Place Value: From Thousandths to Billions

Every digit's job depends only on where it's standing.

🧱 See It

Imagine a long row of houses, one house per digit. Each house can only hold a single digit, 0 through 9 — never more. The houses are grouped into neighborhoods of three: hundreds, tens, ones of whatever the neighborhood is named. Big neighborhoods (thousands, millions, billions) sit to the left of the Ones neighborhood. Tiny neighborhoods (tenths, hundredths, thousandths) sit to the right of the decimal point.

Billions Millions Thousands Ones Decimals
HTO HTO HTO HTO TenthsHundredthsThousandths
0 0 0 0 0 3 2 0 8 4 5 1 . 0 6 7

The number is 3,208,451.067. The five grey zeros aren't really "in" the number at all — they're just empty houses past where the number stops. The bold zeros (ten-thousands, tenths) are part of the number — they're holding a spot open so every other digit stays in the right house.

✏️ Draw It

The same chart, but now look at what happens as you slide one house over. Every house is worth exactly 10 times the house to its right — and exactly one tenth of the house to its left.

⟵ ×10 as you move left

Billions Millions Thousands Ones Decimals
HTO HTO HTO HTO TenthsHundredthsThousandths
0 0 0 0 0 3 2 0 8 4 5 1 . 0 6 7

÷10 as you move right ⟶

The Thousands house (8) is worth 10× the Hundreds house (4). The Tenths house (0) is worth one tenth of the Ones house (1). Same rule, all the way across.

🔢 Write It

Written as one number: 3,208,451.067. Written as "expanded form," every digit gets pulled out into its own house-value and added back together:

3,000,000 + 200,000 + 8,000 + 400 + 50 + 1 + 0.06 + 0.007

Notice there's no "+ 0" term for the ten-thousands or tenths houses — when a house is empty, it just gets skipped in expanded form. Its job was only to hold the other digits in place.

💡 Worked Examples

Example 1 — Expand a number: 45,703.29

  1. Place each digit in its house: Ten Thousands = 4, Thousands = 5, Hundreds = 7, Tens = 0, Ones = 3, Tenths = 2, Hundredths = 9.
  2. Turn each non-empty house into its own value: 40,000, 5,000, 700, 3, 0.2, 0.09.
  3. Skip the empty Tens house — add the rest: 40,000 + 5,000 + 700 + 3 + 0.2 + 0.09
  4. Answer: 45,703.29

Example 2 — Order least to greatest: 3.402, 3.42, 3.399

More digits after the decimal does not mean a bigger number. Line all three up in a place-value chart, filling empty houses with 0, and compare one column at a time starting from the left.

Ones.TenthsHundredthsThousandths
3.402
3.420
3.399
  1. Ones column: all three have 3 — tied, move on.
  2. Tenths column: 3.399 has 3, the other two have 4. Since 3 < 4, 3.399 is the smallest — it's out of the race for biggest.
  3. Hundredths column (only 3.402 vs 3.42 left, both tied on tenths): 3.402 has 0, 3.42 has 2. Since 0 < 2, 3.402 is smaller than 3.42.
  4. Answer, least to greatest: 3.399 < 3.402 < 3.42
The number with more digits after the decimal point isn't automatically bigger — only the leftmost column where they differ decides it.

🧠 Quick Reference

BillionsMillionsThousandsOnes.TenthsHundredthsThousandths
1,000,000,0001,000,0001,0001.0.10.010.001
Moving left = ×10 per step. Moving right = ÷10 per step.
👪 For Parents & Tutors

BC Curriculum (Level 1 / Grade 6, Number): "Place value (thousandths–billions) and operations with those numbers."

Why the chart/grid: A place-value chart externalizes the one thing working memory keeps dropping — which column a digit belongs in. Instead of eyeballing digits into position from memory, he lines them up in a grid where the column itself does the tracking. This is the same principle as the balance-scale and algebra-tile models used elsewhere in these pages: offload the bookkeeping onto a visual structure so the only job left is deciding what goes where.

Watch for: Two specific working-memory traps: (1) misaligning decimal points when comparing or stacking numbers vertically — always line up the decimal points in a grid first, never free-write one under the other by eye; (2) losing count of zeros in large numbers (is it 3,200,000 or 32,000,000?) — always place digits into a labeled chart rather than counting zeros by hand. If either trap shows up, hand him a blank chart before letting him touch the actual digits.

Comes up again: This exact chart resurfaces every time decimals or large numbers appear at a later level — Level 2 decimal operations (multiplying/dividing by powers of ten), and Level 3 percents (converting between percents, decimals, and place value). It's worth keeping this page bookmarked rather than re-teaching the chart from scratch each time.