Counting Factors: Exponent Laws with Variables

We are learning to derive and apply the exponent laws for variables by counting factors, including zero and negative exponents.

Before you startAbout 60 minutes · what success looks like · notes for the helper

About 60 minutes. It's fine to stop after the Quick Check and come back another day: the lesson opens where you left it, at the Quick Check.

You'll know you've got it when

I can explain the product, quotient and power-of-a-power laws by writing out and counting factors
I can use x⁰ = 1 and x⁻ⁿ = 1/xⁿ, and explain why they follow from the quotient law
I can simplify expressions with coefficients, negative signs and more than one variable, and check by substituting a number

For the helperhow to sit beside this lesson, and what this step is for

This step: Spark

Start with the question at the top and chat about it. There is no wrong answer here: it gets your learner thinking about what they already know.

Getting ready: Paper, a pencil and perhaps a dozen small scraps of paper with x written on them, so factors can be laid out and cancelled by hand if that helps (Factor Strips needs about twenty, and scissors). The lesson runs about 60 minutes.
What understanding sounds like: You might listen for an exponent being explained by counting: 'x³ is three x's and x² is two more, so five altogether', and for x⁰ = 1 and x⁻³ = 1/x³ explained from the quotient law rather than recited. A longer simplification with a coefficient, a negative and two letters, checked by putting numbers in, shows the third criterion.
A likely mix-up, and something that helps: The most common slips are multiplying the exponents in x³ × x² (giving x⁶) and reading x⁻² as a negative number. One way to respond is to ask, 'Can you write that out as factors?' or 'Try x = 2 in both versions — do they give the same number?'
If it comes easily: A good puzzle for a confident learner is why the zero-exponent law needs x ≠ 0, or simplifying (x²y⁻³)⁻² and checking the result with numbers.
If it's a struggle: If it is a struggle, one way to step back is to use numbers first — 2³ × 2² written out in full and multiplied — and then swap the 2 for x once the pattern is clear. If your learner says they are 'just bad at algebra' and stops, counting the paper slips together is a way back in that does not feel like a test. Where English is an additional language, base, exponent, coefficient and factor are the words that carry this lesson.

Step 1 of 6 · Spark

Lesson 1 of 5 · Taking Algebra Apart and Putting It Back Together

Five factors or six?

For the helper

Start with the question at the top and chat about it. There is no wrong answer here: it gets your learner thinking about what they already know.

Ask a room of Year 10 students to simplify x³ × x² and you will usually get two answers: x⁵ and x⁶. Both camps feel sure. Both are applying a rule they half remember. Before you read any further, decide which you believe and write down why. Then ask yourself a harder question: if you had forgotten the rule completely, how could you work it out from nothing — and how could you check it?

x³ × x²: is it x⁵ or x⁶? A rule you can rebuild from scratch is a rule you can't misremember.