Standard(s) for Mathematical Practice (MP)
MP 3  Construct viable arguments and critique the reasoning of others. 

MP 7  Look for and make use of structure. 
MP 8  Look for and express regularity in repeated reasoning. 
Grade Level
Grades 8–9
Content Domain
The Real Number System (Number and Quantity Conceptual Category)
Standard(s) for Mathematical Content
NRN.B.3 
Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational. 

Math Topic Keywords
 rational numbers
 irrational numbers
 proof
 proof by contradiction
 indirect proof
The mathematics task is intended to be a problem or question that encourages the use of mathematical practices. The dialogue is meant to show how students might engage in the mathematical practices as they work on the task. Before reading the dialogue, work on the mathematics task. Next reflect on the mathematical practices you engaged in while working on the task. Finally read the student dialogue.
Mathematics Task
Is $\sqrt{2}+\frac{1}{2}$ a rational number?
Student Dialogue
Students have previously learned the difference between rational and irrational numbers and have seen examples of each. They are now investigating what sums of different types of numbers will produce.
(1)  Chris  What’s a rational number again? Isn’t it a fraction? 

(2)  Lee  Yeah, it’s a number that can be written as $\frac{a}{b}$ where $a$ and $b$ are integers and $b$ is not equal to zero. 
(3)  Chris  Ok. So $\sqrt{2}$ is irrational and $\frac{1}{2}$ is rational. What does that tell us about the whole thing, $\sqrt{2}+\frac{1}{2}$? 
(4)  Matei  Well, we basically want to figure out if the following is true: $$ \begin{align*} \sqrt{2}+\frac{1}{2}=\text{rational number} \end{align*} $$ Let’s try rewriting it as: $$ \begin{align*} \sqrt{2}=\text{rational number}\frac{1}{2} \end{align*} $$ 
(5)  Chris  Well… wait! That doesn’t look right! Isn’t the difference of two rational numbers also rational? How can an irrational number equal a rational number?!?! $$ \begin{align*} \underbrace{\sqrt{2}}_{\text{irrational number}}=\underbrace{\text{rational number}\frac{1}{2}}_{\text{rational number}} \end{align*} $$ 
(6)  Lee  You’re right—that doesn’t make sense! So what does that mean? 
(7)  Matei  Well, we assumed $\sqrt{2}+\frac{1}{2}=\text{rational number}$. I guess that must be false. The only other option is for it to be irrational. I wonder though…what would have happened if we started with something different? 
(8)  Chris  Like what? 
(9)  Matei  Well, let’s try $\sqrt{2}\frac{1}{2}$. 
(10)  Lee  So let’s see if $\sqrt{2}\frac{1}{2}=\text{rational number}$. We can rewrite that as $\sqrt{2}=\text{rational number}+\frac{1}{2}$. 
(11)  Chris  But we have the same problem as before. An irrational number can’t be equal to a rational number. So that means $\sqrt{2}\frac{1}{2}$ is irrational, too. 
(12)  Matei  What if we tried $\sqrt{2}+2$? 
(13)  Lee  Well, we can say $\sqrt{2}+2=\text{rational number}$ or $\sqrt{2}=\text{rational number}2$. 
(14)  Chris  That doesn’t work either. $\sqrt{2}+2$ has to be irrational. 
(15)  Matei  Let’s see if $\sqrt{7}+2$ is rational. 
(16)  Chris  We’re just going to end up with $\sqrt{7}=\text{rational number}2$ so $\sqrt{7}+2$ is irrational. 
(17)  Lee  Hmm… I’m starting to see a pattern here. Every time we have: $$ \begin{align*} \text{irrational number}\pm\text{rational number}=\text{rational number} \end{align*} $$ we end up rewriting that as: $$ \begin{align*} \text{irrational number}=\text{rational number}\mp\text{rational number} \end{align*} $$ 
(18)  Chris  You’re right! And that ends up giving us something that doesn’t make sense. The righthand side is always a rational number, and that can’t be equal to an irrational number. 
(19)  Matei  Well, I guess that can only mean one thing: $$ \begin{align*} \text{irrational number}\pm\text{rational number}\neq\text{rational number} \end{align*} $$ So the sum of an irrational and a rational number must be irrational. 
The teacher reflection questions are intended to prompt thinking about 1) the mathematical practices, 2) the mathematical content and extensions, 3) student thinking, and 4) teaching practices. Begin by reflecting on each of the questions, referring to the student dialogue as needed. After you had some time to do your own reflection, read the possible responses and comment at the bottom of the page with questions or thoughts provoked by the reflection questions and/or responses. Please note some of the mathematics extension tasks are meant for teacher exploration, to prompt engagement in the mathematical practices, and may not be appropriate for student use.
Teacher Reflection Questions

What evidence do you see of the students engaging in the Standards for Mathematical Practice?

