Page 2 Mathematics Study Guide for the TASC

Moderately Emphasized Concepts

Building Functions

The elements that go into a function make up a defined set called the domain to produce the elements in the range. One input results in exactly one output value. Functions can be constructed when the question provides input-output pairs, the graph of the function, or a description of the function. In the following example, building a function is necessary to answer the question:

Example: What is the slope of a line perpendicular to the linear function h(x) with the following input-output values?

The function is described as linear, so it must follow the standard form $f(x) = mx + b$. Plug in the first given pair into this form:

$h(2) = m(2) + b = 7$
$2m + b = 7$ (equation 1)

Do the same with the second pair or third pair:

$h(7) = m(7) + b = 17$
$7m + b = 17$ (equation 2)

Subtract equation 1 from equation 2 (elimination method):

$7m + b = 17$
-$2m + b = 7$
$5m = 10$
$m = 2$

m is the slope of function h(x) and any line perpendicular to this linear function must have a slope of $-\frac {1}{m}$ or $-\frac{1}{2}$.

Congruence and Similarity in Geometry

Geometric figures that have the same size, shape, measure of angles, length, area and other dimensions are called congruent. If two shapes are laid on top of each other and all their points coincide completely, then the two shapes are congruent. A shape that initially looks the same as another shape, except that it is oriented differently, is said to be congruent to that shape if it turns to be same shape after rotating, reflecting and translating.

Geometric figures that have the same shape, angles and other properties except the size are said to be similar. Knowing that two figures are similar often helps in solving for unknown values and measurements.

Example:

A triangle has the following side lengths: 6-ft on the short side and 8-ft on the long side. A similar triangle has 9-ft on its short side. What must be the length of the long side?

By ratio and proportion, the length is easily found to be 12 feet. $\frac{6}{8} = \frac{9}{x} -> 6x = 72 -> x = 12$

Right Triangles

Right triangles have special properties that are often useful in math tests. One such property is stated in Pythagoras’ Theorem:

$c^2 = a^2 + b^2$
where:

a and b are the sides of the right-angled triangle, and
c is the hypotenuse

Trigonometry and Geometry Concepts

Familiarity with the following geometric concepts will be useful for this test:

1. Coordinates – midpoint, distances between points, location of points in the Cartesian plane
2. Lines – equations of lines, slopes, parallel and perpendicular lines

Aside from the Pythagoras’ Theorem, special focus is given on Trigonometric Functions and Identities for solving angles and side lengths of rights triangles. The formulas for these are provided in the Math Reference Sheet.

Remember, however, that the trigonometric identities work only for right triangles. For non-right triangles, the following Triangle Identities apply:

Law of Sines:

where: A, B, and C are angles in a non-right triangle, and a, b and c are the length of their opposite sides, respectively.

Law of Cosines:

Take note that it is slightly similar to the Pythagoras’ Theorem.

Law of Tangents:

Quantities

There are various ways of determining quantity (the measure, number or amount) of something. Measure or quantity is usually done to determine length, area, volume or capacity, mass, density, temperature and time. Two major systems of measurement are the Metric System and the US Standard Units. When performing operations with quantities expressed in various units, make sure to convert quantities into a common unit first, before proceeding.

For instance, to add the 3 liters and 550 milliliters of milk, convert 550 ml to liters first, then add: 3 + 0.55 = 3.55 liters.

Inferences and Conclusions in Statistics

TASC questions may touch on statistics and require you to make inferences about a given data set, table or graph.

When you have a set of numbers, scores, or measurements, it is often useful to find the central value or central tendency to be able to infer correctly. The mean, median and mode are measures of central tendency, and it is quite common to find them in math questions.

To calculate the mean, simply add all the values and divide the sum by the number of values. To find the median, arrange all the values in ascending or descending order. The value in the middle of the sorted list is the median. When there is a pair of middle values, get the mean of the pair. For the mode, inspect the occurrence (or frequency) of each of the values; the value that occurs most often is the mode. There are cases when there will be no mode or there will be more than one mode.

Categorical and Quantitative Data

Data is a collection of numbers, facts, observations, quantity, measurements and description of things. Data can be quantitative or numerical information; but it can also be qualitative or categorical information.

A question may go like this: “From the result of a survey shown by this table, what fruit is the least liked by the children?” You have to be able to grasp various ways data is presented. They can be pure numbers shown in tables with several columns and rows, pie graphs with colored sections, bar and line graphs, maps, and many more.

The first thing to do is to go over the title, labels, legend, and scale. The title will describe what all the numbers and figures are about. Then take note how the numbers are presented and how they vary.

Go back to the question and draw your answer from what you understood from the numerical or visual data presented.

Least Emphasized Concepts

Trigonometric Functions

These are the three main trigonometric functions:

Geometry of Circles

The curve formed by connecting all points equidistant to a common point is a geometric shape called a circle.

The common point is called the center. The distance around the curve is the circumference (C). The distance from the center to any point on the circumference is the radius (r).

The formula for circumference is:

The formula for the surface area (A) of a circle is:

Using Equations in Geometry

Geometry makes use of equations in solving for areas and volumes, such as the formulas provided in the reference sheet. The important thing is to be able to interpret these equations and relate equations to graphs and other visualizations.

Equations and formulas can be rearranged to isolate an entity you’re interested in. The formula for the volume of a cone, for instance, can be rearranged to solve for the cone’s height instead.

Complex Number System

Math questions involving complex numbers rarely appear in the TASC, but it is good to have a basic understanding of them. The expression 3 + 4i is a complex number because it is made up of real numbers (3 and 4) and an imaginary number (i). The imaginary unit number i is equivalent to $\sqrt{-1}$, which is unreal but may be useful in certain situations.

Rules of Probability

Probability is a number between 0 and 1, and it gives the likelihood that a thing, occurrence or event will happen. A probability closer to 1 means that there’s a bigger chance that a thing is going to happen. The formula for probability is: