Note that in both cases, there is a multiplication by \(\frac 12\) and another multiplication by 5, which is the same as multiplication by \(\frac 52\). To find out how much is in the whole group, we multiply by 5.) We divide that value by 2 (or multiply by \(\frac12\)) to find one fifth of a group. We mark two-fifths of it as having a value of \(\frac 32\). Where do we see the multiplication by 5 and by \(\frac 12\) in the diagramming process?” (We draw a tape diagram to represent a whole group. “Suppose we interpret \(\frac32 \div \frac 25\) to mean ‘ \(\frac 25\) of what number is \(\frac32\)?’ and use a tape diagram to find the answer.Counting by two-fifths leads to half as many parts. This is the multiplication by \(\frac12\).) Partitioning into fifths gives as 5 times as many parts. This is the multiplication by 5. Where do we see the multiplication by 5 and by \(\frac 12\) in the diagramming process?” (We draw a diagram to represent \(\frac 32\) and draw equal parts, each with a value of \(\frac15\). We count how many groups of \(\frac25\) there are. “Suppose we interpret \(\frac32 \div \frac 25\) to mean ‘how many \(\frac 25\) are in \(\frac 32\)?’ and use a tape diagram to find the answer.(If time permits, consider illustrating each diagram for all to see.) Let's see how this is the same or different than finding the quotient using tape diagrams. We found that to divide \(\frac32\) by \(\frac 25\), for example, we can multiply \(\frac 32\) by 5 and then by \(\frac 12\), or simply multiply \(\frac32\) by \(\frac52\). In this lesson, we noticed a more-efficient way to divide fractions. Explain that although we now have a reliable and efficient method to divide any number by any fraction, sometimes it is still easier and more natural to think of the quotient in terms of a multiplication problem with a missing factor and to use diagrams to find the missing factor. Remind students that in the past few activities, we learned that:ĭividing by a whole number \(n\) is the same as multiplying by a unit fraction \(\frac\). Invite a couple of students to share their conclusion about how to divide a number by any fraction. Then, review the sequence of reasoning that led us to this conclusion using both numerical examples and algebraic statements throughout.
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