How to Prepare for the Numbers and Operations Questions on the SBAC Mathematics Test

General Information

The questions on this test are not divided into skill areas, but span the whole realm of both skills and reasoning abilities covered thus far in your math instruction. Some of them will at least partially involve being able to work with numbers and perform operations on them that produce correct answers when solving problems. You’ll also need to be able to explain why your procedures work and why you chose them in the first place.

Skills to Understand and Be Able to Use

Properties of Exponents

The properties of exponents predict the outcome of operations involving terms with exponents. In this test, you also need to understand why they work in addition to predicting the result.

What Are the Properties and Why Do They Work?

These are the basic properties of exponents:

\[{x^n \cdot \,x^m = x^{(n + m)}}\] \[{(x^n)^m = x^{(n \cdot m)}}\] \[{\frac{x^m}{x^n} = x^{(m - n)}}\] \[{\quad (x \cdot y)^m = x^m \cdot y^m\quad}\] \[{x^{-m} = \frac{1}{x^m} }\] \[{\text{or} \;\frac{1}{x^{-m}} = x^m}\] \[{x^0 = 1}\]

But knowing what they are and even being able to use them is not sufficient to perform well on this test. You’ll need to be able to explain why they work, as follows.

One example of these properties is:

\[a^{m} \cdot a^{n} = a^{(m+n)}\]

This works because the exponent term \(a^{m}\) means \(a \cdot a \cdot a \cdot… \cdot a\) with the a being repeated m times.

So \(a^{m}\) is m copies of a multiplied together, and \(a^{n}\) is n copies of a multiplied together.

Therefore, in total, there are \((m+n)\) copies of a, or \(a^{(m+n)}\).

Similar reasoning applies to the other properties.

Rewriting Expressions

The properties of exponents extend to rational numbers. For example, if \(a^{2}\) is squaring a number, the inverse operation, square root, will use the inverse of 2, which is \(\frac{1}{2}\).

So \(a^{\frac{1}{2}} = \sqrt{a}\) = the square root of a, and by extension \(a^{\frac{1}{n}}= \sqrt[n]{a}\) = the nth root of a. The properties show how to rewrite exponent expressions by performing operations with the exponents. The exponents are rational numbers and follow all rational number properties.

Properties of Rational and Irrational Numbers

Rational numbers can be written as the ratio of two integers, while irrational numbers cannot be written as a ratio of integers. You need to be able to explain why you identify a number as rational or irrational.

What Are the Properties and Why Do They Work?

Properties of rational numbers include closure, commutative, associative, and distributive. Look these up if you do not have a firm understanding of them.

Closure is simply the idea that an operation on two numbers of a similar type results in a number of that same type (rational or irrational).

• Rational numbers are not closed under division because division by zero is not defined.

• Irrational numbers do not have the closure property, because in some cases such as \(\frac{\sqrt{3}}{\sqrt{3}} = 1\), the result is a rational number.

• Irrational numbers are decimals that go on forever without any repeating pattern.

Two Rational Numbers: Adding and Multiplying

Rational numbers are closed under addition and multiplication. The result will always be a rational number. You perform the operations using the arithmetic you have always used in your school career.

Rational and Irrational: Adding

The sum of a rational and irrational number, such as \(3 + \sqrt{5}\), cannot be written as a ratio of integers, so it is just shown as \((3 + \sqrt{5})\), which is irrational. If you need to compute an answer, you use a rational approximation such as \(\sqrt{5} \approx 2.236\), so \((3+ \sqrt{5}) \approx 5.236\).

Non-zero Rational Number and an Irrational Number: Multiplying

If we want to multiply \(3 \times \sqrt{5}\), since we don’t have an exact value of \(\sqrt{5}\), we just express the result as \(3\sqrt{5}\). The product of a non-zero rational and an irrational number is irrational. Again, if we need to compute, we use a rational approximation of the irrational number.

Using These Properties in Math

Since the exact value of an irrational is unknown, we treat them as if they were algebraic variables in mathematical expressions. There are a few properties of radical expressions you should know, such as these:

\[\sqrt{3} \cdot \sqrt{5} = \sqrt{3 \cdot 5} = \sqrt{15}\] \[\sqrt{20} = \sqrt{4} \cdot \sqrt{5} = 2\sqrt{5}\]

Using Units

You can understand some problems by analyzing the units associated with the numbers. For example:

\[\frac{\text{miles}}{\text{hour}} \times \text{hours} = \text{miles}\]

It tells you what to do with the numbers and is very helpful in complicated multi-step problems. This is called dimensional analysis.

Units in Formulas

Units must be chosen and interpreted consistently in a problem. For example, if some times are given in minutes and others in hours, you must convert one of them so all the time units are the same, either all in hours or all in minutes.

Units in Data Displays

When interpreting or creating data displays such as graphs, be careful to notice the values at the origin of the graph and the scale on each axis. Some graphs don’t start at zero, and some graphs have different scales on each axis. These values are often chosen in a way that makes the trends in the data easy to spot.

Quantitative Reasoning

Identifying the quantities in a problem will help you to assign the appropriate units. Quantities are usually the result of a measurement and represent something that can be measured.

Defining Quantities

When attempting to model a system, choose appropriate quantities. For example, modeling the fuel consumption of a car will probably use gallons of fuel, but modelling the liquid consumption of a thirsty mouse would make more sense by using ounces or milliliters.

Level of Accuracy

When reporting the results of measurements, be sure to review and use the concept of significant figures. Every measurement has a limit on accuracy. For example, if the accuracy is \(\pm\) one gram, it makes no sense to report a result down to hundredths of a gram, even if the calculation results in several decimal places.