Download Math 8 Resource Guide for Provo City School District`s Essentials

Math 8 Resource Guide
for Provo City School
District’s Essentials
Summary of Practice Standards
Prompts to develop mathematical thinking
1. Make sense of problems and persevere in solving them.
How would you describe the problem in your own words?
Interpret and make meaning of the problem to find a starting point.
How would you describe what you are trying to find?
Analyze what is given in order to explain to themselves the meaning
of a problem.
What do you notice about . . .?
Plan a solution pathway instead of jumping to a solution.
Describe what you have already tried. What might you change?
Monitor their progress and change the approach if necessary.
Talk me through the steps in the steps you’ve used to this point.
See relationships between various representations.
What steps in the process are you most confident about?
Relate current situations to concepts or skills previously learned and
connect mathematical ideas to one another.
What are some other strategies you might try?
Continually ask themselves, “Does this make sense?”
How might you use one of your previous problems to help you begin?
Can understand various approaches to solutions
How else might you organize . . . represent . . . show . . .?
2. Reason abstractly and quantitatively.
What do the numbers used in the problem represent?
Make sense of quantities and their relationships.
What is the relationship of the quantities?
Decontextualize (represent a situation symbolically and manipulate
the symbols) and contextualize (make meaning of the symbols in a
problem) quantitative relationships.
How is __________ related to ___________?
Understand the meaning of quantities and are flexible in the use of
operations and their properties
Create a logical representation of the problem.
Attend to the meaning of quantities, not just how to compute them.
Describe the relationship between quantities.
What are some other problems that are similar to this one?
What is the relationship between ____________ and ____________?
What does ___________ mean to you? (e.g., symbol, quantity, diagram)
What properties might we use to find a solution?
How did you decide in this task that you needed to use . . .?
Could we have used another operation or property to solve this task? Why or
why not?
3. Construct viable arguments and critique the reasoning
of others.
What mathematical evidence would support your solution?
Analyze problems and use stated mathematical assumptions,
definitions, and established results in constructing arguments.
Will it still work if . . .?
Justify conclusions with mathematical ideas.
Listen to the arguments of others and ask useful questions to
determine if an argument makes sense.
Ask clarifying questions or suggest ideas to improve/revise the
argument.
Compare two arguments and determine correct or flawed logic.
How can we be sure that . . .? How could you prove that . . .?
What were you considering when . . .?
How did you decide to try that strategy?
How did you test whether your approach worked?
How did you decide what the problem was asking you to find? (What was
unknown?)
Did you try a method that did not work? Why didn’t it work? Would it ever
work? Why or why not?
What is the same and what is different about . . .?
How could you demonstrate a counter-example?
4. Model with mathematics.
What number model could you construct to represent the problem?
Understand this is a way to reason quantitatively and abstractly (able
to decontextualize and contextualize, see standard 2 above).
What are some ways to represent the quantities?
Apply the mathematics they know to solve everyday problems.
What is an equation or expression that matches the diagram, number line,
chart, table ?
Are able to simplify a complex problem and identify important
quantities to look at relationships.
Where did you see one of the quantities in the task in your equation or
expression?
Represent mathematics to describe a situation either with an equation
or a diagram and interpret the results of a mathematical situation.
Reflect on whether the results make sense, possibly improving/
revising the model
How would it help to create a diagram, graph, table?
What are some ways to visually represent . . .?
What formula might apply in this situation?
How can I represent this mathematically?
Summary of Practice Standards
5. Use appropriate tools for mathematical practice.
Use available tools recognizing the strengths and limitations of each.
Use estimation and other mathematical knowledge to detect possible
errors.
Prompts to develop mathematical thinking
What mathematical tools could we use to visualize and represent the
situation?
What information do you have?
What do you know that is not stated in the problem?
Identify relevant external mathematical resources to pose and solve
problems.
What approach are you considering trying first?
Use technological tools to deepen their understanding of mathematics
In this situation would it be helpful to use a graph, number line, ruler,
diagram, calculator, manipulative?
What estimate did you make for the solution?
Why was it helpful to use ______?
What can using a _______ show us that _______ may not?
In what situations might it be more informative or helpful to use ________?
6. Attend to precision.
What mathematical terms apply to this situation?
Communicate precisely with others and try to use clear mathematical
language when discussing their reasoning.
How did you know your solution was reasonable?
Understand the meanings of symbols used in mathematics and can
label quantities appropriately.
What would be a more efficient strategy?
Express numerical answers with a degree of precision appropriate for
the problem context.
What symbols or mathematical notations are important in this problem?
Calculate efficiently and accurately.
Explain how you might show that your solution answers the problem?
How are you showing the meaning of the quantities?
What mathematical language, definitions, properties can you use to explain
