Download Level 8 | Unit 2

PowerTeaching Math
®
3rd Edition
Level 8 | Unit 2
Rational and Irrational Numbers
unit guide
With Student and
Assessment Pages
PowerTeaching Math 3rd Edition Unit Guide:
Level 8
© 2015 Success for All Foundation. All rights reserved
Produced by the PowerTeaching Math 3rd Edition Team
Angela Watson
Kate Conway
Nancy Madden
Angie Hale
Kathleen Collins
Nick Leonhardt
Cathy Pascone
Kathy Brune
Patricia Johnson
Debra Branner
Kenly Novotny
Paul Miller
Devon Bouldin
Kimberly Sargeant
Peg Weigel
Erin Toomey
Kris Misage
Rebecca Prell
Irene Baranyk
Laura Alexander
Russell Jozwiak
Irina Mukhutdinova
Laurie Warner
Sarah Eitel
James Bravo
Luke Wiedeman
Sharon Clark
Jane Strausbaugh
Mark Kamberger
Susan Perkins
Janet Wisner
Marti Gastineau
Terri Morrison
Jeffrey Goddard
Meghan Fay
Tina Widzbor
Jennifer Austin
Michael Hummel
Tonia Hawkins
Joseph Wilson
Michelle Hartz
Wanda Jackson
Karen Poe
Michelle Zahler
Wendy Fitchett
We wish to acknowledge the coaches, teachers, and students who piloted the
program and provided valuable feedback.
The Success for All Foundation grants permission to reproduce the blackline
masters of the PowerTeaching Math unit guides on an as-needed basis for
classroom use.
A Nonprofit Education Reform Organization
300 E. Joppa Road, Suite 500, Baltimore, MD 21286
PHONE: (800) 548-4998; FAX: (410) 324-4444
E-MAIL: [email protected]
WEBSITE: www.successforall.org
table of contents
Unit Overview.. .................................................................. 1
Cycle 1
Rational and Irrational Numbers. . .......................................... 3
Student Pages Teamwork, Quick Check, Homework, and Assessments. . ...... 33
Cycle 1..................................................................... 35
This project was developed at the Success for All Foundation under the direction of
Robert E. Slavin and Nancy A. Madden to utilize the power of cooperative learning, frequent
assessment and feedback, and schoolwide collaboration proven in decades of research to
increase student learning.
© 2015 Success for All Foundation
PowerTeaching Math 3rd Edition | Unit Guide
iii
Level 8 | Unit 2: Rational and Irrational Numbers
Level 8 | Unit 2: Rational and Irrational Numbers
Unit Overview
Vocabulary
introduced in
this unit:
rational number
radical sign
square root
perfect square
irrational number
Think Like a
Mathematician
practice(s) used
in this unit:
Make sense of it.
Translate into math.
Defend and review.
Use your math toolkit.
Be precise.
Find the patterns
and structure.
In unit 2 of grade 8, your students will add to their knowledge of the types of
numbers. In grades 6 and 7, your students added negative numbers to their
knowledge and formed a full understanding of rational numbers. In grade 8, your
students will learn about irrational numbers to develop a full view of what real
numbers are. They will define, classify, and approximate the values of various types of
numbers. Unit 2 consists of one cycle: cycle 1—Rational and Irrational Numbers.
Cycle 1—Rational and Irrational Numbers
Lesson 1: Defining Irrational Numbers
Define and explore rational and irrational numbers. (CC 8.NS.A.1 and 2;
TEKS 8.b.2.A; VA SOL 7.1d, 8.2, 8.5a)
Lesson 2: Classifying Numbers
Classify rational and irrational numbers. (CC 8.NS.A.1 and 2; TEKS 8.b.2.A;
VA SOL 8.2)
Lesson 3: Converting a Decimal Expansion
Convert a decimal expansion that repeats eventually into a rational number.
(CC 8.NS.A.1; TEKS 8.b.2.A; VA SOL 8.2)
Lesson 4: Ordering Rational and Irrational Numbers
Use knowledge of perfect squares and the number line to order rational
and irrational numbers. (CC 8.NS.A.2; TEKS 8.b.2.B and D; VA SOL 8.2,
8.5a and b)
Lesson 5: Comparing Irrational Expressions
Use approximations of the value of irrational numbers to estimate and
compare expressions containing irrational numbers. (CC 8.NS.A.2;
TEKS 8.b.2.B and D; VA SOL 8.5b and A.3)
© 2015 Success for All Foundation
PowerTeaching Math 3rd Edition | Unit Guide
1
Access Code: zwdffn
Level 8 | Unit 2: Rational and Irrational Numbers | Cycle 1 Lesson 1
Lesson 1: Defining Irrational Numbers
Vocabulary:
rational number
radical sign
square root
perfect square
irrational number
Materials:
calculators
Lesson Objective: Define and explore rational and irrational numbers.
By the end of this lesson, students will:
• define irrational numbers;
• explore the concept of and values of irrational numbers; and
• determine whether a given number is irrational.
This lesson involves an introduction to square roots with emphasis on the square root
of 2.
opening
(3 minutes)
get the goof
• Ask teams to begin Get the Goof.
TEACHER’S NOTE: Students learned about rational numbers in grade 7. Here they are identifying a
classification error. If students get stuck, refer to grade 7, unit 2, cycle 1, lesson 1.
Maya said that – 2 is a whole number. What’s wrong with her thinking?
Random Reporter Rubric | Possible Answer
Answer: Maya identified the number incorrectly.
Explanation: – 2 is not positive or 0, so it cannot be a whole number . – 2 is an integer .
Math Practice: I know that whole numbers are positive numbers that aren’t fractions or
decimals and include 0. Using the definition of a whole number helped me figure out what
was wrong with Maya’s thinking (TLM #6).
• Use Random Reporter to have teams share. Use the Random Reporter rubric to
evaluate responses and give feedback.
• Record individual scores on the teacher cycle record form.
• Award team celebration points.
© 2015 Success for All Foundation
PowerTeaching Math 3rd Edition | Unit Guide
3
Level 8 | Unit 2: Rational and Irrational Numbers | Cycle 1 Lesson 1 Access Code: zwdffn
active instruction
(10–15 minutes)
set the stage
• Distribute team score sheets. Have students review their scores and set new team
goals in lesson 1.
• Post and present the lesson objective: Today you will define and explore rational and
irrational numbers.
• Ask students to write this cycle’s vocabulary words in their notebooks: rational
number, radical sign, square root, perfect square, irrational number.
• Remind students how to earn team celebration points.
interactive instruction and guided practice
• Use a Think Aloud to model explaining and exploring the differences between
rational and irrational numbers.
Define
irrational numbers.
6 layers
Find the value of each. Is each value a rational or an irrational number?
a) the square roots of 25
b)​
2 ​ 
I know that I cannot claim whether a and/or b are rational or irrational numbers
until I find the value of each. Using my Think Like a Mathematician sheet, I know
that definitions can help me. Being precise and using the correct definition of
terms is very important in math. That’s TLM practice #6.
Let’s start with a: the square roots of 25. What do I know about square roots?
Show layer 1. A square root of a number is a value that can be multiplied by itself
to give the original number. What are the square roots of 25?
Show layer 2. Is this a perfect square? Yes. 25 is a perfect square because 5 times
5 is equal to 25. I also know that – 5 times – 5 is also equal to 25. So the number
25 has two square roots. Both 5 and – 5 are the square roots of 25. Both 5 and
– 5 are rational numbers. Remember, rational numbers are numbers that can be
written in fractional form.
Show layer 3. When the radical sign is used, the only answer is the positive
square root.
Now let’s examine b: the positive square root of 2. What do I know about the
number 2? Is it a perfect square?
Show layer 4. So 2 is not a perfect square. How can I tell whether this is a rational
or an irrational number?
Show layer 5. If I use the calculator to find the square root of 2, the decimal does
not repeat or terminate. So the square root of 2 is an irrational number. Can you
think of another irrational number?
Show layer 6. Pi is another commonly used irrational number. We use the symbol
p to represent the number. Most of the time, we use the estimate 3.14 to
calculate the circumference or the area of a circle.
• Use Think‑Pair‑Share to ask students the following question: Can a length be an
irrational number? How would you measure it?
• Randomly select a few students to share. Possible answer: Yes. A length can be
an irrational number. It is measured as its rational approximate.
4
PowerTeaching Math 3rd Edition | Unit Guide
© 2015 Success for All Foundation
Access Code: zwdffn
Level 8 | Unit 2: Rational and Irrational Numbers | Cycle 1 Lesson 1
• Use Team Huddle to have teams practice defend‑and‑review statements about
rational and irrational numbers.
1) Provide two examples that show that the statement is false.
Lydia said that all square roots are irrational numbers.
Random Reporter Rubric | Possible Answer
4 ​ 5 2 and 
​ 
100 ​ 5 10.
Answer: The statement is false because 
​ 
Explanation: I know the statement is false because the square roots of perfect squares are always rational  numbers .
Math Practice: I examined the statement that Lydia made to determine whether
her argument made sense. I used what I know about the definitions of rational
and irrational numbers to prove her statement as false. I know that any example where
I would find the square root of a perfect square will show that Lydia’s statement is false
(TLM #6).
• Use Random Reporter to have teams share. Use the Random Reporter rubric to
evaluate responses and give specific feedback.
• Record individual scores on the teacher cycle record form.
• Award team celebration points.
team mastery
(10–15 minutes)
• Ask students to follow the Team Mastery student routine.
• Circulate and use the following questions to prompt discussions.
–– What is an irrational number?
–– What’s the difference between a rational and an irrational number?
–– How did you know this was an irrational number?
–– How can a number have two square roots?
–– What is a perfect square?
–– How does the square root of a number relate to its classification?
• When there are 5 minutes left in Team Mastery, prompt teams to prepare for the
Lightning Round. Have teams discuss one Team Mastery problem that the whole
team has completed.
• Award team celebration points for good team discussions that demonstrate
100‑point responses.
© 2015 Success for All Foundation
PowerTeaching Math 3rd Edition | Unit Guide
5
Level 8 | Unit 2: Rational and Irrational Numbers | Cycle 1 Lesson 1 Access Code: zwdffn
lightning round
(10 minutes)
• Tell students the Team Mastery problem that you will use for the Lightning Round.
Classify the number as rational or irrational.
3)​
48 ​ 

