Stuart Donnelly

Article / Updated 08-23-2017

Numbers come in all shapes, sizes, and forms, such as fractions, decimals, and percentages. When a TASC Math question asks you to compare numbers or put a set of numbers in a certain order, it's easiest to do when they're all in the same form. Converting fractions to decimals. To convert a fraction to a decimal, divide the numerator by the denominator. The quotient could result in a terminating, repeating, or non-terminating, non-repeating decimal. To represent repeating decimals, put a bar over the digits that repeat. Converting decimals to fractions. If the decimal is terminating, meaning it ends, you put the given digits over a power of 10. If there are two digits, you put them over 102 and then simplify if possible. Check out this example: You can always check your work by dividing numerator by denominator to make sure you get the same decimal you started with. Here's another conversion example, this one with three digits: If the decimal is repeating, put the given repeating digits over 9s (depends on the number of digits that repeat). If there's one digit that repeats, then it goes over one 9; if there are two digits that repeat, they go over two 9s, so 99. Then simplify if possible. For example: Here's another example: Converting percentages to decimals. To convert a percentage to a decimal, you must consider how you got that percentage in the first place. Percentage literally means per 100, so to go from a percent to a decimal, you move the decimal two places to the left. Mathematically this means you're dividing by 100. Take at look at these examples: Practice questions Which of the following is listed in ascending order? Which of the following is in descending order? Answers and explanations The correct answer is Choice (A). Comparing numbers in different forms can be difficult, so you want to rewrite them all into the same form (usually decimals is the easiest). .6 is already a decimal, and 350% = 3.5. Now that they're all in decimal form, you need to remember what ascending means: smallest to largest. This means the order should be Substituting back in the original values, you get: which is Choice (A). The correct answer is Choice (D). First rewrite each of the numbers into the same form (decimal form is probably the easiest). Now you can order them in descending order, which means from largest to smallest: 2.64, .7, .05, .03 and then substitute your original numbers back in: thus Choice (D) is correct.

Article / Updated 08-23-2017

You will probably encounter questions on the TASC Math exam that involve exponential functions. If the problems appear in the form of a graph or a table, the following instructions will help you navigate through them. An exponential function is when the independent variable is in the exponent of a constant. The base of the function must be greater than 0 and not equal to 1. Some examples of exponential functions are: If the base of the exponential function is a fraction, then the graph falls rapidly to the right, as shown here. Looking for these general shapes when given a graph will indicate whether the graph represents an exponential function. If you're given a table, to determine whether the function is exponential, check if there's a common multiple difference, meaning you can multiply each of the y-values by a number to get to the next y-value. This indicates that the function is exponential and, in fact, that that number is your base. To write an equation for an exponential function, follow these steps: Find the common multiple difference (this is your base, b). Find your y-intercept—this is the coefficient of your exponential function (a). Substitute your values for a and b into the general form of an exponential: For example: First find whether there's a common multiple difference. It's important to note that the x-values are evenly spaced, which allows the common multiple to be identifiable. You now know that your base is 2. Looking for your y-intercept, you see it's (0, 3), so a = 3. Now substitute these values into

Article / Updated 08-22-2017

If you encounter a scatter plot on the TASC Math exam, you’ll probably be asked to show how the two variables in the chart are related to each other—or whether a relationship exists between them at all. A scatter plot shows the relationship between two variables. When looking at a scatter plot, you look at the correlation, which gauges the strength of the relationship and the direction. This means a correlation can be strong or weak and can be positive, negative, or neither. Depending on the strength of the correlation, you can infer a trend in the relationship. The figure shown here illustrates some examples of scatter plots and the types of correlations that can appear. Notice how when there is a correlation, the points tend to line up in one direction. A common example of a scatter plot is the relationship between people’s shoe sizes and their IQs. When a large data collection is analyzed, you see that there’s no correlation. If there were one, you could make a statement like “People with bigger shoe sizes are smarter.” However, there’s a wide range of IQs and shoe size combinations, and you can’t gauge a person’s intelligence based on his or her shoe size (no correlation). Practice question What statement corresponds to the relationship illustrated by the following scatter plot? A. The longer you study, the worse you will do on the test. B. The longer you study, the better you will do on the test. C. The amount of time studying does not affect the score on the test. D. Not enough information can be gathered. Answer and explanation The correct answer is Choice (B). Because the trend line, or line of best fit, of this scatter plot has a positive slope, there is a positive relationship between the variables. This means as one increases the other increases: the longer you study, the better you will do on the test, Choice (B) is correct. For there to be no relationship, the line of best fit would be a “flat” horizontal line. For a negative relationship, the slope would be negative or “pointing down.”