______?
How can you test your solution to see if it answers the problem?
7. Look for and make use of structure.
What observations do you make about _____ ?
Apply general mathematical rules to specific situations.
What do you notice when ______?
Look for the overall structure and pattern in mathematics.
What parts of the problem might you eliminate or simplify?
See complicated things as single objects or as being composed of
several objects.
What patterns do you find in _______ ?
How do you know if something is a pattern?
What ideas that we have learned before were useful in solving this problem?
What are some other problems that are similar to this one?
How does this problem connect to other mathematical concepts?
In what ways does this problem connect to other mathematical concepts?
8. Look for and express regularity in repeated reasoning?
Explain how this strategy will work in other situations.
See repeated calculations and look for generalizations and shortcuts.
Is this always true, sometimes true, or never true?
See the overall process of the problem and still attend to the details.
How would you prove that _______?
Understand the broader application of patterns and see the structure
in similar situations.
What do you notice about ________?
Continually evaluate the reasonableness of immediate results.
What would happen if ________?
What is happening in this situation?
Is there a mathematical rule for _________?
What predictions or generalizations can this pattern support?
What mathematical consistencies do you notice?
In Grade 8, instructional time should focus on three critical areas:
1. Formulating and reasoning about expressions and equations, including modeling an
association in bivariate data with a linear equation, and solving linear equations and systems
of linear equations
2. Grasping the concept of a function and using functions to describe quantitative relationships
3. Analyzing two- and three-dimensional space and figures using distance, angle, similarity,
and congruence, and understanding and applying the Pythagorean Theorem
 1. Students use linear equations and systems of linear equations to represent, analyze, and solve a variety
of problems. Students recognize equations for proportions (y/x = m or y = mx) as special linear equations (y
= mx + b), understanding that the constant of proportionality (m) is the slope, and the graphs are lines
through the origin. They understand that the slope (m) of a line is a constant rate of change, so that if the
input or x-coordinate changes by an amount A, the output or y-coordinate changes by the amount m·A.
Students also use a linear equation to describe the association between two quantities in bivariate data
(such as arm span vs. height for students in a classroom). At this grade, fitting the model, and assessing its fit
to the data are done informally. Interpreting the model in the context of the data requires students to
express a relationship between the two quantities in question and to interpret components of the
relationship (such as slope and y-intercept) in terms of the situation.
Students strategically choose and efficiently implement procedures to solve linear equations in one variable,
understanding that when they use the properties of equality and the concept of logical equivalence, they
maintain the solutions of the original equation. Students solve systems of two linear equations in two
variables and relate the systems to pairs of lines in the plane; these intersect, are parallel, or are the same
line. Students use linear equations, systems of linear equations, linear functions, and their understanding of
slope of a line to analyze situations and solve problems.
 2. Students grasp the concept of a function as a rule that assigns to each input exactly one output. They
understand that functions describe situations where one quantity determines another. They can translate
among representations and partial representations of functions (noting that tabular and graphical
representations may be partial representations), and they describe how aspects of the function are reflected
in the different representations.
 3. Students use ideas about distance and angles, how they behave under translations, rotations, reflections,
and dilations, and ideas about congruence and similarity to describe and analyze two-dimensional figures
and to solve problems. Students show that the sum of the angles in a triangle is the angle formed by a
straight line, and that various configurations of lines give rise to similar triangles because of the angles
created when a transversal cuts parallel lines. Students understand the statement of the Pythagorean
Theorem and its converse, and can explain why the Pythagorean Theorem holds, for example, by
decomposing a square in two different ways. They apply the Pythagorean Theorem to find distances
between points on the coordinate plane, to find lengths, and to analyze polygons. Students complete their
work on volume by solving problems involving cones, cylinders, and spheres
Domain: Expressions and equations
8EE (Quarter 1)
Cluster: Analyze and solve linear equations and pairs of simultaneous linear equations
Standard: 8EE7 Solve linear equations in one variable
a) 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).
b) Solve linear equations with rational number coefficients, including equations whose solutions
require expanding expressions using the distributive property and collecting like terms
Mastery, Patterns of Reasoning:
Conceptual:
 Understand that linear equations in one variable can have a
single solution, infinitely many solutions or no solutions
 Understand how to expand expressions using the distributive
property and collecting like terms
Procedural:
 Identify and provide examples of equations that have one
solution, infinitely many solutions, or no solutions
 Solve multistep linear equations with rational coefficients and
variables on both sides of the equation
Representational:
 Model examples of equations that have a single solution,
infinitely many solutions, or no solutions
Critical Background Knowledge:
Conceptual:

Understand properties of algebra necessary for
simplifying algebraic expressions
Example:
 What are the three possibilities that
describe solutions to linear equations?
 What is another way to write 3(x + 4)?
 Solve for x: 2(3x + 1)= -5(-1 – 4x)
 Solve 6 = x/4 + 2
 Solve -1 = (5 + x)/6
 Find two values of x that make the
statement true: x2 < x
 Which equation has infinitely many
solutions?
a) 2x = 16
b) 2x + 16 = 2(x + 8)
c) 2x + 16 = x + 8
 Find and model the function that adds
one and then multiplies the result by 2
Bridge to previous instruction:
6EE1, 6EE2, 7EE4a
Procedural:
7EE4a
 Solve one- and two-step equations
6EE1
 Use properties of algebra to simplify algebraic expressions
Representational:
6EE2
 Use manipulatives to model the solving of one-step and twostep equations
Common misconceptions:
o Students confuse the operations for the properties of integer exponents, most often due to
memorization of rules rather than internalizing the concepts behind the laws of exponents
o Students sometimes incorrectly assume a value is negative when its exponent is negative
o When simplifying with the quotient of powers rule, students often make subtraction mistakes
o Students sometimes forget there is a negative square root as well as the principal positive root
o Students sometimes mistakenly believe that zero slope is the same as “no slope” and then confuse
zero slope with undefined slope.
Domain: Functions
8F (Quarter 1)
Cluster: Define, evaluate, and compare functions
Standard: 8F1 Understand that a function is a rule that assigns to each input exactly one output. The
graph of a function is the set of ordered pairs consisting of an input and the corresponding output
Mastery, Patterns of Reasoning:
Example:
Conceptual:
 Understand that a function is a rule that assigns to each input
exactly one output
 Does the set of students in the classroom
and their birthdays represent a function?
Procedural:
 Recognize a graph of a function as the set of ordered pairs
consist
 Does the set of ordered pairs (2, 5), (3,
5), (4, 6), (2, 8), and (6, 7) represent a
function?
 Could the set of ordered pairs, (2, 5), (3,
5), (4, 6), (2, 8), and (6, 7) describe the
number of seconds since you left home
and the number of meters you’ve
walked? Is this a function?
 Which of the following are functions?
a)
b)
Representational:
 Model solutions of equations that have a single solution,
infinitely many solutions, or no solutions
c)
Critical Background Knowledge:
Conceptual:
 Understand what a solution to a linear equation is
Bridge to previous instruction:
8EE7
Procedural:
5OA1, 6EE2
 Evaluate expressions for a given value
Representational:
6NS6
 Graph ordered pairs on the coordinate plane
Common misconceptions:
o Students believe a function is a graph
o Students believe that all functions include the notation f(x)
o Students sometimes interchange inputs and outputs causing problems with domain and range as
well as independent v dependent variables
Domain: Expressions and equations
8EE (Quarter 1)
Cluster: Understand the connections between proportional relationships, lines and linear equations
Standard: 8EE5 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.
Mastery, Patterns of Reasoning:
Conceptual:


