Mathematics: Levels E, M, D, and A Study Guide for the TABE Test

Numbers and Quantity

Percentage of Test Level Specifically Assessing Numbers and Quantity (— = Assumed)

L	E	M	D	A
not tested	not tested	not tested	not tested	13%

We certainly have run across exponents and units a number of times in this guide, and this section pushes a bit further into them.

The Properties of Exponents

You may remember from earlier in this guide two important properties are used when multiplying and dividing expressions that have exponents. You’ll recognize the first two listed below, but there is a third that may come into play as you move into more work with exponents.

When multiplying two quantities with the same base, you add their exponents:

\[(x^a)(x^b)= x^{a+b}\ \ \ \ \text{or} \ \ \ \ (x^3)(x^4)=x^7\]

When dividing two quantities with the same base, you subtract their exponents (the top one minus the bottom one):

\[\frac{x^a}{x^b} = x^{a-b} \ \ \ \text{ or } \ \ \ \ \frac{x^5}{x^3} = x^2\]

When you have a power of a power, like \((x^3)^4\), you multiply the exponents:

\[(x^a)^b = x^{ab} \ \ \ \ \text{ or } \ \ \ \ (x^3)^4 = x^{12}\]

Radicals, such as square roots and cube roots can be written using fractional exponents. For example, \(\sqrt{2}\) can be written as \(2^{\frac{1}{2}}\).

Likewise, \(\sqrt{11}\) can be written as \(11^{\frac{1}{2}}\).

Here’s another example, this one with a cube root:

\[\sqrt[3]{22} = 22^{\frac{1}{3}}\]

There’s one more situation with exponents that can come up. What if you have an exponent inside a radical like \(\sqrt{x^3}\)? All you need to do is rewrite the square root as an exponent:

\[\sqrt{x^3} = (x^3)^{\frac{1}{2}}\]

Then you simplify by multiplying the exponents:

\[(x^3)^{\frac{1}{2}} = x^{\frac{3}{2}}\]

A fractional exponent may seem weird to you. Here is why it works out that way:

Suppose there is some exponent, \(x\), such that \(2^x \times 2^x = 2\).

We know \(2=2^1\). We also know that when numbers with the same base are multiplied, the exponents are added. So, if \(2^x \times 2^x = 2^1\), then \(x+x\) must equal \(1\) and \(x=\frac{1}{2}\).

Putting this together, we can add exponents and write \(2^{\frac{1}{2}} \times 2^{\frac{1}{2}} = 2\).

We also know that \(\sqrt{2} \times \sqrt{2} = 2\).

If both of these things are true, then \(2^{\frac{1}{2}}\) is the same thing as \(\sqrt{2}\).

Using Units

In an earlier section of this guide, “Converting Measurements”, we saw a method of changing one unit to another. Here, we will take a look at how the units can tell you how to solve a problem.

In Multi-Step Problems

By carefully letting units ride along with numbers in a problem, they can guide your multiplying and dividing so you don’t multiply when you should have divided or vice versa. Let’s start with just a single conversion of two made-up units.

Suppose \(25 \text{ nerts} = 1 \text{ nit}\). How many nits are in \(275\) nerts? Pretty clearly we need to do something with the \(275\) and the \(25\), but which of these do we do?

\[275 \times 25 \ \ \ \ \text{ or } \ \ \ \ 275 \times \frac{1}{25}\]

Let’s try it with the units of each number and see what cancels out:

\[\require{cancel}\] \[\text{Multiply the numbers and the units: } 275 \text{ nerts} \times 25 \frac{\text{nerts}}{\text{nit}} = 6875 \frac{\text{ nerts}^2}{\text{ nit}}\] \[\text{Multiply the numbers and the units: } 275 \cancel{\text{ nerts}} \times \frac{1 \text{nit}}{25\cancel{\text{ nerts}}} = 11 \text{ nits}\]

The second method will cancel out nerts and leave us with nits, which is exactly what we are looking for. That leaves us with the following game plan for doing unit conversions:

Write down the given quantity that needs to be converted, say \(5\) ft.
Find a unit rate, like \(\frac{ 1\text{ ft}}{12\text{ in}}\), that has the same unit as the given quantity.
Set up the unit rate so it will cancel the given unit: \(5 \text { ft} \times \frac{ 12\text{ in}}{ 1\text{ ft}}\).
Multiply and cancel: \(5 \cancel{\text { ft}} \times \frac{ 12\text{ in}}{ 1 \cancel {\text{ ft}}}=60 \text{ in}\).

