Powers of Ten: What Place Value Has Been Saying All Along

We are learning to write natural numbers in expanded notation using powers of ten, and to connect each exponent to a place-value column.

Before you startAbout 50 minutes · what success looks like · notes for the helper

About 50 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 write a power of ten such as 10³ as a repeated multiplication and as a numeral
I can write a natural number in expanded notation using powers of ten, including numbers with a zero in some columns
I can rebuild a numeral from its expanded notation, keeping every placeholder zero
I can explain why 10⁰ = 1 and why the ones column is the 10⁰ column

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 a place-value chart drawn across a page with columns from ones up to millions. A newspaper, a letter with a big figure in it, or a car's odometer gives a few real large numbers to work with. Two of the choice tasks use more: scissors for the match-up cards, and sticky tape to join sheets into a long strip. The lesson is about 50 minutes, and the guided problems are a natural place to pause if it needs two sittings.
What understanding sounds like: Listen for your young person saying what the small raised number counts: '10³ is three tens multiplied together, so 1000'. Linking each exponent to a column, and saying the ones column is 10⁰, is the understanding this lesson is after.
A likely mix-up, and something that helps: The most common slip is reading 10³ as 10 × 3 = 30, or skipping a zero column so the exponents slide down. You might ask, 'What does the small number count?' and have them write out 10 × 10 × 10 before giving the value.
If it comes easily: For a learner who is flying, Australia's population (about 27 million) and the distance to the Moon (about 384 400 km) are good numbers to write in expanded notation, with a guess at which power of ten each is closest to.
If it's a struggle: If it is a struggle, one way is to stay with four-digit numbers and write each column's value both ways, 1000 and 10³, in the chart heading, before writing any expanded notation. For a learner still building English, base, exponent and power are the words this lesson rests on; a card with 10³ drawn and its parts labelled can stay on the table. If they shut down, it is fine to stop: one number rebuilt together is a real piece of work, and the rest keeps for another day.

Step 1 of 6 · Spark

Lesson 1 of 5 · Inside a Number: Powers of Ten, Primes and Squares

How big is a column?

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.

The average distance from Earth to the Moon is about 384 400 km. You can read that number without thinking, but try explaining why the 3 is worth 300 000 and not 3. Your answer will probably be something like 'because it is in the hundred-thousands column'. That is true, but it only moves the question: why is that column worth 100 000? Today you will see that every column is a power of ten, and that place value has been doing repeated multiplication the whole time.

384 400 km: each step left multiplies the column's value by ten. Six columns, six different powers of ten.