Math Study Guide for the PERT

Working in the Coordinate Plane

The coordinate plane (also known as the Cartesian plane) is a two-dimensional system for displaying graphs. It has a horizontal axis (the \(x\)-axis) and a vertical axis (the \(y\)-axis). These two axes meet at the origin. Points, lines, and other shapes can be plotted on the coordinate plane.

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Points are plotted on the coordinate plane using two numbers (coordinates) that correspond with the number markings on the two axes. The origin is at (\(0,0\)), and each point is identified by its relationship to that origin.

When plotting a point, the first coordinate is plotted along the \(x\)-axis (moving right or left), and the second coordinate is the movement on the \(y\)-axis (moving up or down).

For instance, a point at (\(2,3\)) means that the point is two units to the right of the origin along the \(x\)-axis and three units up from the origin along the \(y\)-axis.

Slope of a Line

The slope of a line is defined as the change in \(y\) over the change in \(x\), often expressed as rise over run. The formula for slope, \(m\), is:

\[m = \frac {(y_2 - y_1)}{(x_2 - x_1)}\]

where \((x_1, y_1)\) and \((x_2, y_2)\) are two points on the line.

What is the slope of a line that passes through the points \((0, 8)\) and \((4, 2)\)?

\[m = \frac {(8 - 2)}{(0 - 4)} = \frac{6}{-4} = - \frac{3}{2}\]

Forms of Linear Equations

Remember that linear equations are equations that have constants and variables of the first degree. The standard form, also called the general form, for writing linear equations is:

\[Ax + By = C\]

where \(A, B\), and \(C\) are integers.

Linear equations can also be written in the slope-intercept form:

\[y = mx + b\]

where \(m\) is the slope and \(b\) is the point where the line crosses the \(y\)-axis (the \(y\) intercept).

Another form for a linear equation is the point-slope form:

\[y - y_1 = m(x - x_1)\]

where \(m\) is the slope, \((y - y_1)\) shows the change in rise or \(y\), and \((x - x_1)\) shows the change in run or \(x\).

Inspecting Equations

Just by looking at a linear equation, you can deduce several facts about it. A positive slope indicates a line that increases from left to right on the Cartesian plane. A negative slope decreases from left to right on the Cartesian plane.

Parallel lines have slopes that are equal. A line perpendicular to another line has a slope that is the negative of the reciprocal of the other line’s slope. For instance, a line with a slope of \(-2\) is perpendicular to a line with a slope of \(\frac{1}{2}\).

To find where a line crosses the \(x\)-axis, solve the equation when \(y = 0\). To solve for the \(y\)-intercept, solve the equation when \(x = 0\).

The linear equation \(2y = -6x + 3\) can be written as \(y = -3x + \frac{3}{2}\), and from that we can use \(y=mx+b\) to determine that the slope of the graph (\(m\)) is \(-3\). That means the line slopes downward from left to right, crossing the \(y\)-axis at \(1.5\) and crossing the \(x\)-axis at \(0.5\).

Creating/Identifying an Equation to Solve a Word Problem

Reading a mathematical problem and being able to translate it into English takes a lot of practice. These tips can help you think clearly during the process:

1. Read the entire question first; reread it if necessary.
2. Note the given information.
3. Identify what is being asked.
4. Assign a variable for the unknown.
5. Make a sketch to help you visualize the given information.
6. Focus on key words that have mathematical meaning.
7. Recall relevant formulas.

Here is an example using the tips above. You are presented with this problem:

George wants to build a pen for his chickens. He would like for the width to be 4 feet and the length to be 6 feet. How much fencing will he need to buy, in feet?

Following the steps above:

1. Read the entire question carefully and reread it if necessary.
2. The given information is: Two sides of the pen will be 4 feet long and two sides will be 6 feet long.
3. You are being asked to find the amount of fencing George needs, so you’ll need to determine the perimeter of the pen.
4. Let’s say \(P\) stands for “perimeter,” the unknown in this problem.
5. Make a sketch:
   
6. Remember that to find the perimeter, you’ll need to add all the sides. You can also use a formula.
7. The formula for the perimeter of a rectangle is:

\[P = (2 \cdot width) + (2 \cdot length)\]

Substitute the information you have into the formula:

\[P = (2 \cdot 4) + (2 \cdot 6)\] \[P = 8 + 12\] \[P = 20\]

The answer is 20 feet of fencing.

Measurement and Data

Using numbers to measure and record data is an important and often used mathematical skill. There are various tools we use to accomplish these tasks.

Tables, Charts, and Graphs

Tables, charts, and graphs are helpful tools for visualizing data. With them, you can extract information from organized data and construct a visual presentation of that data, such as with this pie chart about the subject preferences of students at a school:

Using the provided graph, what percentage of students prefer science?

Percent = \(\frac{part}{whole} \cdot 100\), so first, we have to know how many total students there are (the whole):

\[100+120+60+80=360\]

Now, we’re interested in the part of the whole who prefer science, which is \(80\). So, as a percentage, we have \(\frac{80}{360} \cdot 100\):

\[\frac{80}{360} \cdot 100= 22\%\]

Of all the students at the school, \(22\%\) prefer science to the other subjects.

The line graph below is from a survey of mothers who were asked how many times their teenage son needs to be reminded to clean his room before he actually does it. How many boys needed only two reminders?

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The \(y\)-axis tells us how many boys, from \(0\) to \(16\), needed to be reminded for each number of reminders, from \(0\) to \(6\). Reading across the bottom (the \(x\)-axis), we find \(2\). Then we go up from there to the graph and read a frequency of \(8\). So, \(8\) boys needed only two reminders.

Tips and Tricks

Be sure to review all levels of math concepts, including those that may seem easy to you. If you are currently in a higher-level math course, you may have forgotten the procedures for solving lower-level problems, and it’s a good idea to review them for this test. If you miss lower-level questions, the test will automatically steer you toward even easier questions, and you will not have a chance to prove your skills at the higher levels.