Fractions Practice Order with Printable Worksheets

Before the worksheets

Fraction worksheets start with the operations, so they assume two things. First, a student can read a fraction: in $\tfrac{3}{4}$ , the whole is cut into 4 equal parts and we have 3 of them. Second, the student knows that one fraction can be written in many ways, so $\tfrac{1}{2}$ , $\tfrac{2}{4}$ and $\tfrac{3}{6}$ are the same amount. MathGem has no separate worksheet for naming or comparing fractions, so this part is best done with drawings, fraction bars or paper strips folded into equal parts.

The Fractions page has printable lesson pages next to the worksheet. The Investigate tab asks three questions to work out on paper: why halves and thirds have to be cut into sixths before they can be added, why multiplying by a fraction can make a number smaller, and why dividing "flips" the second fraction. They take a few minutes and make the rules in the steps below easier to remember.

The times tables matter more than they seem. Finding a common denominator means finding a common multiple, and simplifying an answer means spotting a common factor. A student who is still working out $6 \times 8$ will find fractions slow for that reason alone; the times tables guide helps there.

What the settings change

The Fractions worksheet has three settings:

Operation: addition, subtraction, multiplication or division. Each worksheet uses one.
Max Denominator: up to 6 or up to 12. Every fraction on the page is less than 1 and in lowest terms, so "up to 6" uses only 11 different fractions: $\tfrac{1}{2}$ , $\tfrac{1}{3}$ , $\tfrac{2}{3}$ , $\tfrac{1}{4}$ , $\tfrac{3}{4}$ , the fifths and $\tfrac{1}{6}$ , $\tfrac{5}{6}$ . "Up to 12" uses 45.
Mixed Numbers: when this is ticked, every problem uses mixed numbers such as $2\tfrac{1}{3}$ .

The answer sheet gives every answer in simplest form, and an answer greater than 1 as a mixed number: $\tfrac{17}{12}$ appears as 1 5/12.

A practice order

Addition, denominators up to 6. With small denominators, a common denominator can be found by listing multiples; the largest one needed is 30, for fifths and sixths. More than half of these answers are 1 or more, for example $\frac{3}{4} + \frac{2}{3} = \frac{9}{12} + \frac{8}{12} = \frac{17}{12} = 1\tfrac{5}{12},$ so turning an improper fraction into a mixed number is part of the very first page. It is worth showing once before starting.
Subtraction, denominators up to 6. The larger fraction always comes first, so answers are never negative. Every answer is less than 1, which makes this page a little quicker than addition.
Addition and subtraction, denominators up to 12. Listing multiples now takes longer: the common denominator can be as large as 132, for elevenths and twelfths. This is the time to look for the least common multiple. For $\tfrac{5}{8} + \tfrac{7}{12}$ , the common denominator 24 is much easier to work with than $8 \times 12 = 96$ : $\frac{5}{8} + \frac{7}{12} = \frac{15}{24} + \frac{14}{24} = \frac{29}{24} = 1\tfrac{5}{24}$
Multiplication. Start again at denominators up to 6. No common denominator is needed: multiply the numerators and multiply the denominators, then simplify. Every answer is smaller than both fractions in the problem, which gives a quick check. With denominators up to 12, simplifying before multiplying saves work: $\frac{3}{4} \times \frac{2}{9} = \frac{1}{2} \times \frac{1}{3} = \frac{1}{6}$ (3 and 9 share a factor of 3, and 2 and 4 share a factor of 2).
Division. Keep the first fraction, flip the second and multiply. Dividing by a fraction less than 1 always gives an answer larger than the first fraction, and more than half of the answers are 1 or more. Reading a problem as "how many of these fit?" helps it make sense: $\tfrac{2}{3} \div \tfrac{1}{6}$ asks how many sixths fit in two thirds, and $\frac{2}{3} \div \frac{1}{6} = \frac{2}{3} \times \frac{6}{1} = \frac{12}{3} = 4.$
Mixed numbers. Tick Mixed Numbers last, and go through the four operations again, starting with addition. The first number's whole part is from 1 to 5 and the second's from 1 to 4. Changing both numbers to improper fractions first works for every operation, and it is the method the answer sheet follows: $3\tfrac{1}{4} - 1\tfrac{2}{3} = \frac{13}{4} - \frac{5}{3} = \frac{39}{12} - \frac{20}{12} = \frac{19}{12} = 1\tfrac{7}{12}$ If a student has learned to subtract the whole numbers and the fractions separately, regrouping when needed, the answer is the same, and the answer sheet only shows the final answer.

When to move on

Check the unfinished answers separately. An answer such as $\tfrac{6}{15}$ instead of $\tfrac{2}{5}$ is right but not finished. Marking it differently from a wrong answer shows whether the problem is the method or just the last step.
Look at where the mistakes happen. If most mistakes are in finding the common denominator or in simplifying, a few days of times-table pages usually helps more than more fraction pages.
Estimate before calculating. $\tfrac{3}{4} + \tfrac{2}{3}$ is more than 1, because both fractions are more than a half. A student who estimates first will catch the classic mistake of adding the denominators ( $\tfrac{5}{7}$ is less than 1).
Mix the operations once each is secure. Alternating the operation from one page to the next makes students decide which rule to use, which is what a test or a later topic will ask of them.

After fractions

Fractions are the base for the rest of Basic Math 2. Decimals are fractions with denominators of 10, 100 and so on, and percentages are fractions out of 100. Equivalent fractions return in ratios and proportions, and the same rules are used later to simplify and solve equations with fractions in algebra. The grade-by-grade practice roadmap shows where fractions fit in the bigger picture.