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

Operations and Algebraic Thinking: Part 2

Percentage of Test Level Specifically Assessing Operations and Algebraic Thinking (— = Assumed)

L	E	M	D	A
38%	22%	12%	—	—

Different Thinking about the Equal Sign

Here’s an idea that comes up all the time in algebraic thinking. As well as telling you “Here’s the answer” like $3 + 5 - 2 = 6$, an equal sign can also be thought of as a sign of balancing. The total value on the left side of the equal sign must always be the same as the total value on the right side.

Examples:

$9 = 9$
$9 = 10 - 1$
$3 + 6 = 7 + 2$
$4 + 3 + 11 = 9 + 9$

Finding Unknown Numbers

Be able to come up with an unknown number in an equation with two other numbers. Here are a few examples.

\[\text{?} + 7 = 15\] \[12-\text{___} = 3\] \[9 \times \text{___} = 72\] \[40\div \text{___} = 8\]

Equation and Expression Ideas

In the next few sections we will take a wider look at equations and use new skills to solve them. A term that comes up often in algebraic work is “expressions”. An expression is any group of numbers, variables, and math operations like addition, subtraction, multiplication, and division. An equation is also any combination of numbers, variables, and operations, but it has an equal sign and some value or expression on both sides. That value may be zero.

Variables and Constants

In algebraic work, any number you can think of, say $7$, or maybe $44$, is called a constant. That makes sense because they never change in value; $7$ always means the same thing.

On the other hand, there are variables. These are letters that take the place of numbers that are not yet known. $x$ can mean $18$ in one problem and $5$ in another one. You could put a blank line in that place and it would mean the same thing as a variable.

\[\text{___} + 22 = 33 \ \ \ \ \ \ \ \ x + 22 = 33\]

So, the two equations above mean the same thing. The first one is asking you to fill in the blank to make the equation true, and the second one is asking you to figure out the value for $x$ that will make the statement true. To answer the first one, you would just write $11$ in the blank. To answer the second one, you would write $x=11$. Same idea.

Additive Comparisons

Sometimes you will run across problems that say things like, “The height of a box is four more than twelve,” or “A chicken weighs $15$ lb less than a turkey.” You need to recognize that this type of numeric comparison can be used to solve a problem. Look for the phrases “more than” or “less than”.

Example:

A grapefruit weighs one pound and a watermelon weighs $11$ pounds more. How much does the watermelon weigh?

Answer: $1 + 11 = 12$ pounds.

Multiplicative Comparisons

Numeric comparisons can also be given in terms of multiples, as in this:

A squirrel is $14$ inches long, and a desk is $4$ times longer than a squirrel. How long is the desk?

Answer: The desk is $4 \times 14 = 56$ inches long.

Parentheses, Brackets, and Braces

Parentheses, brackets, and braces are symbols used to group expressions.

$\{\}$ are called braces.
$[\ ]$ are called brackets.
$()$ are called parentheses.

Parentheses are always used inside brackets, if there are any, and brackets are always used inside braces, if there are any.

\[\{[(\;)]\}\]

Using parentheses: $31 +(6 -2) +13 -(4+4)$
Using brackets: $31 +[(6-2) + 13] - (4+4)$
Using braces: $\{31 + [(6-2)+13] -(4+4)\}$

Parentheses basically say, “Do inside here first.”
Brackets say, “Do inside here after you’re done with parentheses.”
Braces say, “Do inside here after you’re done with brackets.”

Consider this expression:

\[12 +[3-5 +(8-3)]\]

The first step is inside the parentheses: $8-3 = 5$, so write $5$ in place of $(8-3)$ and get $12 +[3-5+5]$.

The second step is inside the brackets: $3-5+5 = 3$, so write that in place of $[3-5+5]$ and get $12 + 3$.

Then do the final step: $12 - 3 = 9$.

The Order of Operations

Lengthy and seemingly complicated mathematical operations can be simplified by following the PEMDAS rule which states the order of operations: Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right). Here’s an example:

\[5 - (4 + 1)^2 \div (5^2 \cdot \frac{1}{5})\]

Start with performing the operations inside the parentheses:

\[5 - (4 + 1)^2 \div (5^2 \cdot \frac{1}{5})\] \[5-(5)^2 \div (25 \cdot \frac{1}{5})\]

The expression then becomes:

\[5-25 \div 5\] \[5 - \frac{25}{5}\] \[5 - 5\] \[0\]

Relating Word Problems and Equations

For every word problem, there is an equation that can be solved to answer it. Your job will be to use key words, good thinking, and experience to translate the words into equations. The key words to look for have been listed earlier in the section titled Operations. Also, given an equation, we can come up with a word description that fits. Word problems aren’t usually easy, but if you think about it, real-world math problems are all word problems, so they are worth learning.

