4.4: Appropriate Use of the Equal Sign

Always, Sometimes, or Never

We will explore three ways of thinking about statements. There are three possibilities for a statement: always true, sometimes true, or never true.

For example, the statements \(10=10\) and \(11=13-2\) are always true for these whole numbers. However, the statements \(1=2\) and \(20=4+17\) are never true. Statements that are sometimes true can be trickier to think about. For example, \(11=13-x\) could be true only if \(x=2\), but other values like \(x=3\) will make it false. So this means that \(11=13-x\) is sometimes true.

When using the equal sign to solve problems, it is common to misuse it. While you could get a correct answer, the process can still be incorrect. The following discussion discusses an incorrect process to solve \(2.30+3.20-1.50\).

The answer is 4, and that is correct. However, the solution is unclear and incorrect. It is true that

\[2.30 + 3.20 = 5.50 \ldotp \nonumber \]

But it is not true that

\[2.30 + 3.20 = 5.50 - 1.40 \ldotp \nonumber \]

Notice that each equal sign used in a process means that everything is equal to the original expression, so this process would claim that \(2.30 + 3.20 = 4\), which is also not true. The incorrect use of the equal symbol makes a calculation difficult to understand.

Pan-Balance Scale Model

This picture shows a (very simplistic) two-pan balance scale. Such a scale allows you to compare the weight of two objects. Place one object in each pan. If one side is lower than the other, then that side holds heavier objects. If the two sides are balanced, then the objects on each side weigh the same.

In the following discussion and examples, we will explore the idea of keeping the two sides balanced and connect it to algebraic thinking.

Notice that the picture starts by giving you five assumptions, the first four confirm that all figures that look the same will weigh the same, and the last one supposes that the scale is balanced. From the rest of the section and for the assigned problems, please make these assumptions.

In the first representation above, you could remove one triangle on each side and still keep the balance. Which will result in one triangle on one side and two circles on the other. This means that one triangle weighs the same as two circles.

In the second representation, you could group the figures. If you have two stars on the left and four circles on the right, an approach to work on this is to distribute the weight equally for each of the figures. In other words, we could assign two circles to each star. So each star has the weight of two circles.

In the third representation, you could remove one square on each side, which will result in one star on the left and two squares on the right. Therefore, one star weighs the same as two squares.

This foundational discussion will drive the problems and examples for this section.

Notice that while we provided a solution in words, we could also represent this by using equations. Below, we provide a different solution and we also represent it in a different way. We use equations and the equal sign for solving the question from the previous example: How many purple squares will balance with one circle?

Using the first representation we have:

\[2 \text{ triangles}= 6\text{ circles}\]

This is equivalent to saying:

\[1 \text{ triangle}=3 \text{ circles}\]

Now, using the second representation, we have the following equation:

\[3 \text{ triangles}=9 \text{ squares}\]

Which is equivalent to saying:

\[1 \text{ triangle}=3 \text{ squares}\]

Now by using the two results: \(1 \text{ triangle}=3 \text{ circles}\) and that \(1 \text{ triangle}=3 \text{ squares}\), notice that both include \(1\) triangle. So, we have that

\[3 \text{ circles}=1 \text{ triangle}=3 \text{ squares}\]

Which means that:

\[3 \text{ circles}=3 \text{ squares}\]

And we can conclude that:

\[1 \text{ circle}=1 \text{ square}\]

To purposefully connect algebraic processes with balance scales, you could think about figures to represent a specific variable. Let's start by thinking about a scale that is balanced with one square and five circles on the left side and two squares and two circles on the right side. If you think of squares as a representation of \(x\) and circles as units or just whole numbers, then you will have the equation:

\[x+5=2x+2\]

We will expand on this idea in Section 4.6.

Although this representation has limitations, especially with negative numbers, the key idea of keeping an equation balanced is key to understanding algebraic reasoning and the meaning of the equal sign.