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

Patterns, Functions and Algebra (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 Algebra? 

Benchmarks in Patterns, Functions, and Algebra

NOTE: As a general resource for all Algebra benchmarks, students might consult the following resources:

• Algebra from Oswego City School District (NY) Regents Exam Prep Center.  Each sub-section includes lessons and interactive practice exercises. The algebra strand, in particular, addresses variables and expressions, equations and inequalities, patterns/functions/relations, trig functions (sine, cosine, tangent), and the Pythagorean theorem.

• HippoCampus, a project of the Monterey Institute for Technology and Education, provides multimedia content.  Select sections from the topics in the full online texts for Algebra 1A, 1B, or Elementary Algebra.  Each lesson includes multiple-choice assessments and an answer key.  "This curriculum emphasizes a multi-representational approach to algebra, with concepts, results, and problems being expressed graphically, analytically, and verbally."

A.  Generalize and explain patterns and sequences in order to find the next term and the nth term.

Purplemath: Number Patterns

 

Glencoe Online Study Tools, Mathematics, Algebra I, 2005, multiple choice practice:

• Arithmetic Sequences

• Writing Equations from Patterns

Mathguide.com:

• Identifying arithmetic sequences

• Quizmaster for finding the formula for the general term in the arithmetic sequence (fill in and check answer).

Play video at YouTube.com:

• Recognizing Patterns: Review arithmetic, geometric, powers, inductive reasoning patterns

B.  Identify and classify functions as linear or nonlinear, and contrast their properties using tables, graphs or equations.

Purplemath: Functions

Glencoe Online Study Tools, Mathematics, IMPACT Math, Course 3, multiple choice practice: Linear Relationships

 

Play video at PilotMath.com:

• Pilot Math Linear Functions reviews the concept of slope.

C.  Translate information from one representation (words, table, graph or equation) to another representation of a relation or function.

Purplemath: Translating Word Problems Keywords

Glencoe Online Study Tools, Mathematics, Algebra I, 2005, multiple choice practice: Writing Equations

D.  Use algebraic representations, such as tables, graphs, expressions, functions and inequalities, to model and solve problem situations.

Mathguide.com: The quizmaster for each of the following is fill in and check answers:

• Add quadratic trinomials

• Subtract quadratic trinomials

• Multiply binomials with terms in the form ax + b

OnlineMathLearning.com: Types of Algebra Word Problems Among those are complete explanations and how to's for age, average, coin, consecutive integer, distance, fraction, geometry, interest, lever, mixture, number sequences, ratio, proportion, symbol, and work problems.

 

Purplemath: Translating Word Problems Applications

 

Play videos at YouTube.com.  The presenter translates the word problem into the algebraic equation, discusses the problem, shows diagrams as needed, and presents the solution.  He is very detailed--excellent how-to videos.  Students: For a learning experience, pause the video after the problem is presented, then solve.  Use the video to check your solution method:

• Use perimeter of the rectangle.  Find dimensions of the rectangle.

• Sum of three consecutive odd integers.  Find the integers.

• Distance = Rate x Time (Airplanes traveling in opposite directions)

• Mixture (Salt solution)

• Coins (26 coins in your pocket consisting of dimes and quarters)

• Work (How much time does it take to complete the job if two people work together to paint a fence?)

• Investments (Use Interest =Principal x Rate: An investment consisted of stocks and bonds and earned interest in one year.  How much was invested in stocks? in bonds?)

E.  Analyze and compare functions and their graphs using attributes, such as rates of change, intercepts and zeros.

Purplemath: Graphing Overview

 

Shodor Interactivate, Virtual Manipulative: Function Flyer -- plot an algebraic function on the coordinate plane, and use sliders to manipulate the constants of the graphed function.

F.  Solve and graph linear equations and inequalities.

Mathguide.com: Solve linear equations of the form ax + b =c; ax + b + cx + d; and ax + b = cx +d.  The page contains explanations and quizmaster for each type.  Fill in and check answers.

 

Purplemath:

• Graphing Linear Equations

• Graphing Linear Inequalities

The Math Page: Algebra, Section 32, Equation of a straight line, including graphing using intercepts, y = ax, and lines parallel to axes, explanations, plus problems with answers 

 

Glencoe Online Study Tools, Mathematics, Algebra I, 2005, multiple choice practice:

• Graphing Linear Equations

• Graphing Inequalities

• Solve equations for a variable

• Solve multi-step equations

• Solving equations with variables on each side

• Solve multi-step inequalities, including those with variables on both sides

Glencoe Online Study Tools, IMPACT Mathematics, Course 3, multiple choice practice: Solve inequalities symbolically and graphically, including word problems

 

Play videos at YouTube.com:

• Paired Data and the Rectangular Coordinate: graph linear equations using x- and y- intercepts; ordered pairs satisfying the equation.