Word Problem to Equation

Here we’ll take a closer look at some examples of translating basic word problems into equations. Each equation will need a letter (variable) to represent the unknown quantity.

Susie Q. bought a sandwich for $\$3.50$ and a root beer for $\$1.75$. How much did she spend in total?

We see the key word total which means add. What do we add? The sandwich and the root beer. Let’s use the letter $t$ to stand for total, and write an equation, but you don’t need to solve it. We can skip the dollar signs for now.

\[3.50 + 1.75 = t\]

Moses Malone bought three pairs of socks and a baseball. The baseball was $\$6.25$ and the total he spent was $\$13.75$. Let s stand for the price of a pair of socks, and write an equation for this situation.

If the price of a pair of socks was s, then the socks would cost $3$ times s. Add that to $\$6.25$ to get a total of $\$13.75$.

\[3 \times s + 6.25 = 13.75 \ \ \ \ \ \ \ \text{ or }\ \ \ \ \ \ 3s + 6.25 = 13.75\]

If a computer screen is a rectangle with a height of $7$, an unknown width, and an area of $91$, use these facts to write an equation using w for the unknown width.

\[7 \times w = 91 \ \ \ \ \ \ \ \text{ or } \ \ \ \ \ \ 7w = 91\]

Equation to Word Problem

Be able to look at an equation and imagine a situation that the equation could fit. An example will make this clearer.

Think about this equation with multiplication:

T$= 3 \times 12$

It could be used for a lot of different situations. In general, it means you have $12$ groups of $3$ things, or $3$ groups of $12$ things. Here are a few examples:

It could mean, “Find the number of cookies in $3$ dozen cookies.”
It could mean, “If a yardstick is $3$ feet long, how many feet are in $12$ yards?”
It could mean, “A bike moves at $12$ miles per hour. How far will it go in $3$ hours?”

Now, think about this equation with division:

\[N = 66 \div 22\]

It could be written as $N = \frac{66}{22}$. Here we are talking about a large group being divided into smaller groups.
It could mean, “If there are $66$ cookies and $22$ kids, how many cookies will each kid get?
It could mean, “If a chain is $66$ inches long and has $22$ links, how long is each link?
It could mean, “If a barrel is $22$ inches around, how many times can you wrap a $66$ inch string around it?

Other Number Ideas

Here are two other aspects of handling numbers that will be very good to know. Spend a little time with these so you’re able to deal with them when they come up.

Factors and Multiples

Factors:

If you multiply two numbers together to get a third number, the two numbers you multiplied are called factors of the third number. A couple of examples would be good here.

If 3 $\times$ 4 = 12, then we say that $3$ and $4$ are factors of $12$. We call $3$ and $4$ a factor pair.

If $2 \times 6 = 12$, then $2$ and $6$ are also factors of $12$, and $2$ and $6$ are another factor pair.

You can see that $12$ has two different factor pairs: $3$ and $4$ and $2$ and $6$. We could also include the factor pair $1$ and $12$, and conclude that $12$ has three factor pairs. This is something you should be able to do. That is, given a number from $1$ to $100$, be able to list all of its factor pairs. Here’s one more example using a bigger number:

What are all the factor pairs of $48$?

How about $1$ and $48$? $2$ and $24$? $3$ and $16$? $4$ and $12$? $6$ and $8$? They all work.

Multiples:

Remember that $3 \times 4 = 12$ means $4 + 4 + 4 = 12$ or $3 + 3 + 3 + 3 = 12$. Seeing this, we can say that $12$ is a multiple of $3$. It would be just as true to say that $12$ is a multiple of $4$. In short, any time there is a product, that product is a multiple of its factors.

So, not only is $12$ a multiple of $3$ and $4$, it’s also a multiple of $2$ and $6$ because they are all factors of $12$.

This table shows some factors of $30$.

35 Factors of 30.png

Since the factors here are $2, \,3,\, 4, \,6, \,10$ and $15$, we can say that $30$ is a multiple of each one of those numbers. We can’t forget that $1$ and $30$ are also factors of $30$, though they aren’t in the table. In total, $30$ is a multiple of eight numbers; $1, \,2, \,3, \,5,\, 6, \,10, \,15$ and $30$.

Prime and Composite Numbers

Speaking of factors, if a number has only the factors 1 and itself, it’s called a prime number. Let’s look at a few numbers to see if they are prime.

