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# Place value and powers of 10

Number Foundations · Place value and estimating

<!-- Speaker note: Students build numbers on place value columns to see that each column is a power of 10, write numbers in expanded notation with exponents, and convert back (AC9M7N03, nodes 1 and 2). About 40 minutes (Predict and Run 8 min · Investigate 12 min · Modify 12 min · Practise and Make 8 min). -->

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## Today

**We are learning to** each place value column is a power of 10: the thousands column is 10³, the ones column is 10⁰ = 1. So 4052 = 4 × 10³ + 5 × 10¹ + 2 × 10⁰.

- I can say which power of 10 each place value column is.
- I can write a number in expanded notation using powers of 10.
- I can turn expanded notation back into a number.

<!-- Speaker note: Before class: Do the Run board yourself first. Students need exponent notation (Powers lesson) and place value for whole numbers (Year 6). The one idea: each column is ten times the one to its right, so the exponent is the column's position counting from 0 at the ones. The common mistake is counting from 1 (Sam), so every exponent is one too big. The empty column (a 0 digit) may be left out of expanded notation; writing 0 × 10² is also correct, and this lesson's answers leave it out. Two nodes in one lesson: the columns and their powers (Run, Investigate), then expanded notation both ways (Modify, Practise). -->

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## Why it matters

> The 4 in 4052 is worth 4000, but the 5 is worth only 50. What does the column a digit sits in do to its value?

<!-- Speaker note: You already know that the 5 in 4052 is worth 50, not 5. A number is built from its digits, each worth *digit × its column*. Every column is ten times the one to its right, so every column is a **power of 10**: | Column | ones | tens | hundreds | thousands | ten thousands | | --- | --- | --- | --- | --- | --- | | Power of 10 | $10^0$ | $10^1$ | $10^2$ | $10^3$ | $10^4$ | | Value | 1 | 10 | 100 | 1000 | 10 000 | The exponent counts the zeros in the value: $10^3 = 1000$ has three zeros. Writing a number as a sum of *digit × power of 10* is called **expanded notation**. -->

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<span style="background:#dbeafe; color:#1a1a1a; border:1.5px solid #3b82f6; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">💬 Discussion</span>

## Predict

**What is $10^3$?**

- 100
- 1000
- 10 000
- 30

<!-- Speaker note: What happens: 1000. 10³ = 10 × 10 × 10: three 10s multiplied, so 1 followed by three zeros. 30 is 10 × 3. Take two or three predictions before anyone runs it. -->

<!-- Speaker note: Teacher: students who chose 30 read 10³ as 10 × 3. -->

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<span style="background:#fef3c7; color:#1a1a1a; border:1.5px solid #f59e0b; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✍️ Activity</span>

## Try it

{{embed: https://studio.learningwaypoints.com/tutorials/_playground-board/index.html?id=deck%2Fmaths%2Fnumber%2Fexpanded-notation%2Fstep-2&code=widget%3A+placevalue%0Aplaces%3A+5%0An%3A+0%0A&fresh=1}}

[Open this step in the tutorial](https://studio.learningwaypoints.com/tutorials/maths/number/expanded-notation/?full=step-2#step-2)

<!-- Speaker note: Each column's number is its digit times its power of 10. -->

<!-- Speaker note: Teacher: ask where the 0 in 4052 went when we write the expanded notation. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## What happened?

* 1000. 10³ = 10 × 10 × 10: three 10s multiplied, so 1 followed by three zeros. 30 is 10 × 3.

<!-- Speaker note: Reveal after running. Ask who predicted it, and what made them think so. -->

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<span style="background:#dbeafe; color:#1a1a1a; border:1.5px solid #3b82f6; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">💬 Discussion</span>

## Read the powers

The board shows the columns. Here is 4052 written out.

Now try one: which power goes with the 5?

$$4052 = 4000 + 0 + 50 + 2$$

1) $4000 = 4 \times 1000 = 4 \times 10^3$ — The thousands column is 10³.
2) $50 = 5 \times 10 = 5 \times 10^1$ — The tens column is 10¹.
3) $2 = 2 \times 1 = 2 \times 10^0$ — The ones column is 10⁰.
4) $4052 = 4 \times 10^3 + 5 \times 10^1 + 2 \times 10^0$ — The 0 in the hundreds adds nothing, so we leave it out.

<!-- Speaker note: Model this live on the board before revealing anything: write each line and say why, pausing to ask the class what comes next. Then click to reveal the lines and compare. Script. Start: write 4052 = 4000 + 0 + 50 + 2 and say "Each digit is worth its digit times its column." Line 1: write 4000 = 4 × 1000 = 4 × 10^3 and say "The thousands column is 10³." Line 2: write 50 = 5 × 10 = 5 × 10^1 and say "The tens column is 10¹." Line 3: write 2 = 2 × 1 = 2 × 10^0 and say "The ones column is 10⁰." Line 4: write 4052 = 4 × 10^3 + 5 × 10^1 + 2 × 10^0 and say "The 0 in the hundreds adds nothing, so we leave it out." -->

<!-- Speaker note: Teacher: have students predict the value of 10⁰ before the question, from the pattern 1000, 100, 10. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Read the powers

**In 4052, the 5 is in the tens column. What is the exponent of its power of 10? (The 5 is worth $5 \times 10^{?}$.)**

* Answer: 1

<!-- Speaker note: The ones column is 10⁰, the tens column 10¹, the hundreds 10². So 5 tens is 5 × 10¹ = 50. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Find the mistake

**Sam writes 4052 in expanded notation. Click the line where the working first goes wrong.**

```
 1  Write 4052 in expanded notation.
 2  4 is in the thousands: 4 × 10³
 3  0 is in the hundreds: nothing to add
 4  5 is in the tens: 5 × 10²
 5  2 is in the ones: 2 × 10¹
```

* Answer: line 4

<!-- Speaker note: Line 4: the tens column is 10¹, not 10². Sam counted the ones column as 10¹. The ones column is 10⁰. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Why 10⁰?

