Lesson 5

In this lesson, students apply the general procedure they just learned for solving \(px=q\) in order to define what \(\frac{a}{b}\) means when \(a\) and \(b\) are not whole numbers. Up until now, students have likely only seen a fraction bar separating two whole numbers. This is because before grade 6, they couldn't divide arbitrary rational numbers. Now an expression like \(\frac{2.5}{8.9}\) or \(\frac{\frac12}{\frac35}\) can be well-defined. But the definition is not the same as what they learned for, for example, \(\frac25\) in grade 3, where they learned that \(\frac25\) is the number you get by partitioning the interval from 0 to 1 into 5 equal parts and then marking off 2 of the parts.  That definition only works for whole numbers. However, in grade 5, students learned that \(2 \div 3 = \frac23\), so in grade 6 it makes sense to define \(\frac{2.5}{8.9}\) as \(2.5 \div 8.9\).