Article / Updated 04-25-2017

If you encounter a problem on the TASC Math exam that involves conversion of units, it's important that you keep track of the relationships between measurements. For instance, it's common knowledge that there are 12 inches in 1 foot, which is called a unit factor. Unit factors are made up of two measurements that describe the same thing. You can use this fact when translating one rate or measurement. Here's an example involving unit factors where multiple ratios are needed. Imagine that a car is traveling 20 miles per hour (mph) but you need to know how many inches per minute the car is going. This requires knowledge of three unit factors: 1 foot is 12 inches, 5,280 feet is 1 mile, and 60 minutes is 1 hour. Again, start with what you know and then place your converting ratios strategically based on where the units will cancel. Then multiply what's left in the numerator and denominator like regular fractions. Don't forget to simplify at the end. You don't always use unit factors to convert between measurements. Sometimes you'll be told the relationship between two quantities of measurement, such as miles per gallon in a car. Another example is in scale or model drawings. These depictions represent something that's either too large or too small to draw at a normal size, such as a house or plant cell. The process for dimensional analysis stays the same, but it's important to read the questions carefully to pick out the units of conversion and to figure out what the question is asking you to measure. Area is always measured in square units, perimeter is measured in linear units, and volume is measured in cubic units. Practice question Cory is a skateboarder and can travel at a speed of 15 miles per hour. How fast does Cory go in feet per minute? A. 1,320 B. 1,750 C. 1,820 D. 4,752,000 Answer and explanation The correct answer is Choice (A). This a converting units problem, and it's important to remember how many feet are in a mile and how many minutes are in an hour. Setting up your ratios you get: (recall that it's important to place units in opposite places). Simplifying this conversion, you find that Cory travels at 1,320 ft/min, Choice (A). If you misplaced the converting ratios, then you would end up with Choices (B) and (D).

Article / Updated 04-25-2017

While most questions on the TASC Math exam require you to deal with real numbers, you'll probably run into one or two problems that involve complex numbers. The first time most people encounter complex numbers is in algebra, when they find out that it's possible to take the square root of negative numbers. The important thing to remember here is that This means, for example, that Complex numbers aren't just numbers that occur when taking the square root of negative numbers, though. They include any number that can be represented in the form a + bi, where a is the real part and bi is the imaginary part. This means any real number is a complex number when b = 0. Using this definition, the Venn diagram shown here illustrates how complex numbers are the intersection of real numbers and imaginary numbers. Because complex numbers are still numbers, you can perform arithmetic operations with them, such as adding, subtracting, multiplying, and dividing. When you add or subtract two complex numbers, you combine (add or subtract) the real parts together and the complex parts together. Example: (4 + 2i) + (5 + 8i) = (4 + 5) + (2 + 8)i = 9 + 10i Example: (9 + 5i) – (11 – 2i) = (9 – 11) + (5 – –2)i = –2 + 7i When multiplying two complex numbers, treat them more like polynomials than traditional numbers. This means you have to do double distribution. The box method is useful here because it keeps you organized and helps prevent losing terms. To perform multiplication using the box method, separate out each term of the complex number either along the side or on top of the box. To fill in each inside the box, multiply the column header by the row header. Lastly, you need to combine like terms (the two terms that have i in them). Take a look at this example: (2 + 3i)(4 – 5i) Thus (2 + 3i)(4 – 5i) = 8 + 12i – 10i + 15 = 23 + 2i Dividing two complex numbers would look like this: To perform this division problem, you multiply both the top and the bottom of the quotient by the complex conjugate of the denominator. The complex conjugate of the denominator looks like the original denominator but with the opposite sign, so you would multiply the original question by: This results in a rational denominator. Try working through this example: Multiply as if they are regular fractions: Now multiply these two complex numbers: Simplify and you get this solution: This tells you that the real number part of the answer is and the imaginary part is

Article / Updated 04-24-2017

If a question from the TASC Math exam asks you to solve for the solution to a system of equations, one useful approach is to graph the system of equations. Graphically, the solution is the point or points where the lines or curves intersect. This means to solve a system of equations (linear, quadratic, and so on) by graphing, you follow these steps: Graph each function independently but on the same coordinate plane. Look for the point or points where the functions intersect. Test the points you identified by substituting them into all original equations. While this step is optional, it's highly recommended because graphs can be drawn inaccurately if generated by hand. Practice question Which system of equations is represented by the following graph? A. y = 2x – 1; y = x + 3 B. y = –2x + 3; y = x – 1 C. y = –x + 3; y = 2x – 1 D. y = –x –1; y = 2x + 3 Answer and explanation The correct answer is Choice (B). The first thing to do is identify the y-intercepts: 3 and –1. Now find the slopes of the lines associated with each of the y-intercepts; the line with a y-intercept of 3 has a negative slope, which eliminates Choices (A) and (C). Further inspection allows you to conclude that the slope associated with 3 is –2, while the slope of the line with the y-intercept of –1 is 1. This means the equation of the two lines is y = –2x + 3 and y = x – 1, which is Choice (B).