Random Reporter Rubric | Possible Answer
Answer: 
​ 
48 ​ is an irrational number .
Explanation: First, I asked myself if 48 was a perfect square . 6 ? 6 5 36 and 7 ? 7 5 49,
so 48 is not a perfect square. The square root of 48 is an irrational number.
Math Practice: I used the calculator to find the square root of 48 and saw that the
decimal did not terminate or repeat. The calculator helped to confirm my answer (TLM #5).
• Use Random Reporter to have teams share. Use the Random Reporter rubric to
evaluate responses and give specific feedback.
• Record individual scores on the teacher cycle record form.
• Award team celebration points.
celebration
(2 minutes)
• Record team celebration points on the poster.
• Have the top team choose a cheer.
• Assign homework, and remind students about the Vocabulary Vault.
• Ask students to follow the Quick Check student routine. (optional)
Find the square roots for 64.
Possible answer: 8 and – 8
6
PowerTeaching Math 3rd Edition | Unit Guide
© 2015 Success for All Foundation
Access Code: zwdffn
Level 8 | Unit 2: Rational and Irrational Numbers | Cycle 1 Lesson 2
Lesson 2: Classifying Numbers
Vocabulary:
none
Materials:
none
Lesson Objective: Classify rational and irrational numbers.
By the end of this lesson, students will:
• classify rational and irrational numbers; and
• categorize numbers as members of one or more of the following sets:
natural numbers, whole numbers, integers, rational numbers, or irrational numbers.
opening
(3 minutes)
get the goof
• Ask teams to begin Get the Goof.
TEACHER’S NOTE: Students may assume that the square root of any number is rational. Remind them
that the square roots of perfect squares are natural numbers and, therefore, rational. The square
root of a number other than a perfect square is irrational.
Stefan said that 
​ 
110 ​ is rational. What’s wrong with his thinking?
Random Reporter Rubric | Possible Answer
110 ​ incorrectly.
Answer: Stefan classified ​
Explanation: ​
110 ​ is an irrational  number . I asked myself if 110 is a perfect square .
10 ? 10 5 100 and 11 ? 11 5 121, so 110 is not a perfect square.
Math Practice: I checked my thinking with the calculator to find the square root of 110
and saw that the decimal did not terminate or repeat (TLM #5).
• Use Random Reporter to have teams share. Use the Random Reporter rubric to
evaluate responses and give feedback.
• Record individual scores on the teacher cycle record form.
• Award team celebration points.
homework check
• Ask teams to do a homework check.
• Confirm the number of students who completed the homework on each team.
• Poll students on their team’s understanding of the homework.
• Award team celebration points.
• Collect and grade homework once per cycle. Record individual scores on the teacher
cycle record form.
© 2015 Success for All Foundation
PowerTeaching Math 3rd Edition | Unit Guide
7
Level 8 | Unit 2: Rational and Irrational Numbers | Cycle 1 Lesson 2 Access Code: zwdffn
active instruction
(10–15 minutes)
set the stage
• Post and present the lesson objective: Today you will classify rational and
irrational numbers.
• Remind students how to earn team celebration points.
interactive instruction and guided practice
• Use a Think Aloud to review definitions of rational and irrational numbers to
classify a number.
Classify rational and
irrational numbers.
3 layers
Classify 
​ 
121 ​ on the Venn diagram.
What do I know about rational and irrational numbers? Natural numbers are
counting numbers like 1, 2, 3, 4…. Whole numbers are natural numbers and 0.
Integers are whole numbers and their opposites. Rational numbers include
natural numbers, whole numbers, integers, fractions, and terminating and
repeating decimals. An irrational number cannot be written in fractional form.
Show layer 1.
Now I’ll examine the examples I’ve been given.
Rational Numbers
0.5
Integers
5
Irrational Numbers
3
_
10
10
0
Whole
Numbers
Natural
Numbers
1 5 10
I don’t see any examples of irrational numbers, but by definition, I know that
any number that cannot be written in fractional form (such as p and 
​ 
2 ​)  is an
irrational number.
Show layer 2. The problem asks me to classify the square root of 121. Since this
number is under the radical and, I only need the positive square root. Is 121 a
perfect square? Yes, 121 is the product of 11 times 11.
Show layer 3. So the square root of 121 is classified as a natural number, a whole
number, an integer, and a rational number. I used the classifications precisely to
answer this question (TLM #6).
• Use Think‑Pair‑Share to have the students name one number for each section of
the diagram.
8
PowerTeaching Math 3rd Edition | Unit Guide
© 2015 Success for All Foundation
Access Code: zwdffn
Level 8 | Unit 2: Rational and Irrational Numbers | Cycle 1 Lesson 2
• Randomly select a few students to share. Possible answers: Accept reasonable
responses. Natural numbers: 1, 2, 3, 4…. Whole numbers: natural numbers and
0. Integers: …– 4, – 3, – 2, – 1, 0, 1, 2, 3, 4. Rational numbers: natural numbers,
whole numbers, integers, fractions, and terminating and repeating decimals.
Irrational numbers: p, 
​ 
2 ​,  a number that cannot be written in fractional form.
• Use Team Huddle to have teams practice classifying rational and irrational numbers.
Classify the numbers by writing them in the appropriate section of the Venn diagram.
1)
10
– ​ _
  ​,
2
​ 
36 ​,  _
​  0 ​, 7, ​
140 ​ 
,_
​  4 ​, ​
4 ​,  – 8, ​
8 ​,  – 2.89

8
9
Random Reporter Rubric | Possible Answer
Answer:
Rational Numbers
4
_
Irrational Numbers
2.89
9
10
_
2
0
_
8
Integers
140
8
8
Whole
Numbers
Natural
Numbers
36 7
4
Explanation: I started by examining the fractions to see if I could simplify any of them.
I found that one fraction was equivalent to an integer while another was equivalent
to 0. I placed the third fraction in rational numbers . Next, I determined that the square
roots of perfect squares are natural numbers . The square roots of numbers that are not
perfect squares are irrational numbers . I placed the remaining rational numbers in the
appropriate sections.
Math Practice: My classification makes sense because I recalled the definition of each
section before determining which number fit into that section (TLM #1).
• Use Random Reporter to have teams share. Use the Random Reporter rubric to
evaluate responses and give specific feedback.
• Record individual scores on the teacher cycle record form.
• Award team celebration points.
© 2015 Success for All Foundation
PowerTeaching Math 3rd Edition | Unit Guide
9
Level 8 | Unit 2: Rational and Irrational Numbers | Cycle 1 Lesson 2 Access Code: zwdffn
team mastery
(10–15 minutes)
• Ask students to follow the Team Mastery student routine.
• Circulate and use the following questions to prompt discussions.
–– How would you classify this number?
–– How many different ways can you categorize this number?
–– What is the difference between a rational and an irrational number?
–– What is the difference between a terminating and a repeating decimal?
–– How can you distinguish between all the subsets of rational numbers?
• When there are 5 minutes left in Team Mastery, prompt teams to prepare for the
Lightning Round. Have teams discuss one Team Mastery problem that the whole
team has completed.
• Award team celebration points for good team discussions that demonstrate
100‑point responses.
lightning round
(10 minutes)
• Tell students the Team Mastery problem that you will use for the Lightning Round.
4) Classify the numbers by writing them in the appropriate section of the
Venn diagram.
44, _
​  7 ​, ​
144 ​, – 88, _
​  12  ​, 5.4, ​
24 ​ 
3
2
Random Reporter Rubric | Possible Answer
Answer:
Rational Numbers
7
_
3
Irrational Numbers
5.4
24
Integers
88
Whole
Numbers
Natural
Numbers
44
144
12
_
2
10
PowerTeaching Math 3rd Edition | Unit Guide
© 2015 Success for All Foundation
Access Code: zwdffn
Level 8 | Unit 2: Rational and Irrational Numbers | Cycle 1 Lesson 2
Explanation: I started by examining the fractions to see if I could simplify any of
them. I found that one fraction was equivalent to a natural number while another was
a mixed number . Next, I determined that the square roots of perfect squares are natural
numbers. The square roots of numbers that are not perfect squares are irrational numbers .
I placed the remaining rational numbers in the appropriate sections. I used the Venn diagram
to classify the numbers into their appropriate subsets.
Math Practice: My classification makes sense because I recalled the definition of each
section before determining which numbers fit into that section (TLM #1).
• Use Random Reporter to have teams share. Use the Random Reporter rubric to
evaluate responses and give specific feedback.
• Record individual scores on the teacher cycle record form.
• Award team celebration points.
celebration
(2 minutes)
• Record team celebration points on the poster.
• Have the top team choose a cheer.
• Assign homework, and remind students about the Vocabulary Vault.
• Ask students to follow the Quick Check student routine. (optional)
Classify the numbers by writing them in the appropriate section of the
Venn diagram.
9.3, _
​  18  ​, ​
36 ​,  – 13, _
​  55  ​, 1.7, ​
41 ​ 
4
9
Possible answer:
Rational Numbers
Irrational Numbers
55
_
9.3
Integers
4
41
1.7
13
Whole
Numbers
Natural
Numbers
18
_
9
© 2015 Success for All Foundation
36
PowerTeaching Math 3rd Edition | Unit Guide
11
Access Code: zwdffn
Level 8 | Unit 2: Rational and Irrational Numbers | Cycle 1 Lesson 3
Lesson 3: Converting a Decimal Expansion
Vocabulary:
none
Lesson Objective: Convert a decimal expansion that repeats eventually into a
Materials:
none
By the end of this lesson, students will:
rational number.
• convert repeating decimals into fractions; and
• confirm that repeating decimals are rational numbers.
opening
(3 minutes)
get the goof
• Ask teams to begin Get the Goof.
TEACHER’S NOTE: Students may assume that the negative decimal is an integer. Remind them that
integers are whole numbers and their opposites. This does not include negative decimals or
negative fractions.
Omar classified – 8.4 as an integer. What’s wrong with his thinking?
Random Reporter Rubric | Possible Answer
Answer: Omar did not classify the number correctly.
Explanation: Integers are a set of whole numbers and their opposites . Since – 8.4 is not
a whole number, it cannot be classified as an integer. The correct classification of – 8.4 is
a rational number .
Math Practice: I used the definitions of an integer and a rational number to classify the
number correctly (TLM #6).
• Use Random Reporter to have teams share. Use the Random Reporter rubric to
evaluate responses and give feedback.
• Record individual scores on the teacher cycle record form.
• Award team celebration points.
homework check
• Ask teams to do a homework check.
• Confirm the number of students who completed the homework on each team.
• Poll students on their team’s understanding of the homework.
• Award team celebration points.
• Collect and grade homework once per cycle. Record individual scores on the teacher
cycle record form.
© 2015 Success for All Foundation
PowerTeaching Math 3rd Edition | Unit Guide
13
Level 8 | Unit 2: Rational and Irrational Numbers | Cycle 1 Lesson 3 active instruction
Access Code: zwdffn
(10–15 minutes)
set the stage
• Post and present the lesson objective: Today you will convert a decimal expansion
that repeats eventually into a rational number.
• Remind students how to earn team celebration points.
interactive instruction and guided practice
• Use a Think Aloud to introduce the process of converting a repeating decimal into
a fraction by using the students’ prior knowledge.
Convert a decimal
expansion.
6 layers
Write 0.4141…as a fraction.
I need to write this decimal as a fraction. What do I know about fractions?
Fractions are rational numbers. I know from my Think Like a Mathematician
guidelines that I can translate numbers, so I can represent fractions as division
expressions. That’s TLM practice #2. In every fraction, the numerator is the
dividend, and the denominator is the divisor. Therefore, the quotient will either
terminate or repeat. Let’s see if that’s true. I’ll look at one‑half and one‑third.
Show layer 1.
  