If we needed to convert inches to something else, like maybe centimeters, we would just take \(60 \text{ in}\) as the given quantity and multiply by \(\frac{2.54\text{ cm}}{1 \text{ in}}\).

The beauty of this method of converting units is that if you pay attention to the units and canceling, they will tell you how to set up the conversion factors. If you’re good at this stuff and you don’t need that help, you can certainly convert units by just reasoning out the problem.

In Formulas

Another reason to pay attention to units is to compare the units in your answer to the units you would expect to get for that kind of problem. If they don’t match, that’s a big red flag, and you should double check what you’ve done. Let’s look at a few common formulas to see how this works.

What kind of units would you expect in an answer for these formulas? Let’s say all the variables are in meters. The variables are \(l\) for length, \(w\) for width, \(b\) for base, \(h\) for height, and \(r\) for radius.

\[lw \ \ \ \ \ \ \ \frac{1}{2}bh \ \ \ \ \ \ \ \ \pi r^2\]

The formula \(lw\) will give us \(\text{meters} \times \text{meters}\), which gives us square meters (also written \(m^2\)).
The formula \(\frac{1}{2} \times bh\) will give us \(\frac{1}{2} \times \text{meters} \times \text{meters}\), which again gives us square meters (\(m^2\)). We can ignore the \(\frac{1}{2}\) at the moment because it has no units.
The formula \(\pi r^2\) will give us \(\pi \times \text{meters}\times \text{meters}\), which yet again gives us square meters (\(m^2\)). \(pi\) has no units.

You may have recognized the three expressions we started with. Normally they would be part of a formula.

\[A = lw \ \ \ \ \ \ \ \ A=\frac{1}{2}bh\ \ \ \ \ \ \ \ A= \pi r^2\]

The point of this is to show that any time you have two length units multiplied together, you will have squared units in your answer, which is an area unit. To turn this around, if you are calculating area and you carry the units along in the problem, you should get squared units of some kind for the answer. If not, recheck.

We could go through the same kind of discussion for volume units. They are always formed when three length units are multiplied, resulting in cubed units. Notice that each volume formula below has three length units multiplied together. Remember, \(\frac{4}{3}\) and \(\pi\) have no units so they don’t count.

\[V = lwh \ \ \ \ \ \ \ \ V = s^3\ \ \ \ \ \ \ \ V = \frac{4}{3} \pi r^3\] \[V = lwh \ \ \ \ \ \ \ \ V = s \times s \times s\ \ \ \ \ \ \ \ V = \frac{4}{3} \times \pi \times r \times r \times r\]

In Graphs and Data Displays

When making a graph, choose a reasonable data scale for each axis. Using the data in the table below, what scales would be good to use for time and distance?

Time (hours)	Distance (miles)
0	0
1	59
3	177
6	354
7	413
10	590
12	708

Let’s take a look at the time data first. It ranges from \(0\) to \(12\). If you make each square equal to \(1\), that will make the graph \(12\) squares high. That’s reasonable, so let’s go with that. The time scale will be vertical:

\[0, \,1, \,2, \,3, \,4, \,5, \,6, \,7, \,8, \,9, \,10, \,11, \,12\]

Or maybe just label every other square:

\[0, \,2, \,4, \,6, \,8, \,10, \,12\]

How about the distance scale? It has a bigger range, from zero to \(708\). We definitely can’t use one square equals one mile. No graph paper in the world would have enough squares for that. How many squares are reasonable? Usually between \(10\) and \(20\), depending on the graph paper. Let’s try \(15\).

If \(15\) squares have to go from zero to \(708\), we need to divide \(708 \div 15=47.2\) for each square. That would be a weird number of miles for each square, so let’s just round it off to \(50\). Our horizontal scale would be:

\[0, \,50, \,100, \,150, \,200, \,250,\, 300, \,350,\, 400,\, 450, \,500, \,550, \,600, \,650, \,700, \,750\]

As above, for the final scale, we could number every other line:

\[0, 100, 200, 300, 400, 500, 600, 700, 750\]

This scale covers the whole range of distances and it’s reasonably easy to do.

