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Mathematics › 🌉 Grade 5 › The Fractal Number Line: Discovering Decimals
🌉 Grade 5 · Lesson 1 of 11

The Fractal Number Line: Discovering Decimals

Zoom between any two whole numbers and the empty-looking gap splits into tenths, then hundredths — decimals all the way down.

Grade 5Elementary
The Fractal Number Line: Discovering Decimals — illustration
💡
The big idea: Between any two whole numbers there is an entire hidden world of smaller numbers. Decimals are place value extended past the ones place: zoom into the gap between 0 and 1 and it splits into ten equal tenths; zoom into one tenth and it splits into ten equal hundredths. The pattern is exactly the same base-ten pattern you already know, just running in the other direction.
🎯 By the end, you'll be able to
  • Explain that decimals extend place value to parts smaller than one whole
  • Locate and plot decimals (tenths and hundredths) on a number line
  • Compare and order decimals by their place value, not just their digits
  • Convert between a decimal and a fraction with denominator 10 or 100
📎 You should already know
  • Whole-number place value
  • Plotting points on a number line

Zooming into the gap

Between the whole numbers 0 and 1, the number line looks empty — just a bare stretch with nothing marked on it. But imagine a magnifying glass powerful enough to zoom into that gap. It is not empty at all. It is packed with its own evenly spaced numbers, following the very same base-ten pattern you already know from place value.

Those numbers are decimals: a way of writing parts of a whole using the same digits 0 through 9, just shifted one place further to the right of a decimal point.

🔑 Each zoom splits the gap into ten
Zoom in once between any two whole numbers and the gap splits into 10 equal tenths. Zoom in again on just one of those tenths and it splits into 10 equal hundredths. Every zoom multiplies the number of pieces by 10 — place value running in reverse.
\[ 0.1 = \dfrac{1}{10} \]
The first digit after the decimal point counts tenths — the equal-size pieces you get from one zoom.

Reading the tenths place

In the decimal 0.7, the digit 7 sits in the tenths place. It means 7 of the 10 equal pieces between 0 and 1, the same amount as the fraction 7⁄10. The decimal point is just a marker that says “whole numbers stop here, parts of a whole start here.”

🎮 The Decimal Magnifier LIVE
Zoom between two whole numbers to reveal tenths, then hundredths — decimals are just a finer number line.

Zooming again: hundredths

Pick just one of those ten tenths and zoom in on it the same way. It splits into 10 even smaller, equal pieces called hundredths. The second digit after the decimal point counts these. In 0.42, the 4 counts tenths and the 2 counts hundredths — together, 42 of the 100 equal pieces between 0 and 1.

\[ 0.01 = \dfrac{1}{100} \]
The second digit after the decimal point counts hundredths — one more zoom, pieces ten times smaller than tenths.
✨ More digits does not always mean bigger
Which is greater, 0.5 or 0.36? It is tempting to compare 5 and 36 and pick 0.36 — but that is wrong. Line up the place values: 0.5 is the same as 0.50, or 50 hundredths, while 0.36 is only 36 hundredths. So 0.5 is greater than 0.36, even though it is written with only one digit.
📝 Worked example: Order these decimals from least to greatest: 0.3, 0.08, 0.25.
  1. Rewrite each with two decimal places so the place values line up: 0.30, 0.08, 0.25.
  2. Compare the hundredths: 8 hundredths, 25 hundredths, 30 hundredths.
  3. In order from smallest to largest: 0.08, then 0.25, then 0.30.
✓ Least to greatest: <strong>0.08, 0.25, 0.3</strong>.
📝 Worked example: Write 47&frasl;100 as a decimal, and write 0.6 as a fraction.
  1. 47⁄100 means 47 of the 100 equal hundredths pieces, so it is written 0.47.
  2. 0.6 has one digit after the decimal point, so it counts tenths: 6 of the 10 equal pieces.
  3. That is the fraction 6⁄10.
✓ 47&frasl;100 = <strong>0.47</strong>, and 0.6 = <strong>6&frasl;10</strong>.
⚠️ Line up the place values before comparing
A common mistake is comparing 0.9 and 0.30 as if 30 > 9 makes 0.30 the bigger decimal. Rewrite both with the same number of decimal places first: 0.90 versus 0.30. Now it is clear — 0.9 is greater. Zeros at the end of a decimal do not change its value, but they do make comparisons safer.

Check your understanding

1. What is 0.1 as a fraction?
The first digit after the decimal point counts tenths, so 0.1 is 1 of 10 equal pieces: 1/10.
2. Which decimal is greater: 0.5 or 0.36?
0.5 equals 0.50, which is 50 hundredths — more than the 36 hundredths in 0.36.
3. Zooming once between 0 and 1 splits the gap into how many equal tenths?
One zoom always splits a whole-number gap into 10 equal tenths.
4. What is 0.07 as a fraction?
The 0 in the tenths place and 7 in the hundredths place mean 7 of the 100 equal pieces: 7/100.
5. Order from least to greatest: 0.4, 0.04, 0.44.
Rewriting with two places: 0.40, 0.04, 0.44 — so the order is 0.04, then 0.40, then 0.44.
✅ Key takeaways
  • Decimals extend place value to parts smaller than one whole.
  • Zooming into a gap between whole numbers splits it into 10 equal tenths; zooming into one tenth splits it into 10 equal hundredths.
  • The first digit after the decimal point is tenths; the second digit is hundredths.
  • Compare decimals by lining up place values, not by how many digits they have &mdash; 0.5 is greater than 0.36.
  • A decimal like 0.7 is just another way of writing the fraction 7/10.