# RNS3 REAL NUMBER SYSTEM

```Name ___________________________
Period __________
Date ___________
RNS3
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STUDENT PAGES
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REAL NUMBER SYSTEM
Student Pages for Packet 3: Operations with Real Numbers
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RNS3.1 Rational Numbers
• Review concepts of experimental and theoretical probability.
1
a
b
Understand why all quotients of integers ( , b ≠ 0 ) can be written
•
as terminating or repeating decimals.
Learn to change a repeating decimal to an equivalent quotient of
integers.
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•
9
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RNS3.2 Irrational Numbers
• Recognize that there are an infinite number of rational numbers
represented on the real number line.
• Understand that there are real numbers that are not rational, which
are called irrational numbers.
• Explore some properties of irrational numbers.
15
RNS3.4 Vocabulary, Skill Builder, and Review
23
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RNS3.3 Computing with Real Numbers
• Simplify expressions that involve rational numbers.
• Simplify expressions that involve square roots.
• Make conjectures about the sums or products that result from
computing with rational and irrational numbers.
Real Number System (Student Packet)
RNS3 – SP
Operations with Real Numbers
WORD BANK (RNS3)
Definition or Explanation
Picture or Example
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Word or Phrase
integers
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irrational
numbers
natural numbers
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expression
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rational numbers
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N
real numbers
m
repeating
decimal
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terminating
decimal
whole numbers
Real Number System (Student Packet)
RNS3 – SP0
Operations with Real Numbers
3.1 Rational Numbers
RATIONAL NUMBERS
Set (Goals)
We will deepen our understanding of
rational numbers. We will learn that all
rational numbers have decimal expansions
that terminate or repeat. We will learn to
change repeating decimals into equivalent
quotients of integers.
• Review concepts of experimental and
theoretical probability.
• Understand why all quotients of integers
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a
b
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( , b ≠ 0 ) can be written as terminating
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Go (Warmup)
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or repeating decimals.
• Learn to change a repeating decimal to
an equivalent quotient of integers.
Describe each set of numbers using numbers and words.
Number Set
Numerical Description
N
1. Natural numbers
Verbal Description
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o
2. Whole numbers
3. Integers
m
a
A rational number is number that can be written as a quotient of integers  , b ≠ 0  .
b

Write the following as quotients of integers to justify that they are rational.
Sa
4.
0.7
5.
0.53
Real Number System (Student Packet)
6.
-2
7.
1
3
4
RNS3 – SP1
Operations with Real Numbers
3.1 Rational Numbers
TERMINATING AND REPEATING DECIMALS
Examples:
1
= 0.5000… = 0.5;
2
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A terminating decimal is a decimal whose digits are 0 from some point on. The final 0’s
in the expression for a terminating decimal are usually omitted.
3
= 0.75000… = 0.75
4
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2
2
=
0.181818...
= 0.18
Examples:
;
= 0.222222...
= 0.2
11
9
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A repeating decimal is a decimal that ends with repetitions of the same pattern of digits.
(A “repeat bar” can be placed above the digits that repeat.)
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Write each quotient of integers as an equivalent decimal. Use a calculator or scratch
paper if doing long division.
3
5
2.
5
8
3.
17
40
4.
2
3
5.
5
6
6.
4
9
7.
3
11
8.
4
11
9.
List the fractions above that can be represented by terminating decimals.
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1.
10. List the fractions above that can be represented by repeating decimals.
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11. List the fractions above that are NOT rational numbers. Explain.
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12. Predict the decimal values for
13. Predict the decimal values for
Real Number System (Student Packet)
5
6
and .
9
9
5
11
and
6
.
11
RNS3 – SP2
Operations with Real Numbers
3.1 Rational Numbers
ROLL A FRACTION: EXPERIMENTAL
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Terry and Robin are playing a game called “Roll a Fraction.” In this game, they roll two
six-sided number cubes labeled 1-6 and a fraction less than or equal to 1 is formed from
the values on the two number cubes. If the fraction results in a terminating decimal,
Terry gets a point If the fraction results in a repeating decimal, Robin gets a point.
Numbers
Rolled
Fraction
Formed
Winner
Trial #
Fraction
Formed
Winner
11
2
12
3
13
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1
4
14
5
15
6
16
17
N
7
8
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9
10
Numbers
Rolled
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Trial #
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1. With a partner, designate one player to be Terry the Terminator and the other player
to be Robin the Repeater. Then, roll the cubes 20 times with your partner and record
the results in the table.
18
19
20
My Pair’s Game Data (do this now)
Number
of Wins
Proportion Percentage
of Wins
of Wins
Class’ Game Data (do this later)
Number
of Wins
Proportion Percentage
of Wins
of Wins
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Robin
(repeating)
Terry
(terminating)
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2. Based on “My Pair’s Game Data” results, which represents your experimental
probability, do you think this is a fair game? Explain.
3. If you rolled the cubes 1,000 times instead of 20, how many times would you expect
Terry to win?
