Numbers to 10 million & the ×10/÷10 relationship
Read/order numbers to ~10,000,000+; Roman numerals to M (EN Y5); the 10× / ⅒ place-value relationship (the basis for decimals).
Once numbers reach into the millions, the same place-value rule that built two- and three-digit numbers keeps working — and the key insight is that moving one place left multiplies a digit by 10, while moving one place right divides it by 10.
What it means
Every whole number is written with digits in columns, and a digit’s column decides its value. Reading right to left the columns are ones, tens, hundreds, thousands, ten-thousands, hundred-thousands, millions — and on to ten-millions. In 3,540,000 the 3 sits in the millions column, so it means 3,000,000; the 5 means 500,000; the 4 means 40,000. We group digits in threes with commas so long numbers stay readable: 3,540,000 reads “three million, five hundred and forty thousand”.
The heart of this stage is the ×10 / ÷10 relationship. Each column is worth ten times the one to its right. So the same digit is worth ten times as much one step to the left, and one-tenth (1/10, written ⅒) as much one step to the right. Multiplying a whole number by 10 shifts every digit one column left — 52 × 10 = 520. Dividing by 10 shifts every digit one column right — 520 ÷ 10 = 52. This is not a trick of “adding a zero”; it is the columns themselves scaling by ten. That same rule keeps going below the ones column, where the next place to the right is worth ⅒ — the tenths — which is exactly why decimals work. Numbers to ten million are where this pattern becomes unmistakable, ready to extend downward into fractions of one.
Ordering large numbers means comparing column by column from the left: 4,090,000 beats 4,009,000 because, with millions equal, 9 ten-thousands beat 0 ten-thousands.
Roman numerals are a different, older way of writing numbers using letters: I=1, V=5, X=10, L=50, C=100, D=500, M=1000. You add letters left to right (XVI = 10 + 5 + 1 = 16), except when a smaller letter sits before a larger one, when you subtract (IV = 5 − 1 = 4, IX = 9, CM = 900). They have no place value and no zero — which is exactly what makes our column system so powerful by comparison.
Worked examples
Read a seven-digit number. Split 7,208,500 into columns:
7,208,500
7 millions · 2 hundred-thousands · 0 ten-thousands · 8 thousands · 5 hundreds
→ "seven million, two hundred and eight thousand, five hundred"Multiply and divide by 10. Watch the digits slide:
846 × 10 = 8,460 (every digit moves one column LEFT)
846 ÷ 10 = 84.6 (every digit moves one column RIGHT — into tenths)Roman numerals. Write 2024, then read MCMXLIV:
2024 = MM(2000) + XX(20) + IV(4) = MMXXIV
MCMXLIV = M(1000) + CM(900) + XL(40) + IV(4) = 1944The generative-art connection
The ×10 relationship is a rule about scale, and scale is something you can watch happen. In dot-multiplier, adding one arm to the starburst adds a whole fixed group of dots at once — a jump, not a single step. Set the group to ten and each new arm multiplies the count by a visible, countable ten-fold leap, so the abstract “shift one column” becomes a physical burst you can see and count. Base-ten blocks tell the same story more literally: in the Number Pieces app a single unit cube becomes a rod of ten, a rod becomes a flat of a hundred — every step left is the tile itself growing ten times bigger. The mathematics is the geometry of the growth.
Common misconceptions
- “×10 just adds a zero.” True for whole numbers, but it breaks the moment decimals arrive:
4.6 × 10is46, not4.60. The reliable rule is that every digit moves one column, not that a zero appears. - Losing a zero place-holder. Writing “three million and five” as
3,5instead of3,000,005— the zeros hold the empty columns so the 3 stays in the millions place. - Roman numeral order.
IXis 9, butXIis 11: a smaller letter before a larger one subtracts, after it adds.
What a learner should be able to solve
The benchmark for mastering this concept — the problems a child at this stage is expected to be able to work through.
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Write `6,304,070` in words, then say what the digit `3` is worth.
Answer
"Six million, three hundred and four thousand, and seventy." The `3` sits in the hundred-thousands column, so it is worth `300,000`. The zeros hold the empty ten-thousands, hundreds and ones columns so every other digit stays in its place.
Art hook Draw seven stacked column-labels (ones→millions) as a horizontal strip. Type a number and light up each column with a stack of that many dots, tallest stack = biggest place. Watch `6,304,070` render as columns of dots of different heights.
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Put these in order from smallest to largest: `4,090,000`, `4,009,000`, `4,900,000`, `4,090,900`.
Answer
`4,009,000 < 4,090,000 < 4,090,900 < 4,900,000`. Compare column by column from the left: millions all equal (4), then hundred-thousands (0,0,9,0), which lifts `4,900,000` to the top; among the three with 0 hundred-thousands, the ten-thousands (0 vs 9) split them.
Art hook Plot four numbers as dots along a horizontal number line from 4,000,000 to 5,000,000. As the child drags them into order, correct positions snap to a rung and glow green — a visual sorting of place value on a line.
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Complete both: `3,750 × 10 = ?` and `48,200 ÷ 10 = ?`. Explain in one sentence what happens to the digits.
Answer
`3,750 × 10 = 37,500` and `48,200 ÷ 10 = 4,820`. Multiplying by 10 shifts every digit one column LEFT; dividing by 10 shifts every digit one column RIGHT. It is the columns scaling by ten, not just 'adding or removing a zero'.
Art hook Show a number as tiles sitting in labelled columns. A '×10' button slides every tile one cell left with an animation; a '÷10' button slides them right. The child sees the same tiles physically move places.
