Mathematics Study Guide for the ParaPro Assessment

During this section of the test, you will not be allowed to use a calculator. It is a test of skills and knowledge in three areas of mathematics: Number Sense and Algebra, Geometry and Measurement, and Data Analysis. This study guide will help you know what specific concepts you need to review.

Many of the questions will require you to know how to do mathematical operations and problem-solve to find the answer. Some of them, however, may also involve using the best method to help a student with math concepts.

The most productive way to help students is to help them discover a solution instead of just giving them the answer. You want them to be able to do it on their own the next time they see that type of problem. But, in order to do this, you have to understand basic math. That is what we will deal with in this study guide: basic math concepts and skills. Then, when a student is stumped, you can spot the point at which he or she began to have trouble, guiding the student to the next step and/or helping correct procedure mistakes.

Number Sense and Algebra

If your friend invites you to an extreme hike, he or she would expect you to have the common sense not to wear flip-flops. Similarly, If a pair of shoes worth $100 is on sale for 25% off, and you pay $95 at the checkout, you should know something is wrong. This is called number sense: simply, common sense for numbers. Many possible answers to multiple-choice tests can be immediately eliminated using number sense.

Most problems in math deal with unknowns. Rather than guessing and checking, we’ve developed a system called algebra to assign variables to those unknown values. Simply follow a few steps to isolate the variable and, voila, the answer appears!

Types of Numbers

Throughout the history of humanity, we’ve constantly been discovering new numbers. When a new number or set of numbers is found, the ability to express what’s happening around us becomes more precise and descriptive. Therefore, we’ve assigned names to these types of numbers.

Whole, Counting, and Natural Numbers—These were the first numbers we used as a species, and the first numbers you learned as a child. {1, 2, 3, 4, … } is the set of all counting or natural numbers. Surprisingly, it wasn’t until much later in human history that the number zero was added to this set, creating the set of whole numbers: {0, 1, 2, 3, 4, … }

Integers—Next came negative numbers. Add all negative counting numbers to the set of whole numbers and you’ve got the set of integers: { … , -3, -2, -1, 0, 1, 2, 3, … }.

Rational and Irrational Numbers—But not everything in life stays whole. What if you want to split a loaf of bread between two people? Then we have to have a number for the amount of the loaf you each have: $\frac{1}{2}$.

Rational numbers are defined as the set of numbers that can be written as a ratio (fraction) of two integers. Examples of rational numbers: $\frac{2}{3},\; -\frac{5}{92},$ and $\frac{20}{7}$.

Note: Every integer is also a rational number because it can be expressed as a fraction:

{$…, \;-\frac{3}{1}, \;-\frac{2}{1}, \;-\frac{1}{1}, \;\frac{0}{1}, \;\frac{1}{1}, \;\frac{2}{1}, \;\frac{3}{1}, …$}.

Also note: any decimal number that ends or repeats (like 0.5 or 0.33333….) can be written as a fraction and is, therefore, rational.

The ancient Greeks thought that all numbers were rational. They believed it so strongly that when a man called Hippasus challenged it, he was sentenced to death. His claim: that the square root of 2 cannot be written as a fraction of integers. Turns out he was right, and a whole new set of numbers was created. Irrational numbers are just numbers that are not rational.

Common examples include: $\sqrt{2},\; \pi,$ and $e$. If the square root of a number isn’t whole, chances are that the square root is irrational. Any non-repeating decimal that doesn’t end is irrational.

Real Numbers—Put all of the rationals and irrationals together and we’ve collected all the real numbers.

Imaginary Numbers—A question came along: what number multiplied by itself is -1? There is no real answer to this question. So, we created a number called i, the imaginary number, to solve this problem: $i = \sqrt{-1}$. Any multiple of i ($-2i, \;4i, \;\frac{1}{3}i,$etc.) is a member of the set of imaginary numbers.

Complex Numbers—Put all real numbers and imaginary numbers together using addition or subtraction and we’ve created the set of complex numbers. Examples: $2 + 3i,\; -0.35-2i,$and $-4 + \frac{3}{5}i$.

Radicals—You’re probably familiar with the term “square root”. Example: $\sqrt{9} = 3$ because 3 times itself is 9. Cube roots (and higher roots) also exist. Example: $\sqrt[3]{64} = 4$ because 4 times itself 3 times is 64 ($4^3 = 64$). These symbols: $\sqrt{},\; \sqrt[3]{}, \;\sqrt[4]{}$, etc. are radicals.

