

Unit 1 The
Number System Grade
8 Math 


Unit
Length and Description: 20
days The year begins with students
extending their understanding of whole number exponents learned in previous
grades to integer exponents in Unit 1. Students develop the properties of
exponents. The unit progresses to include work with scientific notation which
supports student understanding of positive and negative exponents. Students
also work with radicals as they use square roots and cube roots to represent
solutions to equations of the form x^{2}=p and x^{3}=p. They
evaluate square roots and cube roots of small perfect squares and cubes. Later in the unit, in preparation for work
with geometric contexts such as the Pythagorean Theorem in Unit 3 of this
course, students extend their concept of numbers beyond the system of rationals to include irrational numbers as they estimate
roots of nonperfect squares. They represent these numbers with radical
expressions and approximate these numbers with rationals.
Students use the number line model to support their understanding of the
rational numbers, along with the inclusion of the irrational numbers, to
develop the real number line. 

Standards: 8.NS.A.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. 8.NS.A.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., 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 a
better approximation. 8.EE.A.1 Know and apply the
properties of integer exponents to generate equivalent numerical
expressions. For example, 3^{2} × 3^{5} = 3^{3} = 1/3^{3}
= 1/27. 8.EE.A.2 Use square root and cube root symbols to
represent solutions to equations of the form 𝑥^{2} = 𝑝 and 𝑥^{3} = 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. 8.EE.A.3 Use numbers expressed in
the form of a single digit times an integer power of 10 to estimate very
large or very small quantities, and to express how many times as much one is
than the other. For example, estimate the population of the United States as 3 × 10^{8}
and the population of the world as 7 × 10^{9}, and determine that the
world population is more than 20 times larger. 8.EE.A.4 Perform
operations with numbers expressed in scientific notation, including problems
where both decimal and scientific notation are used. Use scientific notation and choose units of
appropriate size for measurements of very large or very small quantities
(e.g., use millimeters per year for seafloor spreading). Interpret scientific notation that has been
generated by technology. Focus Standards of
Mathematical Practice: MP.1 Make
sense of problems and persevere in solving them. MP.2 Reason
abstractly and quantitatively. MP.3
Construct viable arguments and critique the reasoning of others. MP.4 Model
with mathematics. MP.5 Use
appropriate tools strategically. MP.6 Attend
to precision. MP.7 Look for
and make use of structure. MP.8 Look for
and express regularity in repeated reasoning. Instructional
Outcomes: Full Development of the Major
Clusters, Supporting Clusters, Additional Clusters and Mathematical Practices
for this unit could include the following instructional outcomes: 8.NS.A.1 I can define rational and
irrational numbers. I can show that the decimal
expansion of rational numbers repeats eventually. I can convert a decimal expansion
which repeats eventually into a rational number. I can show informally that every
number has a decimal expansion. 8.NS.A.2 I can approximate irrational
numbers as rational numbers. I can approximately locate
irrational numbers on a number line. I can estimate the value of
expressions involving irrational numbers using rational numbers. Examples: Being able to determine
the value of the √2 on a number line lies between 1 and 2, more
accurately, between 1.4 and 1.5, and more accurately, etc. I can compare the size of
irrational numbers using rational approximations. 8.EE.A.1 I can explain why a zero exponent
produces a value of one. I can explain how a number raised
to an exponent of 1 is the reciprocal of that number. I can explain the properties of
integer exponents to generate equivalent numerical expressions. 1. Multiplying
Exponents of the same base. 2. Dividing
Exponents of the same base. For example, 3²/
31 = 33 = 27 3. Expand an
exponent by another exponent. For example (2²)³ = 64 8.EE.A.2 I can use square root and cube
root symbols to represent solutions to equations of the form x2 = p and x3
=p, where p is a positive rational number. I can evaluate square roots of
small perfect squares. I can evaluate cube roots of small
perfect cubes. I can understand that the square
root of 2 is irrational. 8.EE.A.3
I can express numbers as a single
digit times an integer power of 10 (scientific notation). I can use scientific notation to
estimate very large and/or very small quantities. I can multiply and divide numbers
in scientific notation in order to compare growth or decay. I can compare quantities to
express how much larger one is compared to the other. 8.EE.A.4 I can perform operations using
numbers expressed in scientific notation. I can use scientific notation to
express very large and very small quantities. I can interpret scientific
notation that has been generated by technology. I can choose appropriate units of
measure when using scientific notation. 

Enduring
Understandings: ·
The
number system consists of numbers that are rational and irrational. ·
Irrational
numbers can be represented on a real number line. ·
Every
number has a decimal expansion. The decimal
expansion of a rational number either terminates or repeats. The decimal expansion of an irrational
numbers does not terminate or repeat. ·
The value
of any real number can be represented in relation to other real numbers such
as with decimals converted to fractions and numbers written as exponents or
radical. ·
Properties
of operations with whole and rational numbers also apply to real numbers. ·
The
properties of integer exponents are used to simplify expressions containing
integer exponents. ·
Numbers
can be expressed in scientific notation to compare very large and very small
quantities and to perform computations with those numbers. ·
Perfect
square roots and perfect cube roots can be written as rational numbers. ·
Nonperfect
square roots and nonperfect cube roots are irrational, but can be
approximated as rational numbers. ·
Expressions
are powerful tools for exploring, reasoning about, and representing
situations. ·
Both
rational and irrational numbers can be represented on a real number line. 
Essential
Questions: ·
Why
is it helpful to write numbers in different ways? ·
How
can you determine the difference in a rational and an irrational number? ·
How
can you evaluate positive exponents? ·
How
can you evaluate negative exponents? ·
How
can you develop and use the properties of integer exponents? ·
How
can you use scientific notation to express very large and very small
quantities? ·
Why
are quantities represented in multiple ways? ·
How
is the universal nature of properties applied to real numbers? ·
What
is a square root? ·
What
is a cube root? ·
What
is the difference in a finite decimal and an infinite decimal? ·
Can
a radical be used to represent a finite decimal? ·
Can
a radical be used to represent an infinite decimal? 
