# What Should Students Learn about Fractions?

## ESSENTIAL FRACTION SKILLS

Not everyone loves fractions, but fraction skills are important for a number of reasons.

• It helps to be fluent with fraction skills when you take algebra, trigonometry, and calculus courses. When you solve for an unknown in algebra, the answer doesn’t always turn out to be a whole number. One reason that cross multiplication is difficult for many students is that the problems inherently involve fractions. In calculus, simple polynomial anti-derivatives naturally involve fractions (like the integral of x squared, which is one-third of x cubed plus a constant).
• We also use fractions in science courses like chemistry and physics. So if a student is struggling with fractions in math courses, these struggles also impact science courses. On the other hand, if a student takes the time to master fractions in math, this becomes an asset when fractions appear in other courses. Decimals and percentages (which are basically just other forms of fractions where the denominator is a power of ten) are very common forms of fractions in science and engineering.
• Fractions are actually pretty common. Almost every day, we see something happen a few times, and start to wonder how often that happens. Maybe your cell phone is doing something funny, or maybe you notice a quirk in somebody’s personality. If you give attention to this, you’ll realize that it’s common to wonder, “How often does that happen?” Many people just guess at it, but their guesses aren’t always realistic (especially when something obviously isn’t extremely common, but they say something like 90% of the time). If you have a small sample, like it happened 3 out of the last 8 times, you can use the fraction (in this case, 3/8) as a projection. You might convert 3/8 to a percentage and see that it equates to 37.5%, for example.
• If you have a ruler marked in inches, you will see fractions in many common measurements. The fraction will be something like 5/12, 7/8, 3/4, or 1/2. If you have a good feel for fractions, it helps to interpret these numbers. For example, if you measure the diameters of two different balls, and one ball is 3/8″ while the other is 1/3″, can you tell which is bigger just from the numbers?

With the importance of fractions in mind, following is my list of essential fraction skills, based on my experience helping students learn how to apply their math skills in science classes and laboratories.

• Visual association. Students should be able to draw pie slices to represent fractions, or should be able to write down a fraction to represent a pie slice.
• Terminology. You can’t discuss fractions with anybody or understand a lecture about fractions if you don’t understand what the different words and phrases mean, like numerator, denominator, reciprocal, common denominator, reduced fraction, greatest common factor, decimal, percentage, proper fraction, improper fraction, mixed number, ratio, and proportion.
• Reducing fractions. Students should be fluent in reducing fractions down to their simplest form. For example, 8/12 reduces to 2/3 if you divide the numerator (8) and denominator (12) both by 4.
• Common denominators. Given two different fractions, like 4/7 and 3/5, students should be able to find a common denominator.
• Mixed numbers. Students should be able to convert between improper fractions and mixed numbers.
• Addition and subtraction. Students should be able to find a common denominator in order to add or subtract fractions.
• Multiplication. Students should be able to multiply fractions. (It’s easier than addition or subtraction.)
• Reciprocals. Students should know how to find the reciprocal of any fraction or whole number.
• Division. Students should know that dividing two fractions is equivalent to multiplying by the reciprocal of the second fraction.
• Decimals. Students should be able to convert fractions into decimals or decimals into fractions.
• Percentages. Students should be able to convert decimals into percentages or percentages into decimals.
• Repeating decimals. Students should be familiar with repeating decimals and how they relate to fractions.
• Word problems. You know you understand the concepts well and can apply them when you can solve a variety of word problems that involve fractions. 