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Running to School, Variation 3


Alignments to Content Standards: 6.NS.A.1

Task

Rosa ran $\frac13$ of the way from her home to school. She ran $\frac14$ mile. How far is it between her home and school?

IM Commentary

This task builds on a fifth grade fraction multiplication task, “5.NF Running to School, Variation 1.” “6.NS Running to School, Variation 3” uses the identical context, but asks the corresponding “Group Size Unknown” division problem. See “6.NS Running to School, Variation 2” for the “Number of Groups Unknown” version.

The purpose of this task is to help students extend their understanding of division of whole numbers to division of fractions, and given the simple numbers used, it is most appropriate for students just learning about fraction division because it lends itself easily to a pictorial solution.

Solutions

Solution: Solution

We know that Rosa ran $\frac13$ of the way to school, which is $\frac14$ of a mile.

Sol_1_5aee91025794c24b77b0a143fa8c47e5

So far, all we know is that $\frac13$ of the trip to school is $\frac14$ of a mile. What we are really interested in knowing is the length of the whole trip to school. Thus, the question we're asked to answer is "$\frac13$ of what distance is $\frac14$ mile?" $$\frac13 \times ? = \frac14$$ This is equivalent to the following division problem: $$\frac14 \div \frac13 = ?$$ Since $ \frac14 \div \frac13 = \frac14 \times 3 = \frac34$, the entire trip is $\frac34$ mile.

Alternatively, the whole trip to school is the same as $\frac33$ of the way to school. Thus, we need 3 of the above picture to find the answer.

Sol_2_2a08d7b3378febe7532cc6137f6eda22

Re-labeling the picture will make the answer much easier to see.

Sol_3_7ff31948b033e2bc7bef6b47850b47e0

Solution: A computational approach

This question is equivalent to asking, "$\frac13$ of what distance is $\frac14$ mile?" We can write this symbolically as $$\frac13 \times ? = \frac14$$ which is equivalent to the division problem $$\frac14 \div \frac13=?$$ Since $$\frac14 \div \frac13 = \frac14 \times \frac31 = \frac{3}{4} $$ we see we get the same answer as if we did reasoning about the context in the previous solution.

The distance to Rosa's school is $\frac34$ miles.