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Comparing Sums of Unit Fractions


Alignments to Content Standards: 4.NF.B.3.a

Task

Use <, =, or > to compare the following sums:

  1. $\frac12 + \frac14$ ________ $\frac13 + \frac15$
  2. $\frac13 + \frac12$ ________ $\frac13 + \frac14$

IM Commentary

The purpose of this task is to help develop students' understanding of addition of fractions; it is intended as an instructional task. Notice that students are not asked to find the sum because in grade 4, students are limited to computing sums of fractions with like denominators (i.e. fractions where 1 denominator is a multiple of the other). Rather, they need to apply a firm understanding of unit fractions (fractions with one in the numerator) and reason about their relative size. That understanding begins with area models of fractions in grades one and two and expands to the number line in grade 3. With experience, students come to “know” that $\frac14$ is less (smaller) than $\frac13$ because dividing a whole into 4 rather than 3 pieces means there are more pieces, so they must be smaller.

To be successful with this type of problems, students must be able to easily identify the largest (or smallest) fraction in a group of unit fractions.

Struggling students can be given an easier version that repeats one of the fractions in both sums like:

$\frac12 + \frac15$________$\frac12 + \frac14$

Students who can answer the original problem with ease can be asked about differences in problems like this:

$\frac12 – \frac15$________$\frac12 – \frac13$

$\frac12 – \frac15$________$\frac13 – \frac15$

Solutions

Solution: Compare terms separately

(a) $\frac12$ is greater than $\frac13$ and $\frac14$ is greater than $\frac15$. So

$$\frac12 + \frac14 > \frac13 + \frac15$$

(b) $\frac12$ is greater than $\frac14$, so if I add $\frac13$ to both, I get

$$\frac13 + \frac12 > \frac13 + \frac14$$

Solution: Compare the denominators

(a) By comparing the denominators I know that 1/2 is more than 1/3 because 2 is less than 3. I then compare 1/4 to 1/5 and know that 1/4 is more than 1/5 because 4 is less than 5. The sum of two bigger numbers is greater than the sum of two smaller numbers. $$\frac12 + \frac14 > \frac13 + \frac15$$ (b) I know that 1/2 is more than 1/4 because 2 is less than 4. If I add 1/3 to a smaller number, the result will be less than if I add 1/3 to a bigger number. $$\frac13 + \frac12 > \frac13 + \frac14$$

Solution: Compare visual representations

By fourth grade, students should be able to reason about the relative sizes of unit fractions based on the meaning of the denominators, but for those who need some additional support, the teacher can provide a number line. The issue with student-made number lines is that they may not be drawn precisely, and so the relative sizes may not be apparent. Either students should draw the number lines very carefully, or they should be given number lines with precisely drawn tick marks like those shown below: 1_a9972d190fdb6712d9f60701cefaec0e The number lines should be separated from each other and students should label the tick marks. The colored bars shown below are used to help make the connection between lengths from zero and points on the number line, but the the fractions and their sums should be clearly indicated as points on the number line. 2_d45348fa34ab487ae7354d499888bc5b (a) First, a student marks of 1/2, 1/3, 1/4, and 1/5 on different number lines. Then they mark a point that is 1/4 to the right of 1/2 and label that 1/2+1/4. Similarly, they should mark a point that is 1/5 to the right of 1/3 and label that 1/3+1/5. 3_440d141afc99718e793c63cb98b9310c $$\frac12 + \frac14 > \frac13 + \frac15$$ They should still articulate the reason why this comparison must be valid. For example, the sum of two bigger numbers is greater than the sum of two smaller numbers. (b) Students can start from 1/3 and part 1/4 and then 1/2 to the right of that. 4_f42a441c1f15016438f15336cbc72ec1 $$\frac13 + \frac12 > \frac13 + \frac14$$ They should still articulate the reason why this comparison must be valid. For example, adding a bigger number makes a greater sum than adding a smaller number.

Anthony says:

over 1 year

With the aide of number lines this task would be successful in grade three. Any opportunity to combine visual representations into the context of a problem at earlier ages can only build mastery. Breaking problems up into pieces is also a great method. I like how one solution compares the parts from each problem. I only wish the problem can be turned into a comprehension problem.

Tommy Barbour says:

over 3 years

In the second blue solution block there appears the text: "(b) If you take 1/3 out of each equation you then need to only compare 1/2 to 1/4." But "1/3 + 1/2" and "1/3 + 1/4" are not equations. They are "expressions" in the sense that an expression a string of numbers and operation signs that stands for a number. We could say "take 1/3 out of each member (or each side) of the statement (or sentence) being considered".

In a. of the third blue solution block the text says: "They then put 1/2 and 1/4 side by side and visually compare it to 1/3 and 1/5 put side by side and see which set is longer." Let's reserve the word "set" to its proper mathematical meaning and say instead something like "...side by side to see which sum is greater."

In b. of the third blue solution block the word "equation" should not be used since while solving the problem we do not yet know if the solution will be an equation or inequality. It turns out to be an inequality. My wording above suggests a way to address this.

Clear and correct use of language and symbols is important, and sometimes critical to meaning and understanding.

Kristin says:

over 3 years

Thanks for the suggestions. I actually thought the last solution needed to be completely remodeled. Please let me know what you think.

Jason says:

over 4 years

I think this could be a good problem for a grade 3 classroom, but it should certainly not appear on a summative test of the grade 3 standards.

While it is true that the student need not evaluate sums or differences to solve the problem, the student must still understand the meaning of those sums and differences. That understanding is not required by the grade 3 standards. (But it is required in the grade 4 standards.) So this task falls under the category of things that students might do in grade 3---not things they must do.

A read of the standards for grades 3 and 4 will show that concepts of addition and results of addition are meant to be done together, not in separate grades. Indeed, that holds for whole numbers as well. Whatever the nature of the numbers being operated on, it is never the case that the standards ask students to learn the meaning of an operation before they learn the procedure for carrying it out. (Nor vice versa.)

So again, while this would be an interesting and good task for a grade 3 classroom (depending on the classroom), it would be important not to portray this as a task suitable for summative assessment of the grade 3 standards.

Kristin says:

over 4 years

Thanks for your comments. We decided to move it to grade 4 as it fits 4.NF.3.a much better, and doesn't run into the same issues you mention here.