In this activity, students make the connection between the fraction determined by the original rectangle and the resulting more precise fraction. The two rectangles taken together are designed to help students notice that decomposing rectangles is a geometric way to determine the greatest common factor of two numbers. (This is a geometric version of Euclid’s algorithm for finding the greatest common factor.) Students reason abstractly and quantitatively when switching between numeric and geometric representations (MP2).

The purpose of this discussion is to make sure students understand that using a decomposition model can help turn fractions into mixed numbers and find the greatest common factor of the numerator and denominator. It is not necessary for students to understand a general argument for why chopping rectangles can help you know the greatest common factor of two numbers. Here are some questions for discussion:

• “We call fractions like and equivalent fractions. What would be true about the rectangles we could draw about other equivalent fractions?” (Equivalent fractions would have rectangles that are decomposed in the same way, but each square would be larger or smaller by the same amount.)
• “Suppose there is a pair of equivalent fractions where the numbers in one of the fractions are 5 times the numbers on the other fractions. If both fractions were represented with rectangles decomposed into squares, how would the side lengths of the squares compare?” (The rectangle representing the fraction with the larger numbers would have squares whose side lengths are 5 times those of the smaller rectangle.)

This activity extends the work with rectangles and fractions to continued fractions. Continued fractions are not a part of grade-level work, but they can be reasoned about and rewritten using grade-level skills for operating on fractions. This repeated reasoning allows students to generalize the relationship between the geometric representation and the continued fraction (MP8). In particular, the insight that (a special case of invert and multiply) is helpful.

In this activity, students consolidate their understanding about how the greatest common factor of the numerator and denominator of a fraction can help them write an equivalent fraction whose numerator and denominator have greatest common factor 1—sometimes called “lowest terms.”