 ​  
​
​  0.​3
_
1.0 ​
​  1 ​ 5 1 4 3 or 3​ 
  
​  0.5​ 
_
1.0 ​
​  1 ​ 5 1 4 2 or 2​ 
2
3
In the expression “one‑half,” 1 is divided by 2 in the form a/b. These are integers
written in fractional form where b is not equal to 0. What if I rewrote the
expression as an equation to solve for x?
Show layer 2.
_
​  1 ​
1 5 2x
2
1
​ _
 ​ 5 x
2
To solve this equation, I would isolate the variable by dividing both sides by 2.
So x is equal to one‑half. I have not changed the value of the number.
Let’s use the equation with the decimal form of one‑half to rewrite the decimal
as a fraction.
Show layer 3.
Example 1
Example 2
x 5 0.5
x 5 0.55
​(10)​x 5 0.5​(10)​
(​100)​x 5 0.55​(100)​
10x 5 5
100x 5 55
x5_
​  5  ​ 
55
x 5 ​ _
  ​ 
1
 ​
x 5 ​ _
x5_
​  11  ​
10
2
14
PowerTeaching Math 3rd Edition | Unit Guide
100
20
© 2015 Success for All Foundation
Access Code: zwdffn
Level 8 | Unit 2: Rational and Irrational Numbers | Cycle 1 Lesson 3
I have two examples showing equations with terminating decimals. I want to
write the decimals as fractions. To do this, I need to multiply both sides by a
power of 10. (The power of 10 corresponds to the number of places behind the
decimal point.)
My problem is not a terminating decimal. What kind of decimal is this? It’s
a repeating decimal. Can I use the same method to write this decimal as a
fraction? Let’s find out.
Show layer 4.
x 5 0.4141…
100x 5 0.4141…​(100)​
100x 5 41.4141…
I multiplied both sides by 100 because two numbers repeat in the decimal. Since
I still have a decimal, I cannot treat this the same way I treated the terminating
decimal. I need a different way to isolate the variable. Before, I divided both
sides by the same number. Division is repeated subtraction, so let’s see what
happens when I subtract.
Show layer 5.
100x 5 41.4141…
2 x 5 0.4141…
99x 5 41
Now I can isolate the variable because I’m no longer working with a decimal.
Show layer 6.
_
​  99x  ​ 5 _
​  41  ​
99
99
x 5 _
​  41  ​
99
41
• Use Think‑Pair‑Share to have students check to see whether ​ _
  ​ 5 0.4141….
41
• Randomly select a few students to share. Yes. ​ _
  ​5 0.4141….
99
99
• Use Team Huddle to have teams practice converting a repeating decimal into
a fraction.
1) Find the fractional equivalent of 0.​234 ​
. Show your work.
Random Reporter Rubric | Possible Answer
​ or _
​  78  ​ or _
​  26  ​ 
Answer: _
​  234  
999
333
111
Explanation:
x 5 0.​234
 ​ 
1,000x 5 234.234…
1,000x 5 234.234…
2 x 5 0.234…
999x 5 234
234  
x 5 ​ _
​
999
I started by expressing the repeated decimal as an equation . Then, I multiplied each side of
the equation by 1,000 because three digits repeat. This moved the set of repeating digits to
the left side of the decimal point. Next, I subtracted x from 1,000x to get rid of the decimal.
Finally, I isolated the variable and solved for x.
Math Practice: I used what I know about converting a terminating decimal into an
equation to convert a repeating decimal into an equation (TLM #3).
© 2015 Success for All Foundation
PowerTeaching Math 3rd Edition | Unit Guide
15
Level 8 | Unit 2: Rational and Irrational Numbers | Cycle 1 Lesson 3 Access Code: zwdffn
• Use Random Reporter to have teams share. Use the Random Reporter rubric to
evaluate responses and give specific feedback.
• Record individual scores on the teacher cycle record form.
• Award team celebration points.
team mastery
(10–15 minutes)
• Ask students to follow the Team Mastery student routine.
• Circulate and use the following questions to prompt discussions.
–– How did you write this repeating decimal as a fraction?
–– How do you know that repeating decimals are rational numbers?
–– Why do you multiply each side by a multiple of 10?
• When there are 5 minutes left in Team Mastery, prompt teams to prepare for the
Lightning Round. Have teams discuss one Team Mastery problem that the whole
team has completed.
• Award team celebration points for good team discussions that demonstrate
100‑point responses.
lightning round
(10 minutes)
• Tell students the Team Mastery problem that you will use for the Lightning Round.
5) Find the fractional equivalent of 0.​21 ​
.  Show your work.
Random Reporter Rubric | Possible Answer
​  7  ​ 
Answer: _
​  21  ​ or _
99
Explanation:
33
x 5 0.​21
 ​ 
100x 5 21.21…
100x 5 21.21…
2 x 5 0.21…
99x 5 21
21  ​
x 5 ​ _
99
I started by expressing the repeated decimal as an equation . Then, I multiplied each side of
the equation by 100 because two digits repeat. This moved the set of repeating digits to the
left side of the decimal point. Next, I subtracted x from 100x to get rid of the decimal. Finally,
I isolated the variable and solved for x.
Math Practice: I used what I knew about converting a terminating decimal into an
equation to convert a repeating decimal into an equation (TLM #3).
• Use Random Reporter to have teams share. Use the Random Reporter rubric to
evaluate responses and give feedback.
• Record individual scores on the teacher cycle record form.
• Award team celebration points.
16
PowerTeaching Math 3rd Edition | Unit Guide
© 2015 Success for All Foundation
Access Code: zwdffn
Level 8 | Unit 2: Rational and Irrational Numbers | Cycle 1 Lesson 3
celebration
(2 minutes)
• Record team celebration points on the poster.
• Have the top team choose a cheer.
• Assign homework, and remind students about the Vocabulary Vault.
• Ask students to follow the Quick Check student routine. (optional)
Find the fractional equivalent of 0.​426 ​
. Show your work.
Possible answer: _
​  426  
​ or _
​  142  
​
999
© 2015 Success for All Foundation
333
PowerTeaching Math 3rd Edition | Unit Guide
17
Access Code: zwdffn
Level 8 | Unit 2: Rational and Irrational Numbers | Cycle 1 Lesson 4
Lesson 4: Ordering Rational and Irrational Numbers
Vocabulary:
none
Lesson Objective: Use knowledge of perfect squares and the number line to order rational
Materials:
NO calculators
By the end of this lesson, students will:
and irrational numbers.
• compare rational and irrational numbers, and
• use their knowledge of perfect squares and the number line to order rational and
irrational numbers.
opening
(3 minutes)
get the goof
• Ask teams to begin Get the Goof.
TEACHER’S NOTE: In the previous lesson, students learned to multiply each side of the equation by a
power of 10 that corresponds to the number of decimal numbers that repeat.
Sofia has to write a repeating decimal as a fraction. She multiplied each side of the
equation: x 5 0.636363… by 10. What’s wrong with her thinking?
Random Reporter Rubric | Possible Answer
Answer: Sofia did not multiply each side of the equation by the correct power of 10 .
Explanation: Sofia should have multiplied by the power of 10 with two zeroes, or 100.
Math Practice: My answer makes sense because 0.​63 ​
 has two numbers that repeat behind
the decimal point (TLM #1).
• Use Random Reporter to have teams share. Use the Random Reporter rubric to
evaluate responses and give feedback.
• Record individual scores on the teacher cycle record form.
• Award team celebration points.
homework check
• Ask teams to do a homework check.
• Confirm the number of students who completed the homework on each team.
• Poll students on their team’s understanding of the homework.
• Award team celebration points.
• Collect and grade homework once per cycle. Record individual scores on the teacher
cycle record form.
© 2015 Success for All Foundation
PowerTeaching Math 3rd Edition | Unit Guide
19
Level 8 | Unit 2: Rational and Irrational Numbers | Cycle 1 Lesson 4 Access Code: zwdffn
active instruction
(10–15 minutes)
set the stage
• Post and present the lesson objective: Today you will use knowledge of perfect
squares and the number line to order rational and irrational numbers.
• Remind students how to earn team celebration points.
interactive instruction and guided practice
• Use a Think Aloud to explore the process of organizing and arranging rational and
irrational numbers.
Use knowledge of
perfect squares and
the number line to
order rational and
irrational numbers.
4 layers
Order the numbers from least to greatest.
32
15
4
62 ​,  9.​3
10 ​,  _
​    ​

