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High School — Mathematics
Patterns, Relations and Algebra

From an early age, students develop an awareness of numerical patterns. For example, children may notice how house numbers change while they're walking down the street. As students advance in their study of mathematics, they begin to represent these patterns using variables, algebraic expressions, tables, graphs, equations, inequalities and functions. Students learn to manipulate polynomial expressions and solve linear and quadratic equations. These skills allow them to recognize patterns in real world situations and to represent and solve real world problems algebraically.

Even though students will not receive a score for the Mathematical Processes standard on the Ohio Graduation Test (OGT), it is still an important part of the curriculum. Content and processes should be taught in tandem. To better understand Patterns, Relations and Algebra, 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 OGT. Additionally, these materials are aligned with the National Council of Teachers of Mathematics (NCTM) standards . While various activities are suggested for working with students, this Teaching Tool is designed to complement a rigorous, research-based curriculum, not to substitute for one.



Patterns, Relations and Algebra


1. Patterns, Functions, and Algebra

Click on the following benchmarks for more information and for links to annotated OGT items.

a.

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

At this level students should be able to identify a pattern or a relationship between the terms in a sequence. They should be able to use the relationship to find the next term. Finally, they should be able to express a pattern in a table, graph or equation to find the nth term in the sequence.

Click here for an annotated item from the 2005 Ohio Graduation Test that addresses this benchmark.

b.

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

This benchmark culminates with a description and comparison of nonlinear functions. However, before students can successfully describe and compare the characteristics of nonlinear functions, they should be able to define a function in terms of domain and range and using f(x) notation. They should also be able to distinguish between linear and nonlinear functions that are displayed in tables, graphs or equations.

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.

c.

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

Students should be able to express linear, quadratic and exponential relationships in words, a table, a graph or in symbolic form. They should also be able to explain how these four representations relate and convert from one representation to another.

Click here for an annotated item from the 2005 Ohio Graduation Test that addresses this benchmark.

d.

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

Algebraic representations are most useful when modeling and solving real world problems. Students should not learn these representations without understanding their applications. They should understand situations where one variable depends on another. They should be able to combine monomials and polynomials with and without using physical models and be able to simplify rational expressions. Finally, they should be able to represent and solve problem situations using appropriate equations, inequalities or algebraic representations.

Click here for an annotated item from the 2005 Ohio Graduation Test that addresses this benchmark.

e.

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

In this benchmark, students explore the practical applications of functions and their graphs. They should be able to relate rates of change, intercepts and zeros to graphs of real world situations. They should be able to describe and compare linear, quadratic, and exponential functions using their attributes.

Click here for an annotated item from the 2005 Ohio Graduation Test that addresses this benchmark.

f.

Benchmark F: Solve and graph linear equations and inequalities.

The focus of this benchmark is on linear equations and inequalities. Students should be able to convert problem situations to appropriate equations, inequalities or algebraic representations. They should be able to write linear equations using a variety of information, and they should know several ways to write and solve linear equations and inequalities. Students should be able to combine all these skills to solve real world problems.

Click here for an annotated item from the 2005 Ohio Graduation Test that addresses this benchmark.

g.

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

In this benchmark, students explore various ways to solve quadratic equations. Students move from graphing quadratics to using factoring, the quadratic formula and technology to solve them. Finally, they graph circles and solve problems that can be modeled with linear, quadratic, exponential or square root functions.

h.

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

In this benchmark, students study several methods of solving systems of linear equations. These methods include using graphs, substitution and elimination. Students also learn how the solution relates to intersection of the two lines. Finally, students solve problems using systems of linear equations and inequalities.

Click here for an annotated item from the 2005 Ohio Graduation Test that addresses this benchmark.

i.

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

In this benchmark, students study direct and inverse variation. First, they should be able to distinguish between several types of changes in mathematical relationships. Second, students should be able to apply direct and inverse variation to appropriate problem-solving situations. Finally, students should be able to interpret the graphs of direct and inverse variation.

Click here for an annotated item from the 2005 Ohio Graduation Test that addresses this benchmark.

j.

Benchmark J: Describe and interpret rates of change from graphical and numerical data.

The focus of this benchmark is on the rate of change of a data set. Rate of change is a relationship such as distance over time, often described by using a slope. Students start their explorations within this benchmark by learning to find slope, midpoint and distance. Next they study how graphs of equations change when the numbers in the equation are changed. Finally, they study the relationships between the slopes of parallel and perpendicular lines.

Click here for an annotated item from the 2005 Ohio Graduation Test that addresses this benchmark.

 

ABOUT THE MATH
  • Graphing a function

    To graph a function such as y = 5x - 4, follow these steps:

    1. Create a table. In the left column, list some x-values. Substitute these values for x in the equation and solve for y. Record these values in the right column next to their corresponding x-values.

    2. Write the x- and y-values in each row as coordinate pairs:

    3. Plot the coordinate pairs in the coordinate plane. Each row of the table becomes a point in the plane, and together these points suggest the shape of the graph of the function.

  • The coordinate plane

    Students should have a strong understanding of the coordinate plane. The x-axis (the horizontal number line) and the y-axis (the vertical number line) are perpendicular lines that cross at the point (0, 0), called the origin. The two axes of the coordinate system divide the plane into four separate sections known as quadrants. These are identified as the first, second, third and fourth quadrants.

    Spend time with your students discussing what the points in each quadrant have in common, and what the points located on the axes have in common. The diagram below helps to describe these patterns.

  • Direct and inverse variation

    Direct variation occurs when the values of two variables maintain a constant ratio. The relationship can be expressed as an equation in the form . The graph of such an equation is a line with an x-intercept of 0 and a slope of k.

    Inverse variation occurs when the variables x and y vary inversely. For a constant k, or . When k and x are greater than 0, the graph of an inverse variation shows y decreasing as x increases. When x is less than 0, the graph shows y increasing as x decreases.



Strategies

Help With Fundamentals

Here are a few of the difficulties students might have with this topic, along with suggestions for addressing these difficulties.



Additional Instruction and Practice

If your students need additional practice, supplement your instruction with the following activities:

Activity 1

Ask students to find tables and graphs for which they can determine a variable expression or pattern. They can find these tables and graphs in newspapers or magazines. Have them share their findings.

Activity 2

Have students use computers and graphing calculators to explore the characteristics of square root, cubic, absolute value or basic trigonometric functions. For instance, have students explore the square root functions by graphing on a computer or graphing calculator. Have them explore answers to such questions as: What is the general shape of this type of equation? How does the graph change when a number is added to or subtracted from x ( )? How does the graph change when a number is added to or subtracted from the square root of x ( )? How is the graph affected by positive and negative coefficients? What is the domain of this function? If students explore several families of functions, they can compare their answers to these questions across functions. For instance, does multiplying by a negative coefficient affect all types of functions the same way?



Advanced Work

Solving Quadratic Equations by Completing the Square

Completing the square requires students to factor and multiply polynomials.

Problem 1

Solve by completing the square.

Method

First, we first move the constant term to the right hand side of the equation:

.

We want the left side of the equation to be a trinomial that factors easily, so we take half of the x-term's coefficient (6) and square the result:

.

We add the result, 9, to both sides of the equation:

.

Now the left-hand side can be factored into . Our equation is now:

.

Next, we take the square root of both sides to get:

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Finally, we solve for x. In this case we subtract 3 from both 4 and -4 to get:

.


Answer

Answer Explanation

Extension

Once students have a lot of practice completing the square, ask them to complete the square for the general quadratic equation: . If done correctly, the result is the quadratic formula!