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High School — Mathematics
Geometry and Spatial Sense

Well-developed 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 two-dimensional 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, research-based 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 three-dimensional 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 three-dimensional objects by making, justifying and testing conjectures related to properties of these three-dimensional 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 three-dimensional 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 three-dimensional objects. Second, the benchmark extends the understanding of two-and 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 two-dimensional 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 three-dimensional 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 two-dimensional 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
  • Similarity and congruence

    • Congruent figures can always fit perfectly atop one another because they have exactly the same size and shape. In other words, congruent figures are similar with a 1:1 ratio between the lengths of their corresponding sides and the measures of their corresponding angles. Give your students examples in which one shape must be rotated or reflected before it fits on top of the other. Note that the measures of the corresponding angles are the same and the lengths of the corresponding sides are the same.
    • Similar figures are the same shape but not necessarily the same size. Mathematically, this means that corresponding angles are congruent. Corresponding sides are proportional but not necessarily congruent. Corresponding parts of figures are the pairs of angles or sides in one figure that match with the angles or sides in another figure. Be sure to include examples in which similar shapes are rotated and reflected with respect to one another.

  • Types of geometric transformations

    A transformation is an operation that creates an image from an original figure, or pre-image. 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 three-dimensional figures, and to match three-dimensional objects to their corresponding two-dimensional 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 one-fourth 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 45-45-90 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.