Grade 8 - Lesson Title: Real Numbers
Unit 1: Real Numbers (Lesson 1 of 3)
Time Frame: 1-2 weeks
Essential Question: Why are quantities represented in multiple ways?
Segment 2 – Identifying Rational and Irrational Numbers (8.NS.1)
Adapted from:
http://alex.state.al.us/lesson_view.php?id=24079
http://www.regentsprep.org/Regents/math/ALGEBRA/AOP1/Lrat.htm
http://www.quia.com/pop/37541.html
Approximate Time Frame:
50 minutes
Lesson Format:
Resources:
Whole Group
Small Group
Independent
Irrational Number Power Point
Rational and Irrational Sort. Print the
32 rational and irrational number
cards. You will need one for each
student.
Modeled
Guided
Collaborative
Assessment
Rational and Irrational Identification
Self-Assessment
Focus:
Modalities Represented:
Concrete/Manipulative
Picture/Graph
Table/Chart
Symbolic
Oral/Written Language
Real-Life Situation
The focus of this segment is
conceptual understanding of rational
and irrational numbers and being able
to accurately identify them.
Math Practice Look For(s):
Differentiation for Remediation:
MP.4: Students model rational approximations of
irrational numbers on the number line.
Differentiation for English Language Learners:
Potential Pitfall(s):
Independent Practice (Homework):
Differentiation for Enrichment:
Students may think that irrational means “goes forever”.
You’ll need to remind them that all numbers “go forever”
as we saw in the previous section of the lesson.
Steps:
1. Opening: (5 min) Refer to the opening of the previous concept
and challenge students (if they did not come up with any from the
previous day) to find a decimal that does NOT repeat. Ex: , ,
Teacher Notes/Reflections:
Inquiry Prompt: Can you think of a number
that has a pattern but does not repeat?
Grade 8 - Lesson Title: Real Numbers
Unit 1: Real Numbers (Lesson 1 of 3)
Time Frame: 1-2 weeks
Essential Question: Why are quantities represented in multiple ways?
2. Exploration: (10-15 min) Show the Irrational Numbers Power
Point. The purpose of this is for students to understand the
definition and concept of an irrational number. They should be
able to differentiate between them fluently.
a. Using a few examples, give the class some rational and
irrational numbers and have them hold up 1 finger for
irrational and 2 fingers for rational (or any variation of
student monitoring you chose). When the majority of
students understand the difference between irrational and
rational numbers, continue with the number activity.
Teacher Notes/Reflections:
While the definition of a rational number has
traditionally been about the ratio of integers,
the focus now is more on the idea that a
rational number is any number whose
decimal expansion eventually repeats a
pattern. Notice that a terminating decimal
such as
repeats the digit zero beginning in
the hundredths place. This means an
irrational number does not eventually repeat
a pattern.
3. Card Sort: (5 min) Give every student a number card making sure
that there are at least two irrational number cards handed out.
Ask all the students holding rational numbers to move to one side
of the room, and all the students holding irrational numbers to
move to the opposite side of the room. Students should feel free
to discuss with someone nearby what type of number they have
as long as they can explain why they think so. After students have
chosen a side of the room, make sure they are correct. Discuss
any misconceptions.
Teacher Notes/Reflections:
4. The Number Line: (10-15 min) Ask students holding integers to go
to line up in numerical order creating a human number line.
Teacher Notes/Reflections:
a. Next ask each remaining student to place themselves in their
appropriate location on the human number line. Students
should feel free to discuss their position with each other.
b. Students with irrational number cards may have a difficult
time placing themselves. Discuss with the class why this is
and ask if there is another way to represent these numbers?
c. This will lead to the need to approximate irrational numbers
with rational numbers which is discussed in the next segment
of the lesson. Ask students to determine what integers the
square root is between and have the students with irrational
numbers to stand between those integers.
Materials: Print the Rational and Irrational
Sort cards so that you have enough for each
student.
This connects heavily to the essential
question as we need to represent the
fractions as decimals so that we can more
easily compare the numbers.
Potential Pitfall: Students may have trouble
placing the fractions since they are not in
decimal form. Connect this to the previous
learning about converting fractions to
decimals.
Grade 8 - Lesson Title: Real Numbers
Unit 1: Real Numbers (Lesson 1 of 3)
Time Frame: 1-2 weeks
Essential Question: Why are quantities represented in multiple ways?
5. Self-Assessment: Go to this link and students can answer the
questions on the computer where it will tell them the correct
answer if they’re wrong. You can print out the worksheet to
make copies for students using the “print” link on the right side,
but make sure to include the answers if you want it to be a selfassessment.
http://www.quia.com/pop/37541.html
(If you have no internet access, you may use the Rational and
Irrational Identification Self-Assessment worksheet provided.)
a. Once students have completed the self-assessment, have
them write down any particular type of number that they
struggle with identifying as rational or irrational. As
independent practice, have students write down three
numbers of the type they struggle with and identify them as
rational or irrational. For example, a student may struggle
with remembering that terminating decimals are rational
since they don’t see the repeating zero. That student should
write three terminating decimals and identify them as
rational.
Teacher Notes/Reflections:
Depending on your technology access, you
may want to print out either the Quia
worksheet ahead of time for each student or
the Rational and Irrational Identification SelfAssessment worksheet for each student.
Rational and Irrational Identification Self-Assessment
Identify whether the following are rational or irrational numbers.
1.
2.
3.
4.
5.
6.
7.
8.
9. If the number
is displayed on a calculator that can only display ten digits, do we know whether it is
rational or irrational? In one complete sentence explain why.
10. If the number
is displayed on a calculator that can only display ten digits, do we know
whether it is rational or irrational? In one complete sentence explain why.
Now check your answers with those listed on the back of this paper.

If you missed problems 1 or 2, remember that square roots that simplify to integers are rational, and
square roots that don’t simplify to integers are irrational.

If you missed problems 3 or 4, it may be the negative sign throwing you off. Remember that you can’t
take the square root of a negative value, so problem 4 is not even a real number.

If you missed problems 5 or 6, remember that fractions are one of the definitions of a rational number
because they always make a decimal that repeats a pattern of digits.

If you missed problems 7 or 8, remember that any decimal expansion that repeats a pattern of digits is a
rational number. A number like
would be irrational since we can never put the line over
digits signifying a repeating pattern of digits.

If you missed problems 9 or 10, remember that a calculator can’t tell us for sure whether a number is
rational or irrational unless the decimal clearly terminates. It might give us a best guess if we see a
pattern beginning to repeat, but we can’t know for sure just based on the output of a calculator. We
would have to know what was input into the calculator to give that output.
Identify whether the following are rational or irrational numbers.
1.
2.
Rational
5.
Rational
3.
Irrational
6.
4.
Irrational
7.
Rational
Not a real number
8.
Rational
Rational
9. If the number
is displayed on a calculator that can only display ten digits, do we know whether it is
rational or irrational? In one complete sentence explain why.
The number is rational because the decimal terminates meaning that it repeats the digit zero.
10. If the number
is displayed on a calculator that can only display ten digits, do we know
whether it is rational or irrational? In one complete sentence explain why.
A best guess would be rational since we see a pattern developing, but we can’t know for sure. In fact, we may
notice that since the last digit displayed is a the next digit could not be an . If it were, the calculator would
have rounded the up making the display say
showing us that we can’t know for sure whether
this is rational or irrational. It may yet have a repeating pattern somewhere later in its decimal expansion.
Adapted from Eric Bright, Charleston Middle School Curriculum