The students in this dialogue are able to recognize that the sum of two rational numbers is always rational (line 5 of the dialogue). If students were not able to see this for themselves, how could you help them come to this realization?

What challenges might students encounter if they had tried to show that $\sqrt{2}+\frac{1}{2}$ is irrational? How would you help students overcome that challenge?

If students have trouble understanding what is meant by a contradiction, what other examples of contradictions can you offer that students might be more familiar with?

In line 3, Chris says “$\sqrt{2}$ is irrational.” Prove that this is true.

What is the product of a rational and irrational number? Prove your conjecture

Make a conjecture about the sum of two irrational numbers and support your reasoning.

Write down 20 irrational numbers. What do you notice?
The mathematical overview is intended to 1) explain how students in the dialogue engaged in the mathematical practices and 2) further explore the mathematical content and its extensions. Read the mathematical overview and comment at the bottom of this page with questions or thoughts it provoked.
Mathematical Overview
Mathematical Practice  Evidence 

Construct viable arguments and critique the reasoning of others. 
According to MP 3, “They [students] make conjectures and build a logical progression of statements to explore the truth of their conjectures.” In this dialogue, students make the conjecture that $\sqrt{2}+\frac{1}{2}$ is rational (line 4) and explore to see if that conjecture is true. In the process, they find a contradiction (lines 5–7), thereby “distinguish[ing] correct logic or reasoning from that which is flawed.” Even though the students do not identify their process as such, they are in fact using the same logic as that found in a proof by contradiction (indirect proof). 
Look for and express regularity in repeated reasoning. 
Students in this dialogue “notice if calculations are repeated, and look both for general methods and for shortcuts.” They test several numerical cases (lines 7–16) before they make a generalization about the sum of an irrational and rational number (lines 17–19). Working through several concrete examples allows students to see that the contradiction they found when testing to see if $\sqrt{2}+\frac{1}{2}$ is rational was not limited to that case alone. By seeing that the conjecture was false for several examples, they are able to develop a new, more general conjecture—that the sum of an irrational and rational number is irrational—and begin to understand why that is true. 
Look for and make use of structure. 
In the dialogue, students “can see complicated things… as single objects.” In line 5, Chris is able to see that the expression "$\text{rational number}\frac{1}{2}$” is a rational number by knowing something about the structure of rational numbers and the way they can be added/subtracted together, even though the student never fully explains the thinking used in the dialogue. Chris’s ability to identify expressions as rational is also based on the use of “clear definitions” (MP 6) for a rational number as a fraction of two integers with a nonzero denominator. 
Commentary on the Mathematics
Indirect proofs
Indirect proofs can be challenging at first because of what may appear as backwards logic. Getting a contradiction (a false statement) in the process of trying to prove something and using that contradiction to draw a conclusion may seem bizarre to students. This type of proof is also not always purposeful. Sometimes in the process of trying to prove a statement, you may come across a contradiction that tells you something about the original statement you are trying to prove. Indirect proofs are very beneficial when trying to prove things that are difficult to quantify either due to their inability to be expressed in a generic form, as in the case of irrational numbers, or due to their inability to be measured, as in the case of an infinite number of objects (e.g., prime numbers). In these cases, making an assumption that your number is rational or that prime numbers are finite leads us to a contradiction that helps prove our solution is irrational or our set is infinite.
Student discussion questions and related mathematics tasks are supplementary materials intended for optional classroom use with students. If you choose to use this task and dialogue with your students these discussion questions are intended to stimulate discussion and further exploration of the mathematics. Related mathematics tasks are intended to provide students an opportunity to engage in the mathematical practices as they connect to content that is similar to or an extension of that found in the dialogue. Please note, responses are to be read by teachers only in preparation for using the questions and tasks with their class.
Student Materials
Student Discussion Questions

What is a rational number? What is an irrational number?

In line 5, Chris identifies $\text{rational number}\frac{1}{2}$ as a rational number. How do you think the student knows that it is a rational number?

How do students realize that $\sqrt{2}+\frac{1}{2}$ is an irrational number?

How do students come up with the conclusion that the sum of any rational and irrational number is always irrational?
Related Mathematics Tasks
On odd and even integers

For any integer $n$, what expression can be written to represent all even numbers? What expression can be written to represent all odd numbers? Explain how you came up with your expressions.

Explain why the sum of an even and an even is always even.

Explain why the sum of an odd and an odd is always even.

Explain why the sum of an even and an odd is always odd.

Explain why the product of an odd and even is always even.

Explain why the product of odd and odd is always odd.
On rational and irrational numbers

Find a pair of irrational numbers whose sum is rational. Find a pair of irrational numbers whose sum is irrational.

What can you say about the product of two irrational numbers? Support your answer.

Is $\sqrt{2}\cdot\frac{1}{2}$ a rational number? Explain why or why not.

What is the product of a rational and irrational number? Prove your conjecture.
Comments
a.) As I began this problem I