Listing fractions in increasing size
Task
Order the following fractions from smallest to largest: \frac{3}{8}, \; \;\; \; \frac{1}{3}, \; \; \; \; \frac{5}{9}, \; \; \; \; \frac{2}{5}
IM Commentary
This is a challenging fraction comparison problem. The fractions for this task have been carefully chosen to encourage and reward different methods of comparison. The first solution judiciously uses each of the following strategies when appropriate:
- comparing to benchmark fractions,
- finding a common denominator,
- finding a common numerator.
The second and third solution shown use only either common denominators or numerators. Teachers should encourage multiple approaches to solving the problem. By working through some of the different techniques for comparing fractions, students will develop a deeper "fraction number sense." A discussion should ensue where students compare and share their methods. If all suggested methods shown in the solution given below do not come up in the discussion, the teacher could make a suggestion and let the students work further. In fact, the task is well-suited for helping students develop a strategic sense of when to use which technique.
This task is mostly intended for instructional purposes, although it has value as a formative assessment item as well. Finding the appropriate order for the fractions does not require a common denominator for all fractions, but a student who takes this legitimate approach would face computational difficulty here since the least common denominator for all four is 360, which is quite large. Students who choose a different strategy for a each fraction pair based on which approach would be easier are demonstrating a well developed fraction number sense.
Solutions
Solution: 1 Comparing with benchmark fractions and using common numerators and denominators strategically
The fractions \frac{3}{8}, \frac{1}{3}, and \frac{2}{5} are all less than \frac{1}{2}: this can be seen, for example, by doubling the numerator and observing that twice the numerator is less than the denominator in all three cases. For \frac{5}{9}, on the other hand, twice 5 is bigger than 9 so if this fraction is doubled, it gives more than one and so \frac{5}{9} > \frac{1}{2}. This tells us which of the four fractions is largest, so now we focus on comparing the other three.
Looking at \frac{3}{8}, \frac{2}{5}, and \frac{1}{3}, we can see that \frac{3}{9} \lt \frac{3}{8}
There is no obvious benchmark fraction to compare \frac{3}{8} and \frac{2}{5}. However, with only two fractions we can use a common denominator: 8 \times 5 = 40 is a natural common denominator to use. Each \frac{1}{8} needs to be subdivided into 5 equal parts to make \frac{1}{40}, so \frac{3}{8} = \frac{3\times 5}{8\times 5} = \frac{15}{40}
Notice that we did not need to compare \frac{1}{3} and \frac{2}{5} because our work shows that \frac{3}{8} lies between them. If we had started by comparing \frac{1}{3} to \frac{2}{5} we would have found that \frac{1}{3} \lt \frac{2}{5}: this would not have told us where \frac{3}{8} lies relative to \frac{2}{5} since both are larger than \frac{1}{3}.
Solution: 2 Common Denominator
The denominators for the fractions are 8,3,9, and 5. Since 3 is a factor of 9, 8 \times 9\times 5 = 360
Now that we have a common denominator of 360 the size of the fractions can be compared by looking at the size of the numerators. We have \frac{120}{360} \lt \frac{135}{360} \lt \frac{144}{360} \lt \frac{200}{360}
Solution: 3 Common Numerator
The numerators of these fractions, 3,1,5, and 2, are simpler than the denominators and so finding a common numerator is a good strategy for comparing their size. Since 1 \times 2 \times 3 \times 5 = 30 we can use 30 as a common numerator. We can then rewrite each of these fractions with 30 as a numerator. This will mean multiplying numerator and denominator of each fraction by the appropriate number: 10 for \frac{3}{8}, 30 for \frac{1}{3}, 6 for \frac{5}{9}, and 15 for \frac{2}{5}:
One variation on this theme would be to observe, as in the second solution, that \frac{5}{9} is the largest of the four fractions because it is larger than \frac{1}{2} while the other three are all less than \frac{1}{2}. This makes the rest of the ordering a little easier because now, looking at \frac{1}{3}, \frac{2}{5}, and \frac{3}{8}, we could employ the same method with a smaller common numerator, namely 6. Concretely,
Listing fractions in increasing size
Order the following fractions from smallest to largest: \frac{3}{8}, \; \;\; \; \frac{1}{3}, \; \; \; \; \frac{5}{9}, \; \; \; \; \frac{2}{5}