Real Number System (Student Packet)
RNS3 – SP3
Operations with Real Numbers
3.1 Rational Numbers
ROLL A FRACTION – THEORETICAL
_______ Number Cube
1
3
4
1

→R
3
1

→T
1
2
6
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5
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_______ Number Cube
1
2
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1. Make an outcome grid to determine the theoretical probabilities of Terry the
Terminator winning and of Robin the Repeater winning.
4
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5
6
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2. Determine the theoretical probabilities of wins for Terry and Robin.
P (Terry wins) = __________ = __________ = __________
Decimal
o
Fraction
Percent
P (Robin wins) = __________ = __________ = __________
Decimal
Percent
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Fraction
3. Based on the theoretical probabilities, do you think that this is a fair game? Explain.
4. Based on the theoretical probabilities, out of 1,000 rolls, how many times can we
expect Terry to win?
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5. Go back to “My Pair’s Game Data” on the previous page. How does this
experimental probability compare to the theoretical probability calculated on this
page? Explain.
6. Combine your individual data with others in the class and record on the previous
page to arrive at “Class’ Game Data”. How does this experimental probability
Real Number System (Student Packet)
RNS3 – SP4
Operations with Real Numbers
3.1 Rational Numbers
SOME FRACTION-DECIMAL EQUIVALENTS
1
3
1
7
1
4
1
5
1
9
1
8
1
6
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1
2
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1
11
1
12
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1
10
N
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1
1
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1. Write decimal equivalents for these unit fractions. Use any combination of your own
previous knowledge, number sense, or the long division algorithm.
Sa
2. Are there any unit fractions above whose decimal expansions do not terminate or
repeat? Explain.
Real Number System (Student Packet)
RNS3 – SP5
Operations with Real Numbers
3.1 Rational Numbers
EXPLORING REPEATING DECIMALS
that
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1. From the previous page you found
1
7 = __________________.
3
7
4
7
5
7
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2
7
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division to change one of these
fractions into an equivalent decimal.
3
=
7
4
=
7
5
=
7
N
2
=
7
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2. Compare decimal expansions with
decimal expansions here.
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3. What do you notice about the sequence of digits for the decimal expansions for 7ths?
4. Predict the decimal expansion for
6
. Check your prediction with a calculator.
7
6
=
7
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5. Using the long division work as an example, explain why decimal expansions for
7ths must repeat from some point on.
Real Number System (Student Packet)
RNS3 – SP6
Operations with Real Numbers
3.1 Rational Numbers
EXPLORING REPEATING DECIMALS (continued)
6. From a previous page you found that
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1
= ________________________.
12
3
12
4
12
5
12
2
=
12
3
=
12
4
=
12
5
=
12
6
will have an equivalent decimal expansion that repeats from some
12
N
8. Do you think
o
point on? Explain.
7
will have an equivalent decimal expansion that repeats from some
12
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9. Do you think
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7. Compare decimal expansions with
decimal expansions here.
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2
12
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division to change one of these
fractions into an equivalent decimal.
point on? Explain.
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10. Choose one example of a decimal expansion for 12ths that repeats. Without
performing long division forever, how do you know it must repeat?
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11. Do you think any quotient of integers whose decimal expansions does not terminate
must repeat from some point on? Use the long division process to help you explain.
Real Number System (Student Packet)
RNS3 – SP7
Operations with Real Numbers
3.1 Rational Numbers
A CLEVER PROCEDURE
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The following algebraic process is used to change a repeating decimal to a quotient of
integers.
Change 0.16 = 0.166666... . to a quotient of integers.
10 x = 1.66666...
Let x = 0.16666...
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(2)
(1)
9 x = 1.5
=
x
(3)
1.5 15 1
= =
9 90 6
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Notice that step 2 is above step 1.
• The “trick” is to multiply both sides of the equation
in step 1 by a power of 10 that will “line up” the
repeating portion of the decimal.
• Subtract the expressions in step 1 from step 2.
This results in a terminating decimal (step 3).
• Solve for x and simplify your result into a quotient
of integers (step 4).
(4)
1.
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Use this clever procedure to change each repeating decimal to a quotient of integers.
0.4 = 0.44444...
2.
____ digit(s) repeat(s), so multiply by 10.
Let
x =
0.44444…
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3.
100x =
Let
x =
0.272727…
(_____) • x =
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9x =
x =
____ digit(s) repeat(s), so multiply by ____
N
10x =
0.27 = 0.272727...
=
x
=
9
1.232323…
0.345
____ digit(s) repeat(s), so multiply by ____
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____ digit(s) repeat(s), so multiply by ____
4.
Real Number System (Student Packet)
RNS3 – SP8
Operations with Real Numbers
3.2 Irrational Numbers
IRRATIONAL NUMBERS
Set (Goals)
•
•
Recognize that there are an infinite
number of rational numbers
represented on the real number line.
Understand that there are real numbers
that are not rational, which are called
irrational numbers.