 ​,  – ​
76  ​, ​
– ​ _
I need to make sense of this problem. The problem asks me to order the
numbers from least to greatest.
Show layer 1. To convert a fraction to a decimal, I need to divide the numerator
by the denominator.
Show layer 2. Next I’ll take a look at the irrational numbers. I do not need a
calculator to figure this out. I can use what I know about these numbers to
estimate their position. 62 falls between the perfect squares 49 and 64, while
the negative square root of 10 falls between – 3 and – 4.
Show layer 3. Now I can put this all together and arrange these numbers from
least to greatest.
76 _
15
10 ​,  – ​ _
  ​,  ​    ​,  ​
62 ​ , 9.​3 ​
Show layer 4. My answer is – ​
.  To solve this I converted to
32
4
equivalent forms so my ordering was accurate (TLM #6).
• Use Think‑Pair‑Share to have students answer the following question: Would
the answer be different if we converted to equivalent fractions instead
of decimals?
• Randomly select a few students to share. Possible answer: No. The type of
conversion, whether to decimal or to fraction, would not change the value of the
number. Therefore, the answer would not be different.
• Use a Think Aloud to compare two numbers.
Compare
rational and
irrational numbers.
Use ,, ., or 5 to compare the following numbers.
6.92820323… ____ ​
42 ​ 
Again, before I solve the problem, I have to make sense of the problem. What
do I know about these numbers? I can see that one number is a decimal, but I
cannot tell if the decimal repeats. The decimal does not appear to terminate.
I also know that the square root of 42 is an irrational number.
I’m going to use a number line to help me visualize the position of each number.
That way I can determine whether one number is greater than, less than, or
equal to the other.
20
PowerTeaching Math 3rd Edition | Unit Guide
© 2015 Success for All Foundation
Access Code: zwdffn
Level 8 | Unit 2: Rational and Irrational Numbers | Cycle 1 Lesson 4
Show layer 1. I’ve got a good idea where the decimal is located. I know that
the square root of 42 is between 6 and 7 because 42 falls between 36 and 49.
Show layer 2. So the square root of 42 is about half the distance between
36 and 49. I can put it around the half mark between 6 and 7.
Show layer 3. Now I can make a reasonable conclusion and answer the
original question.
42 ​.  I used a number line as a tool to help me
Show layer 4. 6.92820323… . ​
compare the numbers (TLM #5).
• Use Think‑Pair‑Share to have students answer the following question: Which is
greater: p or 3.14? Explain.
• Randomly select a few students to share. p is greater than 3.14. Possible answer:
The number pi continues on past 3.14 as a never‑ending, nonrepeating decimal.
This is why it is greater than 3.14.
• Use Team Huddle to have teams practice arranging a group of rational numbers on
a number line.
1) Graph each number on the number line below.
2.55, _
​  9 ​, ​
24 ​,  p, 
​ 
9 ​ 
5
–5
–4
–3
–2
–1
0
1
2
3
4
5
Random Reporter Rubric | Possible Answer
Answer:
9
5
–5
–4
–3
–2
–1
0
1
9
2
3
24
4
5
2.55
Explanation: I started by plotting the rational numbers on the number line. I converted _
​  9 ​
5
to 1.8 and ​
9 ​ to 3. Then, I plotted both of them. Next, I approximated 
​ 
24 ​ as close to 5
and p as 3.14 and plotted them also.
Math Practice: My answer makes sense because I used the decimal estimates of each
number to arrange the numbers on the number line (TLM #1).
• Use Random Reporter to have teams share. Use the Random Reporter rubric to
evaluate responses and give feedback.
• Record individual scores on the teacher cycle record form.
• Award team celebration points.
© 2015 Success for All Foundation
PowerTeaching Math 3rd Edition | Unit Guide
21
Level 8 | Unit 2: Rational and Irrational Numbers | Cycle 1 Lesson 4 Access Code: zwdffn
team mastery
(10–15 minutes)
• Ask students to follow the Team Mastery student routine.
• Circulate and use the following questions to prompt discussions.
–– Is this number rational or irrational? How do you know?
–– Between which numbers does this square root fall?
–– How did you determine which number is greater/lesser?
–– How can you compare decimals to fractions?
–– How did you determine the order of this set of numbers?
• When there are 5 minutes left in Team Mastery, prompt teams to prepare for
Lightning Round. Have teams discuss one Team Mastery problem that the whole
team has completed.
• Award team celebration points for good team discussions that demonstrate
100‑point responses.
lightning round
(10 minutes)
• Tell students the Team Mastery problem that you will use for the Lightning Round.
4) Write the following numbers in order from greatest to least.
– 2.3,
1
0, _
​ 23  ​, p, – 7​ _
 ​
7
5
Random Reporter Rubric | Possible Answer
1
 ​
Answer: _
​  23  ​, p, 0, – 2.3, – 7​ _
7
5
Explanation: I started by converting the improper fraction and mixed number into
their decimal forms. I also wrote the decimal approximate for p. Since we are ordering the numbers from greatest to least , I used a number line to approximate each number’s
location. Then, I wrote my answer in the reverse order, starting with the largest positive
number and ending with the negative number at the far left of the number line.
Math Practice: My answer makes sense because on a number line, moving to the right
means the numbers increase in value and moving to the left means the numbers decrease in value (TLM #1).
• Use Random Reporter to have teams share. Use the Random Reporter rubric to
evaluate responses and give feedback.
• Record individual scores on the teacher cycle record form.
• Award team celebration points.
• Tell students that it’s time to power up Random Reporter. Use the layers on the page
to guide discussion.
22
PowerTeaching Math 3rd Edition | Unit Guide
© 2015 Success for All Foundation
Access Code: zwdffn
Level 8 | Unit 2: Rational and Irrational Numbers | Cycle 1 Lesson 4
celebration
(2 minutes)
• Record team celebration points on the poster.
• Have the top team choose a cheer.
• Assign homework, and remind students about the Vocabulary Vault.
• Ask students to follow the Quick Check student routine. (optional)
Use ,, ., or 5 to compare the following numbers.
6.1678203027… _____ ​
32 ​ 
Possible answer: 6.1678203027… . 
​ 
32 ​ 
© 2015 Success for All Foundation
PowerTeaching Math 3rd Edition | Unit Guide
23
Access Code: zwdffn
Level 8 | Unit 2: Rational and Irrational Numbers | Cycle 1 Lesson 5
Lesson 5: Comparing Irrational Expressions
Vocabulary:
none
Lesson Objective: Use approximations of the values of irrational numbers to estimate and
Materials:
none
By the end of this lesson, students will:
compare expressions containing irrational numbers.
• approximate the value of irrational numbers with rational numbers; and
• use number sense and what they know about the number line to estimate
and compare expressions that contain irrational numbers.
opening
(3 minutes)
get the goof
• Ask teams to begin Get the Goof.
Luigi said that the value of 
​ 
93 ​ is between 81 and 100. What’s wrong with
his thinking?
Random Reporter Rubric | Possible Answer
Answer: Luigi did not approximate the value correctly; he didn’t account for
the square root sign.
Explanation: The square root of 93 is a number that can be multiplied by itself to make
93. This means the value is actually between 9 and 10.
Math Practice: I used what I know about the definition of square root to make sense of
this problem (TLM #3). Because 9
​ ​ 2​ 5 81 and 1
​ 0​ 2​ 5 100, 
​ 
93 ​ is between 9 and 10, not
between 81 and 100.
• Use Random Reporter to have teams share. Use the Random Reporter rubric to
evaluate responses and give feedback.
• Record individual scores on the teacher cycle record form.
• Award team celebration points.
homework check
• Ask teams to do a homework check.
• Confirm the number of students who completed the homework on each team.
• Poll students on their team’s understanding of the homework.
• Award team celebration points.
• Collect and grade homework once per cycle. Record individual scores on the teacher
cycle record form.
© 2015 Success for All Foundation
PowerTeaching Math 3rd Edition | Unit Guide
25
Level 8 | Unit 2: Rational and Irrational Numbers | Cycle 1 Lesson 5 Access Code: zwdffn
active instruction
(10–15 minutes)
set the stage
• Post and present the lesson objective: Today you will compare numbers that contain
irrational values.
• Remind students how to earn team celebration points.
interactive instruction and guided practice
• Show the “Think Like a Mathematician” video clip for practice #7, find the patterns
and structure.
• Use Think‑Pair‑Share to have students discuss situations in which it would be
helpful to use this practice and, if possible, to give their own personal usage
examples. Randomly select students to share.
• Use a Think Aloud to model comparing expressions that contain
irrational numbers.
Estimate and
compare the values
of expressions
that contain
irrational numbers.
6 layers
In each pair, decide which number is greater without using a calculator.
What’s going on in this problem? I have to compare these numbers to see which
one is greater, so this is just basic comparison. But I see that the first pair has pi
squared in it, and the second pair has the square root of 30 in it, so I have to do
some approximation here because these are irrational numbers. Show layer 1.
To make sense of the first problem, I have to make sense of the opposite of
pi squared. I know that a good approximation of pi is 3.14, so I’ll start there.
Show layer 2.
Of course, I’m not supposed to use a calculator to solve this, but even if I did,
my answer would be an estimate. Even a calculator can’t accurately represent
an irrational number. So I’ll indicate that I’m making an estimate. Let me think
about the value of pi squared. Squaring any number means using that number
as a factor twice, so I have to estimate 3.14 ? 3.14. Just by thinking about this, I
know that pi squared will be greater than 9 because 3
​ ​ 2​ 5 9, and 3.14 . 3.
26
PowerTeaching Math 3rd Edition | Unit Guide
© 2015 Success for All Foundation
Access Code: zwdffn
Level 8 | Unit 2: Rational and Irrational Numbers | Cycle 1 Lesson 5
Show layer 3.
Now I have to compare the opposite of pi squared with – 6. I’ll use a number line
because sometimes I get confused when comparing negative numbers.
This makes sense. Negative 6 is closer to zero than a value to the left of negative
9, so I know that – 6 is greater in this first pair.
Show layer 4. Now on to the next pair. I have to compare two square roots of 30
and 10. Once again, I have to make sense of an irrational number.
Show layer 5. To take this one step at a time, first I’ll focus on the square root
of 30. I know that value is between the square root of 25 and the square
root of 36. 30 is almost exactly halfway between 25 and 36, so I know a good
approximation is 5.5.
Now, two square roots of 30 is just 2 times that value. 2 ? 5.5 is 11. Even though
I don’t know what 
​ 
30 ​ is exactly, I can just use what I know about the structure
of the expression 2​
30 ​ and multiplication to estimate the value as about 11.
That’s TLM practice #7, finding the patterns and structure. Show layer 6.
Now I can complete my estimate. In this pair, 11 is greater than 10, so two
square roots of 30 is greater. I made sense of the comparison by approximating
the irrational numbers with rational numbers and then using what I know about
multiplication to estimate their values. Once I figured out approximations for the
irrational numbers, I could find products with those approximations to complete
the comparison. That’s TLM practice #7.
• Use Think‑Pair‑Share to have students answer the following question: If 
​ 
45 ​ is
​ 45 ​ 
3
about 6.75, what is _
​ 
   
​? 
• Randomly select a few students to share. Possible answer: It’s about 2.25.
• Use Team Huddle to have teams practice estimating expressions that contain
irrational numbers.
Decide which number is greater without using a calculator.
1)4​
20 ​ or 2​
20 ​ 

