Proper or Improper?

Sort these fractions: $\frac{3}{5}$, $\frac{8}{5}$ and $\frac{5}{5}$. Which are proper and which are improper?

Look at $\frac{3}{5}$: Numerator $3$, denominator $5$: $3 < 5$ = Proper, less than 1

Look at $\frac{8}{5}$: Numerator $8$, denominator $5$: $8 > 5$ = Improper, more than 1

Look at $\frac{5}{5}$: $5 = 5$, so all five fifths together make one whole = Improper, exactly 1

Answer: $\frac{3}{5}$ is proper. $\frac{8}{5}$ and $\frac{5}{5}$ are improper, and $\frac{5}{5} = 1$.

Counting Pizza Slices

Every pizza at a party is cut into 4 equal slices. Lena puts 7 slices on a big plate. What fraction of a pizza is that, and is it more than one pizza?

Find the size of one slice: Each pizza has 4 equal slices = One slice is $\frac{1}{4}$ of a pizza

Count the slices: 7 slices of $\frac{1}{4}$ each = $\frac{7}{4}$

Compare with one pizza: One whole pizza is $\frac{4}{4}$, and $7 > 4$ = $\frac{7}{4} > 1$

See what is left over: $\frac{7}{4} = \frac{4}{4} + \frac{3}{4}$ = 1 whole pizza and 3 more slices

Answer: Lena has $\frac{7}{4}$ of a pizza: one whole pizza plus $\frac{3}{4}$ of another one.

Finding 5/3 on the Number Line

Where does $\frac{5}{3}$ lie on the number line?

Split each unit into thirds: Divide the space from 0 to 1 and from 1 to 2 into 3 equal parts each = Each small jump is $\frac{1}{3}$

Count three jumps: $\frac{1}{3}, \frac{2}{3}, \frac{3}{3}$ = $\frac{3}{3}$ lands exactly on 1

Count two more jumps: $\frac{4}{3}, \frac{5}{3}$ = $\frac{5}{3}$ is 2 jumps past 1

Describe the position: Two of the three jumps between 1 and 2 are done = Between 1 and 2, closer to 2

Answer: $\frac{5}{3}$ lies between 1 and 2, two thirds of the way from 1 to 2.

Improper Fractions That Are Whole Numbers

Which of these fractions are whole numbers: $\frac{6}{3}$, $\frac{10}{4}$, $\frac{12}{4}$?

Divide 6 by 3: $6 \div 3 = 2$ = $\frac{6}{3} = 2$, a whole number

Divide 10 by 4: $4 \times 2 = 8$ and $4 \times 3 = 12$, so 10 is not a multiple of 4 = $\frac{10}{4}$ lies between 2 and 3

Divide 12 by 4: $12 \div 4 = 3$ = $\frac{12}{4} = 3$, a whole number

Answer: $\frac{6}{3} = 2$ and $\frac{12}{4} = 3$ are whole numbers. $\frac{10}{4}$ is not: it lies between 2 and 3.

Comparing Two Improper Fractions

Which is greater: $\frac{9}{4}$ or $\frac{7}{3}$?

Find the wholes in $\frac{9}{4}$: $\frac{8}{4} = 2$, so $\frac{9}{4} = 2 + \frac{1}{4}$ = 2 and one quarter

Find the wholes in $\frac{7}{3}$: $\frac{6}{3} = 2$, so $\frac{7}{3} = 2 + \frac{1}{3}$ = 2 and one third

Compare the extra parts: Thirds are bigger pieces than quarters: $\frac{1}{3} > \frac{1}{4}$ = $\frac{7}{3}$ goes further past 2

Answer: $\frac{7}{3}$ is greater than $\frac{9}{4}$.

Mistake: Thinking a fraction can never be greater than 1

Why: Fractions are often introduced as parts of one whole, so it seems impossible to have more parts than the whole has. But a fraction counts equal parts, and you can count more parts than fit into one whole.

Correct: $\frac{7}{4}$ means seven quarters: one whole (four quarters) plus three more quarters, so $\frac{7}{4} > 1$.

Mistake: Calling $\frac{4}{4}$ a proper fraction

Why: It looks like an ordinary fraction, but a proper fraction must be less than 1, and $\frac{4}{4}$ is exactly 1.

Correct: When the numerator equals the denominator, the fraction is improper and equal to 1: $\frac{4}{4} = \frac{9}{9} = 1$.

Mistake: Putting $\frac{5}{4}$ between 0 and 1 on the number line

Why: Many students stop counting quarters at 1. But after $\frac{4}{4}$ the jumps simply keep going, and $\frac{5}{4}$ is the next mark.

Correct: Keep counting past 1: $\frac{4}{4} = 1$, so $\frac{5}{4}$ is one quarter to the right of 1.

Mistake: Thinking improper means wrong

Why: In everyday language improper sounds like a mistake, but in math it is only the name for fractions that are 1 or more.

Correct: $\frac{9}{4}$ is a perfectly correct answer. Often it is the most useful form for calculating.

Cooking with Measuring Scoops

Recipes are often measured with small scoops, and several scoops easily add up to more than one cup.

Three scoops of $\frac{1}{2}$ cup of milk are $\frac{3}{2}$ cups, which is more than 1 cup.

Time in Quarter Hours

A quarter hour is 15 minutes, so timetables and schedules often count time in quarters of an hour.

A film that lasts 6 quarter hours lasts $\frac{6}{4}$ of an hour, which is 90 minutes.

Water Bottles for a Team

Drinks are often sold in half liter bottles, and a few bottles quickly add up to several liters.

Two half liter bottles hold $\frac{2}{2} = 1$ liter of water.

Q: Is $\frac{4}{4}$ a proper or an improper fraction?

A: It is improper. A proper fraction must be less than 1, but $\frac{4}{4}$ is exactly 1. Every fraction whose numerator equals its denominator is improper and equal to 1.

Q: Why are they called improper?

A: The name is very old. A proper fraction was seen as a true part of one whole, and a fraction that is a whole or more did not fit that idea, so it was called improper. It does not mean wrong.

Q: Can every whole number be written as an improper fraction?

A: Yes. Multiply the whole number by any denominator to get the numerator: $3 = \frac{3}{1} = \frac{6}{2} = \frac{12}{4}$.

Q: Do I always have to change an improper fraction into a mixed number?

A: No. Both forms are correct. Improper fractions are usually easier for calculating, while mixed numbers like $1\frac{3}{4}$ show the size at a glance. Changing between the two forms is the topic of the next lessons.