Explore some properties of irrational
numbers.
Go (Warmup)
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•
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We will learn that there are real numbers
that are not rational, and that these are the
irrational numbers. We will identify some
some famous irrational numbers.
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1. Approximate the placement of the following numbers on the number line. Label the
locations with the capital letters.
(Hint: 0.5 = 0.50 = 0.500 and 0.51 = 0.510; and you know how to count from 1 to 10)
(B) 0.51
(C) 0.505
(D) 0.502
(E) 0.5025
(F) 0.5050505…
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0.5
N
(A) 0.5
0.51
Change each decimal to an equivalent quotient of integers.
0.51
3.
0.505
4.
0.50505050…
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2.
5. Did all numbers in problem 1 fit on the number line? Explain.
Real Number System (Student Packet)
RNS3 – SP9
Operations with Real Numbers
3.2 Irrational Numbers
RATIONAL NUMBERS ON THE NUMBER LINE
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Here is the number line from the previous page. Approximate the placement of points A
and E on the line.
0.5
1. Complete the chart, and place more numbers on the line.
Label the point
E
G
A and
G
H
A and
H
A and
J
A and
K
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A and
Write this rational number
as a decimal
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Any rational number
between points
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0.51
J
N
K
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M
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2. Do you think it is always possible to find a rational number between two rational
numbers? Explain.
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3. Do you think you could list all the rational numbers between point A and point M?
Explain.
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4. Do you think that the rational numbers will fill up the number line without any spaces
or “holes?” In other words, do you think that every point on the line corresponds to a
quotient of integers?
Real Number System (Student Packet)
RNS3 – SP10
Operations with Real Numbers
3.2 Irrational Numbers
IRRATIONAL NUMBERS ON THE NUMBER LINE
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Every rational number has a decimal expansion or “address” and can be represented on
the number line. Here is a magnified version of a portion of the line on the previous
page. Circle this portion on the previous page. Label the rational numbers on the hash
marks.
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(Hint: 0.501= 0.5010 and 0.502=0.5020; and you know how to count from 10 to 20)
0.501
0.502
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1. For the number 0.5010010001…., the decimal expansion follows a pattern, but this
pattern does not repeat. Estimate the location of this number on the number line as
precisely as possible, and label this location N.
N
2. Write another number below that has a pattern but does not repeat (like the
number at N in problem1). Then estimate its location on the number line and
label it P. Be sure to write a number that will fit on this number line.
P: __________________________
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3. Why can the numbers in problems 1-2 NOT be written as quotients of integers? In
other words, why are they NOT rational? (Hint: show that the clever procedure from
the following lesson will not work.)
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The above problems indicate that there are also “addresses” on the number line for
decimal numbers like those at N and P that are not rational. Every point on the number
line that does not represent a rational number represents an irrational number.
The Real Number System
Together the rational numbers and irrational numbers make up the real numbers. Each real
number has a location on the real number line. Every point on the line has a decimal name
Real Number System (Student Packet)
RNS3 – SP11
Operations with Real Numbers
3.2 Irrational Numbers
A FAMOUS IRRATIONAL NUMBER
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Many civilizations over the centuries have observed that the ratio of the circumference to the
diameter of a circle is constant. For example, the Romans observed that the number of paces
around the outer portion of their circular temples was about three times the number of paces
through the center. In mathematics, the Greek letter π (pronounced “pi”) is used to represent
this ratio.
2. Greek:
between
3. Hindu:
3,927
1,250
4. Roman:
377
120
5. Chinese:
355
113
22
223
and
7
71
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25
8
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256
81
Use a calculator to find decimal
approximations for π
(to the nearest ten-thousandth)
N
1. Egyptian:
π
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Fraction used as approximation for
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There is no quotient of integers that represents the exact ratio of a circle’s circumference to its
diameter, or π . Therefore, it is an irrational number. Here are some rational approximations
used by different civilizations over the ages.
6. Babylonian:
The decimal approximation of
π , correct to seven decimal places, is 3.1415926.
7. Round this decimal approximation to the nearest ten-thousandth.
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8. Write the number in problem 7 in words
9. Which civilization(s) had the closest approximation(s) for
Real Number System (Student Packet)
π?
RNS3 – SP12
Operations with Real Numbers
3.2 Irrational Numbers
ANOTHER IRRATIONAL NUMBER
1. Use your calculator to try to find a
decimal such that
If
n = 2
Then n2 = 2
2. Were you able to find an exact value of n
that will satisfy the equation n2 = 2?
Explain.
1
1
low
2
4
high
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Find n
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Is n2
too high or
too low?
Estimate a
decimal
value for n
2
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You are probably already familiar with other irrational numbers. One example is 2 .
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2 is an irrational number because it
cannot be written as a quotient of two
integers.
1.5
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For problems 3 and 4, one unit of length is
defined as illustrated on the right triangle as
well as the number line.
Real Number System (Student Packet)
3. Use the Pythagorean Theorem to find the
hypotenuse of an isosceles right triangle
with leg = 1.
