Fractions: The Mother of All Math Issues (and How to Master Them)

Short answer: A fraction names part of a whole: the bottom number (denominator) is the size of each piece and the top number (numerator) is how many pieces you have. To multiply fractions, multiply across. To divide, keep the first, change ÷ to ×, and flip the second. To add or subtract, get a common denominator, then add the tops.

Fractions cause more long-term trouble than almost any other topic in school math. Students who never fully master them often get by until algebra, precalculus or physics, where a fraction is one small part of a long problem and the whole thing falls apart.

Watch Sean Alexander, founder of Alexander Tutoring, walk through multiplying, dividing and adding fractions in this video lesson.

Key points

• Bottom is slice size: The denominator tells you how big each piece is, not how many you have.
• Multiply straight across: Top times top, bottom times bottom, no common denominator needed.
• Divide: keep, change, flip: Dividing by a fraction is multiplying by its reciprocal.
• Add same-size slices: Find a common denominator first, then add only the numerators.
• Always reduce: Divide the top and bottom by any number that goes into both.
• Fractions predict algebra: Elementary fraction skills predict high school math achievement years later.

Why are fractions called the mother of all math issues?

Because almost everything later depends on them. A large study in Psychological Science analyzed nationally representative, long-term data from the United States and the United Kingdom and found that elementary school students’ knowledge of fractions and division predicted their algebra knowledge and overall math achievement in high school, 5 or 6 years later. The link held even after controlling for other math knowledge, general intellectual ability, working memory and family income and education (Siegler et al., 2012).

The federal National Mathematics Advisory Panel reached the same conclusion in its 2008 final report, Foundations for Success. It named proficiency with fractions a major goal for K–8 math and warned that “difficulty with fractions (including decimals and percent) is pervasive and is a major obstacle to further progress in mathematics, including algebra.”

• 5–6 yrs how far ahead elementary fraction skills predicted math achievement
• 2008 year a federal math panel named fractions a major K–8 goal
• Grades 3–5 when Common Core builds fraction understanding

“Next to fractions, the skill students most often fail to master, and the one that causes them the most grief, is negative numbers. I see it all the way up to high school seniors.”

Sean Alexander, founder of Alexander Tutoring

What is a fraction, and how do you write a number in fraction form?

A fraction is a number. In the Common Core standards, students in grades 3 to 5 learn to understand a fraction as a number on the number line, not just as shaded parts of a shape (Number and Operations: Fractions). In 3/4, the 4 says the whole is cut into four equal pieces and the 3 says you have three of them.

Writing a number “as a fraction” comes up constantly. Three common cases:

• Whole number: put it over 1, so 5 = 5/1. This changes nothing, because 5 ÷ 1 = 5.
• Decimal: use place value, then reduce. 0.75 = 75/100 = 3/4.
• Mixed number: multiply the whole number by the denominator and add the numerator. 1 1/2 = (1 × 2 + 1)/2 = 3/2.

How do you multiply fractions?

Multiplication is the easiest operation, so start there. Multiply the numerators together and the denominators together, then reduce if you can.

1. Multiply the tops. For 2/3 × 4/5: 2 × 4 = 8.
2. Multiply the bottoms. 3 × 5 = 15.
3. Reduce if possible. 8/15 has no common factor, so the answer is 8/15.

To multiply a fraction by a whole number, write the whole number over 1 first: 2/3 × 5 = 2/3 × 5/1 = 10/3, which is 3 1/3 as a mixed number. And yes, you do multiply the denominators when multiplying fractions; you only keep the denominator unchanged when adding or subtracting.

How do you divide fractions?

If you can multiply fractions, you can divide them. Use keep, change, flip:

1. Keep the first fraction. 2/3 ÷ 4/5 starts as 2/3.
2. Change ÷ to ×.
3. Flip the second fraction. 4/5 becomes 5/4, so you have 2/3 × 5/4.
4. Multiply across. 2 × 5 = 10 and 3 × 4 = 12, so 10/12.
5. Reduce. Both are even, so divide by 2: 10/12 = 5/6.

Example: dividing a mixed number

Problem: 1 1/2 ÷ 3/4.

1. Convert the mixed number: 1 1/2 = 3/2.
2. Keep, change, flip: 3/2 × 4/3.
3. Multiply: 12/6 = 2. Check: 3/4 fits into 1 1/2 exactly two times.

Dividing a fraction by a whole number works the same way: 3/4 ÷ 2 = 3/4 × 1/2 = 3/8.

Why do you flip the second fraction?

Division asks “how many times does this fit into that?” Ask how many halves fit into 1. Most students can see the answer is 2: two half-full glasses fill one empty glass. The rule agrees: 1 ÷ 1/2 = 1/1 × 2/1 = 2.

Try thirds. A third fits into 1 three times, so it should fit into 2 six times. The rule gives 2/1 × 3/1 = 6. The flip is just a shortcut that keeps track of this “how many fit” logic.

“At Alexander Tutoring, we derive every formula. Instead of memorizing a formula and plugging numbers in, students see where it comes from.”

Sean Alexander, founder of Alexander Tutoring

How do you add and subtract fractions?

Surprisingly, adding is the hardest operation, because it is the one that needs a common denominator. Picture two pies the same size, one cut into thirds and one into fourths. Two slices of thirds plus one slice of fourths is not “three” of anything, because the slices are different sizes.

Example: 2/3 + 1/4

1. Borrow each other’s denominator. Multiply 2/3 by 4/4 and 1/4 by 3/3. Each of those equals 1, so the values do not change.
2. Multiply. 2/3 × 4/4 = 8/12 and 1/4 × 3/3 = 3/12.
3. Add the tops, keep the bottom. 8/12 + 3/12 = 11/12.

Eight twelfths is exactly the same amount of pie as two thirds; the slices are just smaller. Once every slice is the same size, you can add them.

Example: 1/2 + 1/3 and 5/6 − 1/4

• 1/2 + 1/3 = 3/6 + 2/6 = 5/6.
• 5/6 − 1/4: the smallest common denominator is 12, so 10/12 − 3/12 = 7/12.

Tutor tip: Multiplying the two denominators always gives a common denominator. If both denominators share a factor, like 6 and 4, a smaller one (12 instead of 24) keeps the numbers easier, but either works as long as you reduce at the end.

What are the most common fraction mistakes?

To compare two fractions, such as 1/2 and 2/3, rewrite them with a common denominator: 3/6 and 4/6. Now it is clear that 2/3 is greater.

Tutor tip: Before adding, say out loud what the denominator means: “the size of the slice.” If the slices are not the same size yet, you are not ready to add.

How can you help a child who struggles with fractions?

The U.S. Department of Education’s What Works Clearinghouse practice guide on fractions recommends helping students see fractions as numbers that extend the number line, and helping them understand why computation procedures make sense rather than only memorizing them (IES fractions practice guide). At home, that looks like this:

1. Use real sharing problems. Splitting a pizza, a recipe or a bill makes slice size concrete.
2. Put fractions on a number line. Ask where 3/4, 2/3 and 5/4 go. It builds the idea that fractions are numbers with sizes.
3. Ask “why does that work?” If your child can explain the flip or the common denominator in their own words, the rule will stick.
4. Practice a little, often. Five fraction problems a day for a few weeks beats one long session.
5. Go back a grade if needed. Our 5th grade math guide and 6th grade math essentials show which fraction skills come when.

Fractions show up again in ratios and percents in 7th grade math, and in rational expressions in algebra, where fraction errors drive several of the most common algebra mistakes. Parents of middle schoolers can also read how to help your 7th grader succeed in math.

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