Common Core Learning Standards
GRADE 8 Mathematics
EXPRESSIONS & EQUATONS
Common Core Learning
Standards
Work with radicals and integer
exponents.
8.EE.1.
Know and apply the properties of integer
exponents to generate equivalent numerical
expressions. For example, 32 × 3–5 = 3–3 =
1/33 = 1/27.
Concepts
Embedded Skills
Properties of
exponents
Rewrite numerical expressions with integer
exponents.
Evaluate numerical expressions with integer
exponents.
Rewrite a numerical expression with a negative
exponent to a numerical expression with a positive
exponent (rewrite the base as a fraction).
Add, subtract, multiply, and divide numerical
expressions with integer exponents.
Vocabulary


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
Base
Exponent
Integer
Expression
Monomial
Coefficient
Numerical
expression
SAMPLE TASKS
Copyright (c) 2011 by Erie 1 BOCES- Deep Curriculum Project for Mathematics-- Permission to use (not alter) and reproduce for educational purposes only.
Common Core Learning
Standards
Work with radicals and integer
exponents.
8.EE.2.
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. Evaluate square roots of small perfect
squares and cube roots of small perfect cubes.
Know that √2 is irrational.
Concepts
Embedded Skills
Square roots
and cube
roots
Calculate the solution to x2 = p, where p is a positive
rational number by taking the square root of both
sides.
Calculate the solution to x3 = p, where p is a positive
rational number by taking the cube root of both
sides.
Evaluate a square root of a perfect square.
Identify numbers that are perfect squares.
Evaluate a cube root of a perfect cube.
Identify numbers that are perfect cubes.
Vocabulary



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
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





Square root
Cube root
Squared
Cubed
Solution
Perfect square
Perfect cube
Exponent
Inverse
operation
Index
Rational
Irrational
SAMPLE TASKS
Copyright (c) 2011 by Erie 1 BOCES- Deep Curriculum Project for Mathematics-- Permission to use (not alter) and reproduce for educational purposes only.
Common Core Learning
Standards
Work with radicals and integer
exponents.
8.EE.3.
Use numbers expressed in the form of a single
digit times a whole-number 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 times 108
and the population of the world as 7 times 109,
and determine that the world population is
more than 20 times larger.
Concepts
Scientific
notation
Embedded Skills
Compare the magnitude (size) of numbers written
in scientific notation.
Write numbers in scientific notation.
Vocabulary



Scientific
notation
Magnitude
Standard form
Expand numbers written in scientific notation.
Divide numbers in scientific notation to compare
their sizes.
SAMPLE TASKS
Copyright (c) 2011 by Erie 1 BOCES- Deep Curriculum Project for Mathematics-- Permission to use (not alter) and reproduce for educational purposes only.
Common Core Learning
Standards
Work with radicals and integer
exponents.
8.EE.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.
Concepts
Embedded Skills
Add, subtract, multiply, and divide numbers written
Operations
with scientific in scientific notation, applying laws of exponents.
Find appropriate units for very large and small
notation
quantities.
Demonstrate on the calculator and identify that EE
on the calculator means x 10a.
Vocabulary





Scientific
notation
Decimal
notation
Powers of 10
Standard form
Exponents
SAMPLE TASKS
Copyright (c) 2011 by Erie 1 BOCES- Deep Curriculum Project for Mathematics-- Permission to use (not alter) and reproduce for educational purposes only.
Common Core Learning
Standards
Understand the connections between
proportional relationships, lines, and
linear equations.
8.EE.5.
Graph proportional relationships, interpreting
the unit rate as the slope of the graph.
Compare two different proportional
relationships represented in different ways. For
example, compare a distance-time graph to a
distance time equation to determine which of
two moving objects has greater speed.
Concepts
Graphing
relationships
Embedded Skills
Identify the components of a linear relationship
from equations and graphs.
Create and label an appropriate coordinate plane.
Compare a graph to an equation of a similar
relationship.
Vocabulary




Proportions
Unit rate
Slope
Direct
variation
Describe unit rate as the slope of a graph.
Compare two different proportional relationships
represented in different ways (graph vs. table vs.
equation vs. verbal description).
SAMPLE TASKS
Copyright (c) 2011 by Erie 1 BOCES- Deep Curriculum Project for Mathematics-- Permission to use (not alter) and reproduce for educational purposes only.
Common Core Learning
Standards
Understand the connections between
proportional relationships, lines, and
linear equations.
8.EE.6.
Use similar triangles to explain why the slope
m is the same between any two distinct points
on a non-vertical line in the coordinate plane;
derive the equation y = mx for a line through
the origin and the equation y = mx + b for a line
intercepting the vertical axis at b.
Concepts
Linear
equations
Embedded Skills
Calculate the slope of a line.
Explain why the slope of a line is the same for any
two points on the graph.
Explain slope as a constant rate of change.
Derive the equation y = mx for a line through the
origin (proportional relationship).
Derive the equation y = mx + b for a line not
through the origin.
Vocabulary
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

