Lesson objective: Understand that multiplying a non-unit fraction by a whole number results in a product that is a multiple of the unit fraction.

Students bring prior knowledge of representing the fraction a/b as a sum of fractions 1/b, (4.NF.B.3) and using visual models and equations to represent situational problems involving addition of fractions with like denominators (4.NF.B.3d). This prior knowledge is extended to multiplication of non-unit fractions by whole numbers as students use multiplication to solve repeated addition problems more efficiently. A conceptual challenge students may encounter is recognizing a non-unit fraction as a composite unit in a multiplication problem. For example, when multiplying\(\frac34\) x 2 they need to recognize \(\frac34\) as one amount that is repeated twice. To perform the calculation efficiently, they also need to be able to see this fraction as a certain number of unit fractions that get repeated multiple times, e.g.\(\frac34\) x 2 is also (\(\frac14\) x 3) x 2. Most students will probably be able to see the non-unit fraction in one of these ways. Many students may struggle with toggling between the two ways of looking at the fraction.

The concept is developed through work with a number line, which illustrates that the non-unit fraction can be represented with multiple unit fractions. The number line is also used to show that the same product is reached, whether the non-unit fraction is decomposed into unit fractions and repeated multiple times or the non-unit fraction is taken as a composite unit that gets repeated.

This work helps students deepen their understanding of operations because students learn that the meaning of multiplication remains constant, whether multiplying whole numbers or fractions.

Students engage in Mathematical Practice 8 (look for and express regularity in repeated reasoning) as they notice that instead of repeatedly adding the same non-unit fraction, they can multiply the non-unit fraction, and they begin to develop the algorithm n x a/b = (n xa)/b.

**Key vocabulary:**

- multiple
- non-unit fraction
- unit fraction
- whole number