Ordering Fractions

Imagine cutting two identical cakes. If you cut the first one into 3 pieces, and the second one into 5 pieces, which pieces are bigger? The more pieces you cut, the smaller each piece becomes. Therefore, \( \frac{1}{3} \) (a third of a cake) is larger than \( \frac{1}{5} \).

Mathematically, we can convert to a common denominator: \( \frac{1}{3} = \frac{5}{15} \) and \( \frac{1}{5} = \frac{3}{15} \). Since \( \frac{5}{15} > \frac{3}{15} \), we know \( \frac{1}{3} > \frac{1}{5} \).

We don’t need a common denominator here! Both fractions have the same number of pieces (numerator = 3).

We know that sevenths are larger slices than elevenths (because the whole is shared among fewer people). Therefore, having 3 large slices (\( \frac{3}{7} \)) is more than having 3 small slices (\( \frac{3}{11} \)).

To compare these, we need the pieces to be the same size. We can convert \( \frac{3}{4} \) into eighths by doubling both the numerator and denominator: \( \frac{3}{4} = \frac{6}{8} \).

Now the comparison is obvious: \( \frac{6}{8} \) is clearly greater than \( \frac{5}{8} \). Comparing numerators (3 vs 5) without checking denominators first is a common trap!

Check with decimals: \( \frac{3}{4} = \) 0.75 and \( \frac{5}{8} = \) 0.625.

Both fractions are “one piece away from being whole.” But the missing pieces are different sizes. In \( \frac{8}{9} \), the missing piece is \( \frac{1}{9} \). In \( \frac{6}{7} \), the missing piece is \( \frac{1}{7} \).

Since \( \frac{1}{9} \) is a smaller piece than \( \frac{1}{7} \), the fraction \( \frac{8}{9} \) is closer to 1 because it is missing less. Therefore \( \frac{8}{9} > \frac{6}{7} \).

Using common denominators: \( \frac{8}{9} = \frac{56}{63} \) and \( \frac{6}{7} = \frac{54}{63} \). Since \( \frac{56}{63} > \frac{54}{63} \), we confirm \( \frac{8}{9} > \frac{6}{7} \).

We can use 1 as a benchmark. \( \frac{5}{4} \) is an improper fraction (numerator > denominator), meaning it is greater than 1. On the other hand, \( \frac{9}{10} \) is a proper fraction, meaning it is less than 1.

Since one is more than a whole and the other is less than a whole, \( \frac{5}{4} \) must be the larger number. This strategy is faster than finding a common denominator of 40!

Example: \( \frac{1}{2} \) and \( \frac{2}{4} \)

Another: \( \frac{2}{3} \) and \( \frac{6}{9} \)

Creative: \( \frac{3}{7} \) and \( \frac{12}{28} \) — multiplying both numerator and denominator by 4. Or \( \frac{50}{100} \) and \( \frac{1}{2} \) — using a large denominator that simplifies dramatically.

Trap: \( \frac{1}{3} \) and \( \frac{2}{4} \) — a student might add 1 to both the numerator and denominator of \( \frac{1}{3} \) to get \( \frac{2}{4} \), thinking “I did the same thing to top and bottom.” But adding the same number doesn’t preserve the fraction’s value (only multiplying or dividing does).

Example: \( \frac{3}{10} = \) 0.3, which is between 0.25 and 0.333…

Another: \( \frac{2}{7} \approx \) 0.286

Creative: Convert both to a common denominator of 12: \( \frac{3}{12} \) and \( \frac{4}{12} \). Since there is no whole number between 3 and 4, we multiply by 2 to get \( \frac{6}{24} \) and \( \frac{8}{24} \). Now we can see \( \frac{7}{24} \) fits perfectly.

Trap: \( \frac{1}{5} \) — a student might reason that since 5 is “between” 4 and 3 (or bigger than both), \( \frac{1}{5} \) must be between them. But \( \frac{1}{5} = \) 0.2, which is less than \( \frac{1}{4} \). For unit fractions, a larger denominator means a smaller value.

Example: \( \frac{3}{4} \) and \( \frac{2}{3} \) — denominator 4 > 3, and \( \frac{3}{4} \) (0.75) > \( \frac{2}{3} \) (0.66…).

Another: \( \frac{5}{6} \) and \( \frac{1}{2} \) — denominator 6 > 2, and \( \frac{5}{6} \) > \( \frac{1}{2} \).

