In this lesson, students learn the relationship between fractions and division. In an earlier lesson, students learned that when dividing, the remainder is expressed as a fraction. In this lesson, that understanding is extended to show that the quotient in a division problem can always be expressed as a fraction.
Similarly, any fraction can be written as division. In this lesson, students learn that fraction notation and division notation mean the same thing and are interchangeable. They learn to write fractions in division format and division in fraction format.
In an earlier lesson, students learned to multiply a whole number by a fraction using an array to illustrate the process. In this lesson, multiplying a fraction by a fraction is illustrated using the same kind of arrays used before.
The array is used to illustrate why the standard algorithm is used for multiplication and the concepts underlying it.
A common misconception that students often develop when working with whole numbers is that “multiplication makes bigger” and “division makes smaller.” In this lesson, students learn why that is not the case for multiplication with fractions.
They learn that the outcome of multiplying by a fraction can be the larger than the starting number, smaller than the starting number, or the same as the starting number. The reasoning that explains these outcomes is illustrated using arrays as in prior lessons.
In earlier lessons, students learned how to convert mixed numbers to improper fractions. They also learned how to multiply fractions.
In this lesson, they combine what they’ve learned on these two topics to multiply mixed numbers. They learned to convert the mixed numbers to fractions, then multiply the fractions using the standard algorithm for fraction multiplication.