Year 7 Mathematics · AC9M7N03

Expanded Notation with Powers of 10

We are learning to represent natural numbers in expanded notation using place value and powers of 10

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Learning goalsSay it simply

We are learning to represent natural numbers in expanded notation using place value and powers of 10.

Place value assigns each digit a coefficient of a power of 10. In 7 052 304, the 7 is a coefficient of 10⁶, while the 5 is a coefficient of 10⁴.

Expanded notation records this structure explicitly. Zero coefficients may be omitted from the sum, but their place-value positions must remain intact when the numeral is reconstructed.

The pattern includes 10⁰ = 1, so the ones digit can be written as a coefficient of 10⁰. Converting in both directions checks that every digit is attached to the correct power of 10.

Success criteria

  • I can match each digit with its power of 10.
  • I can express a natural number in expanded notation.
  • I can reconstruct and check a numeral from coefficients and powers of 10.
Key conceptTeach from the board

Use these models to connect the mathematical idea to values, diagrams, coordinates, graphs or structure before moving to symbolic calculation.

Follow each example from representation and setup through calculation/reasoning, interpretation and an independent check.

Example 1

Expand a large natural number

  1. Read 7 052 304 by place value.
  2. Pair each non-zero digit with its power of 10.
  3. Write 7×10⁶ + 5×10⁴ + 2×10³ + 3×10² + 4×10⁰.
  4. Evaluate the terms to confirm their sum is 7 052 304.

Final answer: 7 052 304 = 7×10⁶ + 5×10⁴ + 2×10³ + 3×10² + 4×10⁰.

Check: Adding the expanded terms reconstructs 7 052 304.

Example 2

Reconstruct a numeral

  1. Evaluate 6×10⁵ = 600 000.
  2. Evaluate 4×10³ = 4 000 and 8×10 = 80.
  3. Add the remaining 2×10⁰ = 2.
  4. Combine place values to obtain 604 082.

Final answer: 6×10⁵ + 4×10³ + 8×10 + 2 = 604 082.

Check: Re-expand 604 082 and compare every coefficient.

Example 3

Application problem 1

Problem: A number is written as 6×10⁵ + 4×10³ + 9×10 + 2. Write the numeral and explain the role of the zero place holders.

  1. Plan: Represent the information first, then calculate, interpret and independently check the result.
  2. Work: The numeral is 604 092. Zero placeholders keep the hundred-thousands-to-ones place values aligned where no coefficient is present.
  3. Interpret: This synthesis problem combines the chapter's core representation, calculation and reasoning skills.

Final answer: The numeral is 604 092. Zero placeholders keep the hundred-thousands-to-ones place values aligned where no coefficient is present.

Check: This synthesis problem combines the chapter's core representation, calculation and reasoning skills.

Example 4

Application problem 2

Problem: Compare 9×10⁵ + 8×10² with 8×10⁵ + 9×10⁴. Which is greater?

  1. Plan: Reconstruct both numerals before subtracting.
  2. Work: 900 800 is greater than 890 000 by 10 800.
  3. Interpret: Compare the highest place where coefficients differ.

Final answer: 900 800 is greater than 890 000 by 10 800.

Check: Compare the highest place where coefficients differ.

Clean visual examplesOne-page board

Clean one-page examples

AC9M7N03 - Expanded Notation with Powers of 10
Example 1

Example 1 Expand a large natural number Read 7 052 304 by place value. Pair each non-zero digit with its power of 10. Write 7×10⁶ + 5×10⁴ + 2×10³ + 3×10² + 4×10⁰. Evaluate the terms to confirm their sum is 7 052 304. Final answer: 7 052 304 = 7×10⁶ + 5×10⁴ + 2×10³ + 3×10² + 4×10⁰. Check: Adding the expanded terms reconstructs 7 052 304.

Example 2

Example 2 Reconstruct a numeral Evaluate 6×10⁵ = 600 000. Evaluate 4×10³ = 4 000 and 8×10 = 80. Add the remaining 2×10⁰ = 2. Combine place values to obtain 604 082. Final answer: 6×10⁵ + 4×10³ + 8×10 + 2 = 604 082. Check: Re-expand 604 082 and compare every coefficient.

Example 3

Example 3 Application problem 1 Problem: A number is written as 6×10⁵ + 4×10³ + 9×10 + 2. Write the numeral and explain the role of the zero place holders. Plan: Represent the information first, then calculate, interpret and independently check the result. Work: The numeral is 604 092. Zero placeholders keep the hundred-thousands-to-ones place values aligned where no coefficient is present. Interpret: This synthesis problem combines the chapter's core representation, calculation and reasoning skills. Final answer: The numeral is 604 092. Zero placeholders keep the hundred-thousands-to-ones place values aligned where no coefficient is present. Check: This synthesis problem combines the chapter's core representation, calculation and reasoning skills.

