Chapter 7: Fractions

A whole pizza split between four friends gives each friend one fourth, written $\frac{1}{4}$. A glass full of water poured equally into three cups gives each cup one third, $\frac{1}{3}$. Whenever we share, cut, measure, or compare parts of a whole, we need fractions.

This chapter walks through fractions from scratch. We start with fractional units like halves, thirds, and quarters , the simplest pieces. We then learn to compare fractions: is $\frac{2}{3}$ bigger than $\frac{3}{4}$? Surprisingly, the answer takes a moment of thought, because the denominators are different.

Next we tackle equivalent fractions , pairs like $\frac{1}{2}$ and $\frac{2}{4}$ that look different but are exactly the same. From there it is a short step to adding and subtracting fractions, which only takes one new idea: a common denominator.

Fractions are not a side topic , they are at the heart of mathematics. Without fractions there is no measurement of time ($\frac{1}{4}$ past three), no recipe ($\frac{1}{2}$ cup of milk), and no music ($\frac{1}{4}$-note beats). Becoming comfortable with fractions is one of the most useful things you can do this year.

What's inside

1. Fractional units , halves, thirds, fourths, and the meaning of $\frac{p}{q}$.
2. Equivalent fractions , different names for the same amount.
3. Comparing fractions , which is bigger, and how to tell.
4. Adding and subtracting fractions , the common-denominator trick.
5. Mixed numbers and improper fractions , fractions bigger than $1$.

Key results / Formula card

How to read this chapter

Draw fraction strips , long rectangles divided into equal pieces , for every example. Folding a paper into halves, quarters, and eighths is the most reliable way to "see" what a fraction means.