Sum of Rational and Irrational Is Irrational

Sum of Rational and Irrational Is Irrational

About Illustrations: Illustrations of the Standards for Mathematical Practice (SMP) consist of

several pieces, including a mathematics task, student dialogue, mathematical overview, teacher

reflection questions, and student materials. While the primary use of Illustrations is for teacher

learning about the SMP, some components may be used in the classroom with students. These

include the mathematics task, student dialogue, and student materials. For additional Illustrations

or to learn about a professional development curriculum centered around the use of Illustrations,

please visit mathpractices..

About the Sum of Rational and Irrational Is Irrational Illustration: This Illustration¡¯s student

dialogue shows the conversation among three students who are trying to determine if a sum of a

rational and irrational number is rational. Students determine that the sum is in fact irrational and

after trying several similar sums, they develop a proof for why the sum of a rational and

irrational is always irrational.

Highlighted 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.

Target Grade Level: Grades 8¨C9

Target Content Domain: The Real Number System (Number and Quantity

Conceptual Category)

Highlighted Standard(s) for Mathematical Content

N-RN.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

? 2016 by Education Development Center. Sum of Rational and Irrational Is Irrational is licensed under the

Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License. To view a copy of this

license, visit . To contact the copyright holder email

mathpractices@

This material is based on work supported by the National Science Foundation under Grant No. DRL-1119163. Any

opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do

not necessarily reflect the views of the National Science Foundation.

Sum of Rational and Irrational Is Irrational

Mathematics Task

Suggested Use

This mathematics task is intended to encourage the use of mathematical practices. Keep track of

ideas, strategies, and questions that you pursue as you work on the task. Also reflect on the

mathematical practices you used when working on this task.

Is 2 +

1

a rational number?

2

Sum of Rational and Irrational Is Irrational

Student Dialogue

Suggested Use

The dialogue shows one way that students might engage in the mathematical practices as they

work on the mathematics task from this Illustration. Read the student dialogue and identify the

ideas, strategies, and questions that the students pursue as they work on the task.

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

a

where a and b are integers and b is

b

not equal to zero.

(3) Chris:

Ok. So

thing,

(4) Matei:

2 is irrational and

1

is rational. What does that tell us about the whole

2

1

2+ ?

2

Well, we basically want to figure out if the following is true:

1

2 + = rational number

2

Let¡¯s try rewriting it as:

2 = rational number ?

(5) Chris:

1

2

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?!?!

1

2

= rational number ?

?

2

irrational number

????????

?

rational number

(6) Lee:

You¡¯re right¡ªthat doesn¡¯t make sense! So what does that mean?

(7) Matei:

Well, we assumed

(8) Chris:

Like what?

1

= rational number . I guess that must be false. The only

2

other option is for it to be irrational. I wonder though¡­what would have

happened if we started with something different?

2+

Sum of Rational and Irrational Is Irrational

1

2? .

2

(9) Matei:

Well, let¡¯s try

(10) Lee:

So let¡¯s see if

(11) Chris:

But we have the same problem as before. An irrational number can¡¯t be equal to a

1

rational number. So that means 2 ? is irrational, too.

2

(12) Matei:

What if we tried

(13) Lee:

Well, we can say

(14) Chris:

That doesn¡¯t work either.

(15) Matei:

Let¡¯s see if

(16) Chris:

We¡¯re just going to end up with

(17) Lee:

Hmm¡­ I¡¯m starting to see a pattern here. Every time we have:

1

= rational number . We can rewrite that as

2

1

2 = rational number + .

2

2?

2 +2?

2 + 2 = rational number or

2 = rational number ? 2 .

2 + 2 has to be irrational.

7 + 2 is rational.

7 = rational number ? 2 so

7 + 2 is irrational.

irrational number ¡À rational number = rational number

we end up rewriting that as:

irrational number = rational number ? rational number

(18) Chris:

You¡¯re right! And that ends up giving us something that doesn¡¯t make sense. The

right-hand 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:

irrational number ¡À rational number ¡Ù rational number

So the sum of an irrational and a rational number must be irrational.

Sum of Rational and Irrational Is Irrational

Teacher Reflection Questions

Suggested Use

These teacher reflection questions are intended to prompt thinking about 1) the mathematical

practices, 2) the mathematical content that relates to and extends the mathematics task in this

Illustration, 3) student thinking, and 4) teaching practices. Reflect on each of the questions,

referring to the student dialogue as needed. Please note that some of the mathematics extension

tasks presented in these teacher reflection questions are meant for teacher exploration, to prompt

teacher engagement in the mathematical practices, and may not be appropriate for student use.

1. What evidence do you see of students in the dialogue engaging in the Standards for

Mathematical Practice?

2. 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?

3. What challenges might students encounter if they had tried to show that 2 +

1

is irrational?

2

How would you help students overcome that challenge?

4. 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?

5. In line 3, Chris says ¡° 2 is irrational.¡± Prove that this is true.

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

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

8. Write down 20 irrational numbers. What do you notice?

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