Example 4

Example 4 Application problem 2 Problem: Compare 9×10⁵ + 8×10² with 8×10⁵ + 9×10⁴. Which is greater? Plan: Reconstruct both numerals before subtracting. Work: 900 800 is greater than 890 000 by 10 800. Interpret: Compare the highest place where coefficients differ. Final answer: 900 800 is greater than 890 000 by 10 800. Check: Compare the highest place where coefficients differ.

Curriculum examplesCopied content

The content description and elaborations below show the curriculum ideas taught in this unit. Items marked as teaching context support lesson planning.

  • Content description: represent natural numbers in expanded notation using place value and powers of 10
  • E1: investigating exponent notation for powers of 10 such as “one hundred thousand” is 100 000=10×10×10×10×10=10^5
  • E2: relating the sequences 10, 100, 1000, 10 000 … and 10^1, 10^2, 10^3, 10^4 ...
  • E3: applying and explaining the connections between place value and expanded notations; for example, 7000=7×10^3 and 3750=3×10^3+7×10^2+5×10^1
Questions and answersWith answers

Attempt each problem before opening Check answer. The set moves from core understanding to application and synthesis.

  1. 1. Evaluate 10⁰.

    Check answer

    Answer: 10⁰ = 1.

    Hint: Use the pattern 10², 10¹, 10⁰.

    Why: A non-zero number to power zero equals one.

  2. 2. Write 4 205 in ordinary expanded notation.

    Check answer

    Answer: 4 205 = 4 000 + 200 + 5.

    Hint: Read each digit's place.

    Why: Zero tens contribute no term.

  3. 3. Write 83 010 using powers of 10.

    Check answer

    Answer: 83 010 = 8×10⁴ + 3×10³ + 1×10.

    Hint: Label ten-thousands through ones.

    Why: Each coefficient matches its place-value digit.

  4. 4. Write 7 052 304 using powers of 10.

    Check answer

    Answer: 7×10⁶ + 5×10⁴ + 2×10³ + 3×10² + 4×10⁰.

    Hint: Keep each digit attached to its original power.

    Why: Zero coefficients may be omitted without shifting places.

  5. 5. Reconstruct 5×10⁵ + 9×10² + 6.

    Check answer

    Answer: 500 906.

    Hint: Create a place-value table.

    Why: The missing thousands and tens positions contain zero.

  6. 6. What coefficient of 10³ appears in 2 347 018?

    Check answer

    Answer: The coefficient is 7 because the thousands digit is 7.

    Hint: Locate the thousands column.

    Why: A coefficient is the digit multiplying a power of 10.

  7. 7. Correct: 60 405 = 6×10⁴ + 4×10³ + 5.

    Check answer

    Answer: It should be 6×10⁴ + 4×10² + 5×10⁰.

    Hint: Align digits with powers in a table.

    Why: The digit 4 is in the hundreds place, not thousands.

  8. 8. A number is 3×10⁶ + a×10⁴ + 7×10² + 2. If it is 3 050 702, find a.

    Check answer

    Answer: Matching place values gives a×10⁴=50 000, so a=5; substituting reconstructs 3 050 702.

    Hint: Match coefficients place by place.

    Why: The ten-thousands digit of 3 050 702 is five.

  9. 9. Compare 9×10⁵ + 8×10² with 8×10⁵ + 9×10⁴. Which is greater?

    Check answer

    Answer: 900 800 is greater than 890 000 by 10 800.

    Hint: Reconstruct both numerals before subtracting.

    Why: Compare the highest place where coefficients differ.

  10. 10. A number is written as 6×10⁵ + 4×10³ + 9×10 + 2. Write the numeral and explain the role of the zero place holders.

    Check answer

    Answer: The numeral is 604 092. Zero placeholders keep the hundred-thousands-to-ones place values aligned where no coefficient is present.

    Hint: Represent the information first, then calculate, interpret and independently check the result.

    Why: This synthesis problem combines the chapter's core representation, calculation and reasoning skills.

Practice and reviewReady for practice

Common mistake: 10⁰ treated as zero.

Correction: 10⁰=1.

Common mistake: Exponents counted from one at the units place.

Correction: The units place is 10⁰.

Common mistake: Internal zeros removed from the numeral.

Correction: They preserve place even if their terms are omitted.

Curriculum alignmentStart here

We are learning to represent natural numbers in expanded notation using place value and powers of 10.

Place value assigns each digit a coefficient of a power of 10. In 7 052 304, the 7 is a coefficient of 10⁶, while the 5 is a coefficient of 10⁴.

Expanded notation records this structure explicitly. Zero coefficients may be omitted from the sum, but their place-value positions must remain intact when the numeral is reconstructed.

The pattern includes 10⁰ = 1, so the ones digit can be written as a coefficient of 10⁰. Converting in both directions checks that every digit is attached to the correct power of 10.

Success criteria

  • I can match each digit with its power of 10.
  • I can express a natural number in expanded notation.
  • I can reconstruct and check a numeral from coefficients and powers of 10.
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