1
1
4. How might you use the diagram above to
estimate a location for 2 on this number
line?
0
1
2
RNS3 – SP13
Operations with Real Numbers
3.2 Irrational Numbers
THE REAL NUMBER SYSTEM
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Label the sorting diagram below to show the relationship between the various
subsets of the real number system: the natural numbers (N), whole numbers (W),
integers (Z), rational numbers (Q), irrational numbers (IR), and real numbers (R).
0.12122122212222…
1, 2, 3, …
1.37562987189673…
π
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0
-1,
-2,
-3,
…
N
1. Label the hash marks on the number line.
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-5
2. Complete the table and locate each point on the number line.
Find
a whole number
that is not a natural number
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a rational number
between 1 and 2
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a rational number
between -5 and -4
Label the point(s)
P
Q
R
an irrational number between
3 and 4
V
an irrational number between
-3 and -2
W
Real Number System (Student Packet)
Write the number(s)
RNS3 – SP14
Operations with Real Numbers
3.3 Computing with Real Numbers
COMPUTING WITH REAL NUMBERS
Set (Goals)
•
•
Simplify expressions that involve
rational numbers.
Simplify expressions that contain
square roots.
Make conjectures about the sums or
products that result from computing
with rational numbers and irrational
numbers.
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Go (Warmup)
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•
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We will review techniques for computing
with integers and rational numbers. We will
learn some techniques for simplifying
expressions that involve square roots. We
will explore the closure property for
subsets of real numbers under addition,
subtraction, multiplication, or division.
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1. What is the set of whole numbers?_____________________________________
2. What is the set of integers? ____________________________________________
-12 + 14 – 26
Is result an integer?
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Is result an integer?
4. (-31)(-3)
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3.
N
Each expression below consists of integers. Simplify each expression.
45
6
6.
Is result an integer?
(-6)2
m
9.
Is result an integer?
Sa
12.
4 – (-23)
Is result an integer?
Real Number System (Student Packet)
7.
-3 + 3
Is result an integer?
10.
-62
Is result an integer?
13.
−8 + 24
2
Is result an integer?
5.
45
−3
Is result an integer?
8.
−4
20
Is result an integer?
11.
14 – 25
Is result an integer?
14.
1+ 3( 2)
9 − ( −5 )
Is result an integer?
RNS3 – SP15
Operations with Real Numbers
3.3 Computing with Real Numbers
THE CLOSURE PROPERTY
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A set of numbers is said to be closed, or to have the closure property, under a given
operation (such as addition, subtraction, multiplication, or division), if the result of this
operation on any numbers in the set is also a number in that set.
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Answer each question. If the answer is “yes”, then explain what the statement means in
your own words and give an example. If the answer is “no”, then, give a counterexample.
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1. Is the set of whole numbers closed under
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b. Subtraction?
c. Multiplication?
N
d. Division?
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2. Is the set of integers closed under
b. Subtraction?
c. Multiplication?
m
d. Division?
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3. Under what condition is half of an integer an integer? Support your conjecture with
examples.
Real Number System (Student Packet)
RNS3 – SP16
Operations with Real Numbers
3.3 Computing with Real Numbers
OPERATIONS WITH RATIONAL NUMBERS
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1. What is the set of rational numbers?_____________________________________
Each expression below consists of rational numbers. Simplify each expression.
3.
Is result a rational number?
6.
Is result a rational number?
Is result a rational number?
2
9.
2
−  
5
1 18

2
-5
2
Is result a rational number?
10.
−5 + 14
2
Is result a rational number?
Is result a rational number?
o
Is result a rational number?
Is result a rational number?
7.
N
 1
2
 
8.
2 -1
÷
3 3
1
of (-25)
2
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2
÷3
3
5.
4.
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Is result a rational number?
-1 5
−
3 6
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-2 5
+
3 3
2.
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Answer each question. If the answer is “yes”, then explain what the statement means in
your own words and give an example. If the answer is “no”, explain the statement and
give a counterexample.
11. Is the set of rational numbers closed under
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b. multiplication and division?
12. Is half of a rational number always a rational number? Support your conjecture with
examples.
Real Number System (Student Packet)
RNS3 – SP17
Operations with Real Numbers
3.3 Computing with Real Numbers
OPERATIONS WITH IRRATIONAL NUMBERS 1
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1. What is the set of irrational numbers? _____________________________
Irrational numbers may occur as expressions containing square roots. Often we rewrite
the expression to preserve its exact value using the multiplication property of square
roots.
For every number a ≥ 0 and b ≥ 0 ,
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Multiplication Property of Square Roots
ab = a b
( 7)
=
25
=
( )
2
= ____
3
2
3
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=
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Rewrite each expression by removing any perfect squares from under the radical sign.
2.
3.
4.
75 =
25  3
50
49 = 7 
= _________
Is result rational or irrational?
Is result rational or irrational?
Is result rational or irrational?
5.
