Subtest II: Mathematics Study Guide for the CSET Multiple Subjects Test

Algebra and Functions

Algebra is largely about working with quantities that are related to each other. The relationships generally use symbols called variables to write expressions or equations, and there are numerous rules for manipulating them. As in a lot of mathematics, it helps to have a good eye for noticing and remembering patterns.

Patterns and Relationships

Numerical patterns can range from the obvious to the kind that make you want to pull your hair out. To help decipher them we can use tables, graphs, word rules, and symbolic rules. For example: You are riding in your car at a speed of 60 mph. What is the relationship between the distance you go and the time it takes? You could make a table to show it.

\[\begin{array}{|c|c|c|c|c|} \hline \text{d (miles)} & \text{15} & \text{30} & \text{60} & \text{150} \\ \hline \text{t (hours} & \text{0.25} & \text{0.50} & \text{1.0} & \text{2.5} \\ \hline \end{array}\]

Or you could graph the values, in this case giving a straight line.

State the rule in words: distance \(\div\) time = 60.

State the rule symbolically: \(d/t=60\).

Some relations are called functions and you should know how to recognize one.

Proportional Reasoning

An equation like \(\frac{x}{4} = \frac{7}{12}\) is called a proportion and you can use it to deal with quantities that are directly proportional to each other.

It’s nothing more than a statement of two ratios that are equal, also known as equivalent fractions. The example in the previous section is a perfect example of a direct proportion:

\[\frac{d}{t} = \frac{60}{1}\]

If you know t is 3 hours, you can easily calculate d.

\[\frac{d}{3} = \frac{60}{1}\] \[d = 180\]

Any problem involving proportional reasoning can be set up as a proportion and solved. Review similar triangles to get a geometric view of proportional reasoning.

Dependent and Independent Variables

Be able to recognize, represent, and analyze relationships between dependent and independent variables. The independent variable is the one you are directly changing, and the dependent variable is the result of that change.

Suppose you put a pressure gauge on a tank of nitrogen and put it in the oven at 200 ℉ and read the pressure gauge. Then do the same at 300 ℉ and 400 ℉ and 500 ℉. The pressure readings keep going up each time. You are directly setting the temperature (input) and observing the pressure (output). The temperature is controlling the pressure, or you could say pressure is dependent on the temperature. Pressure is the dependent variable, and temperature is the independent variable. On a graph, the independent variable is on the horizontal axis and the dependent one is on the vertical axis.

Equations and Inequalities

Equations tell us only one thing: The thing on the left side equals the thing on the right side.
Inequalities can tell us one of four things:

\(a \lt b\) tells us that a is less than b.
\(a \le b\) tells us that a is less or equal to b.
\(a \gt b\) tells us that a is greater than b.
\(a \ge b\) tells us that a is greater than or equal to b.

Except for one case, solving inequalities is like solving equations.

\[\begin{array}{ll} \text{3x+4} \lt \text{ 28} & \text{Given problem} \\ \text{3x} \lt \text{24} & \text{Subtracted 4 from both sides} \\ \text{x} \lt \text{8} & \text{Divided both sides by 3} \\ \end{array}\]

The exception is that when you multiply or divide by a negative number, you have to reverse the inequality sign.

\[\begin{array}{ll} \text{5-2x} \lt \text{ 25} & \text{Given problem} \\ \text{-2x} \lt \text{20} & \text{Subtracted 5 from both sides} \\ \text{x} \gt \text{-10} & \text{Divided both sides by -2} \\ \end{array}\]

Equivalent Expressions

Equivalent expressions are expressions that express the same quantity, though they may look different. For example:

The expressions \(x(3x-2) + 3x\) and \(x(3x+1)\) are equivalent because they are both the same as \(3x^2 +x\) when they are simplified.

Simplifying is the key If you completely simplify equivalent expressions, they will end up being identical.

Matching Expressions to Situations

Be able to translate symbolic expressions such as \(7x-4\) into words such as 4 less than 7 times x or turn words into a symbolic expression. From geometry, for example, we know that in a right triangle the sum of the squares of the legs is equal to the square of the hypotenuse. Noticing key words like sum, square and equal to will help you to end up with \(a^{2} + b^{2} = c^{2}\).

Expressing Problems Algebraically

Word problems can strike fear into the hearts of math students, and for good reason. They can be hard. Watch for key words or phrases such as added to, increased by, less, less than, of, times, and others to help you on your way. For example:

Suppose Max had a certain number of links on his website and Claire had 11 fewer links.
If together they have 25 links, how many does Max have?

First, assign variables:

Max’s links \(= x\)
Fewer means less, so Claire’s links \(= x-11\)
Together implies added and they have implies equal to, so:

\[(x)+(x-11)=25\]

These kinds of problems also can show up in geometry, dealing with topics like area, perimeter, similar triangles, and volumes.

Properties of Linear Equations

Linear equations are pretty easy to recognize:

You will see no powers of x and y higher than 1. An easy example is \(y=2x-5\).
Also, their graphs are always straight lines. Be able to determine the slope of a graph by looking at it (rise/run) or from the equation (get the equation in the form \(y=mx+b\) and m is the slope). In our easy example above, the slope would be \(2\).

If two lines are parallel, they will have the same slope.
If the two slopes are negative reciprocals of each other, they are perpendicular.
A graph perpendicular to our easy example above would have a slope of \(-\frac{1}{2}\).

Working with Polynomials

An expression with two or more terms is a polynomial, although the term polynomial is often used to mean an expression with more than two terms, such as \(4x^{3}-3x^{2} -x +7\). You should know how to multiply, divide, and factor them.

Working with Quadratic Equations

Any equation that fits this pattern \(y=ax^2+bx+c\) is a quadratic equation. a, b, and c are constants that could be positive, negative or zero. Here are a few examples:

\[y=2x^{2}+5x+6\] \[y= x^{2} -4\] \[y=-2x^{2}\]

There are a few different techniques to use in solving quadratics.

Factoring— Follow these steps:

Factor the quadratic:
\(x^{2}+5x+6 = 0\) gives \((x+3)(x+2)\)
Set each factor equal to zero:
\(x+3=0\) and \(x+2=0\)
Solve each little equation:
\(x=-3\) and \(x=-2\)

Completing the square—If you can’t solve the quadratic by factoring the equation, you can complete the square, but it’s unlikely you will have to. Go to the next method instead.

The Quadratic Formula—Look back to the general form of a quadratic.equation above: \(y=ax^2+bx+c\). Unlike factoring, the quadratic formula can solve any quadratic equation. Unlike completing the square, it doesn’t take a lot of steps to get to the answer(s).

This is it:

\[x= \frac{-b \pm \sqrt{b^{2} -4ac}}{2a}\]

Solve:

\[x^{2} + 4x +1 = 0\] \[a=1, b=4, c=1\]

Substitute into the quadratic formula.

\[x= \frac{-4 \pm \sqrt{4^{2} -4\cdot 1\cdot1}}{2 \cdot 1}\]

Simplifying will give you \(-2 \pm \sqrt{3}\)

Interpreting Graphs

Be able to interpret graphs of:

Linear equations (straight lines)
Quadratic equations (parabolas)
Inequalities (straight line + all area to one side of the line)
Systems of equations (two plots on the same axes, often intersecting)

In systems of equations, the point where the graphs meet is the solution to the system.