Explain your thinking.
Random Reporter Rubric | Possible Answer
20 ​ . 2​
20 ​ 
Answer: 4​
Explanation:
Estimate:
4​(4.5)​or 2​(6.4)​
18 . 12.8
Math Practice: I made sense of the comparison by approximating the irrational numbers
with rational numbers and then using what I know about multiplication to estimate their
values. Once I figured out that 
​ 
20 ​ is about halfway between 4 and 5, I could find 4 times
that value (TLM #7).
© 2015 Success for All Foundation
PowerTeaching Math 3rd Edition | Unit Guide
27
Level 8 | Unit 2: Rational and Irrational Numbers | Cycle 1 Lesson 5 Access Code: zwdffn
• Use Random Reporter to have teams share. Use the Random Reporter rubric to
evaluate responses and give feedback.
• Record individual scores on the teacher cycle record form.
• Award team celebration points.
team mastery
(10–15 minutes)
• Ask students to follow the Team Mastery student routine.
• Circulate and use the following questions to prompt discussions.
–– What information do you need to solve this problem?
–– How did you approximate the square root of this number?
–– Which perfect squares does the square fall directly between?
–– How did you figure out which expression was greater?
• When there are 5 minutes left in Team Mastery, prompt teams to prepare for the
Lightning Round. Have teams discuss one Team Mastery problem that the whole
team has completed.
• Award team celebration points for good team discussions that demonstrate
100‑point responses.
lightning round
(10 minutes)
• Tell students the Team Mastery problem that you will use for the Lightning Round.
Decide which number is greater without using a calculator.
4) – 5​
5 ​ or – 25
Explain your thinking.
Random Reporter Rubric | Possible Answer
5 ​ is greater
Answer: – 5​
Explanation:
Estimate:
– 5(2.25)
or – 25
– 11.25 or – 25
– 11.25 . – 25
Math Practice: I made sense of the comparison by approximating the irrational number
with a rational number and then using what I know about multiplication to estimate the
value. Once I figured out that 
​ 
5 ​ is about a quarter of the way from 2 to 3, I could find
– 5 times that value (TLM #7).
• Use Random Reporter to have teams share. Use the Random Reporter rubric to
evaluate responses and give feedback.
• Record individual scores on the teacher cycle record form.
• Award team celebration points.
28
PowerTeaching Math 3rd Edition | Unit Guide
© 2015 Success for All Foundation
Access Code: zwdffn
Level 8 | Unit 2: Rational and Irrational Numbers | Cycle 1 Lesson 5
celebration
(2 minutes)
• Record team celebration points on the poster.
• Have the top team choose a cheer.
• Assign homework, and remind students about the Vocabulary Vault.
• Ask students to follow the Quick Check student routine. (optional)
Decide which number is greater without using a calculator.
3​
12 ​ or 3​
11 ​ 
Possible answer: 3​
12 ​ is greater.
© 2015 Success for All Foundation
PowerTeaching Math 3rd Edition | Unit Guide
29
Access Code: brdpsg
Level 8 | Unit 2: Rational and Irrational Numbers | Cycle 1 Assessment Day
Assessment Day: Unit Check on Rational and
Irrational Numbers
Materials:
extra blank copies
of the assessment
Lesson Objective: Demonstrate mastery of unit content.
assessment
(20–30 minutes)
• Confirm the number of students who completed the homework on each team.
Award team celebration points.
• Remind the students that the test is independent work.
• Distribute the tests so students can preview the questions.
• Tell students how much time they have for the test and that they may begin. Give
students a 5‑minute warning before the end of the test.
• Collect the tests.
team reflection
(5 minutes)
• Display or hand out blank copies of the test.
• Explain or review, if necessary, the student routine for team discussions after
the test.
• Award team celebration points.
prep points
(5–10 minutes)
• Assign prep points for each team for the five questions indicated (#s 2, 6, 7, 11, 14).
• Score individual tests when convenient.
vocabulary vault
(2 minutes)
• Randomly select Vocabulary Vouchers, and award team celebration points.
• Ask students to record the words that they explain on their team score sheets.
team scoring
(5 minutes)
• Lead the class in completing the team scoring on their team score sheets.
© 2015 Success for All Foundation
PowerTeaching Math 3rd Edition | Unit Guide
31
Level 8 | Unit 2: Rational and Irrational Numbers | Cycle 1 Assessment Day celebration
Access Code: brdpsg
(2 minutes)
• Announce team statuses, and celebrate.
• Poll teams about how many times they have been super teams. Celebrate those
teams, and encourage all teams to work toward super team status during the
next cycle.
• Play the video “Practice Active Listening.”
• Use Think-Pair-Share to have students discuss how this goal can help them reach
super team status. Randomly select a few students to share.
32
PowerTeaching Math 3rd Edition | Unit Guide
© 2015 Success for All Foundation
student pages
Level 8
8 || Unit 2: Rational and Irrational Numbers
Numbers || Cycle 1
1 Lesson
Lesson 1
1
teamwork
1) Provide two examples that show that the statement is false.
Lydia said that all square roots are irrational numbers.
Directions for questions 2 and 3: Classify the numbers as rational or irrational.
2) 
77  
3) 
48  
Directions for questions 4 and 5: Find the square roots for each number.
4) 121
5) 81
Directions for questions 6–8: Classify the numbers as rational or irrational.
6)
– 8.875
7) 
16  
8) 2.67034165508…
9) Provide two examples that show that the statement is false.
Sebastian said that if a number is a perfect square, then the number is even.
Directions for questions 10–12: Find the square roots for each number.
10) 25
11) 49
12) 144
Challenge
13) Is the product of a rational and an irrational number rational or irrational? Give an example to support
your answer.
14) Do the expressions 
100 2 64  
and 
100  2 
64  have the same value? Explain your thinking.
© 2015 Success for All Foundation
Foundation
Math
3rd Edition
PowerTeaching Math PowerTeaching
3rd Edition | Unit
Guide
35
Level 8
8 || Unit 2: Rational and Irrational Numbers
Numbers || Cycle 1
1 Lesson
Lesson 1 1
teamwork answers
1) Possible answers:
Example 1: 
4 
Example 2: 
100  
2) irrational
3) irrational
4) 11 and – 11
5) 9 and – 9
6) rational
7) rational
8) irrational
9) Possible answers:
Example 1: 49
Example 2: 121
10) 5 and – 5
11) 7 and – 7
12) 12 and – 12
13) The product of a rational and an irrational number is irrational. Accept reasonable explanations.
14) No, these expressions do have not the same value.
Possible explanation: To solve the first expression, I subtracted 64 from 100 and got 36. The
square root of 36 is equal to 6. For the second expression, I found the square roots of 100 and 64 and
got 10 and 8. Then, I subtracted 8 from 10 and got 2. Therefore, the values of the expressions are not
the same.
PowerTeaching
Math 3rdMath
Edition
36
PowerTeaching
3rd Edition | Unit Guide
© 2015 Success for All Foundation
quick check
Level 8
8 || Unit 2: Rational and Irrational Numbers
Numbers || Cycle 1
1 Lesson
Lesson 1
1
Name
Find the square roots for 64.
© 2015 Success for All Foundation
quick check
PowerTeaching Math 3rd Edition
Name
Find the square roots for 64.
© 2015 Success for All Foundation
Foundation
Math
3rd Edition
PowerTeaching Math PowerTeaching
3rd Edition | Unit
Guide
37
Level 8 | Unit 2: Rational and Irrational Numbers | Cycle 1 Lesson
Level 81 | Unit 2: Rational and Irrational Numbers | Cycle 1 Lesson 1
homework
Quick Look
Vocabulary words introduced in this cycle:
rational number, radical sign, square root, perfect square, irrational number
Today we defined and explored irrational numbers. An irrational number is a number that cannot be
written in fractional form. We know a number is irrational if it is a decimal number that is infinitely long
and has no repeating pattern. We also learned the difference between rational and irrational numbers.
For example:
Number
Type and Explanation
2 

Irrational; 2 is not a perfect square.
9 

Rational; 9 is a perfect square, 
9  5 3.
0.0101010101…
0.01001000100001…
Rational; repeating decimal, it has a pattern
Irrational; nonrepeating, nonterminating decimal
We learned that the square root of a number is a number that when multiplied by itself, equals the
original number. Square roots include both positive and negative numbers. For example, the square root of
2
2
25 is 5 because (5) 5 25, and (– 5) 5 25. However, if we write it as 
25 ,  then we are only talking about
the positive root, 5.
1) Provide two examples that show the statement is false. Explain your thinking.
Zoe said that an irrational number can be expressed as a terminating decimal.
Directions for questions 2–6: Classify the numbers as rational or irrational.
2) p
3) 
110  
4) 
81  
5) 
14  
6)
2
–_
3
Directions for questions 7 and 8: Find the square roots for each number.
7) 100
8) 49
©
for All Foundation
382015 Success
PowerTeaching
Math 3rd Edition | Unit Guide
Math
3rd Edition
©PowerTeaching
2015 Success for
All Foundation
Level
8 | Unit 2: Rational and Irrational Numbers | Cycle 1 Lesson
Level 8 1 | Unit 2: Rational and Irrational Numbers | Cycle 1 Homework
Lesson 1
Mixed Practice
9) Evaluate the expression.
229?123?9
10) Solve for x.
6x 2 5 5 59
11) You have a number cube labeled 1–6. What is the probability of rolling an even number?
12) What is the measure of the radius of the circle whose circumference is 21.98 inches? Use 3.14 for p.
Word Problem
13) Tell what an irrational number is in your own words. Give an example of an irrational number and a
rational number.
PowerTeaching
Math
Edition
© 2015 Success for
All3rd
Foundation
2015
Success
for Guide
All Foundation
PowerTeaching Math©3rd
Edition
| Unit
39
Level 8 | Unit 2: Rational and Irrational Numbers | Cycle 1 Lesson
Homework
Level 81 | Unit 2: Rational and Irrational Numbers | Cycle 1 Lesson 1
For the Guide on the Side
Today your student defined irrational numbers. An irrational number is a number that cannot be written in
fractional form. Any infinite nonrepeating decimal is an irrational number.
Your student also discussed square roots by learning to identify perfect squares and nonperfect squares
and to tell when a nonperfect square is irrational. Your student learned that the square root of a perfect
square is a rational number. Furthermore, the square root of a number that is not a perfect square is an
irrational number.
Your student should be able to answer the following questions about irrational numbers:
1) What is an irrational number?
2) What’s the difference between a rational and an irrational number?
3) How did you know this was an irrational number?
4) How can a number have two square roots?
5) What is a perfect square?
6) How does the square root of a number relate to its classification?
Here are some ideas to use to practice working with irrational numbers:
1) Understand and apply the definition of irrational numbers:
http://learnzillion.com/lessons/220‑understand‑and‑apply‑the‑definition‑of‑irrational‑numbers
2) What’s an irrational number?:
http://virtualnerd.com/pre‑algebra/real‑numbers‑right‑triangles/real‑and‑irrational/define‑real‑numbers/
irrational‑number‑definition
3) Understanding Square Roots:
www.khanacademy.org/math/arithmetic/exponents‑radicals/radical‑radicals/v/understanding‑square‑roots
4) Think of two integers on a number line. What rational numbers can you think of that fall between the
two integers? What irrational numbers fall between them?
©
for All Foundation
402015 Success
PowerTeaching
Math 3rd Edition | Unit Guide
Math
3rd Edition
©PowerTeaching
2015 Success for
All Foundation
Level
8 | Unit 2: Rational and Irrational Numbers | Cycle 1 Lesson
Level 8 1 | Unit 2: Rational and Irrational Numbers | Cycle 1 Lesson 1
homework answers
1) Possible answers:
Example 1: p
Example 2: 
2 
Possible explanation: I know the statement is false because an irrational number cannot be expressed
in fractional form. A terminating decimal is a number that can be written over a power of 10 in
fractional form. I examined the statement Zoe made to determine whether her argument made sense.
I used what I know about the definitions of irrational numbers and terminating decimals to prove her
statement false (TLM #3). I know that any example of a terminating decimal cannot be an irrational
number, and an irrational number (since its decimal does not repeat or terminate) cannot be expressed as
a terminating decimal.
2) irrational
3) irrational
4) rational
5) irrational
6) rational
7) 10 and – 10
8) 7 and – 7
Mixed Practice
9)
– 34
2
10) 10 _
3
1
11) 0.5 or _
2
12) r 5 3.5 in.
Word Problem
13) Possible answer: An irrational number is a number that cannot be written in fractional form. It includes
the square roots of nonperfect squares and decimals that keep going without any pattern. Accept
appropriate examples.
PowerTeaching
Math
Edition
© 2015 Success for
All3rd
Foundation
2015
Success
for Guide
All Foundation
PowerTeaching Math©3rd
Edition
| Unit
41
Level 8 | Unit 2: Rational and Irrational Numbers | Cycle 1 Lesson
Level 82 | Unit 2: Rational and Irrational Numbers | Cycle 1 Lesson 2
teamwork
Classify the numbers by writing them in the appropriate section of the Venn diagram.
1)
10 ,
–_
2
0
8
4
9
36 ,  _, 7, 
140 , _, 
4 ,  – 8, 
8 ,  – 2.89

