In the previous lesson, students generalized strategies to find the product of any 2 fractions, including some problems with factors greater than 1. This lesson begins with area diagrams with whole numbers that students saw in an earlier grade. The diagrams support students making connections to use the distributive property to multiply mixed numbers by decomposing the mixed number into a whole number and a fraction, then finding the sum of the partial products. Students analyze strategies and solve problems in ways that make sense to them. Building on prior work, students may:

• Use an area diagram.
• Decompose the mixed numbers as a sum of a whole number and a fraction and use the distributive property, supported by a more abstract partial products diagram.
• Express the mixed numbers as fractions and find the product of those fractions.

While the area diagram is more concrete and shows the individual parts that make up the product, the more abstract diagram helps relate the calculation to one that students have seen before and works well for more complex numbers. The choice of values in the problems in this lesson allows students to find the sum of the partial products with reasoning and sense making. It’s acceptable for students to write an expression with the sum of partial products as their solution. In a later unit, students will learn an algorithm for adding fractions with unlike denominators.

What strategies did you anticipate today? Which did you not anticipate? What did you learn about student understanding from the strategies you did not anticipate?