Creative: \( \frac{99}{100} \) and \( \frac{1}{2} \) — denominator 100 is vastly larger than 2, yet \( \frac{99}{100} = \) 0.99 is almost double \( \frac{1}{2} \). This proves that a larger denominator does not always equal a smaller fraction.

Trap: \( \frac{1}{4} \) and \( \frac{1}{3} \) — a student might offer this, thinking “4 is bigger than 3.” But here the fraction with the larger denominator is smaller. The condition only works when the numerator of the larger-denominator fraction is large enough to compensate.

Example: \( \frac{3}{2} = \) 1.5

Another: \( \frac{5}{3} \approx \) 1.667

Creative: \( \frac{10}{9} \approx \) 1.111 — only just greater than 1. Or \( \frac{7}{4} \), which connects to the mixed number \( 1\frac{3}{4} \).

Trap: \( \frac{3}{3} \) — a student might think “\( \frac{3}{3} \) is improper because the top is not smaller than the bottom.” However, \( \frac{3}{3} = 1 \) exactly, so it is not between 1 and 2. To be between 1 and 2, the numerator must be larger than the denominator but less than double the denominator.

When fractions share a denominator, the pieces are the same size — so more pieces simply means more. For example, \( \frac{5}{7} > \frac{3}{7} \) because five-sevenths is more than three-sevenths.

This principle is the foundation of the common-denominator comparison method. It always works because once denominators match, the only variable is how many equal-sized pieces you have.

True for unit fractions: \( \frac{1}{5} < \frac{1}{3} \). Dividing a whole into more pieces makes each piece smaller.

False when numerators differ significantly: \( \frac{3}{4} \) and \( \frac{2}{3} \). Here, 4 > 3, but \( \frac{3}{4} \) is larger than \( \frac{2}{3} \). The larger denominator (4) creates smaller pieces, but we have more of them (3 vs 2), and that compensates for the smaller size.

Simplifying a fraction means dividing both numerator and denominator by the same number. This is equivalent to dividing by 1 (in the form \( \frac{n}{n} \)), so the value stays exactly the same.

Example: \( \frac{6}{8} \) simplified to \( \frac{3}{4} \). Both represent 0.75 or 75% of a whole. Some students confuse the numbers getting smaller with the value getting smaller, but simplifying changes the form, not the value.

This is only true when the fraction already equals 1. For example, \( \frac{3}{3} = 1 \), and adding 2 gives \( \frac{5}{5} = 1 \) — the value is unchanged.

But for any other fraction, the value changes: \( \frac{1}{2} \) with 1 added to each gives \( \frac{2}{3} \), and \( \frac{1}{2} \neq \frac{2}{3} \).

Students often confuse this with multiplying both top and bottom by the same number, which does preserve equivalence (scaling). Adding changes the ratio.

The student has the order exactly backwards. They are applying the rule for whole numbers (“6 is bigger than 2”) to denominators. But for fractions, a bigger denominator means smaller pieces.

Because \( \frac{1}{6} \) is the smallest slice, the correct order is \( \frac{1}{6}, \frac{1}{4}, \frac{1}{2} \).

The student got the right answer for the wrong reason (a “False Positive”). The “gap method” is unreliable because it ignores the size of the pieces.

Counter-example: Compare \( \frac{1}{2} \) (gap of 1) and \( \frac{9}{10} \) (gap of 1). The method says they should be equal, but \( \frac{9}{10} \) is clearly much larger!

The correct method is common denominators: \( \frac{5}{9} = \frac{35}{63} \) and \( \frac{4}{7} = \frac{36}{63} \), confirming \( \frac{4}{7} \) is larger.

The student is only comparing numerators. This is like saying “I have 3 pennies and you have 2 fifty-pound notes, so I have more money.” You have more coins, but mine are worth more!

\( \frac{2}{5} \) has fewer pieces than \( \frac{3}{8} \), but the pieces are significantly larger.

Common denominator (40): \( \frac{2}{5} = \frac{16}{40} \) and \( \frac{3}{8} = \frac{15}{40} \). Since \( 16 > 15 \), \( \frac{2}{5} \) is the winner.

The student is treating fractions as separate whole numbers. Crucially, they missed that \( \frac{2}{3} \) and \( \frac{4}{6} \) are equivalent — they have the exact same value!

To compare \( \frac{2}{3} \) and \( \frac{5}{8} \), convert to 24ths: \( \frac{2}{3} = \frac{16}{24} \) and \( \frac{5}{8} = \frac{15}{24} \).

So \( \frac{5}{8} \) is actually the smallest.