6.
7.
− 80
16 + 9
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2 • 200
Is result rational or irrational?
Is result rational or irrational?
Is result rational or irrational?
8.
9.
10.
m
5 • 20
Sa
Is result rational or irrational?
( −6) 2 − 4(1)(7)
Is result rational or irrational?
− 4+9
Is result rational or irrational?
11. Do you think that the irrational numbers are closed under multiplication? Explain.
Real Number System (Student Packet)
RNS3 – SP18
Operations with Real Numbers
3.3 Computing with Real Numbers
OPERATIONS WITH IRRATIONAL NUMBERS 2
Division Property of Square Roots
a
a
=
For every number a ≥ 0 and b > 0 ,
b
b
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Here is another property commonly used to rewrite square roots.
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=
=
• 2
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Rewrite each expression by removing any perfect squares from under the radical sign.
Note whether the result is a rational or an irrational number.
1.
2.
3.
88
− 6
•
• 2
18
=
=
11
3
25
•
Is result rational or irrational?
Is result rational or irrational?
Is result rational or irrational?
4.
5.
6.
44
48
75
N
−
121
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9
120
Is result rational or irrational?
Is result rational or irrational?
Is result rational or irrational?
7.
8.
9.
3 12
m
9
Is result rational or irrational?
13
49
Is result rational or irrational?
16 − 421
8
Is result rational or irrational?
Sa
10. Do you think that the irrational numbers are closed under division? Explain.
Real Number System (Student Packet)
RNS3 – SP19
Operations with Real Numbers
3.3 Computing with Real Numbers
GENERALIZING RULES
Simplify using properties of square roots.
1.
2.
7 8 6
7 3 6
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3.
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73 6
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4. In your own words, generalize a rule for multiplying expressions that contain rational
numbers and irrational numbers in the form of square roots.
Rewrite using properties of square roots
5.
6.
6
N
4 6
2
4 6
2
8.
6
2
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2
7.
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9. In your own words, generalize a rule for dividing expressions that contain rational
numbers and irrational numbers in the form of square roots.
Real Number System (Student Packet)
RNS3 – SP20
Operations with Real Numbers
3.3 Computing with Real Numbers
EVALUATING EXPRESSIONS WITH ROOTS
a = 1, b = 6, c = 8
2.
a = 2, b = 9, c = -5
3.
4.
a = 1, b = 6, c = 7
5.
a = 1, b = 2, c = 1
6.
a = 3, b = 4, c = 3
a = 2, b = -5, c = -8
ot
R
ep
r
1.
uc
e
b 2 − 4ac if:
od
Evaluate the expression
a = 2, b = 12, c = 10
8.
a = 1, b = -6, c = -7
9.
a = 1, b = 0, c = -9
12.
a = 6, b = 12, c = 0
pl
e:
D
o
7.
N
− b 2 − 4ac
if:
Evaluate the expression
2a
a = 2, b = -1, c = -4
11.
a = 5, b = 2, c = -2
Sa
m
10.
Real Number System (Student Packet)
RNS3 – SP21
Operations with Real Numbers
3.3 Computing with Real Numbers
RATIONALIZING THE DENOMINATOR
An expression containing square roots is in “simplest radical form” when:
• The radicand has no perfect square factors other than 1.
• The radicand has no fractions.
• The denominator of the fraction does not contain a radical.
uc
e
2
7
in
od
Use the process called “rationalizing the denominator” to write the fraction
2
7
• 7
7
7
7
and write in simplest radical form →
ot
R
2. Multiply the fraction by
ep
r
1. Is the denominator rational or irrational? ______________________________
=
N
What property of arithmetic is being used here? _____________________________
3. Is the new denominator rational or irrational? _______________________________
pl
e:
D
o
Write each expression in simplest radical form. If it already is, then circle the
expression.
4.
5.
6.
2
5 11
5
5
Sa
m
7.
13
4
Real Number System (Student Packet)
18
8.
20
9.
5 30
6
18
RNS3 – SP22
Operations with Real Numbers
3.4 Vocabulary, Skill Builders, and Review
FOCUS ON VOCABULARY (RNS3)
b. rational number (Q)
c. irrational number
d. real number (R)
e. natural number (N)
f. repeating decimal
h. terminating decimal
i.
j.
2.
25
whole number (W)
3.
ot
R
1.
ep
r
od
a. Integer (Z)
0
5.
0.101010101…
6.
25
0.25
pl
e:
D
o
0.101001000100001…
N
4.
uc
e
Choose words from this list. Match each number below with all words that could be
used to describe it.
7.
8.
“25” in
25
11.
m
10.
3
11
11
2
12.
0.33
0.3
Sa
- 25
9.
Real Number System (Student Packet)
RNS3 – SP23
Operations with Real Numbers
3.4 Vocabulary, Skill Builders, and Review
 1
-  ÷
 6
 2
- 
 3
3.
5
6
1
3
Solve for x:
-x + 5 = -2(x – 6) +
5.
