Download The Number System (NS) Know that there are numbers that are not

The Number System (NS)
Know that there are numbers that are not rational, and approximate them by rational numbers.
STANDARD: CC.8.NS.1. 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.
The intent of this standard is…Students understand that Real numbers are either rational or irrational.
Learning Targets
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Notes for Teacher
Students can use graphic organizers to show the
relationship between the subsets of the real number
system.
I can convert any number
type to a fraction of A/B.
I can convert any number
to decimal form.
I can explain that every
number has a decimal
expansion.
I can recognize decimal
expansions as rational or
irrational numbers.
I can identify a decimal
expansion that repeats or
terminates as a rational
number.
I can convert a repeating
decimal into rational
number in A/B form.
I can identify a decimal
expansion that does not
repeat or terminate as an
irrational number.
Some students are surprised that the
decimal representation of π does not
repeat. Some students believe that if
only we keep looking at digits farther
and farther to the right, eventually a
pattern will emerge.
Some students will think that a
negative fraction is an integer.
Vocabulary
 Rational Numbers
 Irrational numbers
 Terminating decimals
 Repeating
decimals
Madison
County Schools
Common Misconceptions/
Vocabulary not to use
Samples
Mathematical Practices
8.MP.2. Reason abstractly and quantitatively.
8.MP.4. Model with mathematics.
8.MP.7. Look for and make use of structure.
8.MP.8. Look for and express regularity in repeated reasoning.
Spring 2014
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Fractions in A/B form
Bar notation
Resources
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Graphing calculators
Dynamic geometry software
The Number System (NS)
Know that there are numbers that are not rational, and approximate them by rational numbers.
STANDARD: CC.8.NS.2. 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., 2). For example, by truncating the decimal expansion of √2, show that √2 is
between 1and 2, then between 1.4 and 1.5, and explain how to continue on to get better approximations.
The intent of this standard is…Students locate rational and irrational numbers on the number line. Students compare and order rational and
irrational numbers. Students also recognize that square roots may be negative and written as - 28 .
Learning Targets
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I can find a number’s
square and its square root.
I can explain that squaring
and square roots are
inverse operations.
I can categorize the square
root of a number as
irrational or rational.
I can explain why an
irrational number does not
have a position on a
number line diagram.
I can compare the size of
irrational numbers using
rational numbers on a
number line diagram.
Notes for Teacher…
Students understand that
the value of a square root
can be approximated
between integers and that
non-perfect square roots
are irrational.
Students can approximate square
 roots by iterative
processes.
Examples:
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Approximate the value of
hundredth.
5 to the nearest
Solution: Students start with a rough estimate based
upon perfect squares. 5 falls between 2 and 3
because 5 falls between 22 = 4 and 32 = 9. The value
will be closer to 2 than to 3. Students continue the
A few irrational numbers are given
special names (pi and e), and much
attention is given to A sqrt(2). Because
we name so few irrational numbers,
students sometimes conclude that
irrational numbers are unusual and
rare. In fact, irrational numbers are
much more plentiful than rational
numbers, in the sense that they are
“denser” in the real line.
iterative process with the tenths place value. 5 falls
between 2.2 and 2.3 because 5 falls between 2.22 =
4.84 and 2.32 = 5.29. The value is closer to 2.2. Further
iteration shows that the value of
Madison County Schools
Common Misconceptions/
Vocabulary not to use
Samples
5 is between 2.23
Spring 2014
and 2.24 since 2.232 is 4.9729 and 2.242 is 5.0176.
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Compare √2 and √3 by estimating their values,
plotting them on a number line, and making
comparative statements.
Solution: Statements for the comparison could include:
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√2 is approximately 0.3 less than √3
√2 is between the whole numbers 1 and 2
√3 is between 1.7 and 1.8
Vocabulary
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Square root
Truncating
Estimation
Approximation
Π (Pi) as an irrational number
Degree of accuracy
Mathematical Practices
8.MP.2. Reason abstractly and quantitatively.
8.MP.4. Model with mathematics.
8.MP.7. Look for and make use of structure.
8.MP.8. Look for and express regularity in repeated reasoning.
Resources


Graphing calculators
Dynamic geometry software
Madison County Schools
Spring 2014