What an Exponent Really Says

We are learning to read, write and expand powers, and to explain why the base and the exponent do different jobs.

Before you startAbout 55 minutes · what success looks like · notes for the helper

About 55 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 name the base and the exponent of a power, and write a repeated multiplication in exponent notation
I can expand a power into repeated multiplication and evaluate it without confusing 2³ with 2 × 3
I can compare powers such as 2¹⁰ and 10² and explain why swapping the base and the exponent changes the value

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 and a pencil are enough for most of it, and about 55 minutes; the lesson splits cleanly after the guided problems if a second sitting suits better. A calculator is optional, because the lesson is designed to be done by hand. The Swap Test wants a ruler for the grid; the cube task wants a couple of dozen dice, sugar cubes or small blocks (3³ takes 27), or grid paper instead.
What understanding sounds like: Listen for the exponent described as a count of factors, such as 'three twos multiplied together', rather than 'two times three', and for a repeated product turned into a power with the right base and exponent. Saying which number is repeated and which number counts, and why 2¹⁰ and 10² end up so far apart, is the understanding this lesson is after.
A likely mix-up, and something that helps: The most common slip is reading 2³ as 2 × 3 = 6. One way to respond is to ask, 'Can you write it out the long way to check?' It tends to land best as a habit worth keeping rather than as being caught out.
If it comes easily: For a learner who is flying, one question to offer is whether any pair of different whole numbers other than 2 and 4 gives the same value when the base and exponent swap, and how they could convince you there isn't one below 10.
If it's a struggle: If it's a struggle, one way is to say every power aloud in full ('two, multiplied by itself, three twos in all') and write the expanded form before evaluating, building the answer one multiplication at a time. The words base, exponent, squared and cubed carry this lesson, so a learner still building English may find it helps to label one power with arrows and keep it in view. If they shut down, it is fine to stop after one power done well and come back to the rest another day.

Step 1 of 6 · Spark

Lesson 1 of 4 · Powers: The Shorthand for Repeated Multiplication

Same two digits, very different numbers

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.

Here are two numbers made from the same two digits: 2¹⁰ and 10². Before working anything out, make a quick guess. Are they about the same? Is one double the other? Write your guess down. Most people guess they're close, because both are built from a 2 and a 10. By the end of this lesson you'll know exactly how far apart they are, and why the gap is no fluke.

2¹⁰ versus 10². Same digits, different jobs. Which number is being repeated, and which one is doing the counting?