Ordering Fractions
The goal of this activity is to practice using a variety of strategies to compare fractional quantities and place them on a number line.
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| Sequence 1: Use the red points on the arrows to place the fractions in the correct order and position on the number line. When you are finished, click on the "Show Answers" button. |
Reflecting on the example:
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Did you get the correct order?
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Did you get the correct position on the number line? How close do you think one should be to the actual position indicated in the answer?
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What was similar about each of the fractions in this example?
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Summary of sequence 1: In a "Same Denominator" strategy, if the denominators of all fractions are the same, simply compare their numerators. The fractions will end up in the order of their numerators. For example, with fractions 4/6, and 7/6, 2/6 is the least and 7/6 is the greatest.
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Reflection on sequence 2:
As noted above, the first sequence contained fractions that had a common denominator. In this sequence, the fractions all have a common numerator. Which of the two series of examples was easier to compare? Why?
Summary of sequence 2: In a "Same Numerator" strategy, if the numerators of the fractions are the same, simply compare their denominators. The fractions will be in reverse order of their denominators. Another way to consider this strategy is to think of each fraction as something being shared by a group that has the same number in it as the denominator, and then asking yourself which group would get the largest pieces since each group gets exactly the same number of pieces. For example, if the fractions are 2/5 and 2/8, then the largest pieces - and the greatest fraction - go to the group that is only sharing with 5. The smallest pieces - the least fraction - will go to the group sharing with 8.
| Sequence 3: In this sequence, all of the fractional amounts are near a benchmark. |
Reflections for sequence 3:
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In this sequence, all of the fractions are close to which benchmark?
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Did you use their proximity to the benchmark in your comparison strategy?
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Did you reveal just one answer at a time, or did you reveal them altogether?
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Summary of Sequence 3: In a "Close to 1/2" strategy, you can compare the fractions by comparing the numerator to the denominator. If you see a fraction that has a numerator that is close to being 1/2 the denominator, you can use that to determine whether it is greater than or less than 1/2. For example, 2/5 is less than 1/2 because 2 is less than 2 1/2. It would take 2 1/2 as a numerator to make half of 5 which would make the fraction 1/2. Conversely, 3/5 is greater than 1/2 because 3 as a numerator is greater than 2 1/2. In these examples, we are always comparing to 1/2 to determine the size of the fraction.
| Sequence 4: In this sequence, all of the fractional values are again near a benchmark. |
Reflections:
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How did you compare these fractions?
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Around which benchmark do all of these fractions appear?
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Did you measure their distance from 1 when comparing their values and relative positions? If so, how? If not, what strategy did you use?
Summary of sequence 4: In a "Close to 1" strategy, you can again compare the fractions by comparing the numerator to the denominator. With fractions whose numerators are all one less than their denominators, the fractions will be in the same order as the denominators. This is because you are comparing the distance of each fraction from 1. The greater the denominator, the smaller the distance to 1. For example, 4/5 is less than 7/8 because both are missing one piece, and 1/8 is a smaller distance to 1 than 1/5.
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*** NOTE: in this activity, you may notice that x is limited to positive values. What would you conjecture if x < 0?
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