The unit Fractions: Multiply and Divide starts out with simplifying fractions. Since this process is the opposite of making equivalent fractions, it should be relatively simple for students to understand. We also use the same visual model as we would for making equivalent fractions, just backwards: The pie pieces are joined together instead of split apart.
Next comes the topic of multiplying a fraction by a whole number. Since this can be solved by repeated addition, it is not a difficult concept at all.
Multiplying a fraction by a fraction is first explained as taking a certain part of a fraction. After that, students are shown the usual shortcut for the multiplication of fractions.
Simplifying before multiplying is a process that is not absolutely necessary for fifth graders. I have included it here because it prepares students for the same process in future algebra studies and because it makes fraction multiplication easier. I have also tried to include explanations of why we are allowed to simplify before multiplying. These explanations are actually proofs. I feel it is a great advantage for students to get used to mathematical reasoning and proof methods well before they start high school geometry.
Then, we find the area of a rectangle with fractional side lengths, and show that the area is the same as it would be found by multiplying the side lengths. Students multiply fractional side lengths to find areas of rectangles, and represent fraction products as rectangular areas.
Students also multiply mixed numbers, and study how multiplication can be seen as resizing or scaling. This means, for example, that the multiplication (2/3) × 18 km can be thought of as finding two-thirds of 18 km.
Next, we study division of fractions in special cases. The first one is seeing fractions as divisions; in other words recognizing that 5/3 is the same as 5 ÷ 3. This of course gives us a means of dividing whole numbers and getting fractional answers (for example, 20 ÷ 6 = 3 2/6).
Then students encounter sharing divisions with fractions. For example, if two people share equally 4/5 of a pizza, how much will each person get? This is represented by the division (4/5) ÷ 2 = 2/5. Another case we study is dividing unit fractions by whole numbers (such as (1/2) ÷ 4)
We also divide whole numbers by unit fractions, such as 6 ÷ (1/3). Students will solve these thinking how many times the divisor "fits into" the dividend.
The last lesson is an introduction to ratios, and is optional. Ratios will be studied a lot in 6th and 7th grade, especially in connection with proportions. We are laying the groundwork for that here.