Slope
Y-intercept
Slope
intercept form
Similar
triangles
Non-vertical
line
Origin
Constant rate
of change
SAMPLE TASKS
Copyright (c) 2011 by Erie 1 BOCES- Deep Curriculum Project for Mathematics-- Permission to use (not alter) and reproduce for educational purposes only.
Common Core Learning
Standards
Analyze and solve linear equations
and pairs of simultaneous linear
equations.
8.EE.7a.
Give examples of linear equations in one
variable with one solution, infinitely many
solutions, or no solutions. Show which of these
possibilities is the case by successively
transforming the given equation into simpler
forms, until an equivalent equation of the form
x = a, a = a, or a = b results (where a and b are
different numbers).
Concepts
Solving linear
equations
Embedded Skills
Solve a linear equation with one solution.
Solve a linear equation with infinitely many
solutions.
Solve a linear equation with no solution.
Check the solution to an equation.
Explain the differences between one solution, no
solution, and infinitely many.
Vocabulary
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
Equation
Variable
Infinite
solution
Linear
No solution
Inverse
operation
Distributive
property
Combine like
terms
SAMPLE TASKS
I.
II.
III.
IV.
V.
VI.
Does this equation have no solution, one solution or infinitely many solutions? Explain your response.
 x+1=x+2
Does this equation have no solution, one solution or infinitely many solutions? Explain your response.
 2x + 3 = 7x + 13
Does this equation have no solution, one solution or infinitely many solutions? Explain your response.
 4 + 12x = 6 + 3x – 2 + 9x
Solve for x in this equation. Check your solution.
𝑥
 − 6 + 9 = −1
Explain, using numbers and words, how to identify whether an equation has no solution, one solution, or infinitely many
solutions.
Write 3 equations that have a variable on both sides of the equal sign. Write the equations so that one equation will have no
solution, one equation will have one solution, and one equation will have infinitely many solutions. Solve each equation to show
each example.
.
Copyright (c) 2011 by Erie 1 BOCES- Deep Curriculum Project for Mathematics-- Permission to use (not alter) and reproduce for educational purposes only.
Common Core Learning
Standards
Concepts
Multi-step
equations
Analyze and solve linear equations
and pairs of simultaneous linear
equations.
8.EE.7b.
Solve linear equations with rational number
coefficients, including equations whose
solutions require expanding expressions using
the distributive property and collecting like
terms.
Embedded Skills
Simplify equations using the distributive property,
combining like terms, and inverse operations.
Solve multi-step linear equations with rational
number coefficients.
Add, subtract, multiply, and divide rational
numbers.
Vocabulary






Distributive
property
Combine like
terms
Coefficient
Inverse
operations
Equations
Rational
numbers
SAMPLE TASKS
I.
II.
III.
Does this equation have no solution, one solution or infinitely many solutions? Explain your response.
1
 𝑛 + 2 = (4𝑛 + 8)
4
Solve and check the equation
 6 + 6𝑥 = 2(5 − 𝑥)
Did Josh solve the following equation correctly. If not, find the correct solution and explain his mistake using numbers and
words.
1
3

− ( − 𝑥) = 3𝑥 −

− 3 − 𝑥 = 3𝑥 −






1
1
−3 −
4
+3
𝑥 = 3𝑥
1 − 𝑥 = 3𝑥
+𝑥 +𝑥
1 = 4𝑥
1
=𝑥
4
4
3
4
− 3
4
+3
4
3
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Copyright (c) 2011 by Erie 1 BOCES- Deep Curriculum Project for Mathematics-- Permission to use (not alter) and reproduce for educational purposes only.
IV.
A pizza delivery man earns $7.50 per hour and $2.00 per delivery. Each week the driver spends $28.00 on car maintenance and
$86.00 on gas. If the driver works 40 hours per week, how many deliveries are needed each week to earn $300.00 after
expenses? Set up an equation to solve the problem above.
Copyright (c) 2011 by Erie 1 BOCES- Deep Curriculum Project for Mathematics-- Permission to use (not alter) and reproduce for educational purposes only.
Common Core Learning
Standards
Analyze and solve linear equations
and pairs of simultaneous linear
equations.
8.EE.8a.
Understand that solutions to a system of two
linear equations in two variables correspond to
points of intersection of their graphs, because
points of intersection satisfy both equations
simultaneously.
Concepts
Systems of
linear
equations
Embedded Skills
Define the solution to a linear system of equations
as the intersection point on a graph.
Graph a system of linear equations.
Identify the point of intersection to a system of
linear equations.
Vocabulary