Solve for x in terms of y:
3x – 2y = 15
N
ot
R
4.
ep
r
od
Compute with rational numbers.
3 1  1
1.
+ •  -1 
2.
4 2  4
uc
e
SKILL BUILDER 1
6.
7.
x is less than -3
x is greater than or equal to 2
the opposite of x is less
than or equal to -3
m
8.
pl
e:
D
o
Write each statement using symbols. If the variable is on the right side, change it to the left
side using appropriate properties. Then graph each.
1 is greater than x
Sa
9.
10.
-2 is less than the opposite of x
Real Number System (Student Packet)
RNS3 – SP24
Operations with Real Numbers
3.4 Vocabulary, Skill Builders, and Review
1
2
x3
2.
x2
3.
x1
5.
x-1
6.
x-2
7.
x-3
4.
x0
8.
x-4
ep
r
1.
od
Evaluate each expression if x =
uc
e
SKILL BUILDER 2
Why only one has a value equal to 1.
ot
R
9.
10. Why three have a value less than 1.
12.
o
N
11. Why four have a value greater than 1.
Given two points on a line, (-3, -6) and (3, -2):
b. Find the x-intercept.
c. Find the slope.
d. Write the equation of
the line in slopeintercept form.
Sa
m
pl
e:
D
a. Find the y-intercept.
Real Number System (Student Packet)
RNS3 – SP25
Operations with Real Numbers
3.4 Vocabulary, Skill Builders, and Review
SKILLBUILDER 3
Compute with rational numbers.
24 – 52 +
2.
36
-3
5.
Rewrite each number as indicated.
192,800
8.
6.
N
Product of a number between 1
and 10, and a multiple of 10
pl
e:
D
7.
-8 - 2(4 - 6)
4 - 23
3.
 5
 6
 1
  3
  5
 -8 
• 1 
 

  4
-101 – 99 + 4(-25)
Scientific notation
o
Number
 1  1 1
-  ÷  + 
 6  8 4
uc
e
 3
• 1 
 4
od
1 2
3 5
ep
r
4.
−
ot
R
1.
0.0007
Approximate the square roots.
m
Number
34
10.
58
Sa
9.
Between
square roots of
perfect squares:
Real Number System (Student Packet)
Between
two
consecutive
integers:
(as a
fraction):
(as a
decimal):
Calculator
check
(nearest
tenth):
RNS3 – SP26
Operations with Real Numbers
3.4 Vocabulary, Skill Builders, and Review
SKILLBUILDER 4
Simplify each expression. Leave results in exponent form (x ≠ 0).
35
105
3.
1.
2.
36
10 2
5.
(10 ) (10 )
1
7.
2
3
2
8.
uc
e
od
(5 )
3
x4
x4
6.
(4 )
3
2
2
9.
48
57
(x )
7
2
x3
pl
e:
D
x3 • x5
x10
o
N
10 2
5
ep
r
4.
(3)
ot
R
5 6 • 53
5 2 • 511
34
m
10.
11.
(2x )
5
(2x )
3
5
12.
x5
x3 • x 6
15.
18
50
Find the square roots.
Sa
13.
1
121
14.
Real Number System (Student Packet)
64
9
RNS3 – SP27
Operations with Real Numbers
3.4 Vocabulary, Skill Builders, and Review
Write using square root or cube root notation. Then compute.
92
2.
−27
1
1
3
3.
Write using exponent notation. Then compute.
169
3
125
6.
3
−216
ep
r
5.
Compute.
7.
9 + 16
10.
 20 

 +
 5 
25 + 144
1
8
9.
3
12.
2 16 + 3 4
15.
3
2
11.
pl
e:
D
4
81
o
N
8.
ot
R
4.
343 3
od
1
1.
uc
e
SKILL BUILDER 5
3
-1
14.
54
6
( 3 )( 3 )( 3 )
3
3
3
8
27
Sa
m
13.
4
Real Number System (Student Packet)
RNS3 – SP28
Operations with Real Numbers
3.4 Vocabulary, Skill Builders, and Review
SKILL BUILDER 6
Compute.
13
)
9+ 9
5.
9− 9
3
8.
64
9.
)(
64 7 − 64
)
3
−64
ep
r
7. 3 49
(7 +
6.
uc
e
)(
od
13
ot
R
(
4.
10. Write these numbers in order from least to greatest.
0.6
2
3
0.67
0.6666
. 0.6
N
0.7
_______,
o
_______,
pl
e:
D
_______,
_______,
_______,
_______
Write each rational number as a quotient of integers.
-7
12.
0.23
13.
1
1
3
14.
-4
3
4
m
11.
Sa
Evaluate if x = -3
12.
(x)2
Real Number System (Student Packet)
13. (x)-2
14. x3
RNS3 – SP29
Operations with Real Numbers
3.4 Vocabulary, Skill Builders, and Review
1. Write these numbers in order from least to greatest.
0.3
B.
0.3131131113…
C.
