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Which is Closer to 1?

Alignments to Content Standards: 3.NF.A.2

Which is closer to 1 on the number line, $\frac45$ or $\frac54$? Explain.

IM Commentary

This can be seen as a multi-step problem for grade 3:

• compare $\frac45$ to $\frac55$ (like denominators)
• compare $\frac44$ to $\frac54$ (like denominators)
• compare $\frac14$ to $\frac15$ (like numerators)
This task is also a natural fit for 4.NF.2 as a lower-level task in a set that would illustrate that standard. Thus, this task provides a nice transition between grade levels.

Although a number line diagram is not provided, many students may choose to draw one; teachers might suggest doing so if a student is struggling with the problem. For many students, creating a number line will help them recognize that $\frac45$ is only $\frac15$ from 1 and that $\frac54$ is only $\frac14$ from 1 even though the relative size of the fractions is similar and the two fractions are plotted on different sides of 1 on the number line. Some students may choose to create two number lines, so they can partition one into fourths and one into fifths, and then compare them.

It is possible to solve this problem using common denominators. While not technically incorrect, that solution is not shown because first, students do not work with common denominators until fourth grade, and second, even when they know how to find common denominators, they should recognize in a case like this that straight-forward reasoning about the relative sizes of unit fractions is more efficient.

Below is a list of related tasks in this set in order of sophistication:

• Locating Fractions Less than One on the Number Line
• Locating Fractions Greater than One on the Number Line
• Closest to $\frac12$
• Find 1
• Find $\frac23$
• Which is Closer to 1?

Solutions

Solution: 3.NF.2 Plot Points to Compare

Many students will sketch something like the following number line diagram:

Fifths are closer together than fourths. $\frac45$ is $\frac15$ from 1. $\frac54$ is $\frac14$ from 1. So $\frac45$ is closer to 1 than $\frac54$.

Solution: 3.NF.2 Recognize and Reason

A few students may simply reason in the following way (stated in student language rather than the more formal language used here):

If we divide the segment from 0 to 1 in four equal-length segments (so each has a length of $\frac14$), and we also divide the segment from 0 to 1 into five equal-length segments (so each has a length of $\frac15$), the segments of length $\frac14$ will be longer than the segments of length $\frac15$ because there are fewer of them.

The distance between $\frac45$ and 1 is $\frac15$, and the distance between $\frac54$ and 1 is $\frac14$.

So $\frac45$ is closer to 1 than $\frac54$

A correct solution does require that the student explain how he or she figured out that $\frac45$ is closer to 1.

drombola says:

over 6 years

Great comment about exposing students to additional denominators. I saw a curriculum outline limiting fractions to parts of a whole. What about parts of a set? Even if that is not assessed shouldn't students be exposed to "12 out of 30 skittles are yellow".

Kristin says:

over 6 years

Hi drombola,

The fractions progression is pretty clear that the focus on fractions in grade 3 does not include sets:

"In grade 3... the whole can be a shape such as a circle or rectangle, a line segment, or any one finite entity susceptible to subdivision and measurement. In Grade 4, this is extended to include wholes that are collections of objects."

The difference I see here is that exposing students to other denominators besides the ones listed is not introducing a different idea but giving more examples of the same idea. In the case where the whole is a single entity vs. a set, these are actually quite different ideas. When the whole is a set of objects, it actually requires students to hold an additional piece of information in working memory (namely the individual objects in the set in addition to the set as a thing in and of itself). I think the idea here is for students to develop a solid understanding of a fraction before introducing that extra layer of complexity, and so it is probably better to leave sets as wholes to fourth grade.

almost 7 years

Kristin says:

over 6 years

Hi katiek,

I think you are exactly right about "proper" vs "improper" fractions. The word "improper" fraction does not appear in the standards precisely because the meaning of a/b is the same whether or not the numerator is greater than or less than the denominator--in fact, this insight is exactly captured in 3.NF.1 and 3.NF.2. Students should begin to build an understanding(based on the meaning of a unit fraction) that if the numerator is less than the denominator, then the fraction is less than one, and if the numerator is greater than the denominator, then the fraction is greater than 1. However, this idea builds over several years, culminating in the understanding described in 5.NF.5 related to fraction multiplication.

I think that facility with mixed numbers is slightly more nuanced. The idea with Common Core is to first understand the meaning of a/b (which is independent of whether a<b or a>b), then understand the idea of fraction equivalence, and then to finally graduate to an understanding of the equivalence of fractions and mixed numbers. In third grade, students should build an understanding of the equivalence of whole numbers and various equivalent fraction representations of them, but the full understanding of mixed numbers does not come until 4th grade. The Number and Operations—Fractions progression might be of some help here:

http://commoncoretools.me/2011/08/12/drafty-draft-of-fractions-progression/

I think for instructional purposes, you are exactly right. The restrictions on allowable denominators are most salient for assessment developers. In other words, student might very well benefit from thinking about fractions with different denominators than are listed, but we want to be sure that they have enough experience with fractions with the listed denominators that they will be comfortable and successful if they are asked to reason about them in an assessment situation (especially high-stakes assessments).