Prime Numbers—These are the building blocks of the integers. Try to find what numbers divide into 12 (other than 1 and 12). The numbers 2, 3, 4, and 6 do, and we call these numbers factors of 12. Now try to find factors of 13 (other than 1 and 13). There are none. We call the number prime because it has no factors other than 1 and itself. Example: 2, 3, 5, 7, 11, 13, …

Composite Numbers—If an integer isn’t prime, it is composite. This means it has other factors in addition to 1 and itself. Examples: 4, 6, 8, 9, 10, 12, …

Properties of Numbers

Now that we have an understanding of types of numbers, we need to agree on how they behave.

Commutative Property

$a + b = b + a$ (addition)
$ab = ba$ (multiplication)

Think about your morning and evening commute to and from work or school, and imagine you bring a coworker or friend along with you. The total drive to pick them up at their house (say 2 miles) followed by the remaining distance (say 3 miles) is the same as the total drive to drop them off (3 miles) followed by the distance to your home (2 miles). 2 + 3 = 3 + 2 is an example of the commutative property, and this story should help you remember the name.

The commutative property also works for multiplication and states that the order you add (or multiply) two numbers shouldn’t matter. The equation $5 \cdot 3 = 3 \cdot 5$ is another example of the commutative property.

Associative Property

\[a + (b + c) = (a + b) + c\]

Try it with $2+(3+5)=(2+3)+5$. This is an example of the associative property. Simply put, in addition (or multiplication) of 3 or more values, the way you group them shouldn’t change the result. Try $(3 \cdot 5) \cdot 2$ and then try $3 \cdot (5 \cdot 2)$ and you’ll find both answers to be the same. To remember the name: this property is about grouping. Think about a group of business “associates”.

Distributive Property

\[a(b + c) = ab + ac\]

The distributive property gives two different ways to simplify an expression where a sum is to be multiplied by a value. Example: $3 \cdot (2+5)$. The order of operations would simplify the sum first and then multiply by 3: $3 \cdot (2+5) = 3 \cdot 7 = 21$. The distributive property allows you to distribute the 3 to the two values, and then add the results: $3 \cdot (2+5) = 3 \cdot 2 + 3 \cdot 5 = 6 + 15 = 21$. Usually this property is used in algebra to simplify expressions with variables, such as here: 3(4x + 2) = 12x + 6.

Identity Properties

$a + 0 = a$
$a \cdot 1 = a$

What number can you add to 3 and not change the value? The answer is 0, and we call the number zero the additive identity. The property that you can add 0 to anything and its identity doesn’t change is called the identity property of addition.

What number can you multiply 3 by and not change its value?

This answer is 1, and we call the number one the multiplicative identity. The property that you can multiply anything by 1 and not change its identity is called the identity property of multiplication.

Place Value

Sometimes you’ll be asked to identify what place value a digit is in. Let’s take the number 8,765.432 for example. This table shows the place values for the particular digits.

\[\begin{array}{|c|c|c|c|c|c|c|} \hline \text{8} & \text{7} & \text{6} & \text{5} & \text{.4} & \text{3} & \text{2} \\ \hline \text{thousands} & \text{hundreds} & \text{tens} & \text{ones} & \text{tenths} & \text{hundredths} & \text{thousandths} \\ \hline \end{array}\]

What is the place value for the digit 3? Hundredths.
What number is in the thousands place value? 8.

The names continue to the left (after thousands): ten thousands, hundred thousands, millions, ten millions, etc. Similarly, on the right (after thousandths): ten thousandths, hundred thousandths, millionths, ten millionths, etc.

Expanded vs Standard Form

Written numbers are always in standard form. For example, 13, 506, and 12,347 are in standard form.

Expanded form demonstrates which place value each digit is in. Let’s say you’re asked to put the number 12,347 into expanded form. Reference this table:

\[\begin{array}{|c|c|c|c|c|} \hline \text{1} & \text{2} & \text{3} & \text{4} & \text{7} \\ \hline \text{ten thousands} & \text{thousands} & \text{hundreds} & \text{tens} & \text{ones} \\ \hline \end{array}\]

The response would be: $1 \cdot 10,000 + 2 \cdot 1,000 + 3 \cdot 100 + 4 \cdot 10 + 7 \cdot 1$

Number Value

You may be asked to put numbers in order to tell whether two numbers have the same value. This is pretty easy to do if we use integers, but if we mix types of numbers (fractions, decimals, exponents, etc.) the problem gets a little more challenging.