Article / Updated 04-24-2017

If you're given the standard form of a quadratic equation on the TASC Math exam, the equation provides valuable information when you're asked to graph it. A quadratic function is a polynomial in which the highest degree is two. The shape of the graph representing the quadratic function is called a parabola, which is a U shape. You can create a table, select values for the independent variable (x), and substitute in and solve for the values of the dependent variable (y). Similar to how it takes two points to construct a line, you need a minimum of three points to make a parabola. The standard form of a quadratic function is y = ax2 + bx + c. To graph a quadratic function in standard form, follow these steps: Determine the axis of symmetry (this is also the x-coordinate of the vertex). Substitute the value obtained in Step 1 back into the original formula to determine the y-coordinate of the vertex. Pick two points that are equidistant from the x-coordinate of the vertex. Substitute these values into the original formula (these resulting y-coordinates should be the same if the x-coordinates are the same distance away). Try to pick when x = 0 as one of the values because doing so simplifies the algebra. Plot the three points (vertex and two points from Step 4) on the coordinate plane. Connect the points you plotted in a smooth curve (careful, you don't want to make the graph pointy or V-shaped). Here's an example: y = 2x2 – 4x + 3 a = 2 b = –4 c = 3 y coordinate: y = 2(1)2 – 4(1) + 3 y = 2(1) – 4 + 3 y = 1 vertex: (1, 1) Because 0 is one away from 1, you pick 2 as the other point. To obtain the y-values, substitute 0 and 2 back into y = 2x2 – 4x + 3 and solve.

Article / Updated 03-22-2017

Some questions on the TASC Math exam involve linear functions. A linear function represents a relationship between two variables in which one variable influences the other. In a linear function, x is usually considered to be the independent variable and y to be the dependent variable (x influences y). The independent variable (x) runs horizontally, while the dependent variable (y) runs vertically. The minimum number of points you need to construct a line is two. The common difference of the points that describes both the steepness and direction of the line is called the slope. This is also referred to as the ratio of the rate of change in the dependent variable to the rate of change in the independent variable. The letter associated with slope is m; if m is positive, then the line rises to the right, and if m is negative, then the line falls to the right. To determine the slope of a line, you need two points: (x1, y1) and (x2, y2). Then substitute into the formula: Slope-intercept is the most commonly used formula to represent a linear function. Just as the name implies, this formula tells you the slope (m) of the line as well as the y-intercept (b). Recall that the y-intercept is the point at which the graph crosses the y-axis. Slope-intercept form: y = mx + b (m is the slope and b is the y-intercept) Practice questions The equation of the line perpendicular to Which line would be parallel to the line y = –3x + 4? Answers and explanations The correct answer is Choice (A). Because you're looking for a line perpendicular to the given line, the new equation would have a negative reciprocal of the given slope: so the new slope is m = 2. Because Choice (A) is the only equation with that slope, it is the correct answer. Choice (C) represents a line parallel to the given line, while Choices (B) and (D) have slopes that have no specific relationship with the given slope. The correct answer is Choice (A). Because you're looking for a line parallel to the given line, the new equation would have the same slope: m = –3. Because Choice (A) is the only equation with that slope, it's the correct answer. Choice (C) represents a line perpendicular to the given line, while Choices (B) and (D) have slopes that have no specific relationship with the given slope.

Article / Updated 03-22-2017

If you run across a system of equations problem on the TASC Math exam, two ways to solve it—if you decide to avoid graphing—are through substitution and elimination. Using the substitution method To solve a system of two variables using the substitution method, follow these steps: Solve one or both equations for one of the variables. Substitute what one variable equals into the other equation. Solve the resulting equation for that one variable. Substitute the found value back into either original equation to solve for the second value. You can also use this method to solve more than two variables, but it may require a bit more algebra because there will be more equations involved. Try this example: y = 2x + 5 3x – 4y = 10 Because the first equation is already solved for y, you substitute what y equals into the second equation. Now that you have a value for x, you can substitute it into either of the original equations. So the solution to the system is (–6, –7). You can check this solution by substituting both values into both original equations. Using the elimination method To solve a system with two variables using the elimination method, follow these steps: Rewrite both equations so their variables are in the same order. One pair of variables must have the same coefficients but with opposite signs. For example, if one equation has 2x, then the other equation needs to have –2x. If there isn't an existing pair of coefficients that meet this requirement, you must multiply or divide one or both equations. Add the equations together; the pair identified in Step 2 should cancel. Solve the produced equation for the remaining variable. Substitute the value obtained in Step 4 back into one of the original equations. Solve for the other variable. Try this example: Notice how the y's fit the requirement in Step 2. Add the two equations together. Now that you have a value for x, you can substitute it back into either of the original equations. The solution to the system is (3, –1/3).

Article / Updated 03-22-2017

Not all equations that you'll deal with on the TASC Math exam will be in single variables. There are common formulas that you may need to rearrange to find a certain variable or combination of variables. The following examples illustrate how to isolate a particular variable given an equation or formula. Example 1: In this example, the formula is how force is calculated: Force equals mass times acceleration. As indicated, you're solving for mass, m. F = ma To isolate m, notice that it's being multiplied by a. This means you must divide by a on both sides. Example 2: This formula is used to calculate the volume of a cone given its height and radius. As indicated, you're solving for the radius, r. First, multiply by 3 to "get rid of" the fraction. Don't be scared of pi; it's just a number that can be divided. Lastly, take the square root of both sides to isolate r. Example 3: Because you're looking for xy, you need to simplify both sides and get any terms with xy onto the same side. Distribute 4x. Notice that xy is on one side, and all that's left is to divide by 2. This means that the solution to this problem is 4 – 10x.