Math // Number Representation & Place Value

Place Value × 10 and ÷ 10

Recognise that in a multi-digit number, a digit in one place represents 10 times as much as in the place to its right and 1/10 of what it represents in the place to its left

Ages
10 to 11
Type
Concept
Builds on
2 topics
Leads to
4 topics
Standards
1 code

Quick check

Can the learner explain that in 4.56, the digit 5 is worth 5 tenths — ten times less than the 5 in 54, and ten times more than the 5 in 0.056?

Ask this to see whether the topic is secure.

Topic // Mastery

What mastery looks like

What a learner can do once this topic is secure.
  • Explain that the 4 in 0.04 is 1/10 of the 4 in 0.4
  • State that a digit moving one place left is ×10 and one place right is ÷10
  • Compare the value of the 6 in 6,000 and in 0.006

Topic // Map

Where this fits

What to learn first and what this topic opens up, with the reason for each link. Required links come first.
  • RequiredMust come first
  • HelpfulHelps, but not essential

Before this

2 topics
  • Place Value × 10 PatternRequired

    Number Representation & Place Value // Ages 9 to 10

    ×10 both directions extends Y5 ×10 relationship

  • Tenths (age 9+)Helpful

    Fractions // Ages 9 to 10

    Understanding 1/10 of requires familiarity with thousandths

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Place Value × 10 and ÷ 10

Ages 10 to 11 // Concept

Leads to

4 topics

Topic // Research-informed

Teaching notes

From the Marble Skill Taxonomy. These notes have not been validated in classrooms, so treat them as a starting point for your own judgement.

The approach

For ages 10–11: understand that each place in a number is worth ten times the place to its right and one tenth of the place to its left, and that this ×10/÷10 structure continues unbroken through the decimal point. The core goal is for a learner to state multiplicative relationships ('ten times', 'one tenth of') between the same digit sitting in different places, and to justify them by position rather than by face value or decimal length.

Common misconceptions

  • Believing 'moving a place left just adds a zero' — a rule that breaks for decimals (0.4 → 4, not 0.40)
  • Treating the decimal point as moving rather than the DIGITS moving relative to a fixed point
  • Thinking the gap between places is additive (0.4 is '0.36 more than' 0.04) rather than multiplicative (ten times)
  • Assuming longer decimals are always larger, ignoring place value (0.004 seen as bigger than 0.4)
  • Losing the self-similar structure at the decimal point — seeing tenths as unrelated to ones

Useful representations

  • Gattegno / place-value chart (digit slides between columns, value scales ×10 or ÷10)
  • Place-value chart spanning millions to thousandths with the decimal point fixed and digits mobile
  • Arrow/relationship cards labelling adjacent columns '×10 →' and '← ÷10'
  • Base-ten / Dienes extended to decimal flats-longs-units where the 'one' can be reassigned

Teaching practices

  • Present place value as one continuous ×10/÷10 chain that runs unbroken through the decimal point — do not teach whole-number and decimal place value as separate systems
  • Use a Gattegno chart or place-value slider so a digit physically moves between columns and its value visibly recomputes (×10 left, ÷10 right)
  • Contrast the SAME face-value digit sitting in different places (4 in 4000 vs 0.4 vs 0.04) and require a multiplicative statement ('ten times', 'one tenth of'), never 'bigger/smaller'
  • Anchor the ÷10 direction in concrete partitioning — one tenth of a tenth is a hundredth — before abstracting to the rule
  • Vary the non-example: ask why moving a place is NOT 'add a zero' when decimals are involved, exposing the multiplicative meaning

Activities to try

  1. 1

    Slide the same digit across the places

    Why it works: A digit's worth changes multiplicatively (×10 each step left, ÷10 each step right) according to its place; the decimal point stays fixed while the digit moves, and the same self-similar rule runs through the decimal point.

    1. Work on a place-value chart with columns from millions down to thousandths and a fixed decimal point.
    2. Take a single digit, for example a 4, and place it in the tenths column so it reads 0.4.
    3. Move the same 4 one column to the left to read 4, then 40, then 400, then 4000; each move, say aloud 'ten times as much' and write the ×10 relationship.
    4. Before moving right, split one tenth into ten equal parts and check that each part is one hundredth (and one hundredth into ten parts gives thousandths). Then move the same 4 one column to the right from 0.4 to 0.04 to 0.004; each move, say aloud 'one tenth of' and write the ÷10 relationship.
    5. Park the same 4 three columns apart (for example 4000 and 4, or 4 and 0.004 — each three steps, so ×1000) and state how many times larger one is than the other.
  2. 2

    Why isn't it just 'add a zero'?

    Why it works: The underlying operation is multiply/divide by ten, not a digit-appending trick; 'add a zero' is a surface pattern that breaks for decimals, exposing the multiplicative meaning.

    1. Consider the claim 'moving a digit one place left just adds a zero.'
    2. Test it on whole numbers: 4 → 40 (works) and then on a decimal: 0.4 → 4 (not 0.40).
    3. Explain in your own words why 'add a zero' fails for decimals but ×10 always works.
    4. Write a corrected rule that holds for both whole numbers and decimals.
  3. 3

    Compare across many places and across zero

    Why it works: The place-value relationship holds across several steps and across the decimal point; value decreases to the right and increases to the left, regardless of decimal length.

    1. Compare the 6 in 6,000 with the 6 in 0.006: count how many ×10 steps separate the two places (six steps).
    2. State and justify how many times larger the 6 in 6,000 is (6,000 is 0.006 × 1,000,000).
    3. Place 0.4 and 0.04 on a 0–1 number line (0.4 near the middle, 0.04 close to zero) to see the value shrinking to the right.
    4. Explain which stays the same and which changes when a digit moves place — the digit, or its worth.

Topic // Standards

Curriculum standards

Codes for the curriculum statements this topic maps to. Each publisher owns its standard text.
Common Core State Standards for Mathematics
5.NBT.1

Math // Number Representation & Place Value

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