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

Operations and Algebraic Thinking: Part 1

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

L	E	M	D	A
38%	22%	12%	—	—

To ease into algebra, we will take a look at different ways of thinking about numbers.

Using Objects, Models, and Equations

You can use objects and models as helpful ways to think about numbers and why certain mathematics operations work. One of the most basic things in algebra is the equation, and we will see lots of examples of them in the sections to follow.

Simple Adding

You need to be able to add up to three whole numbers that have a sum not exceeding \(20\). It can help to use something physical such as an object or drawing to see the logic of addition. Suppose Angela owned three bracelets, Claire owned six, and Maria owned four. How many bracelets did they own together?

You could model these numbers by using disks, as shown. With that model before you, you could then just count the disks. Let’s start from the left.

\[3 \text{ disks } --- 6 \text{ disks } --- 4\text{ disks }\] \[1, 2, 3 \ \ \ \ 4, 5, 6, 7, 8, 9 \ \ \ \ 10, 11, 12, 13\] \[\text{The sum is }3+6+4=13\]

With Addition and Subtraction

Be able to use addition and subtraction using numbers less than \(100\) to solve one- or two-step word problems using unknown symbols (the most common unknown symbol by far is the letter \(x\)). Let’s look at a couple examples.

A one-step problem:

Speedy Joe bought a steering wheel cover and a sun visor for his car for \(\$36\). If the sun visor cost \(\$19\), how much did the steering wheel cover cost? We can write an equation for this situation. We know that something plus \(\$19\) equals \(\$36\), so we could write this:

\[\text{____} + \$19 = \$36\]

It would mean the same thing if we used the symbol \(x\) to represent the missing number:

\[x + \$19 = \$36\]

Either way, we need a number that adds with \(\$19\) to make \(\$36\). This is a case for subtraction: \(\$36 - \$19 = \$17\). That’s the price of the steering wheel cover.

Using a more algebraic method, we can do the problem this way:

\[x + \$19 = \$36\]

Subtract \(\$19\) from both sides:

\[x + \$19 - \$19= \$36- \$19\] \[x = \$17\]

A Two-Step Problem

Johnny Applepicker picked \(27\) apples on Saturday, and \(41\) apples on Sunday. If he used \(9\) apples to make pies on Monday, how many apples does he have left?

Step 1—Add the apples picked: \(27+41=68\).

Step 2—Subtract the apples used: \(68 - 9 = 59\) apples left.

With Multiplication and Division

Be able to use multiplication and division, where appropriate, with numbers up to \(100\) to solve word problems like those below.

An office building has \(18\) floors. Floors \(1\) and \(2\) are occupied by a store and the other floors have \(21\) offices on each floor. How many offices are there in the building?

This problem has a number of equal-sized groups (each floor has \(21\) offices) to add together. This is perfect for multiplication. How many of the groups are there? There are 16 because there are \(16\) floors. (Don’t get fooled and use \(18\). The bottom two floors aren’t offices.)

So, we multiply the number of groups, \(16\), by the number in each group, \(21\):

\[16 \times 21 = 336 \text{ offices}\]

The offices have a total of \(48\) copy machines. If the copiers are spread evenly, how many of them are on each floor?

Here we are dividing a group of \(48\) among \(16\) floors, which means we need to divide \(48\) by \(16\):

\[48 \div 16 = 3\]

If \(n\) stands for the number of boxes, complete the calculation for the number of boxes in the array below.

\[n =\] \[\begin {array}{cccccc} &\Box&\Box&\Box&\Box&\Box&\Box&\\ &\Box&\Box&\Box&\Box&\Box&\Box&\\ &\Box&\Box&\Box&\Box&\Box&\Box&\\ &\Box&\Box&\Box&\Box&\Box&\Box&\\ &\Box&\Box&\Box&\Box&\Box&\Box&\\ &\Box&\Box&\Box&\Box&\Box&\Box&\\ &\Box&\Box&\Box&\Box&\Box&\Box&\\ \end{array}\]

There are six boxes in each row and seven rows, so you would multiply and write this: \(n = 6 \times 7\).

Counting On and Counting Back

If the numbers you’re working with aren’t too big, you can use a strategy called “counting on” or “counting back”. Counting on is for addition and it’s almost exactly like adding on a number line that we saw earlier in this review.