0.30
D.
E.
1
3
F.
0.3333
G.
0.3
H.
______
______
______
______
1
______
3
______
______
ep
r
______
0.33
od
A.
uc
e
SKILLBUILDER 7
ot
R
2. Which of the numbers above are terminating decimals?
3. Which of the numbers above are repeating decimals?
N
4. Which of the numbers above are rational numbers?
pl
e:
D
o
5. Which of the numbers above are irrational numbers?
Sa
m
6. Two trains start traveling in the same direction at the same time. The Red train
starts 100 miles west of the Yellow train. The Yellow train travels at a rate of 65
miles per hour. The Red train travels at a rate of 50 miles per hour. How long does
it take for the Yellow train to catch the Red train?
Real Number System (Student Packet)
RNS3 – SP30
Operations with Real Numbers
3.4 Vocabulary, Skill Builders, and Review
SKILL BUILDER 8
Fill in the blanks.
=
decimal
2. Are the fraction and decimal values at
the bottom of problem 1 equivalent?
Explain.
uc
e
2
3
______________
______________
fraction
od
=
decimal
ep
r
+
1
3
decimal
ot
R
1.
Use the clever procedure from Lesson 3.2 to change each repeating decimal into an
equivalent quotient of integers.
0.666…
4.
0.4666…
pl
e:
D
o
N
3.
1.30303030…
6.
0.99999…
Sa
m
5.
Real Number System (Student Packet)
RNS3 – SP31
Operations with Real Numbers
3.4 Vocabulary, Skill Builders, and Review
SKILLBUILDER 9
2
A. Place a natural number. (example above for A is 2)
C. Place an integer that is not a whole number.
ep
r
B. Place a whole number that is not a natural number.
od
A
uc
e
1. Use the letter next to each direction when placing the number on the number line.
D. Place a rational number that is not an integer
ot
R
E. Place a rational number that can be renamed as a terminating decimal.
F. Place a rational number that can be renamed as a repeating decimal.
G. Place two fractions between 0 and 1. Find another fraction between them.
N
H. Place an irrational number between 3 and 4.
o
3
to a decimal using a calculator and saw
13
0.2307692308
pl
e:
D
2. Roger changed
Since this number is rational, it must either terminate or repeat. Explain to Roger
Sa
m
3. In a triangle, the second side is one inch smaller than the first, and the third side is 2
inches larger than the first. The total perimeter of the triangle is 17.5 inches. Find the
length of the longest side.
Real Number System (Student Packet)
RNS3 – SP32
Operations with Real Numbers
3.4 Vocabulary, Skill Builders, and Review
SKILLBUILDER 10
uc
e
Write one example and one counterexample for each statement below. If no example or
counterexample is possible for a statement, write “not possible.”
One
One example
Statement
counterexample
(if possible)
(if possible)
od
1. The sum of two integers is an integer.
ot
R
3. The product of two integers is an integer.
ep
r
2. The difference of two integers is an integer.
4. The quotient of two integers is an integer.
N
5. The square root of an integer is an integer
o
6. One-fourth of an integer is an integer.
pl
e:
D
Simplify each expression by removing any perfect squares from under the radical sign.
Note whether the result is a rational or irrational number.
7.
32
m
Is result rational or irrational?
3 8
4
Sa
10.
Is result rational or irrational?
Real Number System (Student Packet)
8.
24
-2
Is result rational or irrational?
11.
4+9
Is result rational or irrational?
9.
3 • 12
Is result rational or irrational?
12.
81+19
Is result rational or irrational?
RNS3 – SP33
Operations with Real Numbers
3.4 Vocabulary, Skill Builders, and Review
SKILLBUILDER 11
Simplify using properties of square roots.
3 24
-6
3 24
-6
5.
3 27
36
3.
6.
6 • 18
uc
e
4.
2.
od
5 28
56
7
ot
R
ep
r
1.
b 2 − 4ac if:
Evaluate the expression -b +
a = 1, b = 4, c = 3
8.
a = 2, b = -5, c = -7
o
N
7.
pl
e:
D
Evaluate the expression -b –
a = 1, b = -2, c = -3
10.
a = 2 , b = 0, c = -6
m
9.
b 2 − 4ac if:
Sa
Write each expression in simplest radical form by rationalizing the denominator.
11.
3
3
Real Number System (Student Packet)
12.
2 3
20
RNS3 – SP34
Operations with Real Numbers
3.4 Vocabulary, Skill Builders, and Review
TEST PREPARATION (RNS3
uc
e
Show your work on a separate sheet of paper and choose the best answers.
1
.
6
1. Choose all phrases that apply to the number
is a rational number
B.
is an irrational number
C.
can be written as a terminating
decimal
D.
can be written as a repeating decimal
B.
All rational numbers are integers.
All integers are rational numbers.
C.
All irrational numbers are real
numbers
ot
R
A.
ep
r
2. Choose all statements that are true.
od
A.
D.
All integers are irrational numbers.
