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# Ordering Fractions

Alignments to Content Standards: 3.NF.A.3

Arrange the fractions in order from least to greatest. Explain your answer with a picture.

1. $\frac{1}{5}, \frac{1}{7}, \frac{1}{3}$

2. $\frac{2}{5}, \frac{2}{7}, \frac{2}{3}$

3. $\frac{5}{6}, \frac{3}{6}, \frac{1}{6}$

4. $\frac{5}{12}, \frac{8}{12}, \frac{4}{12}$

## IM Commentary

The purpose of this task is to extend students' understanding of fraction comparison and is intended for an instructional setting. While the conceptual components of this task fit squarely in the 3rd grade (ordering fractions with either like numerators or like denominators), the fractions that are given in the task involve denominators beyond those expected at a mastery level in 3rd grade ("Grade 3 expectations in this domain are limited to fractions with denominators 2, 3, 4, 6, and 8"). This makes the task inappropriate for high-stakes assessment, but it could be used for e.g. 3rd students who are ready for more challenging problems or as a bridge for 4th grade students.

## Solution

1. The more pieces the whole is divided into, the smaller the pieces will be. So a unit fraction with a larger denominator will represent a smaller number. We can see this if we shade in one piece of each bar as shown below:

The diagram shows that we can order the fractions from least to greatest as follows: $$\frac{1}{7} \text{, } \frac{1}{5} \text{, } \frac{1}{3}.$$

2. When the numerators are the same, that means we have the same number of pieces. A larger denominator means a smaller piece (see above). If we have the same number of pieces but the pieces are smaller, we will have a smaller total amount. We can shade in each bar as shown below to illustrate:

The diagram shows that we can order the fractions from least to greatest as follows: $$\frac{2}{7} \text{, } \frac{2}{5} \text{, } \frac{2}{3}.$$

3. When the denominators are the same, the sizes of the pieces are the same. If we have a larger numerator, we have more pieces. We can illustrate this by shading in each bar as shown below:

The diagram shows that we can order the fractions from least to greatest as follows: $$\frac{1}{6}\text{, }\frac{3}{6}\text{, }\frac{5}{6}.$$

4. The reasoning here is the same as in part (c). Shade in each bar as shown below:

The diagram shows that we can order the fractions from least to greatest as follows: $$\frac{4}{12}\text{, }\frac{5}{12}\text{, }\frac{8}{12}.$$

#### BArbara says:

over 4 years

This task (arranging fractions from least to greatest) should be ongoing practice for students. Maybe every week or so for ten minute math, students could do one in their math notebooks.

#### Chris says:

With respect to the limited denominators in 3rd grade: I think using 12 is ok because I think in that question students should be looking at the numerators to make their decisions. Not necessary to do it that way but it is and reasoning about the denominators. something I would point out to a student. Likewise in the questions with 5ths and 7ths, the 3rd grade student should be looking at the numerators. Just a thought.

#### Kristin says:

almost 5 years

Thanks for that idea--much appreciated.

#### weilandl says:

over 6 years

The standard says compare two fractions. Why does the assessment ask the students to compare three?

#### Cam says:

over 6 years

Thanks for the comment. A couple of points:

1. The language of 3.NF.3 itself (as opposed to its various subparts) does not reference a limitation to two fractions, but to generically "compare fractions by reasoning about their size."
2. As with a previous comment on this task, we shouldn't construe the standard as explicitly limiting the kinds of tasks we ask students about.
3. To order three fractions is basically to compare two fractions several times.

#### ARutledge says:

over 6 years

This task exceeds the content limits for grade 3. According to the CCSS, "Grade 3 expectations in this domain are limited to fractions with denominators 2, 3, 4, 6, 8."

#### Cam says:

over 6 years

Nice catch -- it's quite possible the variety of denominators makes this too difficult a task for 3rd graders. We'd be interested in collecting feedback or opinions on this from teachers who use this or related tasks in the classroom. Anyone?

But it's also worth mentioning that the standards, this footnote in particular, shouldn't ever be construed as limiting what a teacher can('t) do in the classroom. Rather, this is a note to assessment-writers that they shouldn't expect students to have knowledge of denominators beyond 2, 3, 4, 6, and 8.