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?
................
................
In order to avoid copyright disputes, this page is only a partial summary.
To fulfill the demand for quickly locating and searching documents.
It is intelligent file search solution for home and business.
Related download
- rational and irrational numbers review
- 5 7 rational and
- rational and irrational numbers 1 wasatch
- classifying rational and irrational numbers
- rational and irrational numbers 1
- notes rational vs irrational algebra i
- concept 13 rational irrational numbers
- rational and irrational numbers notes
- rational and irrational numbers 2
- identifying rational and irrational numbers
Related searches
- adding rational and irrational numbers
- identifying rational and irrational numbers calculator
- identifying rational and irrational numbers
- rational and irrational numbers worksheet pdf
- rational and irrational numbers pdf
- rational and irrational calculator
- rational and irrational worksheets pdf
- rational and irrational number worksheet
- rational and irrational numbers notes
- rational and irrational number relationships
- what are rational and irrational number
- rational and irrational numbers worksheet