2)
– 25,
72
21
_
, 
14 ,  6.5, _
, 
81  
9
6
6
3
12 , 
3) 
53  , – _
19 ,  8, _
, p, 0, _
7
12
4) 44, _
, 
144 , – 88, _
, 5.4, 
24  
80
45
0
5) _
, 
72 ,  _
, 
50 ,  _
, 
49  
36
2
6) 
4 ,  – 7, _
, 
97 ,  _
, 3, 
30 ,  p
45
6
1
6
2
94
©
for All Foundation
422015 Success
PowerTeaching
Math 3rd Edition | Unit Guide
3
2
9
9
Math
3rd Edition
©PowerTeaching
2015 Success for
All Foundation
Level
8 | Unit 2: Rational and Irrational Numbers | Cycle 1 Lesson
Level 8 2 | Unit 2: Rational and Irrational Numbers | Cycle 1 Teamwork
Lesson 2
3
7
7) 1, _
, 
37  , _
, 
23 ,  
16  
8)
25
2
9) 83, – 9.3, 12, _
, 
90 ,  9.8, _
, 
63  
0
7
1
10) _
, 32, 6.4, _
, 
87 ,  – 30, _
45
19
26
2
9 , – 88,
–_
1
6
2
9.2, 26, 
71 ,  – 67, _
, 
1 
8
41
3
Challenge
Find the square roots.
11) 0.25
1
12) _
4
PowerTeaching
Math
Edition
© 2015 Success for
All3rd
Foundation
2015
Success
for Guide
All Foundation
PowerTeaching Math©3rd
Edition
| Unit
43
Level 8 | Unit 2: Rational and Irrational Numbers | Cycle 1 Lesson
Level 82 | Unit 2: Rational and Irrational Numbers | Cycle 1 Lesson 2
teamwork answers
1)
2)
3)
4)
3)
6)
7)
8)
©
for All Foundation
442015 Success
PowerTeaching
Math 3rd Edition | Unit Guide
Math
3rd Edition
©PowerTeaching
2015 Success for
All Foundation
Level
8 | Unit 2: Rational and Irrational Numbers | Cycle 1 Lesson
Level 8 2 | Unit 2: Rational and Irrational Numbers Teamwork
| Cycle 1 Lesson
Answers
2
9)
10)
11) 0.5 and – 0.5
1
1
12) _
and – _
2
2
PowerTeaching
Math
Edition
© 2015 Success for
All3rd
Foundation
2015
Success
for Guide
All Foundation
PowerTeaching Math©3rd
Edition
| Unit
45
Level 8 | Unit 2: Rational and Irrational Numbers | Cycle 1 Lesson
Level 82 | Unit 2: Rational and Irrational Numbers | Cycle 1 Lesson 2
quick check
Name
Classify the numbers by writing them in the appropriate section of the Venn Diagram.
18
55
9.3, _
, 
36 ,  – 13, _
, 1.7, 
41
 
9
4
© 2015 Success for All Foundation
PowerTeaching Math 3rd Edition
quick check
Name
Classify the numbers by writing them in the appropriate section of the Venn Diagram.
18
55
9.3, _
, 
36 ,  – 13, _
, 1.7, 
41
 
9
4
©
for All Foundation
462015 Success
PowerTeaching
Math 3rd Edition | Unit Guide
Math
3rd Edition
©PowerTeaching
2015 Success for
All Foundation
Level 8
8 || Unit 2: Rational and Irrational Numbers
Numbers || Cycle 1
1 Lesson
Lesson 2
2
homework
Quick Look
Vocabulary words introduced in this cycle:
rational number, radical sign, square root, perfect square, irrational number
Today we delved further into understanding rational versus irrational numbers. We also used a Venn diagram
to help us classify rational and irrational numbers and see the relationships between classifications.
For example, the Venn diagram shows how we classify numbers.
Directions for questions 1–5: Classify the numbers by writing them in the appropriate section of the
Venn diagram.
8
9
, 
47  , – 7, _
, 
81 ,  10.6, 
2 
1) _
3
3
© 2015 Success for All Foundation
Foundation
14 –
4
2) _
, 63, 2.8, 0, 
25 ,  – 3.9, _
, 
75 ,  24
27
9
Math
3rd Edition
PowerTeaching Math PowerTeaching
3rd Edition | Unit
Guide
47
Level 8
8 || Unit 2: Rational and Irrational Numbers
Numbers || Cycle 1
1 Lesson
Lesson 2 2
Homework
8 –
0
24
2
3) _
, 80, 
5 ,  _
, 55, _
, 
67 ,  – 48, _
, 
91  
7
16
6
6
Explain your thinking.
5
12 –
, 
96 ,  – 87, _
, 8.4, 
49  
4) 14, _
6
2
4 –
4
5) _
, 2.6, 11, 
2 ,  – 82, _
, 
32  
4
9
Mixed Practice
6) Write 0.35 as a fraction in simplest form.
7) Anita purchased 3 pairs of shoes for a total of $89.13. Calculate the average cost that Anita spent for
1 pair of shoes.
8) Is 26,877,240 divisible by 2, 3, 4, 5, 6, 7, 8, 9, and 10?
6
2
9) _
4_
7
3
Word Problem
10) Siddiquah said that zero is not a rational number because you cannot write zero in fractional form.
Is Siddiquah correct? Explain your thinking.
PowerTeaching
Math 3rdMath
Edition
48
PowerTeaching
3rd Edition | Unit Guide
© 2015 Success for All Foundation
Homework
Level 8
8 || Unit 2: Rational and Irrational Numbers
Numbers || Cycle 1
1 Lesson
Lesson 2
2
For the Guide on the Side
Today your student classified numbers as rational, integers, whole, natural, or irrational. In a previous lesson,
your student learned that irrational numbers are numbers that cannot be written in fractional form and that
an infinite nonrepeating decimal is an irrational number. He or she recalled the following:
• Natural numbers 5 counting numbers: 1, 2, 3, …
• Whole numbers 5 natural numbers and zero: 0, 1, 2, 3, …
• Integers 5 positive and negative whole numbers: – 3, – 2, – 1, 0, 1, 2, 3, …
72
• Rational numbers 5 numbers that can be written in fractional form: – 1.2, _
,4
5
In the next lesson, your student will confirm that repeating decimals are rational numbers.
Your student should be able to answer the following questions about classifying numbers:
1) How would you classify this number?
2) How many different ways can you categorize this number?
3) What is the difference between a rational and an irrational number?
4) What is the difference between a terminating and a repeating decimal?
5) How can you distinguish between all the subsets of rational numbers?
Here are some ideas to practice classifying numbers:
1) Learn Zillion: Distinguish Between Rational and Irrational Numbers:
http://learnzillion.com/lessons/221‑distinguish‑between‑rational‑and‑irrational‑numbers
2) Virtual Nerd: How do different categories of numbers compare to each other?
http://virtualnerd.com/algebra‑1/algebra‑foundations/number‑category‑comparison.php
© 2015 Success for All Foundation
Foundation
Math
3rd Edition
PowerTeaching Math PowerTeaching
3rd Edition | Unit
Guide
49
Level 8
8 || Unit 2: Rational and Irrational Numbers
Numbers || Cycle 1
1 Lesson
Lesson 2 2
homework answers
1)
2)
3)
Possible explanation: I started by examining
the fractions to see if I could simplify any of them. I
found that one fraction was equivalent to a natural
number while another was equivalent to 0. I placed
the other fractions in the section for rational numbers.
Next, I determined that the square roots were not
for perfect squares and, therefore, were irrational
numbers. I placed the remaining rational numbers in
the appropriate sections based on their definitions.
My classification makes sense because I recalled the
definition of each section before determining which
number fit into that section (TLM #1).
4)
5)
PowerTeaching
Math 3rdMath
Edition
50
PowerTeaching
3rd Edition | Unit Guide
© 2015 Success for All Foundation
Homework
Answers
Level 8
8 || Unit 2: Rational and Irrational Numbers
Numbers || Cycle 1
1 Lesson
Lesson 2
2
Mixed Practice
7
6) _
20
7) Anita spent $29.71 on 1 pair of shoes.
8) 26,867,240 is divisible by 2, 3, 4, 5, 6, 8, 9, and 10.
9
2
9) _
or 1_
7
7
Word Problem
10) No. Siddiquah is not correct.
0
Possible explanation: Zero can be written in fractional form as _
x , where x stands for any number
other than 0 because zero divided by any number other than zero is equal to zero. When zero is the
denominator (or divisor) the answer is undefined.
© 2015 Success for All Foundation
Foundation
Math
3rd Edition
PowerTeaching Math PowerTeaching
3rd Edition | Unit
Guide
51
Level 8 | Unit 2: Rational and Irrational Numbers | Cycle 1 Lesson
Level 83 | Unit 2: Rational and Irrational Numbers | Cycle 1 Lesson 3
teamwork
1) Find the fractional equivalent of 0.234
. Show your work.
2) Find the fractional equivalent of 0.91
.  Show your work.
3) Find the fractional equivalent of 0.4
.  Show your work.
4) Find the fractional equivalent of 0.5
.  Show your work.
5) Find the fractional equivalent of 0.21
.  Show your work.
6) Find the fractional equivalent of 0.37
.  Show your work.
7) Find the fractional equivalent of 0.719
. Show your work.
8) Find the fractional equivalent of 0.372
. Show your work.
Challenge
9) Find the fractional equivalent of 1.2345
 . Show your work.
©
for All Foundation
522015 Success
PowerTeaching
Math 3rd Edition | Unit Guide
Math
3rd Edition
©PowerTeaching
2015 Success for
All Foundation
Level
8 | Unit 2: Rational and Irrational Numbers | Cycle 1 Lesson
Level 8 3 | Unit 2: Rational and Irrational Numbers | Cycle 1 Lesson 3
teamwork answers
234
78
26
1) _
or _
or _
999
333
111
91
2) _
99
4
3) _
9
5
4) _
9
21
7
5) _
or _
99
33
37
6) _
99
719
7) _
999
372
124
8) _
or _
999
333
12,222
2,322
1,161
387
129
9) __ or 1 _ or 1 _ or 1 _
or 1 _
9,900
9,900
PowerTeaching
Math
Edition
© 2015 Success for
All3rd
Foundation
4,940
1,650
550
2015
Success
for Guide
All Foundation
PowerTeaching Math©3rd
Edition
| Unit
53
Level 8 | Unit 2: Rational and Irrational Numbers | Cycle 1 Lesson 3 quick check
Level X | Unit X: UnitTitle | Cycle X Lesson X
Name
Find the fractional equivalent of 0.426
. Show your work.
© 2015 Success for All Foundation
quick check
PowerTeaching Math 3rd Edition
Name
Find the fractional equivalent of 0.426
. Show your work.
©
for All Foundation
542015 Success
PowerTeaching
Math 3rd Edition | Unit Guide
Math
3rd Edition
©PowerTeaching
2015 Success for
All Foundation
Level 8
8 || Unit 2: Rational and Irrational Numbers
Numbers || Cycle 1
1 Lesson
Lesson 3
3
homework
Quick Look
Vocabulary words introduced in this cycle:
rational number, radical sign, square root, perfect square, irrational number
Today we explored why repeating decimals are rational numbers and how to convert them from repeating
decimals to their fractional equivalents.
For example:
To find the fractional equivalent of 0.158
 ,
x 5 0.158
 