• Solve and graph the linear inequality in one variable: Reviews rule for division by a negative integer when solving

• Graph inequalities in two variables

G.  Solve quadratic equations with real roots by graphing, formula and factoring.

Mathguide.com:

• Solving quadratic equations with the formula.  Formula is presented.  Quizmaster presents the quadratic equation.  Fill in answers rounded to nearest tenth and check answers.

• Practice factoring quadratic trinomials with integer coefficients
  ax² + bx + c where a = 1 or a >1.  Fill in and check answers.

The Math Page: Algebra, Section 36, Solve Quadratic Equations by graphing, formula, factoring, explanations, plus problems with answers 

 

Purplemath: Solving Quadratic Equations

 

Edmonton Public Schools (CA), Jim Reed, Math 9: Interactive Quadratic Formula and Sum/Product Method 

 

Mercer Community College, Debbie Kell, multiple choice practice: Solve Quadratic Equations by Factoring and Applications 

 

Glencoe Online Study Tools, Mathematics, IMPACT Math, Course 3, multiple choice practice:

• Quadratic Formula Use

• Solve Quadratics by Factoring 

Glencoe Online Study Tools, Algebra I, 2005, multiple choice practice: Solving Quadratic Equations by Graphing

 

Play videos at YouTube.com:

• Solve a simple Quadratic Equation with no linear term (solve by square root method)

• Solve Quadratics by Factoring #1(common factor is present)

• Solve Quadratics by Factoring #2 (reminder to set equation =0 as a first step)

• The Quadratic Formula: How to solve a quadratic equation using the formula

 

Play videos at Mefeedia.com [Note, while completing the square is one method for solving the quadratic equation, it is not one that is expected for mastery on this benchmark]:

• GCSE maths Tutor no.3 completing the square part 1 --how to solve quadratics by completing the square method

• GCSE Maths Tutor no.4 completing the square part 2 --common examples of the  use of completing the square for the solution of quadratic equations
   

H.  Solve systems of linear equations involving two variables graphically and symbolically.

Mathguide.com: Solve systems of linear equations with addition, multiplication/addition, and substitution.  Lesson and then quizmaster.  Fill in and check answers.

 

Purplemath: Systems of Linear Equations

 

The Math Page: Algebra, Section 33, Simultaneous Linear Equations,  explanations for graphical, addition, substitution methods, plus problems with answers

 

Glencoe Online Study Tools, Algebra I, 2005, multiple choice practice:

• Graphing Systems of Equations

• Graphing Systems of Linear Inequalities

Glencoe Online Study Tools, IMPACT Mathematics, Course 3, multiple choice practice: Solve symbolically Systems of Linear Equations

 

Systems of Linear Equations drill by Elias Saab and Daniel Saab.  Enter the ordered pair for the answer.  Feedback tells correct answer, but not the complete details to solution.  Select from five levels of difficulty with either two, three, or four variables in the problem.

 

Play videos at YouTube.com:

• Solving Systems of Linear Equations: an overview of the three methods. 

• Solve Systems of Linear Equations by Substitution

• Solve Systems of Linear Equations by Elimination (Part 1: you can add or subtract at step 1)

• Solve Systems of Linear Equations by Elimination  (Part 2: first find a common multiple for one variable in both equations, then add or subtract)

• Solve Systems of Linear Equations by Graphing

I.  Model and solve problem situations involving direct and inverse variation.

Purplemath: Direct, Joint, and Inverse Variation

 

Glencoe Online Study Tools, Algebra I, 2005, multiple choice practice: 

• Slope and Direct Variation

• Inverse Variation

 

Play videos at YouTube.com for solving algebra word problems.  The presenter translates the word problem into the algebraic equation, discusses the problem, shows diagrams as needed, and presents the solution.  He is very detailed--excellent how-to videos.  Students: For a learning experience, pause the video after the problem is presented, then solve.  Use the video to check your solution method:

• Direct Variation (Application to wages earned)

• Inverse Variation (F varies inversely with the square of M)

• Direct and Inverse Variation in same problem (P varies directly with X and inversely with Y)

J.  Describe and interpret rates of change from graphical and numerical data. (Includes: Compute and interpret slope, midpoint, and distance given a set of ordered pairs.)

Purplemath: Meaning of Slope and Y-Intercept 

Glencoe Online Study Tools, Algebra I, 2005, multiple choice practice: Slope

AlgebraLab.org: Algebra I: Distance and Midpoint Formulas.  Study aids with examples and practice problems (short answer and T/F).  Given two ordered pairs, determine length of a segment and midpoint.  Given three ordered pairs, determine if the vertices form a right triangle.