Download Common Core Learning Standards GRADE 8 Mathematics

Common Core Learning Standards
GRADE 8 Mathematics
EXPRESSIONS & EQUATONS
Common Core Learning
Standards
Work with radicals and integer
exponents.
Concepts
Embedded Skills
Properties of
exponents
Divide the numerical expressions with integer
exponents with like bases by subtracting the
exponents.
Evaluate numerical expressions with integer
exponents.
Write a numerical expression with a negative
exponent as an equivalent numerical expression
with a positive exponent (write the base as a
fraction).
Multiply numerical expressions with integer
exponents with like bases by adding the exponents.
8.EE.1.
Know and apply the properties of integer
exponents to generate equivalent numerical
expressions. For example,
=
Vocabulary







Base
Exponent
Integer
Expression
Monomial
Coefficient
Numerical
expression
Add and subtract and divide numerical expressions
with integer exponents.
SAMPLE TASKS
I.) Explain why the given expressions are equivalent:
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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.
II.) Using your knowledge of the laws of exponents, explain why each student is correct or incorrect
Jose’s simplified expression)
= x12
Charlotte’s simplified expression)
= x12
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III.) Complete the following table of values:
Column A
Column B
What effect does the sign of the exponent in column A have on the answer in column B?
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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.
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
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.
Name numbers that are perfect squares and nonperfect squares.
Evaluate a cube root of a perfect cube.
Name numbers that are perfect cubes and nonperfect cubes.
Vocabulary


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




Square root
Cube root
Squared
Cubed
Solution
Perfect square
Perfect cube
Exponent
Inverse
operation
Index
Rational
Irrational
Identify that non-perfect squares and non-perfect
cubes are irrational numbers (focus on ).
Copyright (c) 2011 by Erie 1 BOCES- Deep Curriculum Project for Mathematics-- Permission to use (not alter) and reproduce for educational purposes only.
SAMPLE TASKS
I.) Sally incorrectly solved the following equation on her last math test. Where is Sally’s mistake and on the lines below how should she
correctly solve the equation?
Step 1 
Step 2 
Step 3 
Step 4 
=
x=8
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_____________________________________________________________________________________________________
_____________________________________________________________________________________________________
II.) Classify each number below as a rational or irrational number:
a)
= __________________
d)
= ____________________
b)
= _________________
e)
= ____________________
c)
= _________________
III.) Solve for x in the following equation and justify 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.
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 X
and
the population of the world as 7 X
, and
determine that the world population is more
than 20 times larger.
Concepts
Scientific
notation
Embedded Skills
Vocabulary
Compare the magnitude (size) of 2 or more
numbers written in scientific notation.
Rewrite numbers in standard form in scientific
notation.

Expand numbers written in scientific notation into
standard form.
Divide numbers in scientific notation to compare
their sizes.




Scientific
Notation
Magnitude
Standard
Form
Estimate
Expand
SAMPLE TASKS
I.) The estimated circumference of the Earth is 300,000 miles. The estimated width of the United States is 3,000 miles.
Convert each measurement into Scientific Notation and determine approximately how many times larger the circumference of
the Earth is compared to the width of the United States.
II.) A liter of organic lemon water has
molecules in it. If Wegmans sells 2,500 liters of organic lemon water a day, how many
molecules would be sold after one week? Give your answer using scientific notation.
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.
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 E
on the calculator means x 10a.
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.
Vocabulary





Scientific
notation
Decimal
notation
Powers of 10
Standard form
Exponents
SAMPLE TASKS
I.) The continent of North America consists of Canada, the United States, and Mexico. Find the approximate population of North
America in scientific notation if the approximate population of Canada is 30 million, Mexico is 100 million, and the United States
is 3 x 108 people.
II.) The equation to find the density of an object is
equal to
grams and its volume is
where m = mass and V = volume. The mass of one sample of a substance is
liters. Find the density of the given substance in scientific notation.
Copyright (c) 2011 by Erie 1 BOCES- Deep Curriculum Project for Mathematics-- Permission to use (not alter) and reproduce for educational purposes only.
III.) Alex was using his scientific calculator to do his math homework and he got the number 1.34E28. Explain what this value means and
write the answer in both scientific notation and standard form.
IV.) A fern grows
inches per second. How much will it have grown after three and one-half hours? Give your answer in
scientific notation and label it appropriately.
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 slope of a linear relationship from
equations, tables, and graphs.
Create and label an appropriate quadrant I graph.
Describe unit rate as the slope of a graph.
Vocabulary
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