1,000x 5 158.158
 
1,000x 5 158.158
 
2x5
0 .158
 
Multiply by a power of 10.
Subtract.
999x 5 158
158
x5_
999
Divide to isolate the variable.
1) Find the fractional equivalent of 0.57
.  Show your work.
2) Find the fractional equivalent of 0.238
. Show your work.
3) Find the fractional equivalent of 0.63
.  Show your work. Explain your thinking.
4) Find the fractional equivalent of 0.7
.  Show your work.
5) Find the fractional equivalent of 0.374
. Show your work.
Mixed Practice
6) Simplify the expression: 3t  4t  5t.
7) Evaluate the expression.
49442528
8) Classify 
120  as rational or irrational.
9) Find the surface area of a cube whose side length is 1.85 in.
Word Problem
10) In your own words, describe how to convert a repeating decimal into a fraction.
© 2015 Success for All Foundation
Foundation
Math
3rd Edition
PowerTeaching Math PowerTeaching
3rd Edition | Unit
Guide
55
Level 8
8 || Unit 2: Rational and Irrational Numbers
Numbers || Cycle 1
1 Lesson
Lesson 3 3
Homework
For the Guide on the Side
Today your student confirmed that repeating decimals are rational numbers because they can be written in
fractional form. To develop a clear concept of the differences between rational and irrational numbers, your
student will examine, compare, estimate, and order numbers and use his or her number sense.
In the next lesson, your student will approximate the decimal value of irrational numbers.
Your student should be able to answer the following questions about classifying numbers:
1) How did you write this repeating decimal as a fraction?
2) How do you know repeating decimals are rational numbers?
3) Why do you multiply each side by a multiple of 10?
Here are some ideas to practice classifying numbers:
1) Learn Zillion: Convert repeating decimals to fractions:
http://learnzillion.com/lessons/223‑convert‑repeating‑decimals‑into‑fractions
2) Virtual Nerd: How do you turn a repeating decimal into a fraction?:
http://virtualnerd.com/pre‑algebra/rational‑numbers/repeating‑decimal‑to‑fraction‑conversion.php
PowerTeaching
Math 3rdMath
Edition
56
PowerTeaching
3rd Edition | Unit Guide
© 2015 Success for All Foundation
Level 8
8 || Unit 2: Rational and Irrational Numbers
Numbers || Cycle 1
1 Lesson
Lesson 3
3
homework answers
57
19
1) _
or _
99
33
238
2) _
999
63
7
3) _
or _
99
11
Possible explanation: I started by expressing the repeating decimal as an equation. Then, I multiplied
each side of the equation by 100 because two digits repeat. This moved the set of repeating digits
to the left side of the decimal point. Next, I subtracted x from 100x to get rid of the decimal. Finally, I
63
isolated the variable and solved for x. So 0.63
 is equivalent to _. I used what I knew about converting a
99
terminating decimal into an equation to convert a repeating decimal into an equation (TLM #1).
7
4) _
9
374
5) _
999
Mixed Practice
6) 12t
7)
– 6.75
8) irrational
9) 20.535 in.
2
Word Problem
10) Accept reasonable answers.
© 2015 Success for All Foundation
Foundation
Math
3rd Edition
PowerTeaching Math PowerTeaching
3rd Edition | Unit
Guide
57
Level 8 | Unit 2: Rational and Irrational Numbers | Cycle 1 Lesson
Level 84 | Unit 2: Rational and Irrational Numbers | Cycle 1 Lesson 4
teamwork
1) Graph each number on the number line below.
9
2.55, _
, 24 , , 9
5
–5
–4
–3
–2
–1
0
1
2
3
4
5
2) Use , , or to compare the following numbers.
55 _______ 7.874007874…
3) Write the following numbers in order from least to greatest. Explain your thinking.
9,
–_
6
6
8.9, – 12 , _
, 7 , – 8.3
3
4) Write the following numbers in order from greatest to least.
– 2.3,
23
1
0, _
, , – 7 _
7
5
5) Graph each number on the number line below.
10
_
, 10 , 4.75, 16
4
6) Use , , or to compare the following numbers.
40 _______ 5.916079783…
7) Write the following numbers in order from least to greatest.
3,
–_
9
9
6.6, – 54 , _
, 75 , 5
8) Write the following numbers in order from greatest to least.
9
9
2
5
34 , – 4.2, 9.6, _, – 85 , – 7 _
9) Graph each number on the number line below.
24
3
85 , _, 8.85, 100
10) Use , , or to compare the following numbers.
99 ________ 9.38083152…
Challenge
11) Find five irrational numbers and five rational numbers between 5 and 6. Explain your thinking.
©
for All Foundation
582015 Success
PowerTeaching
Math 3rd Edition | Unit Guide
Math
3rd Edition
©PowerTeaching
2015 Success for
All Foundation
Level
8 | Unit 2: Rational and Irrational Numbers | Cycle 1 Lesson
Level 8 4 | Unit 2: Rational and Irrational Numbers | Cycle 1 Lesson 4
teamwork answers
1)
2)
3)
55  , 7.874007874…

– 8.3, –
6
6 3
9 , _,  7 ,  8.9
12 ,  – _


23
1
4) _
, p, 0, – 2.3, – 7 _
5
3
5)
6)
7)
40  . 5.916079783…

–
9
9 5
3 , _, p, 6.6,  75  
54  , – _


9 –
2 –
8) 9.6, 
34 ,  _
, 4.2, – 7 _
, 
85  
9
5
9)
10) 
99  . 9.38083152…
11) Possible irrational answers: 
26 ,  
27 ,  
28 ,  
29 ,  
30 ,  
31 ,  
32 ,  
33 ,  
34 ,  
35  
26
339
987
2
Possible rational answers: 5.1, _
, 5.313, 5 _
, 5.58, 5 _
, 5.7, 5.81, 5 _
5
5
500
1,000
Possible explanation: 5 is the square root of 25 while 6 is the square root of 36. Therefore, I know
that I’m looking for square roots between 25 and 36. None of these numbers are perfect squares, so
their square roots are irrational.
PowerTeaching
Math
Edition
© 2015 Success for
All3rd
Foundation
2015
Success
for Guide
All Foundation
PowerTeaching Math©3rd
Edition
| Unit
59
Level 8 | Unit 2: Rational and Irrational Numbers | Cycle 1 Lesson
Level 84 | Unit 2: Rational and Irrational Numbers | Cycle 1 Lesson 4
quick check
Name
Use ,, ., or 5 to compare the following numbers.
6.1678203027… _____ 
32  
© 2015 Success for All Foundation
quick check
PowerTeaching Math 3rd Edition
Name
Use ,, ., or 5 to compare the following numbers.
6.1678203027… _____ 
32  
©
for All Foundation
602015 Success
PowerTeaching
Math 3rd Edition | Unit Guide
Math
3rd Edition
©PowerTeaching
2015 Success for
All Foundation
Level 8
8 || Unit 2: Rational and Irrational Numbers
Numbers || Cycle 1
1 Lesson
Lesson 4
4
homework
Quick Look
Vocabulary words introduced in this cycle:
rational number, radical sign, square root, perfect square, irrational number
Today we arranged rational and irrational numbers in order from least to greatest and/or from greatest to
least. We also compared numbers, determining which value was the least and which was the greatest. We
used a number line to approximate an irrational number’s location.
For example:
9
We can compare _
to 
9  by approximating each numbers location on a number line.
5
9
So _
. 
9 . 
5
1) Graph each number on the number line below.
17
5.5, 
40 ,  _
, 
55  
4
2) Graph each number on the number line below.
78
7.25, 
45 ,  _
, 
70  
8
3) Use ,, ., or 5 to compare the following numbers.
– 10
____ – 
104  
4) Use ,, ., or 5 to compare the following numbers.
33  _____ 5.19615242…

5) Write the following numbers in order from least to greatest. Explain your thinking.
9 –
5
_
, 
9 ,  _
, 
24 ,  p
6
5 5
1,
–_
© 2015 Success for All Foundation
Foundation
Math
3rd Edition
PowerTeaching Math PowerTeaching
3rd Edition | Unit
Guide
61
Level 8
8 || Unit 2: Rational and Irrational Numbers
Numbers || Cycle 1
1 Lesson
Lesson 4 4
Homework
6) Write the following numbers in order from greatest to least.
– 6.3,
8
4
0, _
, 7.1, – 6 _
7
5
Mixed Practice
7) Find the unit rate.
15 books in 3 days
8)
– 68.86
 85.18
9) What is 75% of 96?
4
2
10) Multiply. 3 _
 8_
7
5
Word Problem
11) A local television station recorded the daily low temperature for a week in January.
Sunday
Monday
Tuesday
Wednesday
Thursday
Friday
Saturday
– 15C
– 16.7C
– 17.8C
– 21.1C
– 20.6C
– 18.9C
– 17.8C
Which temperature is the highest? Which is the lowest?
For the Guide on the Side
Today your student learned how to compare and order numbers and approximate their locations on a
number line. Your student worked with both rational and irrational numbers.
Your student should be able to answer the following questions about comparing and ordering numbers:
1) Is this number rational or irrational? How do you know?
2) Between which numbers does this square root fall?
3) How did you determine which number is greater/lesser?
4) How can you compare decimals to fractions?
5) How did you determine the order of this set of numbers?
Here are some ideas to practice comparing and ordering numbers:
1) Learn Zillion: Comparing Irrational and Rational Numbers:
http://learnzillion.com/lessons/222‑compare‑irrational‑and‑rational‑numbers
2) Virtual Nerd: How do you put real numbers in order?:
http://virtualnerd.com/common‑core/grade‑8/8_NS‑number‑system/A/2/order‑real‑numbers‑example
PowerTeaching
Math 3rdMath
Edition
62
PowerTeaching
3rd Edition | Unit Guide
© 2015 Success for All Foundation
Level 8
8 || Unit 2: Rational and Irrational Numbers
Numbers || Cycle 1
1 Lesson
Lesson 4
4
homework answers
1)
2)
3)
– 10
. – 
104  
4) 
33  . 5.19615242…
5)
–
5 9
5 6 5
1 , _, _, p, 
9 ,  – _
24  

