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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.