Match fraction forms to compare sizes

3.NF.A.3 · take · grade 3

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

How many whole numbers can go in the $\star$ ?

$4\frac{3}{8} < \frac{\star}{8} < 5\frac{1}{8}$

Show solution

Understand

I need to count how many whole numbers can replace the star so that the improper fraction star/8 sits strictly between the mixed numbers 4 and 3/8 and 5 and 1/8.

Givens

The inequality is 4 3/8 < star/8 < 5 1/8.
The middle quantity is star/8, where star is a whole number.
Both ends are mixed numbers with denominator 8.

Unknowns

How many whole numbers star make the inequality true.

Constraints

star must be a whole number.
The inequality is strict, so star/8 cannot equal either endpoint.

Plan

#15 Organize Information in More Ways · also uses: #2 Make a Systematic List

To compare fairly, rewrite both mixed-number endpoints as improper fractions over 8 so every quantity has the same denominator. Then the numerators can be compared directly and the valid whole numbers listed.

Execute

#15 Organize Information in More Ways 3.NF.A.3

Convert 4 3/8 to eighths: 4 wholes is 32 eighths, plus 3 eighths.

4\tfrac{3}{8} = \frac{4\times 8 + 3}{8} = \frac{35}{8}

Matching denominators lets us compare fractions just by their numerators.

#15 Organize Information in More Ways 3.NF.A.3

Convert 5 1/8 to eighths: 5 wholes is 40 eighths, plus 1 eighth.

5\tfrac{1}{8} = \frac{5\times 8 + 1}{8} = \frac{41}{8}

Now all three quantities are eighths, so only the top numbers matter.

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

The inequality becomes 35/8 < star/8 < 41/8, so star must be a whole number strictly between 35 and 41: that is 36, 37, 38, 39, 40.

35 < \star < 41 \Rightarrow \star \in \{36,37,38,39,40\}

With equal denominators, comparing fractions is the same as comparing whole-number numerators.

#2 Make a Systematic List 3.OA.A.1

There are five whole numbers in the list.

\{36,37,38,39,40\} \Rightarrow 5

Just count the items in the systematic list.

Answer: 5

Review

The endpoints 35 and 41 are 6 apart; excluding both ends leaves the 5 whole numbers in between, which matches the count. Each candidate, like 36/8 = 4.5, indeed lands between 4 3/8 and 5 1/8.

Number-line reasoning (tool 1): mark 35/8 and 41/8 on a line of eighths; the whole-number numerators strictly between them are 36 through 40, again giving 5 values.

Standards · min grade 3

3.NF.A.3 Explain equivalence of fractions and compare fractions by reasoning — Converting mixed numbers to eighths so the fractions can be compared by numerator.
3.OA.A.1 Interpret products of whole numbers as total number of objects in groups — Converting whole parts to eighths (4 x 8, 5 x 8) and counting the valid values.

💡 This only needs Grade 3 fraction sense: make the bottoms match, then just compare the top numbers!