PrepShorts · Teaching notes · Class 8 Mathematics · Chapter 2, Power PlayPrepShorts

Chapter 2 · Power Play

Powers of 10, and place value written out for whole numbers and decimals

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What the video covers, what to say before it, where a class usually goes wrong, and what to set afterwards. An account is free, and it opens every chapter of every book.

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What to assume they know

What they should be able to do

  • Write a whole number in expanded form using the multipliers 10, 100, 1000
  • Rewrite that expansion with each multiplier as a power of 10
  • Say which power of 10 belongs to a given digit's place, without counting from the left
  • Explain why the units digit is multiplied by 10⁰, and why that requires 10⁰ = 1
  • Write a decimal number in expanded form using negative powers of 10
  • Read the exponent off a digit's position, and the position off the exponent
  • Account for a zero digit in an expansion, and say what its term contributes
  • Explain what the decimal point marks, once the places are labelled by exponents

Where it usually goes wrong

  • "The units digit has no multiplier." It has 10⁰, which is 1, which is why it can be written bare. The chapter itself prints it both ways within four lines. That is not carelessness — it is the point.
  • "10⁰ = 1 is a special convention for place value." It was forced two pages earlier by the division rule (Part I p.28). Place value uses it; it does not create it.
  • "The first digit after the point is the tenth place, so it goes with 10¹." It goes with 10⁻¹. Students reading "one-tenth" hear "ten". Show the fraction and the power side by side until they separate.
  • "Negative exponents extend place value past the point." Nothing is being extended. The pattern of exponents was already running 4, 3, 2, 1, 0 and simply continues to −1, −2, −3. The decimal point is a reading aid, not a boundary.
  • "A zero digit can be left out of the number." Not from the numeral: 561.903 without its zero reads 561.93, a different number. In the expansion the term (0 × 10⁻²) adds nothing and could be dropped, because every other term carries its own power of ten; the chapter writes it anyway, which keeps every place visible.
  • "Every number has a fixed set of places." The three numbers the chapter leaves open — 172, 5642, 6374 — have different highest exponents. The expansion is read off the number, not recited from a table.
  • "This is only about the number ten." It works for ten because there are ten digits. Worth one sentence at the end: the base of the notation and the count of available symbols are the same number.

Questions to check understanding

  • Write a given 3-, 4- or 5-digit number in expanded form using powers of 10
  • Write a given decimal to three places in expanded form using powers of 10
  • Reconstruct the number from an expansion given in powers of 10
  • Supply the missing power in a partially completed expansion
  • State which power of 10 a named digit of a given number is multiplied by
  • Explain what the term 0 × 10⁻² contributes, and why it is written
  • Compare two decimals by comparing their highest differing power
  • Explain in words why the units digit carries the exponent 0

Examples worth working on the board

  • The worked whole number (Part I p.30). The chapter takes 47561 and writes it as (4 × 10000) + (7 × 1000) + (5 × 100) + (6 × 10) + 1. Note the last term: the units digit is printed bare, with no multiplier at all.
  • The same line with exponents (Part I p.30): (4 × 10⁴) + (7 × 10³) + (5 × 10²) + (6 × 10¹) + (1 × 10⁰). Here the units digit does get a multiplier, and it is 10⁰. Put the two lines one above the other — the difference between them is the whole reason this section follows the one on zero exponents.
  • Three to do (Part I p.30), left open: write 172, 5642 and 6374 the same way. Note that 5642 and 6374 are four-digit and 172 is three-digit, so the highest exponent changes between them; that is the point of giving three.
  • The worked decimal (Part I p.30). For 561.903 the chapter writes (5 × 100) + (6 × 10) + 1 + (9 × 1/10) + (0 × 1/100) + (3 × 1/1000). Again the units term is printed bare, and the three fractional terms are printed as stacked fractions, not as decimals and not in words.
  • The same decimal with exponents (Part I p.30): (5 × 10²) + (6 × 10¹) + (1 × 10⁰) + (9 × 10⁻¹) + (0 × 10⁻²) + (3 × 10⁻³).
  • The zero term. The hundredths digit of 561.903 is 0, so its term is 0 × 10⁻², which contributes nothing to the sum and everything to the notation. An explanation that quietly drops it has taught that places can be skipped. The chapter prints the term; keep it.
  • A cross-check. Every exponent in both expansions equals the number of places that digit sits from the units place, positive to the left and negative to the right. Count them off once on 47561 and once on 561.903 and the rule needs no statement.

Figures to have open

  • One exponent strip running from 10⁴ down to 10⁻³, with slots for digits above it and the decimal point drawn as a small mark between the 10⁰ and 10⁻¹ slots rather than as a wall. Every section of this topic hangs off this figure, and the chapter draws nothing like it — not in the book, and the reason to make this an explanation rather than a page.
  • A two-line comparison card: the chapter's plain expansion above, the same expansion with exponents below, aligned term by term. Standard schematic.
  • A counting overlay showing the number of places each digit sits from the units position, so the exponent can be seen being counted. Not in the book.
  • No photograph or textbook art is needed; Part I p.30 carries none in this section.

Where this sits in the book

The book

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