Equivalent Forms of Numbers

Reducing Fractions: This is the first skill you’ll need to remember to tell if two numbers are equivalent or not. To reduce a fraction, simply find the greatest common factor (GCF) of the numerator (top number) and the denominator (bottom number). Next, divide both numbers by the GCF. Example: Reduce $\frac{12}{42}$. Both $12$ and $42$ have $6$ as their GCF.

\[12 \div 6 = 2\] \[42 \div 6 = 7\]

So,

\[\frac{12}{42} = \frac{2}{7}\]

Note: If you didn’t know that $6$ was the GCF of $12$ and $42$, don’t worry! Just keep reducing a fraction until there are no common factors. Let’s say you knew $2$ was a common factor of $12$ and $42$.

\[12 \div 2 = 6\] \[42 \div 2 = 21\]

Now you should notice that $6$ and $21$ have a common factor of $3$. So, continue the process.

\[6 \div 3 = 2\] \[21 \div 3 = 7\]

Finally, you arrive at the same number. Along the process, we found another number that is equivalent to $\frac{12}{42}$ and $\frac{2}{7}$: $\frac{6}{21}$. ($\frac{12}{42} = \frac{6}{21} = \frac{2}{7}$). So, when asked if any fractions are equivalent, the best thing to do would be reduce each of them. Here is an example:

Which, if any, of these numbers are equivalent?

\[\frac{2}{3} \quad\quad\frac{5}{8}\quad \quad\frac{16}{18}\quad \quad\frac{15}{24}\]

Solution:

$\frac{2}{3}$ and $\frac{5}{8}$ are already reduced.

Reduce $\frac{16}{18}$ (divide both by $2$) and you have $\frac{8}{9}$.

Reduce $\frac{15}{24}$ (divide both by $3$) and you have $\frac{5}{8}$.

So, $\frac{15}{24}$ and $\frac{5}{8}$ are equivalent.

Converting Between Decimals and Fractions

How do we know that $0.5$ and $\frac{1}{2}$ are the same thing?

There are two main ways to show equality.

Turn $0.5$ into a fraction and reduce. The place value of the “$5$” digit is “tenths”. You write “tenths” as $\frac{1}{10}$. So, “$5$ tenths” can be written $\frac{5}{10}$. Reduce this and you’ll get $\frac{1}{2}$.
Turn $\frac{1}{2}$ into a decimal. Since you won’t have a calculator, use long division to divide $1$ by $2$. The answer will be $0.5$[.]

Example:

Are any of these numbers equivalent?

\[\frac{3}{4}\quad \quad0.7 \quad \quad \frac{6}{9} \quad \quad 0.75\]

Let’s try to turn them all into reduced fractions.

Already reduced: $\frac{3}{4}$

$0.7$ can be said “$7$ tenths” which can be written $\frac{7}{10}$, which is reduced.

$6$ and $9$ are divisible by $3$, so the reduced form is $\frac{2}{3}$

$0.75$ can be said, “$75$ hundredths” which can be written $\frac{75}{100}$. Divide both by $25$ and you get $\frac{3}{4}$.

So, in reduced form, the fractions above equal $\frac{3}{4}$, $\frac{7}{10}$, $\frac{2}{3}$, and $\frac{3}{4}$. The first and last fractions are equal.

Evaluating Exponents

An exponent (or power) tells how many times you multiply a number by itself. For example:

\[2^3 = 2 \cdot 2 \cdot 2 = 8\]

Do this to any value with an exponent when asked to compare number values.

Ordering Numbers by Value

So we can now tell if numbers are equivalent or not. But how do we tell which numbers are bigger than others? The key is to think of a number line. Convert all your numbers into standard decimal form and place them on the number line. Order these numbers from least to greatest:

\[3^2 \quad\frac{1}{4}\quad -\frac{3}{8} \quad-0.4\quad 0.3\]

To convert:

\[3^2 = 3 \cdot 3 = 9\]

$\frac{1}{4} = 0.25$ (Think $1$ quarter which is $25$ cents.)

\[-\frac{3}{8} = 3 \div 8 = -0.375\] \[-0.4\] \[0.3\]

So we have: $9$, $0.25$, $-0.375$, $-0.4$, and $0.3$.

Put them in order of value: $-0.4$, $-0.375$, $0.25$, $0.3$, and $9$.