6•10
B.
6 • 10
C.
2 15
D.
120
2
D.
60
3
pl
e:
D
o
A.
N
3. Choose all expressions that are equivalent to 60 .
4. Choose all expressions that are equivalent to
A.
4 5
3
B.
40
3
2 20
.
3
4 15
3
C.
Sa
m
5. Which one statement disproves the following: “irrational numbers are closed under
multiplication.”?
A.
C.
9• 2 = 3 2
4 • 25 = 2• 5 = 10
Real Number System (Student Packet)
B.
3 • 12 = 3•12 =
D.
2• 7 =
36 = 6
2• 7 = 14
RNS3 – SP35
Operations with Real Numbers
3.4 Vocabulary, Skill Builders, and Review
KNOWLEDGE CHECK (RNS3)
3.1 Rational Numbers
1. State in your own words how you know whether a number is rational.
uc
e
2
5
b.
3
8
c.
5
6
d.
9
11
ep
r
a.
od
2. Write the decimal equivalent of each rational number below. Note whether it repeats
or terminates.
ot
R
3. Change 0.34 to an equivalent quotient of integers.
3.2 Irrational Numbers
N
4. Write two different irrational numbers with patterns in the decimal expansions, using
dots at the end (…).
pl
e:
D
o
5. Using a calculator, write 3 as a decimal to as many places as your calculator
shows, and explain why this number is irrational.
3.3 Computing with Real Numbers
Simplify using properties of square roots. Rationalize the denominator as needed.
5 • 10
7.
8 44
8.
54
6
9.
18
5
Sa
m
6.
Real Number System (Student Packet)
RNS3 – SP36
Operations with Real Numbers
HOME-SCHOOL CONNECTION (RNS1)
ep
r
od
uc
e
1. Explain what the set of real numbers is, including the difference between rational
and irrational numbers.
2. Change each decimal to an equivalent quotient of integers.
0.7
b.
0.7
N
ot
R
a.
8 • 20
b.
56
2 7
Sa
m
pl
e:
D
a.
o
3. Simplify the following using properties of square roots.
Parent (or Guardian) signature __________________________________________
Real Number System (Student Packet)
RNS3 – SP37
Operations with Real Numbers
COMMON CORE STATE STANDARDS – MATHEMATICS
STANDARDS FOR MATHEMATICAL CONTENT
Approximate the probability of a chance event by collecting data on the chance process that
produces it and observing its long-run relative frequency, and predict the approximate relative
frequency given the probability. For example, when rolling a number cube 600 times, predict that
a 3 or 6 would be rolled roughly 200 times, but probably not exactly200 times.
Develop a probability model and use it to find probabilities of events. Compare probabilities from
a model to observed frequencies; if the agreement is not good, explain possible sources of the
discrepancy: Develop a uniform probability model by assigning equal probability to all outcomes,
and use the model to determine probabilities of events. For example, if a student is selected at
random from a class, find the probability that Jane will be selected and the probability that a girl
will be selected.
Find probabilities of compound events using organized lists, tables, tree diagrams, and
simulation: Represent sample spaces for compound events using methods such as organized
lists, tables and tree diagrams. For an event described in everyday language (e.g., “rolling
double sixes”), identify the outcomes in the sample space which compose the event.
uc
e
7.SP.6
od
7.SP.7a
ep
r
7.SP.8b
Know that numbers that are not rational are called irrational. Understand informally that every
number has a decimal expansion; for rational numbers show that the decimal expansion repeats
eventually, and convert a decimal expansion which repeats eventually into a rational number.
Use rational approximations of irrational numbers to compare the size of irrational numbers,
locate them approximately on a number line diagram, and estimate the value of expressions
(e.g., π 2 ). For example, by truncating the decimal expansion of 2 , show that 2 is between 1
and 2, then between 1.4 and 1.5, and explain how to continue on to get better approximations.
2
Use square root and cube root symbols to represent solutions to equations of the form x = p
3
and x = p, where p is a positive rational number. Evaluate square roots of small perfect squares
and cube roots of small perfect cubes. Know that √2 is irrational.
Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in realworld and mathematical problems in two and three dimensions.
ot
R
8.NS.1
8.NS.2
N
8.EE.2
8.G.7
Explain why the sum or product of two rational numbers is rational; that the sum of a rational
number and an irrational number is irrational; and that the product of a nonzero rational number
and an irrational number is irrational.
pl
e:
D
o
N-RN-3
STANDARDS FOR MATHEMATICAL PRACTICE
Make sense of problems and persevere in solving them.
MP2
Reason abstractly and quantitatively.
MP3
Construct viable arguments and critique the reasoning of others.
MP4
Model with mathematics.
MP5
Use appropriate tools strategically.
MP6
Attend to precision.
MP7
Look for and make use of structure.
Sa
m
MP1
MP8
Look for and express regularity in repeated reasoning.
First Printing
Real Number System (Student Packet)
DO NOT DUPLICATE
RNS3 – SP38
```