Possible explanation: I started by converting the fractions and square roots into decimal form. I also
wrote the decimal approximation for p. Since I am ordering the numbers from least to greatest, I used a
number line to approximate each number’s location. My answer makes sense because as you go left, the
values decrease, and as you go right, the values increase.
8
4
6) 7.1, _
, 0, – 6.3, – 6 _
7
5
Mixed Practice
7) 5 books per day
8) 16.32
9) 72
10) 30
Word Problem
11) The highest temperature is – 15C while the lowest temperature is – 21.1C.
© 2015 Success for All Foundation
Foundation
Math
3rd Edition
PowerTeaching Math PowerTeaching
3rd Edition | Unit
Guide
63
Level 8 | Unit 2: Rational and Irrational Numbers | Cycle 1 Lesson
Level 85 | Unit 2: Rational and Irrational Numbers | Cycle 1 Lesson 5
teamwork
Directions: Decide which number is greater without using a calculator.
1) 4
20  or 2
40  
Explain your thinking.
2)
– 12
or – 8 
8 
3) 3
5  or 
88  
4)
–5
5  or – 25

Explain your thinking.
5) 2
30  or 3
20  
6)
–7
7  or – 27

7) 6
50  or 5
60  
Challenge
8) Without using the square or cube root button on your calculator, approximate the cubic root of 25 to the
nearest hundredth.
©
for All Foundation
642015 Success
PowerTeaching
Math 3rd Edition | Unit Guide
Math
3rd Edition
©PowerTeaching
2015 Success for
All Foundation
Level
8 | Unit 2: Rational and Irrational Numbers | Cycle 1 Lesson
Level 8 5 | Unit 2: Rational and Irrational Numbers | Cycle 1 Lesson 5
teamwork answers
1) 4
20  . 2
40  
Possible explanation:
Estimate:
4(4.5) or 2(6.4)
18 . 12.8
I made sense of the comparison by approximating the irrational numbers with rational numbers and then
using what I know about multiplication to estimate their values. Once I figured out that 
20  is about
halfway between 4 and 5, I could find 4 times that value (TLM #7).
2)
– 12
is greater.
3) 3
5  , 
88  
4)
–5
5  is greater.

Possible explanation:
Estimate:
– 5(2.25)
or – 25
– 11.25 or – 25
– 11.25 . – 25
I made sense of the comparison by approximating the irrational number with a rational number and then
using what I know about multiplication to estimate the value. Once I figured out that 
5  is about a
quarter of the way from 2 to 3, I could find – 5 times that value (TLM #7).
5) 2
30  , 3
20  
6)
–7
7  is greater.

7) 6
50  . 5
60  
8) 2.92
PowerTeaching
Math
Edition
© 2015 Success for
All3rd
Foundation
2015
Success
for Guide
All Foundation
PowerTeaching Math©3rd
Edition
| Unit
65
Level 8 | Unit 2: Rational and Irrational Numbers | Cycle 1 Lesson
Level 85 | Unit 2: Rational and Irrational Numbers | Cycle 1 Lesson 5
quick check
Name
Decide which number is greater without using a calculator.
3
12  or 3
11  
© 2015 Success for All Foundation
quick check
PowerTeaching Math 3rd Edition
Name
Decide which number is greater without using a calculator.
3
12  or 3
11  
©
for All Foundation
662015 Success
PowerTeaching
Math 3rd Edition | Unit Guide
Math
3rd Edition
©PowerTeaching
2015 Success for
All Foundation
Level 8
8 || Unit 2: Rational and Irrational Numbers
Numbers || Cycle 1
1 Lesson
Lesson 5
5
homework
Quick Look
Vocabulary words introduced in this cycle:
rational number, radical sign, square root, perfect square, irrational number
Today we estimated and compared the value of expressions that contained irrational numbers.
For example:
In each of these pairs, we were able to determine which number was greater without using a calculator.
Directions for questions 1–4: Determine which number is greater without using a calculator.
1) 7
80  or 8
70  
2) 3
50  or 5
30  
Explain your thinking.
3) 40 or 12
10  
4)
–1
30  or – 2

Mixed Practice
5) The ratio of boys to girls in grade 8 at Robinson Middle School is 5:7. If there are 600 students in
grade 8, how many are girls?
6) Find both square roots for 81.
7) Find the area of a square whose side is 8.82 centimeters.
8) Classify 
100  as: natural number, whole number, integer, rational number, irrational number. (Use all
that apply.)
Word Problem
9) A square‑shaped pool has an area of 234 square meters. Find the approximate length of each side of
the pool.
© 2015 Success for All Foundation
Foundation
Math
3rd Edition
PowerTeaching Math PowerTeaching
3rd Edition | Unit
Guide
67
Level 8
8 || Unit 2: Rational and Irrational Numbers
Numbers || Cycle 1
1 Lesson
Lesson 5 5
Homework
For the Guide on the Side
Today your student learned how to approximate the square root of irrational numbers. Your student used his
or her number sense to estimate a good rational approximation of the square root of an irrational number. He
or she started with the perfect squares that the number falls between. Next, he or she focused on the square
roots of the perfect squares to narrow the range of the approximation. Even though we cannot calculate an
exact rational equivalent for the square root of an irrational number (because the decimal does not terminate
or repeat), your student was able to calculate a good approximation.
In the next lesson, your student will compare and order rational and irrational numbers.
Your student should be able to answer the following questions about approximating irrational numbers:
1) What information do you need to solve this problem?
2) How did you approximate the square root of this number?
3) Which perfect squares does the square fall directly between?
4) How did you figure out which expression was greater?
Here are some ideas to practice approximating irrational numbers:
1) Khan Academy: Approximating Square Roots:
www.khanacademy.org/math/arithmetic/exponents‑radicals/radical‑radicals/v/approximating‑square‑roots
2) Learn Zillion: Estimate the value of square roots:
http://learnzillion.com/lessons/3136‑estimate‑the‑value‑of‑square‑roots
PowerTeaching
Math 3rdMath
Edition
68
PowerTeaching
3rd Edition | Unit Guide
© 2015 Success for All Foundation
Level 8
8 || Unit 2: Rational and Irrational Numbers
Numbers || Cycle 1
1 Lesson
Lesson 5
5
homework answers
1) 7
80  , 8
70  
2) 3
50  , 5
30  
Possible explanation:
Estimate:
3(7.1) or 5(5.5)
21.3 , 27.5
I made sense of the comparison by approximating the irrational numbers with rational numbers and then
using what I know about multiplication to estimate their values. Once I figured out that 
30  is about
halfway between 5 and 6, I could find 5 times that value. That’s TLM practice #7.
3) 40 is greater.
4)
–2
is greater.
Mixed Practice
5) There are 350 girls in grade 8.
6)
9
and – 9
2
7) 77.7924 cm
8) natural number, whole number, integer, rational number
Word Problem
9) The length of each side of the pool is approximately 15.29 meters.
© 2015 Success for All Foundation
Foundation
Math
3rd Edition
PowerTeaching Math PowerTeaching
3rd Edition | Unit
Guide
69
Level 8 | Unit 2: Rational and Irrational Numbers | Cycle 1 AssessmentLevel
Day 8 | Unit 2: Rational and Irrational Numbers Cycle 1
unit check
Directions for question 1 and 2: Find the square roots for each number.
1) 49
2) 81
Directions for question 3–5: Classify each number as rational or irrational.
3) 63.88
 
4) 
35  
5) 
36  
Directions for question 6: Classify the numbers by writing them in the appropriate section of the
Venn diagram.
6)
3
0, 
144 , 1.1, 
2 ,  _
, 
8 ,  p,
2
16 ,  – 64, 
128  

Directions for questions 7 and 8: Find the fraction equivalent of each number. Show your work.
7) 0.27
 
8) 0.11…
Directions for questions 9–13: On the number line, approximate the locations of the numbers.
9) 
53  
10) 
22  
11) 
74  
12) 
39  
13) 
89  
14) Determine which number is greater without using a calculator. Explain your thinking.
3
20  or 5
10  
©
for All Foundation
702015 Success
PowerTeaching
Math 3rd Edition | Unit Guide
Math
3rd Edition
©PowerTeaching
2015 Success for
All Foundation
Level
8 | Unit 2: Rational and Irrational Numbers Cycle
Level
1 8 | Unit 2: Rational and Irrational Numbers | Cycle 1 Assessment Day
unit check answers
Lesson 1: Define and explore rational and irrational numbers. [20 points]
1)
7
and – 7 [4 points]
2)
9
and – 9 [4 points]
3) rational [4 points]
4) irrational [4 points]
5) rational [4 points]
Lesson 2: Classify rational and irrational numbers. [20 points]
6)
[2 points each]
Lesson 3: Convert a decimal expansion that repeats eventually into a rational number. [20 points]
7)
3
0.27
 5 _
[5 points]
Possible work:
[5 points]
11
8)
1
0.11… 5 _
[5 points]
Possible work:
[5 points]
9
x 5 0.2727…
100x 5 27.2727…
x 5 0.11…
10x 5 1.11…
100x 5 27.2727…
2 x 5 0.2727…
2
99x 5 27
9x 5 1
3
27 5 _
x5_
99
10x 5 1.11…
x 5 0.11…
11
PowerTeaching
Math
Edition
© 2015 Success for
All3rd
Foundation
1
x5_
9
2015
Success
for Guide
All Foundation
PowerTeaching Math©3rd
Edition
| Unit
71
Level
Cycle 8 Check
| Unit
Answers
2: Rational and Irrational Numbers | Cycle 1 AssessmentLevel
Day 8 | Unit 2: Rational and Irrational Numbers Cycle 1
Lesson 4: Use knowledge of perfect squares and the number line to order rational and irrational numbers.
[20 points]
9–13) [4 points each]
Lesson 5: Use approximations of the value of irrational numbers to estimate and compare expressions
containing irrational numbers. [20 points]
14) 3
20  , 5
10  [10 points]
Possible explanation: [10 points]
Estimate:
3(4.5) or 5(3.1)
13 or 15.5
13 , 15.5
I made sense of the comparison by approximating the irrational numbers with a rational number and
then using what I know about multiplication to estimate the value. That’s TLM practice #7.
Prep Points Analysis
Question
Number
Team Scores
(out of 20 points)
Core Objective
2
Define and explore rational and irrational numbers.
6
Classify rational and irrational numbers.
7
Convert a decimal expansion that repeats eventually into a
rational number.
11
Use knowledge of perfect squares and the number line to
order rational and irrational numbers.
14
Use approximations of the value of irrational numbers
to estimate and compare expressions containing
irrational numbers.
©
for All Foundation
722015 Success
PowerTeaching
Math 3rd Edition | Unit Guide
1
2
3
4
5
Class Results
(check if 16 out
of 20 points
or better)
Math
3rd Edition
©PowerTeaching
2015 Success for
All Foundation