Compare fractions sharing numerator or denominator

3.NF.A.3 · take · grade 3

Archetype: Compare Fractions and Decimals by Structure · step in a 7-type progression

Among the numbers from 1 to 9, how many numbers can go in the $\bigstar$ ?

$\dfrac{1}{7} < \dfrac{1}{\bigstar}$

Show solution

Understand

I need to count how many of the digits 1 through 9 can be placed in the star so that the fraction 1/7 is less than 1/star.

Givens

The inequality is 1/7 < 1/(star).
The star must be a whole number from 1 to 9.

Unknowns

How many values of the star make the inequality true.

Constraints

Both fractions are unit fractions (numerator 1).
For unit fractions, a smaller denominator means a larger fraction.

Plan

#5 Look for a Pattern · also uses: #2 Make a Systematic List

Both fractions have numerator 1, so the comparison depends only on the denominators: the unit fraction with the smaller denominator is larger. That pattern instantly tells me 1/star > 1/7 exactly when star < 7, and I can list those digits to count them.

Execute

#5 Look for a Pattern 3.NF.A.3

Both fractions split one whole into equal parts. The more parts you cut a whole into, the smaller each part. So with numerator 1, the fraction is larger when the denominator is smaller.

\dfrac{1}{7} < \dfrac{1}{\bigstar} \iff \bigstar < 7

Sharing a pizza among fewer people gives each person a bigger slice, so smaller denominator = bigger unit fraction.

#2 Make a Systematic List 3.NF.A.3

We need star < 7, with star chosen from 1 through 9. The digits less than 7 are 1, 2, 3, 4, 5, and 6.

\bigstar \in \{1, 2, 3, 4, 5, 6\}

Only denominators smaller than 7 make a slice bigger than 1/7; 7 itself ties and 8, 9 are smaller.

#2 Make a Systematic List 3.NF.A.3

Count the digits in the list 1, 2, 3, 4, 5, 6.

6 \text{ numbers}

There are exactly six whole numbers between 1 and 6, so six choices work.

Answer: 6 numbers

Review

Spot-check the boundaries: 1/6 > 1/7 (true, 6 counts) and 1/7 < 1/7 is false (7 excluded) and 1/8 < 1/7 so 1/7 < 1/8 is false (8 excluded). Only 1-6 qualify, giving 6 - consistent with the answer.

Guess and check (tool 6): test each star from 1 to 9 in 1/7 < 1/star; it holds for 1,2,3,4,5,6 and fails for 7,8,9, again giving 6 values.

Standards · min grade 3

3.NF.A.3 Explain equivalence of fractions and compare fractions by reasoning — Comparing unit fractions by their denominators to determine which stars satisfy 1/7 < 1/star.

💡 For 1-over-something fractions, fewer parts means bigger pieces - so just count the denominators smaller than 7!