**Why is $10^0 = 1$?**

- It is not allowed
- It is the ones column: 1 one, so the power is 1
- Any number to the power 0 is 0
- 10⁰ means 10 × 0

* Answer: It is the ones column: 1 one, so the power is 1

<!-- Speaker note: Each step down divides by 10: 1000, 100, 10, then 1. So 10⁰ = 1, the value of the ones column. -->

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<span style="background:#fef3c7; color:#1a1a1a; border:1.5px solid #f59e0b; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✍️ Activity</span>

## Build, read, convert



- **Far apart:** 
- **Back to a number:** 
- **Which exponent?:** 
- **Shift the digits:** 

[Open this step](https://studio.learningwaypoints.com/tutorials/maths/number/expanded-notation/?full=step-4#step-4)

<!-- Speaker note: What this step is for: Building a number with gaps, reading the value of a digit, converting expanded notation back, and shifting by ×10 or ×100 connect the exponent to the number of zeros. For students who need more support: Keep the long form (4 × 1000 + 5 × 10 + 2) beside the powers until the exponent is automatic. For early finishers: What would the column to the right of the ones be worth? Continue the pattern 1000, 100, 10, 1, … and write the next two columns as powers of 10 with negative exponents. The hard part, on purpose: Starting the exponent count at the ones column as 0, not 1. Have students say 'ones is ten to the nothing' once, and point at the column while they write the exponent. When students get stuck: A student writes 5 × 10² for the 5 in the tens (Sam's mistake). Ask: What power of 10 is the ones column? Fix: Count from 0 at the ones: ones 0, tens 1, hundreds 2. / A student writes 10³ = 30. Ask: How many 10s are multiplied? Fix: Write 10 × 10 × 10 the long way, then count the zeros. / A student writes 4 × 10³ + 0 × 10² + 5 × 10¹ + 2 × 10⁰ and asks if they can drop the 0 term. Ask: What does 0 × 100 add? Fix: Nothing, so it can be left out. Both forms are correct. -->

<!-- Speaker note: Teacher: ask what × 10 does to every exponent. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## The value of a digit

**What is the value of the 2 in 52 300?**

* Answer: 2000

<!-- Speaker note: The 2 is in the thousands column: 2 × 10³ = 2000. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Back to a number

**Write $6 \times 10^5 + 4 \times 10^2$ as a number.**

* Answer: 600400

<!-- Speaker note: 600 000 + 400 = 600 400. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Times 100

**Why is $6300 \times 100$ the same as $6300 \times 10^2$, and why is it easy?**

- × 10² means × 10 × 2 = × 20
- It is not the same
- 10² = 100, and × 10² moves every digit two columns left (add two zeros)
- 10² = 20, so it doubles

* Answer: 10² = 100, and × 10² moves every digit two columns left (add two zeros)

<!-- Speaker note: 10² = 100. Multiplying by 100 moves each digit two columns left: 630 000. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Review: powers

**Review: what is $2^5$?**

* Answer: 32

<!-- Speaker note: 2 × 2 × 2 × 2 × 2 = 32. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Millions

**Write 3 000 000 as $3 \times 10^{?}$. What is the exponent?**

* Answer: 6

<!-- Speaker note: The exponent counts the zeros: 3 000 000 = 3 × 10⁶, and 10⁶ is the millions column. -->

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<span style="background:#fef3c7; color:#1a1a1a; border:1.5px solid #f59e0b; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✍️ Activity</span>

## Make: three terms

Then read your number out in expanded notation.

[Open this step](https://studio.learningwaypoints.com/tutorials/maths/number/expanded-notation/?full=step-6#step-6)

<!-- Speaker note: What this step is for: Building any number with exactly three non-zero terms and a top power of 10⁴ shows that the top term decides the size and the zeros are columns. Hints, in order: The top power is 10⁴, so the first digit is in the ten thousands column. / Three columns with digits and two with 0. -->

<!-- Speaker note: Teacher: collect numbers and swap: partners write the expanded notation. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Exit ticket

**Write 7 304 in expanded notation using powers of 10.**

* 7 × 10³ + 3 × 10² + 4 × 10⁰. (The 0 in the tens column is left out.)

<!-- Speaker note: Students answer on paper or a mini whiteboard before the reveal. Anyone who cannot answer needs the main idea again next lesson. Where this leads: Next: Rounding (to a given place) uses the same column names. Where it leads: scientific notation in Year 9 and 10, and decimal place value, where the columns continue as 10⁻¹, 10⁻², …. -->