The notion of equality is fundamental in mathematics and especially in algebra and algebraic thinking. The symbol “=”’ expresses a relationship. It is not an operation in the way that + and are × operations. It should not be read left-to-right, and it definitely does not mean “… and the answer is …”.

For your work to be clear and easily understood by others, it is essential that you use the symbol = appropriately. For your future students to understand the meaning of the = symbol and use it correctly, it is essential that you are clear and precise in your use of it.

Examples

Example 1.3.1

Consider the following problem.

Kim solved the problem as follows:

2.30 + 3.20 = 5.50 – 1.50 = 4. So the answer is 4

a) What do you think about Kim’s solution? Did she get the correct answer?

b) Is her solution clear? How could it be better ?

c) Rewrite Kim’s solution making correct use of the equal sign. Show your work below .

An equation is a statement asserting that two expressions have the same numerical value (represent the same real number). Like all statements, equations may be true or false. All equations and inequalities fall into one of three categories:

Always true: for example 2𝑥 − 3 = 1 + 𝑥 − 4 + 𝑥

Never true: for example 2x = 2x + 5

Sometimes true: for example 3x – 5 = 10

Example 1.3.2

Examine the following equations. Decide whether the statement is always true, never true or sometimes true. Record your answers below and explain how you know.

a) [latex]5 + 3 = 8[/latex]

b) [latex]5k = 5k + 1[/latex]

c) [latex]5 + 3 = y[/latex]

d) [latex]\frac{2}{3} + \frac{1}{2}=\frac{3}{5}[/latex]

e) [latex]n+3=m[/latex]

f) [latex]\frac{a}{5}=\frac{5}{a}[/latex]

g) [latex]3x=2x+x[/latex]

Example 1.3.3

Consider the equation [latex]18 - 7 = \rule{1cm}{0.4pt}[/latex]

a) What can you put in the blank that will make the equation always true?

b) What can you put in the blank that makes the equation always false?

c) What can you put in the blank that makes the equation sometimes true and sometimes false?

A solution to an equation is a value for the variable or variables that makes the equation true.

Illustration

1. The solution to  2𝑥 − 3 = 1 + 𝑥 − 4 + 𝑥 is any real number x
2. 2x = 2x + 5 does not have any solutions
3. The solution to 3x – 5 = 10 is x = 5.

In general English, a ‘solution’ is a way to fix a problem. And even in math, if you are not talking about an equation, the word ‘solution’ is often used to refer to the ‘answer’ or more accurately ‘a method for finding the answer’ to a problem. But, in the context of equations, the ‘solution’ to an equation does not mean “the answer” or “a method for finding an answer,”but rather it means “the values for the variables that make the equation true.”

Example 1.3.4

Consider the following problem: [latex]6+4 = \rule{1cm}{0.4pt}+3[/latex].

a) Jose thinks that 10 goes in the blank. Why might he say this, and what would you say to him as his teacher ?

b) Mari argues that 7 goes in the blank because 6 and 4 is 10. And you have to add 7 to three to get 10.  Peter argues that 7 goes in the blank because 3 is one less than 4 and so you have to fill in the blank with one more than 6.   How are Mari and Peter’s arguments different? As the teacher of these children, what do these two responses tell you about what they know?  Record your answers below .

As a teacher you will need to help your students understand that ‘=’ does not simply mean ‘write the answer,’ and instead help them to see the equation as a balance.

Example 1.3.5

How might you use a balanced scale to demonstrate for your class a solution to the problem: [latex]6+4 = \rule{1cm}{0.4pt}+3[/latex]. Draw a corresponding picture below.

Your use of the ‘=’ sign is important; your students will be watching and learning from the way you write and talk about mathematics. Remind them that an equation is like a balanced scale: whatever you put on or take away from one side, you must do to the other side to maintain the balance.

Give your students experiences that will help them think this way. Write equations (with blanks for them to fill in) in a variety of ways.

Illustration

Different ways to ask fill in the blank questions with equalities

1. [latex]\rule{1cm}{0.4pt}=6+4[/latex] may be rewritten as [latex]10=\rule{1cm}{0.4pt}+6[/latex] or [latex]10-\rule{1cm}{0.4pt}=4[/latex]
2. [latex]\rule{1cm}{0.4pt}\times3=24[/latex] may be rewritten as [latex]24\div\rule{1cm}{0.4pt}=8[/latex]

Example 1.3.6

Fill in the blank for each equation and then rewrite the problem with the blank in a different place .

a) [latex]13-7=\rule{1cm}{0.4pt}[/latex]

b)  [latex]7 \times 4 = \rule{1cm}{0.4pt}[/latex]

c) [latex]15 \div 3 = \rule{1cm}{0.4pt}[/latex]

d)  [latex]12 + 5 = \rule{1cm}{0.4pt}[/latex]