Up until now, students have used visual representations or other strategies to reason about and generate equivalent fractions. Along the way, they are likely to have noticed patterns in the numerators and denominators of equivalent fractions. While some students may have generalized and applied those observations intuitively, this is the first lesson in which students are prompted to reason numerically about the numbers in equivalent fractions. 

Students notice that a fraction has the same location on the number line as a fraction , so we can generate fractions that are equivalent to by multiplying both and by the same factor . In other words, students can use multiples of and to generate fractions that are equivalent to . Sample responses are shown in the form , but students do not need to use this notation.

In an upcoming lesson, students will reason in the other direction: using factors that are common to and to write equivalent fractions. They will see that dividing and by the same factor gives a fraction equivalent to .

To reason numerically, the goal is for students to begin to describe number relationships without visual representations. Did it seem that students were doing this in today’s lesson? Which diagrams are they still holding on to?