System of
equations
Solution
Point of
intersection
SAMPLE TASKS
I.
Graph the system of linear equations below. Name the point they have in common.
𝒙 + 𝒚 = 𝟏𝟎
𝒙−𝒚=𝟔
Copyright (c) 2011 by Erie 1 BOCES- Deep Curriculum Project for Mathematics-- Permission to use (not alter) and reproduce for educational purposes only.
II.
Solve the following system of equations graphically.
𝟐𝒙 − 𝒚 = 𝟒
𝟏
𝒚 = 𝒙+𝟓
𝟒
III.
Stacey believes the solution to this system of equations 𝟐𝒙 + 𝟑𝒚 = 𝟓 and 𝟑𝒙 − 𝟐𝒚 = 𝟏𝟒 is (-5,5). See her response below.
Explain why she is incorrect.
𝟐(−𝟓) + 𝟑(𝟓) = 𝟓 ________________________________________________________________________________
−𝟏𝟎 + 𝟏𝟓 = 𝟓
________________________________________________________________________________
𝟓=𝟓
________________________________________________________________________________
IV.
What is the solution to the system of linear equations shown in the graph.
ANSWER: ______________
Copyright (c) 2011 by Erie 1 BOCES- Deep Curriculum Project for Mathematics-- Permission to use (not alter) and reproduce for educational purposes only.
Common Core Learning
Standards
Concepts
Embedded Skills
Analyze and solve linear equations
and pairs of simultaneous linear
equations.
Systems of
equations
algebraically
Solve a system of linear equations algebraically
with one solution.
Solve a system of linear equations algebraically with
no solution.
Solve a system of linear equations algebraically with
infinitely many solutions.
Estimate the solution of a system of linear
equations by graphing.
Solve simple systems of linear equations by
inspection.
8.EE.8b.
Solve systems of two linear equations in two
variables algebraically, and estimate solutions
by graphing the equations. Solve simple cases
by inspection. For example, 3x + 2y = 5 and 3x +
2y = 6 have no solution because 3x + 2y cannot
simultaneously be 5 and 6.
Vocabulary








Elimination
Substitution
Algebraically
Graphically
One solution
No solution
Infinitely
many
solutions
Systems of
equations
SAMPLE TASKS
I.
Identify the number of solutions from each graph below.
a.)
b.)
________________________
________________________
c.)
________________________
Copyright (c) 2011 by Erie 1 BOCES- Deep Curriculum Project for Mathematics-- Permission to use (not alter) and reproduce for educational purposes only.
II.
Adam graphs a system of equations. He notices that the lines of the graphs are parallel. Given that the lines are parallel, what
does it mean for the system of equations?
_______________________________________________________________________________________________________
_______________________________________________________________________________________________________
_______________________________________________________________________________________________________
_______________________________________________________________________________________________________
_______________________________________________________________________________________________________
_______________________________________________________________________________________________________
III.
Solve the system of equations ALGEBRAICALLY. Tell what you did.
𝒚=x
𝟑𝒙 − 𝟐𝒚 = 𝟒
________________________________________________________________________________________________
________________________________________________________________________________________________
________________________________________________________________________________________________
________________________________________________________________________________________________
IV.
Give three reasons why it is more accurate to solve a system of equations containing decimals by using algebra than by graphing
the equations.
1.) ________________________________________________________________________________________________
________________________________________________________________________________________________
2.) ________________________________________________________________________________________________
________________________________________________________________________________________________
3.) ________________________________________________________________________________________________
________________________________________________________________________________________________
Copyright (c) 2011 by Erie 1 BOCES- Deep Curriculum Project for Mathematics-- Permission to use (not alter) and reproduce for educational purposes only.
Common Core Learning
Standards
Analyze and solve linear equations
and pairs of simultaneous linear
equations.
8.EE.8c.
Solve real-world and mathematical problems
leading to two linear equations in two
variables. For example, given coordinates for
two pairs of points, determine whether the line
through the first pair of points intersects the
line through the second pair.
Concepts
Real world
situations
Embedded Skills
Write a system of linear equations from a word
problem.
Solve a system of linear equations created from a
word problem.
Graph points on a coordinate plane.
Calculate the slope of a line using the slope formula.
Create an appropriate coordinate plane based on
the constraints of a word problem.
Vocabulary








Intersect
Linear
equation
Solution
Variable
Slope/rate of
change
Y-intercept
Constraints
Coordinate
plane
SAMPLE TASKS
I.
Emily sells pretzels and lemonade at a concession stand. One morning she sells 10 pretzels and 5 lemonades and makes a total of
$40.00. In the afternoon, she sells 25 pretzels and 20 lemonades and makes $115.00. How much does Emily charge for 1 pretzel
and how much does she charge for lemonade?
II.
A line passes through points (-3, -1) and (2, 4). Another line passes through (-3, 2) and (-1, -2). Do the lines intersect? Explain
how you arrived at your answer.
______________________________________________________________________________________________________
______________________________________________________________________________________________________
______________________________________________________________________________________________________
Copyright (c) 2011 by Erie 1 BOCES- Deep Curriculum Project for Mathematics-- Permission to use (not alter) and reproduce for educational purposes only.