Scaling isn’t an exact science. It’s more an estimate of what would be reasonable, and will not give you two many or too few squares.

Level of Accuracy

When you are doing calculations with measurements, how you report your answer becomes important. We’ll do an example.

The formula to calculate density is \(D=\frac{m}{v}\), where \(m\) is mass and \(v\) is volume.

Suppose a student in a science class uses a graduated cylinder to measure a volume of \(24.5\) mL of alcohol. She then uses a balance and finds that the mass of the alcohol to be \(17.8\) g. Calculate the density of the alcohol.

\[D = \frac{m}{v} = \frac{17.8 \ g}{24.5 \ mL} = 0.730612 \frac{g}{mL}\]

Now, here is what the section is all about: The answer written above, with all those decimals, looks like it must be a super accurate number. On the other hand, \(24.5\) and \(17.8\) don’t appear to be that accurate, with only three digits each. So how can our answer be more accurate than the numbers it came from? It can’t.

So, what do we do? It’s pretty simple. All we do is round off our answer so that it doesn’t have so many digits. How many digits are reasonable? A good rule of thumb is to round so that your answer has the same number of digits as the numbers you started with, not counting the first zero in \(0.730612\).

That gives us a value of \(0.731\).

Throw the units in there and get a density of \(0.731 \frac{g}{mL}\).

There can be a lot more to this, but the point of this section is to do your best to make sure that the answer you report doesn’t look more accurate than the data that it is based on. Keep in mind that this only applies to calculations with measurements. If you’re working with only numbers (no units), there are no rules about rounding off your answer. Just do whatever your instructor wants you to do.

BONUS: Problem Solving Tips

Much of the TABE allows you to use a calculator, so it is not testing your computing skill. Rather, you will be assessed on how well you can take given numbers and design a means for finding a solution to a problem, including which operation(s) to use and in what order to use them.

Here are some procedures with which you will need to be fluent. We have just given the basic information about them and you should pursue more information and practice, especially for areas in which you are unsure.

Skills to Practice

Many students cringe at the thought of word problems and the math section of the TABE is made up of these. Not only do you have to do a given operation with numbers, but you first have to figure out which operation, or operations, to do. But there are clues in every word problem to help you out and understanding these will be valuable.

Choosing the Right Operation(s)

Be familiar with buzzwords and clues because these will make it easier for you to translate words and phrases into numbers and mathematical operations.

A mathematical sentence can express an equality or inequality.

The words is and equal to suggest an equality of two statements, and are represented by the equal sign (=).
Greater than and less than suggest an inequality, and are represented by the greater than sign (>) and less than sign (<), respectively.
Then there are the phrases greater than or equal to and less than or equal to, which are written mathematically as \(\ge\) and \(\le\), respectively.

Watch out for these words and the operations they indicate:

Plus, combine, total, sum, together, more, and increase are used to indicate addition.
Minus, less, difference, left, take away, decreased, and fewer are used to indicate subtraction.
Times, product, twice, thrice, and of are used to indicate multiplication.
Per, quotient, and ratio are used to indicate division.

Identifying Unimportant Information

Irrelevant information may be deliberately inserted in math questions to make simple questions seem complicated or to mislead test-takers. Familiarity and constant practice in solving word problems will help you find what’s useful information in a math question. It helps to know formulas. A given number could be irrelevant if it does not fit in the formula. Sketch and label as you read the question. With a visual guide, an irrelevant piece of information will be easier to spot.

Manipulate Formulas

Math questions may provide the areas or volumes and ask for the other elements instead. In this case, plug in the known values, represent the unknown values with letters, isolate the unknown value to one side of the equation, and solve for the unknown.

For instance, a TABE math question may ask for the height of a cylindrical tank if it has a capacity of 30 cu m of water and has a radius of 1.5 m.

Write the appropriate formula and plug in the known values:

\[V = \pi r^2 h\] \[30 = \pi (1.5)^2 h\]

Isolate the unknown value and perform the required operations:

\[h = 30 \div \pi(1.5)^2 = 4.2 m\]