Suppose we want to add 4 to 2.

We start with the 2 and count onward over the next 4 higher numbers: 3, 4, 5, 6. The number we end on, 6, is the sum of 2 and 4. It is just like counting 4 past 2 on the number line, except you don’t use a number line.

Counting back is also a lot like subtracting on a number line.

To subtract 2 from 4, start at 4 and count back the next two lower numbers. From 4 you would count 3, 2. The number you end on is the result of 4 - 2.

These work on bigger numbers than we have used here, say maybe 25 + 8, but you probably wouldn’t use it for 25 + 31.

Using Operation Properties

The common operations of addition, subtraction, multiplication, and subtraction have certain patterns that are pointed out below. Being able to use these patterns, called properties, will help guide you in handling these operations.

Addition and Subtraction Properties

Commutative property for addition: When adding, the order of numbers makes no difference.

Example:

\[15 + 9 = 9 + 15\]

There is no commutative property for subtraction, because there the order matters: \(13-8\) doesn’t equal \(8-13\).

Associative property for addition: When adding three or more numbers, how they are grouped makes no difference.

Example:

\[(19 + 9) +11 = 19 + (9 + 11)\] \[28 + 11 = 19 + 20\] \[39 = 39\]

There is no associative property for subtraction, because in subtraction, grouping matters.

There are a couple of small tricks to help with your addition skills. One of them is called making tens. Here’s how it works.

Suppose you had the problem \(9 + 8 + 2\). If you add the \(8\) and \(2\) first, you will get \(9 + 10\), which is pretty easy to add and get \(19\). That’s a bit easier to do than adding \(9+8\) first. This is a good trick, but you won’t be able to use it in every problem. Look for numbers that will add up to \(10\), such as \(1\) and \(9\), \(2\) and \(8\), \(3\) and \(7\), \(4\) and \(6\), and \(5\) and \(5\).

Another mental trick doesn’t have an official name, but you might think of it as making twins. Adding two of the same number is easier for many of us than adding two different numbers. Make use of that. If you have \(8+9\), mentally break up the \(9\) into \(8\) and \(1\). Now you have \(8 + 8 + 1\). Now, \(8 + 8 = 16\), and add the \(1\) and get \(17\).

Multiplication and Division Properties

In this section we’ll take a look at properties of multiplication and division to see how they can be used to help work out problems.

Commutative property for multiplication: In multiplication, the order of numbers doesn’t matter.

Example:

\[12 \times 15 = 15 \times 12\]

Associative property of multiplication: In multiplication, how you group numbers doesn’t matter.

Example:

\[(7 \times 3) \times 4 = 7 \times (3 \times 4)\] \[21 \times 4 = 7 \times 12\] \[84 = 84\]

Distributive property of multiplication over addition: Think of distributing one number over two others.

Example:

\[7 \times (3 + 5) = 7 \times 3 + 7 \times 5\] \[7 \times 8 = 21 + 35\] \[56=56\]

Using Inverse Operations

Certain operations undo each other. For example:

\[12 + 9 = 21 \text{ and } 21 - 9 = 12\]

Adding \(9\) to \(12\) gets you \(21\), but subtracting \(9\) from \(21\) gets you back to \(12\). Here’s another example:

\[6 \times 9 = 54 \text{ and } 54 \div 9 = 6\]

Multiplying \(6\) by \(9\) gets you \(54\), but dividing \(54\) by \(9\) gets you back to \(6\).

These kinds of undoings will pop up all the time in algebra. Solving equations often involves moving numbers and variables around in an equation using exactly these ideas. When one operation undoes another operation, they are called inverse operations. Subtraction is the inverse of addition, and division is the inverse of multiplication.

Addition and Subtraction

Since addition and subtraction are inverse operations, thinking of both sides of a problem this way can help in understanding it. For example, the result of the problem \(14 - 8 = \text{__}\) can also be thought of as \(8 + \text{__} = 14\); what number makes \(14\) when added to \(8\)?

Multiplication and Division

Be able to multiply any two two-digit numbers. It can help to think of both the multiplication and division sides of these kinds of problems. For example, if you know that \(6\times 7 = 42\), then the other side of that is \(42/7 = 6\) or \(42/6 = 7\). You need to have memorized the products of any two numbers up to \(10 \times 10\).