Standardized Test Preparation and Tips for Success
Preparing Your Students for the Ohio Graduation Test
in Mathematics

Students should be able to review on their own using these resources.  If you are not an Ohio educator, these resources can still benefit your test prep efforts, as benchmarks in other states are similarly stated.

Download the Test Prep Strand Booklet

Geometry and Spatial Sense (pdf)  This test prep booklet (about 200 KB) is designed to help you monitor your progress toward mastery of each benchmark in this strand.  You can print the entire booklet, or just those pages for benchmarks you need to work on.  Write notes in the booklet next to each resource you use in your review.

If needed, download Adobe Acrobat Reader, free software for viewing and printing PDF files.

You might need to download Java, a free plug-in from Sun Microsystems to interact with virtual manipulatives on this page.

Other strand resources:

[ Number, Number Sense, and Operations ] [ Measurement ] [ Geometry and Spatial Sense ] [ Patterns, Functions, and Algebra ] [ Data Analysis and Probability ]

What is Geometry?

Benchmarks for Geometry and Spatial Sense

NOTE: As a general resource for all Geometry and Spatial Sense benchmarks, students might consult the following resources:

• Geometry from Oswego City School District (NY) Regents Exam Prep Center.  Each sub-section includes lessons and interactive practice exercises. Sections address geometric relationships, constructions, locus, formal and informal proofs, transformations, and coordinate geometry.

A.  Formally define geometric figures.

Learner.org: Glossary of key geometry terms

 

Math.com: Geometry basics.  Figures are defined with lessons, examples, workout quizzes, and unit quizzes

 

Mathwords.com: interactive math dictionary with words, terms, formulas, pictures, diagrams, tables, and examples

 

Explore Learning Gizmos (Virtual Manipulatives): Quadrilaterals and other Polygons.

 

Geometry 3-D Shapes is one of the Interactives from Annenberg Media's Learner.org.  Learn about three-dimensional shapes, calculate surface area and volume, and discover some of the mathematical properties of shapes. Use the tutorials and take the interactive test.

B.  Describe and apply the properties of similar and congruent figures; and justify conjectures involving similarity and congruence.

Edmonton Public Schools (CA), Jim Reed, Math 9: Similar and Congruent Triangles

AlgebraLab.org:  Lesson and fill in the blank practice problems, including word problems.  Be sure to show the related AlgebraLab documents for additional practice problems on each topic.

• Congruent Triangles

• Similar Triangles

Mathleague.com: Describe similar and congruent figures

 

Math.com: Congruent and similar figures including lesson and workout problems

 

Utah State University, National Library of Virtual Manipulatives: investigate methods for determining congruent triangles.  Explore SSS, SAS, ASA, SSA.  Note SSA does not guarantee congruent triangles.

 

Play videos at YouTube.com:

• Similar Triangles illustrates the definition.  Example is used to solve a problem of two triangles formed from by two parallel lines cut by two intersecting lines. This was posted from Nutshellmath.com

• Proving Triangles Congruent explores 5 methods to show congruence: SSS, SAS, ASA, AAS, and HL in right triangles.  This shows use of the two-column proof method: Statement/Reason.  This was posted from Nutshellmath.com

C.  Recognize and apply angle relationships in situations involving intersecting lines, perpendicular lines and parallel lines.

AlgebraLab.org:  Lesson and fill in the blank practice problems.  Be sure to show the related AlgebraLab documents for additional practice problems on each topic.

• Parallel Lines and Transversals

• Slope of Parallel and Perpendicular Lines

Math.com: Angles and Intersecting Lines including lesson and workout problems

AmbleWeb, Virtual Manipulative: 20 Virtual Protractor Angle Activities -- investigate sum of interior and exterior angles in polygons, supplementary angles, angles created by parallel lines and a transversal, vertical angles, and so on.

Shodor Interactivate, Virtual Manipulative: Angles formed by parallel lines and one or two transversals

W. Fendt's Sum of Angles in a Triangle, Virtual Manipulative

D.  Use coordinate geometry to represent and examine the properties of geometric figures.

Holt, Rinehart, Winston, Geometry, section 5.7 homework help: Proofs in Coordinate Geometrydetermine when to use distance, slope, or midpoint formula. 

 

Skills for coordinate geometry:

• MathsNet: What is meant by a midpoint? A virtual manipulative.

• Bottomless worksheets, free from CafeGenius web site. Problems generated randomly with answers.

  • Length of a line segment. 

  • Midpoint of a line segment. 

  • Slope of a line.

SAT Math Pro: Coordinate Geometry Practice Set with videos

E.  Draw and construct representations of two- and three-dimensional geometric objects using a variety of tools, such as straightedge, compass and technology.

MathsNet: Construction  This is an interactive to learn to construct mathematical shapes out of points, lines and circles. Then you can change the diagrams yourself using the mouse and see the effect these changes have on the mathematical properties of the shapes.

 

Mathematics Open Reference: Visual demonstrations of how to do various Euclidean constructions with compass and straightedge.  (highly recommended).

 

Oswego City School District (NY), Regents Prep, new Geometry section: Constructions

 

Virtual manipulative: Explore the net for a pyramid with rectangular base.  You can modify the length, width, and height in this JAVA applet by Pavel Safronov (MSTE division at University of Illinois-Urbana-Champaign).

 

Virtual manipulative: Explore building "houses" with cubes, given the top, front, and right side views.  This activity builds 3-D spatial abilities.  This JAVA applet is from the Freudenthal Institute.

F.  Represent and model transformations in a coordinate plane and describe the results.

Edmonton Public Schools (CA), Jim Reed, Math 9: Transformations (Translations, Rotations, Reflections, Dilations)

Glencoe Online Study Tools, Geometry Concepts and Applications 2004, multiple choice practice:

• Translations

• Reflections

• Rotations

• Dilations

Utah State University, National Library of Virtual Manipulatives: show axes, create objects with pattern block pieces, then apply and explore the following:

• Translations

• Reflections

• Rotations

• Effect of two transformations (Translations, Reflections, or Rotations) in any order.  

W. Fendt's Simple Geometric Transformations, Virtual Manipulative

 

Play video at YouTube.com:

• Transformations: Review reflections and translations. This was posted from Nutshellmath.com

G.  Prove or disprove conjectures and solve problems involving two- and three-dimensional objects represented within a coordinate system.

Glencoe Online Study Tools, Geometry Concepts and Applications 2004, multiple choice practice:

• Distance on the Coordinate Plane

• Midpoints

• Coordinate Proofs

Utah State University, National Library of Virtual Manipulatives: Platonic Solids, Verify Vertices, Faces, Edges of five Solids

H.  Establish the validity of conjectures about geometric objects, their properties and relationships by counter-example, inductive and deductive reasoning, and critiquing arguments made by others.

Glencoe Online Study Tools, Geometry Concepts and Applications 2004, multiple choice practice:

• Deductive Reasoning

• Inductive Reasoning

Utah State University, National Library of Virtual Manipulatives: walk through a visual proof of the Pythagorean theorem.

I.  Use right triangle trigonometric relationships to determine lengths and angle measures.