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Comparing Rational and Irrational Numbers
Task
For each pair of numbers, decide which is greater without using a calculator. Explain your choices.

$\pi^2$ or $9$

$ \sqrt{50}$ or $\sqrt{51}$

$\sqrt{50}$ or $8$

$2\pi$ or $6$
IM Commentary
This task can be used to either build or assess initial understandings related to rational approximations of irrational numbers.
The Standards for Mathematical Practice focus on the nature of the learning experiences by attending to the thinking processes and habits of mind that students need to develop in order to attain a deep and flexible understanding of mathematics. Certain tasks lend themselves to the demonstration of specific practices by students. The practices that are observable during exploration of a task depend on how instruction unfolds in the classroom. While it is possible that tasks may be connected to several practices, only one practice connection will be discussed in depth. Possible secondary practice connections may be discussed but not in the same degree of detail.
This task allows students to construct viable arguments by identifying and justifying the greater of two expressions in each part. As students reason through determining which amount is greater, they will use “previously established results,” their knowledge of products, squares, and square roots of benchmark numbers, to reason about slightly larger or smaller numbers. In order to identify a greater expression without making an exact numerical calculation, students will build a logical progression of statements, starting with previous knowledge, and building from there. Students might reason, “I know that three squared is nine, and pi is slightly larger than three, therefore pi squared should be slightly larger than 9.”
The teacher could have students individually solve each of the problems and then discuss their solutions in small group to promote collaboration and refinement of their solutions. They would then verbalize their final answers in large group defending or again refining their answers. The teacher could promote the development of number sense and further get to the meaning of the standard by posing questions such as “How would these comparisons look on a number line?” This approach of reasoning about quantities without making exact calculations ties to MP.7, Look for and make use of structure. This is also a prime opportunity for the teacher to make sure that students address MP.6 and use clear and precise language when justifying their conclusions.
Solution

$\pi \gt 3$ so $\pi^2 \gt 9$.

$ \sqrt{50} \lt \sqrt{51}$ because $50 = (\sqrt{50})^2 \lt (\sqrt{51})^2 = 51.$

$7^2 = 49$ and $8^2 = 64$. Thus we have that $\sqrt{49} \lt \sqrt{50} \lt \sqrt{64}$. So $\sqrt{50} \lt 8$.

$\pi \gt 3$ so $2\pi \gt 2\cdot3$. If you look at these numbers on the number line, that means that $2\pi$ is farther to the right than $6$. When you look at their opposites, $2\pi$ will be farther to the left than $6$, so $2\pi \lt 6$.
Comparing Rational and Irrational Numbers
For each pair of numbers, decide which is greater without using a calculator. Explain your choices.

$\pi^2$ or $9$

$ \sqrt{50}$ or $\sqrt{51}$

$\sqrt{50}$ or $8$

$2\pi$ or $6$
Comments
Log in to commentCam says:
over 2 yearsAgreed. I think the two are often used interchangeably, but it couldn't hurt to be more precise when given the chance. Thanks!
Jessica says:
over 2 yearsDo you think that the word larger could be misleading? I could certainly make the argument that 2pi is larger than 6 because it takes up more space. Would greater be a better word choice?
Ken Mullen says:
over 4 yearsSome of these solutions seem more about properties of square roots than about the concept of irrational numbers.
Cam says:
over 4 yearsWell, square roots of integers certainly make up a large proportion of the irrational numbers we expect middle school students to be conscious of. That said, I don't disagree with your general point. While it's true that $\sqrt{50}\approx \sqrt{49}=7$, and this is pretty convincing that $\sqrt{50}<8$, I don't think this (or the more careful solution provided for this fact) is the type of reasoning being requested under the heading "rational approximation" in the standard. If the task disappears for a while, it's because we've pulled it offline for fixing up.
Kristin says:
over 4 yearsIt seems to me that noting that $\sqrt{50}\approx \sqrt{49} = 7$ is equivalent to saying that 7 is a decent rational approximation of $\sqrt{50}$ in the right context, so in that sense, I think it does fit the standard nicely. Perhaps part (b) is less clearly connected to the standard, but I think it helps bridge the perspectives on square roots encapsulated in 8.EE.A.2 and 8.NS.A.2. In general, I think this task is about helping students develop/demonstrate "irrational number sense" which is a main reason to find/recognize rational approximations of irrational numbers. I think this task an appropriate member of a set illustrating the standard, although we need more. Any suggestions?