Place value and powers of 10

Classroom resources

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Students will be able to

• 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.

Timing

Predict and Run 8 min · Investigate 12 min · Modify 12 min · Practise and Make 8 min

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).

Running the lesson

1. Predict: students who chose 30 read 10³ as 10 × 3.
2. Run: ask where the 0 in 4052 went when we write the expanded notation.
3. Investigate: have students predict the value of 10⁰ before the question, from the pattern 1000, 100, 10.
4. Modify: ask what × 10 does to every exponent.
5. Make: collect numbers and swap: partners write the expanded notation.

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

Answers to the checks

1. how much is 10³?: What is $10^3$?
   1000. 10³ = 10 × 10 × 10: three 10s multiplied, so 1 followed by three zeros. 30 is 10 × 3.
2. build 4052: Build 4052: 4 thousands, no hundreds, 5 tens and 2 ones. Which power of 10 goes with the 5?
   
   4052 = 4 × 10³ + 5 × 10¹ + 2 × 10⁰. The 5 is in the tens column, so its power is 10¹. The hundreds column holds a 0, so it adds nothing.
3. 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 $5 \times 10^{?}$.)
   1
   The ones column is 10⁰, the tens column 10¹, the hundreds 10². So 5 tens is 5 × 10¹ = 50.
4. Find the mistake: Sam writes 4052 in expanded notation. Click the line where the working first goes wrong.
   line 4
   Line 4: the tens column is 10¹, not 10². Sam counted the ones column as 10¹. The ones column is 10⁰.
5. Why 10⁰?: Why is $10^0 = 1$?
   It is the ones column: 1 one, so the power is 1
   Each step down divides by 10: 1000, 100, 10, then 1. So 10⁰ = 1, the value of the ones column.
6. Far apart: Build a number with a 6 in the ten thousands column and a 9 in the ones column, and zeros between.
   
   60 009 = 6 × 10⁴ + 9 × 10⁰.
7. Back to a number: Write $7 \times 10^4 + 2 \times 10^2 + 5$ as a number.
   70205
   70 000 + 200 + 5 = 70 205. The empty tens column is a 0.
8. Which exponent?: In 380 000, the 8 is worth $8 \times 10^{?}$. What is the exponent?
   4
   The 3 is in the hundred thousands (10⁵) and the 8 in the ten thousands (10⁴): 380 000 = 3 × 10⁵ + 8 × 10⁴.
9. Shift the digits: Build 345, then press × 10 twice. What number do you get, and what happened to each digit?
   
   345 × 100 = 34 500. Every digit moved two columns left, so each power of 10 went up by 2, and two zeros filled the gaps.
10. The value of a digit: What is the value of the 2 in 52 300?
    2000
    The 2 is in the thousands column: 2 × 10³ = 2000.
11. Back to a number: Write $6 \times 10^5 + 4 \times 10^2$ as a number.
    600400
    600 000 + 400 = 600 400.
12. Times 100: Why is $6300 \times 100$ the same as $6300 \times 10^2$, and why is it easy?
    10² = 100, and × 10² moves every digit two columns left (add two zeros)
    10² = 100. Multiplying by 100 moves each digit two columns left: 630 000.
13. Review: powers: Review: what is $2^5$?
    32
    2 × 2 × 2 × 2 × 2 = 32.
14. Millions: Write 3 000 000 as $3 \times 10^{?}$. What is the exponent?
    6
    The exponent counts the zeros: 3 000 000 = 3 × 10⁶, and 10⁶ is the millions column.
15. three terms: Build a number with three digits that are not 0, and a top power of 10⁴.
    
    For example 30 502 = 3 × 10⁴ + 5 × 10² + 2 × 10⁰.

Solutions

One possible solution for each task. Other correct programs are fine.

build, read, convert › Far apart

Open task: any program that does what the step asks.

build, read, convert › Back to a number

Open task: any program that does what the step asks.

build, read, convert › Which exponent?

Open task: any program that does what the step asks.

build, read, convert › Shift the digits

Open task: any program that does what the step asks.

three terms

Open task: any program that does what the step asks.

Stretch and support

Stretch: 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.

Support: Keep the long form (4 × 1000 + 5 × 10 + 2) beside the powers until the exponent is automatic.

Coach prompt (optional)

For a student who has done the lesson and is still stuck. They paste it into an AI tool your school allows; it asks questions about this lesson's one idea and doesn't give answers. Students see it on the course recap page only through a link ending in ?coach=1: recap with coach prompts.

You are a patient maths coach for a Year 7 student in Australia. Help me understand the ideas in ONE lesson only, from the lesson "Place value and powers of 10".

The ideas:
- Powers of 10 as place value columns: I can Students connect each place value column to a power of 10 and write powers of 10 in exponent notation (10⁰=1, 10¹=10, 10²=100, etc.).
- Expanded notation using powers of 10: I can Students write any natural number in expanded notation using powers of 10 (e.g. 7000 = 7 × 10³) and convert back from expanded to standard form.

What understanding looks like:
- Correctly states the power of 10 corresponding to each place value column up to millions.
- Writes 10⁰ = 1 and explains why.
- Does not confuse the exponent with the value (e.g. does not say 10³ = 30).
- Correctly writes any given natural number in expanded notation using powers of 10.
- Reconstructs a number correctly from its expanded notation.
- Explains what each term in the expanded form represents.

Mistakes students often make here:
- A student writes 5 × 10² for the 5 in the tens (Sam's mistake).
- A student writes 10³ = 30.
- A student writes 4 × 10³ + 0 × 10² + 5 × 10¹ + 2 × 10⁰ and asks if they can drop the 0 term.

How to coach me (rules from the Waypoints Kit coach):
- Ask me one short question at a time (2 to 4 sentences) and wait for my answer. Never give me the answer before you have heard my thinking, and don't hide the answer inside your question.
- Start in Discover: ask me to explain the idea in my own words, or to predict what happens in a simple example. I can switch any time: "teach it back" (you play a beginner and I explain), "spot the flaw" (you show me an answer with one common mistake and I find it), or "check" (I explain, and you tell me what is solid and what to revisit).
- If I am stuck or say "just tell me", ask what I think first, then give a small hint or a simpler example. Explain only after I have had a real go.
- Stay in this idea: don't teach later ideas or add extra content. If I ask about something else, say it is for another lesson and bring me back.
- After you help, ask me one reflective question, such as how I know my answer is right or what I would check.
- Praise my effort and method, not my ability.
- You can be wrong: if I rely on something you said, remind me to check it against my notes or the lesson. If you don't know, say so.
- Use simple words.
- Stop when I can explain the idea in my own words and get one new example right. Then tell me to go back to the lesson "Place value and powers of 10".

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.)

What each step is for

• how much is 10³?: Committing to a value for 10³ shows whether they read the exponent as a multiplier (30) or as the number of zeros (1000).
• build 4052: Raising the thousands digit to 4 and the tens digit to 5 shows each digit multiplied by its column's power of 10, and the empty hundreds column left as 0.
• columns and powers: The exponent is the column's position counting from 0 at the ones; Sam counts from 1 and every power is one too big. 10⁰ = 1 is the ones column, one 10 to no power.
• build, read, convert: 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.
• expanded notation: The value of a digit, the number from its expanded notation, and why times 100 adds two zeros; the review keeps powers fresh.
• three terms: 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.

Where it 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, ….