First of all, think about 1, 2, and 3. Each is prime because each one has no factors other than 1 and itself.

How about 4? It has the factors 1 and 4, but also 2 and 2, so it’s not prime.

Numbers that are not prime are called composite numbers.

To tell if a number is prime, then, you need to see if you can factor it other than using 1 multiplied by itself.

18 = 3 x 6, so 18 is not prime. It’s composite. You need to try dividing each number that could possibly be a factor. If there are none (except 1 and itself) it is prime.

Tip: Factors will never be higher than half the number. For example, no factor of 100 will ever be higher than 50.

What about 22? It’s an even number, so it has a factor of 2 for one. It’s not prime. No even number will be prime (except for 2).

One more: Try 31. Is 2 a factor? No. Is 3 a factor? No. In fact there is no factor other than 1 and 31, so 31 is prime.

Multi-Step Word Problems

You need to be able to solve word problems with more than one step, staying with only whole numbers. Do these by using a letter (a variable) that stands for the quantity you’re looking for and writing an equation based on what the problem gives you. See if you can use rounding and estimation to help you judge how correct your answer is.

Patterns

A math question may show a series of numbers that has a pattern and ask you to identify the $n$th element. First, you need to know whether the pattern shows an arithmetic sequence or a geometric sequence.

You know that a pattern is an arithmetic sequence if the same value is added to an element to get the next element. The series $5, \,8, \,11, \,14, \,17, \,20, …$ is an arithmetic sequence because $3$ is added to get the next element. A question may go about asking you to give the next, the $100$th, or the missing element in a pattern.

Finding Patterns

Here is a formula to help you predict a specific element without going through repetitive addition. The formula for computing the $n$th element, $x_n$ is:

\[x_n = x_1 + d (n -1)\]

where $x_n$ is the $n$th element $x_1$ is the first element $d$ is the common difference between elements

In the given arithmetic sequence, we can solve for the $100$th element using the formula:

\[x_{100} = 5 + 3(100-1) = 5 + 297 = 302\]

The elements may also be added using this formula for the sum of arithmetic sequences:

\[\text{Sum } = \frac {n}{2} [(2 \cdot x_1) + d(n -1)]\]

To take the sum of the first six elements of the above sequence without adding them one by one:

\[\text{Sum} = \frac {6}{2} [(2 \cdot 5) + 3(5)] = 3 (10 + 15) = 75\]

This series of numbers, $3, \,6,\, 12, \,24, \,48, \,96, …$ is obviously not an arithmetic sequence because the difference between elements is not constant. This is a geometric sequence. You get the next element by multiplying the previous element by a constant; in this sequence, the constant multiplier is $2$.

The formula for finding the $n$th element is the following:

\[x_n = x_1 \cdot r^{(n-1)}\]

where $x_n$ is the $n$th element
$x_1$ is the first element
$r$ is the common ratio, or the common divisor

For the given series, the $12$th element can be computed using the formula this way:

\[x_{12} = 3 \cdot 2^{11} = 6\text{,}144\]

The formula for the sum of a geometric sequence is:

\[Sum = x_1 \cdot \frac{(1 - r^n)}{(1 - r)}\]

To find the sum of the first six elements using the formula:

\[Sum = 3 \cdot \frac{(1-2^6)}{(1-2)} = 3 \cdot \frac{(1-64)}{(1-2)} = 189\]

There are more complicated patterns that you may also explore,such as square number sequences, Fibonacci sequences, triangular sequences, etc. The patterns presented here, however, are the basics.

Creating Patterns

Be able to make up a sequence rule and generate terms of the sequence. Try an arithmetic sequence first. As we just saw above, an arithmetic sequence can start with any number and add any number over and over to get a sequence. So, pick a starting number, say $6$, and generate an arithmetic sequence by using $4$ to add to each term:

\[6, \,10, \,14,\, 18, \,22, \,26,\, 30, …\text{ Notice: all even numbers}\]

Start with $7$ and use $4$ to add to each term:

\[7, \,11,\, 15, \,19, \,23,\, 27, \,31,\, … \text { Notice: all odd numbers}\]

What about a geometric sequence? Start with $3$ and use $-3$ to multiply by each term:

\[3,\, -9, \,27,\,- 81,\, 243,\, … \text{Notice: alternating negative numbers}\]

Can you explain the alternating negative signs in this sequence? Maybe something like this: Any time you multiply by a negative number, the sign changes. Since you are always multiplying by a $-3$, the signs will always alternate.