Proportions
Unit rate
Slope
Direct
variation
Compare two different proportional relationships
represented in different ways (graph vs. table vs.
equation vs. verbal description).
SAMPLE TASKS
See Below
Copyright (c) 2011 by Erie 1 BOCES- Deep Curriculum Project for Mathematics-- Permission to use (not alter) and reproduce for educational purposes only.
I.) Part A: Given the following table of values, illustrate the relationships for each car on the graph provided.
Time Elapsed (in hours)
Car A Distance (in miles)
Car B Distance (in miles)
0
0
0
2
100
104
4
200
208
6
300
312
8
400
416
Part B: Based on your graph, which of the following
cars have the greater rate of speed? Explain how you
determined your answer.
____________________________________________
____________________________________________
____________________________________________
____________________________________________
Part C: Write the equation of a line that represents
Car A if d is the distance and t is the time.
_____________________________
Copyright (c) 2011 by Erie 1 BOCES- Deep Curriculum Project for Mathematics-- Permission to use (not alter) and reproduce for educational purposes only.
II.)
John is going on a trip driving his motorcycle to
Albany. His friend, Jeff, is going and driving his
motorcycle too. Which friend is using more gas per
mile at a faster rate? Explain how you found your
answer.
John’s Trip
Jeff’s Trip
Miles Traveled
0
50
100
150
Gas in Tank (in gal.)
7
5
3
1
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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.
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
Embedded Skills
Linear
equations
Determine the slope of a line by counting the rise
over the run of the given line.
Explain why the slope of a line is the same for any
two points on the graph using rise over run.
Explain slope as a constant rate of change (rise over
run).
Given a line that passes through the origin, write
the equation for the line in the form y = mx, where
the slope is found using rise over run.
Given a line that passes the vertical axis at point
other than the origin, write the equation for the line
in the form y = mx + b, where the slope is found
using rise over run and b is where the line
intercepts the vertical axis.
Vocabulary
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
Slope
Y-intercept
Slope
intercept form
Similar
triangles
Non-vertical
line
Origin
Constant rate
of change
SAMPLE TASKS
I.)
Using any two points on the given line find the slope.
Show your work.
Does the slope of the line change depending on which
two points you use? Show an example to justify your
answer.
__________________________________________
What is the equation of the line?
______________________________
Copyright (c) 2011 by Erie 1 BOCES- Deep Curriculum Project for Mathematics-- Permission to use (not alter) and reproduce for educational purposes only.
II.)
Given the following diagram, explain how the slope is
found between the coordinates (-4, 4) and (8, -2).
If you pick two different points on the same line,
explain what happens to the slope.
__________________________________________
__________________________________________
III.)
The equation of the line in graph d. is y = 2x, what 2
characteristics about this line makes this equation
true?
__________________________________________
__________________________________________
__________________________________________
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.
Does this equation have no solution, one solution or infinitely many solutions? Explain your response.
 x+1=x+2
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___________________________________________________________________________
II. Does this equation have no solution, one solution or infinitely many solutions? Explain your response.
 2x + 3 = 7x + 13
___________________________________________________________________________
___________________________________________________________________________
Copyright (c) 2011 by Erie 1 BOCES- Deep Curriculum Project for Mathematics-- Permission to use (not alter) and reproduce for educational purposes only.
III. Does this equation have no solution, one solution or infinitely many solutions? Explain your response.
 4 + 12x = 6 + 3x – 2 + 9x
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IV. Solve for x in this equation. Check your solution.

V. 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
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.
Concepts
Multi-step
equations
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.
Does this equation have no solution, one solution or infinitely many solutions? Explain your response.

____________________________________________
____________________________________________
____________________________________________
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 and check the equation

III. When Josh solved the following equation it did not check. Correct Josh’s mistake and explain his mistake using numbers and words.
_________________________________________________________
_________________________________________________________
_________________________________________________________
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+
+
_________________________________________________________
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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. What is the solution of the system of equations?
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.
III. Stacey believes the solution to this system of equations
answer.
and
is (-5,5). Is Stacey correct? Justify your
Copyright (c) 2011 by Erie 1 BOCES- Deep Curriculum Project for Mathematics-- Permission to use (not alter) and reproduce for educational purposes only.
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
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
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
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
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?
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III. Solve the system of equations ALGEBRAICALLY.
=x
IV. Explain why it is more accurate to solve a system of equations containing decimals by using algebra than by graphing the equations.
________________________________________________________________________________________________
________________________________________________________________________________________________
____________________________________________________________________________________________________________________________________
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?
Copyright (c) 2011 by Erie 1 BOCES- Deep Curriculum Project for Mathematics-- Permission to use (not alter) and reproduce for educational purposes only.
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.
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III. The table shows the initial height and growth rate of an apple tree and a cherry tree. Write a system of equations the gardener could
use to determine the time t when height h of the two trees will be the same.
Initial Height (ft)
Growth rate (ft/yr)
Apple
2.75
1.2
Cherry
2.12
1.38
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IV. Two brothers decide to save money. One starts with $10, and saves $2 each day. The other starts with none, but saves $3 each day.
Write two equations to represent both brothers’ savings plans.
Will they ever have the same amount of money? If not explain why. If they will, after how many days they have the same
Copyright (c) 2011 by Erie 1 BOCES- Deep Curriculum Project for Mathematics-- Permission to use (not alter) and reproduce for educational purposes only.
amount? ____________________________________________________________________________________________
____________________________________________________________________________________________________
____________________________________________________________________________________________________
Copyright (c) 2011 by Erie 1 BOCES- Deep Curriculum Project for Mathematics-- Permission to use (not alter) and reproduce for educational purposes only.