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Write the year `2049` in Roman numerals, then read `MCMXCIX`.
Answer
`2049 = MM(2000) + XL(40) + IX(9) = MMXLIX`. `MCMXCIX = M(1000) + CM(900) + XC(90) + IX(9) = 1999`. Remember a smaller letter before a larger one subtracts.
Art hook Build a Roman-numeral clock face: place the letters I, V, X etc. around a circle and, for a chosen number, draw arcs joining the letters used, subtractive pairs (like IX) shown as an arc pointing backwards.
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The distance to a far star is written `5,208,000` km. What is the value of the `2`, and what would the number become if you divided it by 10?
Answer
The `2` is in the hundred-thousands column, worth `200,000`. Dividing by 10 shifts every digit one column right: `5,208,000 ÷ 10 = 520,800`.
Art hook A dot travels along a spiral labelled with place-value columns; pressing ÷10 pulls the dot one loop inward (one column right), showing scale shrinking ten-fold each loop.
Training exercises
Practice problems, easier first, that build toward the bar above. Each doubles as a seed for an interactive artwork.
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Read this number out loud: `2,400,000`. Write it in words.
Answer
"Two million, four hundred thousand." The 2 is millions, the 4 is hundred-thousands, and the rest of the columns are empty (held by zeros).
Art hook Type a millions number; the app fills a row of seven dot-columns, one per place, so you can literally count the dots in each labelled column.
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What is the value of the digit `7` in `7,050,000`? Choose: 7, 700, 7,000, or 7,000,000.
Answer
`7,000,000` (seven million). The `7` sits in the millions column.
Art hook Highlight one digit of a big number on click; a burst of that many 'thousands/millions' dots pops up to show the digit's true value.
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Fill in the blanks: `620 × 10 = ____` and `6,200 ÷ 10 = ____`.
Answer
`620 × 10 = 6,200` and `6,200 ÷ 10 = 620`. Multiplying shifts digits one column left; dividing shifts them one column right — they undo each other.
Art hook A see-saw: ×10 tips digits one column left, ÷10 tips them back right, animating the two operations as opposites.
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Use `<`, `>` or `=` to compare: `1,300,000 ___ 1,030,000`.
Answer
`1,300,000 > 1,030,000`. Millions are equal (1), but the first has 3 hundred-thousands versus 0, so it is larger.
Art hook Two bars grow from left to right sized by each number; the taller bar's symbol (> or <) lights up between them.
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Which is bigger, `9,999` or `10,001`? Explain how you know without lining up every digit.
Answer
`10,001` is bigger. It has 5 digits (reaching the ten-thousands column) while `9,999` has only 4, so it must be larger — count the columns first.
Art hook Show both numbers as columns of dots; the one that reaches into an extra column glows, teaching 'more columns = bigger'.
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A child works out `3.7 × 10` and writes `3.70`. Is that right? If not, give the correct answer and say why.
Answer
Not right — 'adding a zero' fails for decimals, and `3.70` is just `3.7` again. Every digit must shift one column LEFT, so `3.7 × 10 = 37`: the 3 moves from ones to tens and the 7 moves from tenths to ones.
Art hook Animate the decimal point staying put while all the digit-tiles slide one column left, showing why the answer becomes 37, not 3.70.
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Write these Roman numerals as ordinary numbers: `XL`, `XC`, `CD`.
Answer
`XL = 50 − 10 = 40`, `XC = 100 − 10 = 90`, `CD = 500 − 100 = 400`. Each is a subtractive pair (smaller letter before larger).
Art hook Draw each pair as two arcs on a circle: the small letter's arc points backward to show subtraction, the total shown in the middle.
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Order from largest to smallest: `3,600,000`, `3,060,000`, `3,006,000`.
Answer
`3,600,000 > 3,060,000 > 3,006,000`. Millions equal (3); compare hundred-thousands (6 > 0 > 0), then ten-thousands (6 > 0) to split the last two.
Art hook Three dots on a shared number line slide into a descending staircase as the learner sorts them.
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A stadium holds `85,000` people. Ten identical stadiums together hold how many? Write the calculation.
Answer
`85,000 × 10 = 850,000`. Every digit shifts one column left, so 85 thousands become 850 thousands.
Art hook Tile ten small stadium icons in a grid; a counter ticks up by 85,000 each icon, ending at 850,000 with all digits shifted left.
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Read the year `MMXLVIII` and check it makes sense as a future year.
Answer
`MMXLVIII = M+M(2000) + XL(40) + V(5) + III(3) = 2048`. It reads as 'two thousand and forty-eight' — a sensible future year.
Art hook A timeline arc where entering Roman numerals drops a marker at the matching year, letters lighting up as arcs as you type.
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Open-ended: find a 7-digit number where the digit `5` is worth `500,000` and the digit `2` is worth `20`. Write one that works.
Answer
You need a 5 in the hundred-thousands column and a 2 in the tens column; the other five digits can be anything. For example `1,500,024` — its 5 is worth `500,000` and its 2 is worth `20`. Many answers are correct.
Art hook A grid of seven empty column-slots the child fills with digit-dots; when the 5-column and the tens-column hold the right digits, both slots glow to confirm.
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Reasoning: I multiply a whole number by 10 twice. By how much has it grown in total, and what is `430 × 10 × 10`?
Answer
Multiplying by 10 twice multiplies by `100` in total (each digit shifts two columns left). `430 × 10 × 10 = 43,000`.
Art hook Two ×10 taps each slide the digit-tiles one column left; the total shift of two columns is highlighted as a ×100 jump.