify the equation to see that line segment CD is inches
or approximately 3.54 inches.
Second, since AC and CB are have the same length, angles CAB and CBA are
each 45 degrees. Therefore, triangle ACD is a 454590 triangle
and the length of each of the legs is .
CD is a leg of triangle ACD, so its length is or
3.54 inches.

Answer
Answer Explanation
The length of line segment CD is inches
or 3.54 inches.


Extension
Ask students to try to make up their own problems that use at least three
properties from different areas of geometry.

