# Properties of equality – do you need them to solve equations?

July 27, 2010 3 Comments

Should students learn the properties of equality before we ask them to solve equations in one variable? Would it be too hard for them to solve the equation 2(x + 7) = 4x without knowing the properties of equality?

In the grades, pupils learn to find equivalent ways of expressing a number. For example 8 can be written as 4 + 4, 3 + 5, 4 x 2, 10 – 2. Now, what has the pupils previous experience of expressing numbers in different ways got to do with solving equations in one variable?

Let us take this problem. What value of x will make the statement 2(x-5) = 20 true?. The technique is to express the terms in equivalent forms.

2(x-5) = 20 can be expressed as 2(x-5) = 2(10).

2(x-5) = 2(10) implies (x – 5) = 10

x-5 = 10 can be expresses as x-5 = 15 – 5. Thus x = 15.

This way of thinking can be used to solve the equation 2(x + 7) = 4x. (Note: Apart from applying the properties of equality, it can also be solved by graphing.)

2(x+7) = 2(2x)

=> (x+7) = 2x

=> x + 7 = x + x

=> x = 7.

Of course not all equations can be solved by this method efficiently. So why not teach them the properties of equality first before asking them to solve equations like these? What will students learn from this method, anyway? Here are some benefits:

1. It makes students focus on the structure of the equation. Noticing equivalent structure is very useful in doing mathematics.

2. It makes the equations like 3x = 18, x + 15 = 5, which are used to introduce how the properties are applied, problems for babies.

3. It is easier to do mentally. Try solving equations using the properties of equality mentally so you’ll know what I mean.

4. I hope you also notice that the technique has similarities for proving identities.

So when do we teach the properties of equality? In my opinion, after the students have been exposed to this way of solving and thinking.

Click here on how to introduce the properties of equality via problem solving.

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I like it! I have encouraged many students to replace quantities with equivalent quantities… but it never occurred to me to have them solve an equation solely by creating similar structures (other than with exponential equations before they have been introduced to logarithms). Good puzzle-solving approach which, as you say, is also very useful in later mathematics all the way through Calculus and beyond.

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