Problem
Find fractions between two on the number line
On a number line from 0 to 1 divided into 12 equal parts, mark the fractions 5/6, 11/12, 1/2, and 4/6, then list every one of them that lies between 5/12 and 3/4.
Fractions
Your answer
Watch out It's easy to check only that a fraction is greater than 5/12 and forget it must also be less than 3/4 — that would wrongly include 5/6 and 11/12 too.
How to solve
Strategy Make a Systematic List — Rewriting every fraction over the common denominator 12 (matching the number-line ticks) lets us list each one's position and read off which lie between the two endpoints. The number line is the diagram that anchors the comparison.
1STEP 1
Express the endpoints in twelfths
Write the boundaries over 12: 5/12 stays 5/12, and 3/4 = 9/12 — so look for fractions between 5/12 and 9/12.
3/4 = 9/12
Matching the tick spacing makes every position a count of twelfths.
3.NF.A.3Draw A Diagram2STEP 2
Convert the four fractions to twelfths
Rewrite each fraction with denominator 12: 5/6 = 10/12, 11/12 stays 11/12, 1/2 = 6/12, and 4/6 = 8/12.
5/6 = 10/12, 1/2 = 6/12, 4/6 = 8/12
Equivalent fractions over 12 line up exactly with the marked ticks.
3.NF.A.3Make A Systematic List3STEP 3
List which fall between 5/12 and 9/12
In twelfths, 6/12 and 8/12 fall inside (5/12, 9/12), but 10/12 and 11/12 do not — so 1/2 and 4/6 lie between.
6/12 and 8/12 are between 5/12 and 9/12; 10/12 and 11/12 are greater than 9/12
On the number line, a fraction is between two others exactly when its tick sits between their ticks.
3.NF.A.2Make A Systematic ListWritten over twelfths, a fraction lies between 5/12 and 9/12 exactly when its count of twelfths is more than five and fewer than nine.
Why?
Rewriting a fraction over 12 does not move its point on the line — 3/4 marks the very same place as 9/12 — so the endpoints and all four fractions can be read as counts of twelfths.
Why?
Cutting each of the 4 fourths into 3 equal pieces makes 12 equal twelfths, and 3 of those small pieces exactly refill one fourth, so 9 of them reach where 3 fourths reached.
🧱Whole is the sum of its partsBreak a thing into pieces with no gaps or overlaps and the pieces add back to the whole.
Why?
When every fraction is built from the same-size twelfths, the one with more twelfths sits farther to the right, so comparing positions becomes comparing counts.
Why?
Each twelfth is one equal step, and the distance from 0 to a tick is those equal steps laid end to end, so more steps reach a farther point.
🧱Whole is the sum of its partsBreak a thing into pieces with no gaps or overlaps and the pieces add back to the whole.
Answer
1/2 and 4/6
In twelfths the target window is 6/12 through 8/12; 1/2 = 6/12 and 4/6 = 8/12 sit inside it, while 5/6 = 10/12 and 11/12 sit past 9/12, so they are correctly excluded. The marks on the number line agree.
Takeaway
Put every fraction over the same 12 ticks, and you can see at a glance which ones sit in between!
- Express the endpoints in twelfths
- Convert the four fractions to twelfths
- List which fall between 5/12 and 9/12
Where next?
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