Mathagogy: how to introduce the balance method for solving equations

The balance method:

Why teach this method? After all, there are many others which can bring much quicker success at a question like ‘4x + 8 = 24’, e.g. the ‘undoing’ method:

  • 4x + 8 = 24
  • 24 – 8 = 16
  • 16 ÷ 4 = 4
  • x = 4

To succeed with this method, a student simply has to apply BIDMAS and know which inverse functions are which. Yet, related to my previous post, this method makes things too easy: it minimizes cognitive load at the expense of wider algebraic fluency and understanding. Even more practically, the undoing method is inapplicable to solving equations like ‘4x + 8 = 24 + 2x’ and rearranging complicated formulae.

So, the balance method it is. But before you can get onto teaching it, many students lack these key prerequisite understandings:

  • 1. The equals sign means ‘the same as’ – as opposed to just ‘put the answer after this sign’.
  • 2. Inverse operations cancel each other out and thus don’t need to be calculated.
  • 3. Equations can be manipulated on either side.

This means any attempt to teach the balance method (where inverse operations are applied to both sides of an equation) seem completely alien to many students. They have never viewed equations as objects that can be manipulated. They do not understand why inverse operations make things disappear. They do not see each side of the equation as a mathematical object with meaning in itself. (That’s exactly why it might be tempting to teach an easier method – but don’t!) So, what else can you do? Here are some things that have worked for my classes.

Teaching strategies

Before you ever introduce the balance method, do the following:

Equals sign: asking students for their own definition; you’ll get some classic ‘it means here is the answer’. Push the definition of ‘the same as’.

Show examples of simple number facts using the equals sign in interesting ways: e.g. 6 = 6; or 6 = 7-1; or 2 x 6 = 9 + 3.

Ask whether these statements are true. Students should start to say ‘because 6 is the same as 7-1’. Stress such good responses. ‘Equals’ =  ‘the same as’. Slowly you’ll see lightbulbs click.

Activities: To get students accustomed to seeing expressions and operations on the right hand side of an equals sign:

equals sign

Inverse operations

Start by showing lots of examples of questions like ’36 + 2 – 2 =’ that get trickier and trickier, until calculating them is too difficult. Students will then see the pattern that they cancel out. Test them, and interleave understandings of the equals sign, with quick questions like:

inverse operations

Manipulating the equals sign

Start with numbers, such as:

manipulating

Let them calculate and see if it’s true; it should be obvious what they need to do to make it true.

Then move onto equations with unknowns; add something to one side, then ask what needs to be done.

Language

When you get onto equations, talk about ‘cancelling out’ various operations, to draw the link with inverse operations; remind students that they must then be done to both sides. Don’t talk about ‘moving over’.

Forcing the method

Many younger students will have some experience solving one/two-step equations using arithmetic methods (e.g. doing inverse operations on the number on the right). Furthermore, students with good number sense often just work out the answers from the meaning of the equation (‘2a+6=23… what times two, plus 7, gives me a 23?’). They will thus refuse to use the balance method and stick with what they know. This is where desirable difficulties are helpful. Use equations where the unknown is a negative number, or a fraction; this usually prevents the latter difficulty, and helps with the former – they aren’t sure what to do as the numbers ‘feel all wrong’, so will listen to what you tell them to do. (If you have fraction solutions, remember to teach them that divisions can be expressed as fractions.)

From my limited experience, 30-60 minutes spent on these explicit facts makes teaching the balance method – which is really just putting all these things together – go a lot smoother.

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2 thoughts on “Mathagogy: how to introduce the balance method for solving equations

  1. Pingback: Efficacy, efficiency, and mastery | mathagogy

  2. Pingback: The descriptive principle that learners minimize their cognitive load | mathagogy

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