In previous lessons, students have compared the size of a product to the size of one factor by reasoning about the size of the other factor. They have done this, using calculations, area diagrams, and number-line diagrams. The goal of this lesson is to use the distributive property to explain why the comparisons work, in all cases, without calculating. The key observation is that a fraction greater than 1, such as , can be written as ‘the whole part plus the fractional part,’ in our example, . So multiplying by a fraction greater than 1 increases any number by the part of the fraction greater than the whole. This means multiplying by increases any number by of that number. In the same way, a fraction less than 1, such as , can be written as 'the whole minus the difference between the whole and the fraction,' in our example,   or . So multiplying by a fraction less than 1 decreases any number by the difference between the whole and the fraction. This means multiplying by decreases any number by of that number.  

This lesson has a Student Section Summary.