Understand the connections between proportional
relationships, lines and linear equations
Understand that the unit rate is the slope of a linear graph
Procedural:



Recognize unit rate as slope and interpret the meaning of the
slope in context
Recognize that proportional relationships include the point
(0,0)
Compare different representations of two proportional
relationships represented as contextual situations, graphs, or
equations
Representational:


Represent proportional relationships graphically when given a
table, equation or contextual situation
Model proportional relationships with manipulatives
Critical Background Knowledge:
Conceptual:

Understand unit rates
Example:
 Assuming the relationship between minutes and
phone calls is directly proportional, if Sam spends 6
minutes on the phone for 3 phone calls. How many
phone calls does Sam need to make to be on the
phone 10 minutes?
 If Gordin has 16 cards in 4 packages and 6 packages
has 24 cards, which description of the graph would
show this?
a) a straight line that drops as it moves to the right
b) a straight line that rises as it moves to the right
c) a curve that grows steeper as it moves to the
right
 50 plates in 5 stacks = _____ plates per stack
 Solve for x: 15:6 = x:4
 Do these ratios form a proportion? 8 tents: 32
campers and 5 tents: 20 campers. (Yes or No)
 Use h to represent heartbeats and t to represent
time. Tiffany counted her heartbeats every 10
seconds for one minute and got the following values
(15, 30, 45, 60, 75, 90). Graph these values and find
an equation to represent the relationship.
Bridge to previous instruction:
6RP2, 6RP3
Procedural:
6EE9, 7RP2
 Use an equation to create a table
6RP3
 Calculate unit rates
Representational:
 Represent values by plotting them on the coordinate 5G1, 6G3, 6NS8, 6NS6
axes
Common misconceptions:
o Students do not understand the relationship of the wording so proportions are incorrectly written
o Students struggle with ratios that do not have the same units
o Students will occasionally reduce the significance of ratio to simply being a fraction and a proportion
is the equality of two ratios. This eliminates the importance of the constant relation between
quantities
Domain: Statistics and Probability
8SP (Quarter 1)
Cluster: Investigate patterns of association in bivariate data
Standard: 8SP1 Construct and interpret scatter plots for bivariate measurement data to investigate
patterns of association between two quantities. Describe patterns such as clustering, outliers,
positive or negative association, linear association, and nonlinear association
Mastery, Patterns of Reasoning:
Conceptual:
 Understand clustering patterns of positive or negative
association, linear association, and nonlinear association
 Know what outliers are
Procedural:
 Collect, record, and construct a set of bivariate data using a
scatter plot
 Interpret patterns on a scatter plot such as clustering, outliers,
and positive, negative or not association
Representational:
 Graphically represent the values of a bivariate data set with a
scatter plot
Critical Background Knowledge:
Conceptual:
 Understand graphing of linear values and points
 Understand the meaning of linear and nonlinear
Example:
 Create and describe examples of scatter
plots that are positive-, negative- and
non-correlation
 Measure and record the height and arm
span of all class members. Then create a
scatter plot of the data. Is there a
relationship between a student’s height
and their arm span?
 Construct a scatter plot and describe any
association you observe for the data:
Height hand span
70 in
10 in
72 in 9.5 in
61 in 8 in
62 in 9.5 in
68 in 9 in
Bridge to previous instruction:
5G1
Procedural:
7EE1
 Graph points on a coordinate system
Representational:
8EE7
 Represent linear relationships graphically
Common misconceptions:
o Students sometimes attempt to connect all points on a scatter plot
o Students often believe that correlation between two variables automatically implies causation
o Students sometimes believe that bivariate data is only displayed in scatter plots
Domain: Expressions and equations
8EE (Quarter 2)
Cluster: Understand the connections between proportional relationships, lines and linear equations
Standard: 8EE6 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.
Mastery, Patterns of Reasoning:
Example:
Conceptual:
 Understand why the slope is the same between any two
distinct points on a non-vertical line
 What does a 7% slope mean? How can it
be represented with different measures?
Procedural:
 Explain why the slope is the same between any two distinct
points on a non-vertical line using similar right triangles
 Write an equation in the form y = mx + b from a graph of a line
on the coordinate plane
 Determine the slope of a line as the ratio of the leg lengths of
similar right triangles
Representational:
 Represent similar right triangles on a coordinate plane to
show equivalent slopes
 Write the equation of the line containing
points A and D
Critical Background Knowledge:
Conceptual:
 Understand triangle similarity requires proportionality
 Graph y = 2x
 Points A, D, B and E are collinear. Show
that segment AB and segment DE have
the same slope
Bridge to previous instruction:
7RP2
Procedural:
4G2, 6G4, 7G2
 Recognize similar triangles
Representational:
5G2, 6NS6, 6G3
 Model similar triangles on a coordinate plane
Common misconceptions:
o Students sometimes cannot visualize the corresponding parts of similar triangles because of
orientation
o Students sometimes forget that congruent triangles are also similar
Domain: Functions
8F (Quarter 2)
Cluster: Use functions to model relationships between quantities
Standard: 8F4 Construct a function to model a linear relationship between two quantities. Determine the
rate of change and initial value of the function from a description of a relationship or from two (x, y) values,
including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear
function in terms of the situation it models, and in terms of its graph or a table of values
Mastery, Patterns of Reasoning:
Conceptual:
 Know how to determine the initial value and rate of change
given two points, a graph, a table of values, a geometric
representation, or a story problem
Example:
 How would you find the rate of change
on the graph below?
 Find the equation of the line that goes
Procedural:
through (3, 5) and (-5, 7)
 Determine the initial value and rate of change given two
 What is the initial value and rate of
points, a graph, a table of values, a geometric representation,
change if we know that during a run,
or a story problem
sally was 2 km from her starting point
 Write the equation of a line given two points, a graph a table of
after 2.7 minutes and then at 11.5
minutes she was at 7.7 km?
values, a geometric representation, or a story problem (verbal
description) of a linear relationship
 The student council is planning a ski trip
Representational:
to Sundance. There is a $200 deposit for
 Model relationships between quantities
the lodge and the tickets will cost $70
per student. Construct a function, build a
table, and graph the data showing how
much it will cost for the students’ trip
Critical Background Knowledge:
Conceptual:

Understand the meaning of slope and y-intercept
Bridge to previous instruction:
8EE5,
Procedural:
8EE6
 Write an equation as y = mx + b given a graph
Representational:
8EE5
 Graphically represent linear equations
Common misconceptions:
o Students sometimes confuse the two axes of the graph
o Students sometimes do not understand the significance of points in the same location relative to one
of the axes
o Students often believe the graph is a picture of situations rather than an abstract representation
o Students often believe graphs must go through the origin
o Students often think graphs must go through both axes
o Students often believe all relationships are linear
Domain: Expressions and equations
8EE (Quarter 2)
Cluster: Analyze and solve linear equations and pairs of simultaneous linear equations
Standard: 8EE8 Analyze and solve pairs of simultaneous linear equations
a)
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
b) Solve systems of two linear equations in two variables algebraically, and estimate solutions by
graphing the equations. Solve simple cases by inspections. For example, 3x + 2y = 5 and 3x + 2y
= 6 have no solution because 3x + 2y cannot simultaneously be 5 and 6.
Mastery, Patterns of Reasoning:
Conceptual:

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
Procedural:
 Solve systems of two linear equations in two
variables algebraically
 Estimate solutions by graphing the equations
 Solve simple cases by inspections
 Solve real-world and mathematical problems
leading to two linear equations in two variables
Representational:
 Model solutions of equations that have a single
solution, infinitely many solutions, or no solutions
Critical Background Knowledge:
Conceptual:
 Understand what a solution to a linear equation is
Example:
 You are solving a system of two linear equations in two
variables. You have found more than one solution that
satisfies the system. Which of the following is true?
a) there are exactly two solutions to the system
b) there are exactly three solutions to the system
c) there are infinitely many solutions to the system
d) there isn’t enough information to tell
 Solve the systems of equations:
2x + 3y = 4 and –x + 4y = -13
 When trying to find the solutions to the system
4x – 2y = 4 and 2x – y = 3, you complete several correct
steps and get a result 4 = 6. Which statement is true?
a) x = 6 and y = 4
b) y = 6 and x = 4
c) the system has no solution
d) the system has infinitely many solutions
 You have been hired by a cell phone company to create
two rate plans for customers, one that benefits
customers with low usage and that benefits customers
with high usage. At 500 minutes, both plans should be
within $5 of each other. Design a presentation showing
two plans that will meet these requirements, including
graphs and equations
Bridge to previous instruction:
6EE6, 8EE7
Procedural:
5OA1, 6EE2, 6EE6, 7EE4, 8EE7
 Solve a one variable equation
 Solve for a specified variable in an equation
Representational:
6NS6
 Represent linear equations graphically
Common misconceptions:
o Students sometimes do not know what “solution” means, they know it as an answer, but not what it
represents.
Domain: The Number System
8NS (Quarter 3)
Cluster: Know that there are numbers that are not rational, and approximate them by rational numbers
Standard: 8NS1Know 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
Mastery, Patterns of Reasoning:
Conceptual:
 Know that there are numbers that are not rational
 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
Procedural:
 Convert a decimal expansion which repeats into a rational
number
Representational:
 Graph the approximate value of an irrational number on a
number line
Critical Background Knowledge:
Conceptual:
 Understand the subsets of the real number system (natural
numbers, whole numbers, integers, rational numbers)
Example:
 Group the following numbers based on
your understanding of the number
system:
5.3
1.7 where the 7 repeats infinitely
square root of 10
2
pi
4.01001000100001. . .
 Convert 0.352 (where the 2 repeats
infinitely) to a fraction
 Graph the values or approximate values
of the square roots of 1, 2, 3 and 4 on a
single number line
Bridge to previous instruction:
6NS6, 7EE3, 7NS2,
Procedural:
7NS2d
 Convert rational numbers to decimals using long division
(terminating and repeating)
Representational:
6NS6
 Graph rational numbers on a number line
Common misconceptions:
o Students sometimes think that non-common numbers that do not terminate but repeat infinitely are
not rational for example, 1.1666666. . .
o Students sometimes think that a square root sign automatically identifies an irrational number (even
the square root of 4)
o Students often think that all fractions are rational (square root of six over 3)
Domain: Expressions and equations
8EE (Quarter 3)
Cluster: Work with radicals and integer exponents
Know and apply the properties of integer exponents to generate equivalent numerical
expressions. For example:
Standard: 8EE1
32 x 3-5 = 3-3 = 1/33 = 1/27
Mastery, Patterns of Reasoning:
Example:
Conceptual:
 Know the properties of integer exponents
 Write the expression 4•4•4•4 using
exponents
Procedural:
 Apply the properties of integer exponents to simplify and
evaluate numerical expressions
 Which equation has more than one
solution, but not infinitely many
solutions?
a) 2x = 16
b) x2 = 16

c) 2x + 16 = x + 8
 Caleb has a job that pays $39,000
annually with a promise of a 5% raise
each year if his work remains
satisfactory. Determine his salary for
the next ten years.
Representational:
 Model the properties of integer exponents as multiple
multiplications
Critical Background Knowledge:
Conceptual:

Understand exponents as repeated multiplication
Procedural:
 Compute fluently with integers (add, subtract, and
Bridge to previous instruction:
6EE1
4NBT4, 5NBT5, 6NS2, 6NS3,
multiply)
Representational:
4OA1, 4NBT6
 Model multiplication of integers
Common misconceptions:
o Students confuse the operations for the properties of integer exponents, most often due to
memorization of rules rather than internalizing the concepts behind the laws of exponents
o Students sometimes incorrectly assume a value is negative when its exponent is negative
o When simplifying with the quotient of powers rule, students often make subtraction mistakes
o Students sometimes forget there is a negative square root as well as the principal positive root
o Students sometimes mistakenly believe that zero slope is the same as “no slope” and then confuse
zero slope with undefined slope.
Domain: Geometry
8G (Quarter 3)
Cluster: Understand congruence and similarity using physical models, transparencies, or geometry
software
Standard: 8G3 Describe the effect of dilations, translations, rotations, and reflections on two-dimensional
figures using coordinates
Mastery, Patterns of Reasoning:
Conceptual:
 Understand how to dilate, translate, rotate, and reflect twodimensional figures on the coordinate plane
Procedural:
 Describe the effects of dilations, reflections, translations and
rotations using coordinate notation
 Given an image and its transformed image, use coordinate
notation to describe the transformation
Representational:
 Model transformations on a coordinate plane
Critical Background Knowledge:
Conceptual:

Know coordinate notation
Procedural:
 Plot points on a coordinate plane

Example:
 The vertices of triangle A are (1, 0), (1,1),
(0, 0) and triangle A’ are (2, 1), (2, 2),
(3, 1). Describe the series of
transformations performed on triangle A
that result in triangle A’
 Given a triangle with vertices at (5, 2), (-7,
8) and (0, 4) find the new vertices of the
triangle after undergoing the transformation
described as follows:
 Given a triangle with vertices at (4, 3), (8, 7) and (-1, 5), show on a coordinate
plane the transformation of
(x, y) –> (x + 1, y -1)
Bridge to previous instruction:
5OA3, 5G1, 5G2
5G1, 5G2
Identify points on a coordinate plane
Representational:
5G1, 5G2, 6NS6
 Represent location on a coordinate plane
Common misconceptions:
o Students often confuse horizontal and vertical
o Students sometimes use a corner of an object being rotated with the center of rotation
Domain: Geometry
8G (Quarter 3)
Cluster: Understand congruence and similarity using physical models, transparencies, or geometry software
Standard: 8G4 Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a
sequence of rotations, reflections, translations, and dilations; given two similar two-dimensional figures, describe a sequence that
exhibits the similarity between them
Mastery, Patterns of Reasoning:
Example:
Conceptual:
 Understand that any combination of
transformations will result in similar figures
 What combination of transformations would make
triangle ABC be similar to triangle A’B’C’?
Procedural:
 Describe the sequence of transformations needed
to show how one figure is similar to another
 Point A was reflected about the x-axis. What is the next
transformation needed to map point A to point A’?
Representational:
 Model dilations of figures by a given scale factor
 If the measure of segment GA is 12 units, and the
measure of segment GE is 6 units then what is the scale
factor of triangle EHJ to triangle ABC?
Critical Background Knowledge:
Conceptual:

8.G.1, 8.G.2
Rotate, translate, reflect and dilate two-dimensional figures
Representational:

6.RP.1, 7.RP.1, 7.RP.2, 7.RP.3
Understand ratios and proportions
Procedural:

Bridge to previous instruction:
8.G.3
Represent rotations, reflections, translations, and dilations
graphically
Common misconceptions:
o
o
o
o
Students sometimes do not understand that congruence is not dependent upon orientation.
Students sometimes apply congruence requirements to similarity. They believe similar shapes must have congruent
sides.
Students might not recognize that the ratio of the perimeters of similar polygons is the same as the scale factor of
corresponding side lengths
Students might not recognize that the ratio of the areas of similar polygons is the square of the scale factor of
corresponding side lengths
Domain: Geometry
8G (Quarter 4)
Cluster: Understand congruence and similarity using physical models, transparencies, or geometry software
Standard: 8G5 Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles
created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles. For example, arrange
three copies of the same triangle so that the sum of the three angles appears to form a line, and give an argument in terms of
transversals why this is so
Mastery, Patterns of Reasoning:
Conceptual:
Example:
 Are these triangles similar?
 Understand that the measure of an exterior angle of triangle is equal to the
sum of the measures of the non-adjacent angles
 Know that the sum of the angles of a triangle equals 180 degrees.
 Recognize that if two triangles have two congruent angles, they are similar
(A-A similarity)
 Know what a transversal is and its properties in relation to parallel lines and
pairs of angles
Procedural:
 If line l || m, what is the measure of angle 4?
 Determine the relationship between corresponding angles, alternate interior
angles, alternate exterior angles, vertical angle pairs, and supplementary
pairs when parallel lines are cut by a transversal
 Use transversals and their properties to argue three angles of a triangle
create a line
Representational:
 Model A-A similarity
 Model the sum of three angles of a triangle form a line
Critical Background Knowledge:
 Using a paper triangle, show the three
angles of the triangle from a line.
Bridge to previous instruction:
Conceptual:

N/A
Procedural:

Representational:

4.MD.5
Measure angles
7.G.5
Model adjacent angles
Common misconceptions:
o Students sometimes think the numbering of angles created by a transversal cutting parallel lines
must always be the same and attempt to memorize the relationship between the numbers rather
than the relationship of position
Domain: Geometry
8G (Quarter 4)
Cluster: Understand and apply the Pythagorean Theorem
Standard: 8G7 Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in
real-world and mathematical problems in two and three dimensions
Mastery, Patterns of Reasoning:
Example:
Conceptual:
 Know the Pythagorean Theorem
 Can you use the Pythagorean Theorem to
find the length of an unknown side of a
non-right triangle?
Procedural:
 Use the Pythagorean Theorem to solve for a missing side of a
right triangle given the other two sides
 Use the Pythagorean Theorem to solve problems in real-world
contexts, including three-dimensional contexts.
 What is the length of b?
Representational:
 Use manipulatives to represent the Pythagorean Theorem to
find missing sides of a right triangle
Critical Background Knowledge:
 If the height of a cone is 10 m and the
radius is 6 m, what is the slant height?
 TV’s are measured along their diagonal
to report their dimension. How does a
52 in. HD (wide screen) TV compare to a
traditional 52 in. (full screen) TV?
Bridge to previous instruction:
Conceptual:
 Know approximate values of irrational numbers
8.NS.2
Procedural:
 Solve an equation using squares and square roots
 Use rational approximations of irrational numbers to express
answers
Representational:
 Represent approximate values of irrational numbers on a
number line
Common misconceptions:
8.EE.2, 8.NS.2
o
o
o
8.NS.2
Students sometimes misinterpret the relationship of the number 2 in squares and square roots and then multiply or
divide by 2 rather than squaring or taking the square root.
Students often combine numbers under the radicand when they should be combining like terms (e.g., 2√3 +4√3 = 6√6)
Students sometimes over extend order of operations without regard to rules of exponents.
e.g.,
Domain: Geometry
8G (Quarter 4)
Cluster: Solve real-world and mathematical problems involving volume of cylinders, cones, and spheres
Standard: 8G9 Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve
real-world and mathematical problems
Mastery, Patterns of Reasoning:
Example:
Conceptual:
 Know the formulas for the volumes of cones,
cylinders, and spheres
 What is the formula for the volume of a cylinder?
 What is the formula for the volume of a sphere?
Procedural:
 Use the formulas for volume to find the volumes of
cones, cylinders, and spheres
 A silo has 1500 ft3 of grain. The grain fills up the silo 20
ft .high. What is the radius of the silo?
 What is the relationship between the volume of a
cylinder and a cone with the same radius and height?
 What does the height of the cone need to be so that one
spherical scoop of ice cream with the same radius as
the cone won’t overflow if it melts?
 Find the volume of a given tin can. After calculating the
volume, attempt to fill the can with the amount of
water to verify your calculation.
Representational:
 Use manipulatives to represent the volumes of
cones and cylinders
Critical Background Knowledge:
Conceptual:
 Know what π is and how to derive it
 Understand that volume is measured in cubic units
 Understand exponential notation for squares and
cubes
Procedural:
 Solve equations involving square roots and cube
roots
Representational:
 Represent rational approximations of irrational
numbers such as pi
Common misconceptions:
o
o
Bridge to previous instruction:
8.NS.2
5.MD.3, 5.MD.4, 5.MD.5, 6.G.2, 7.G.6
5.NBT.2, 6.EE.1
8.EE.2
8.NS.2
Students learning volume sometimes do not understand the volume of an object is independent of the material it is made
of, they confuse mass and volume.
Students often ignore the relationship of the height and radius on volume, for example, if we create two cylinders with
one piece of 8.5” •11” each, one that is made with the top and bottom connected and one with the left side connected to
the right side, do they have the same volume? Many student will say yes or think the taller cylinder has more volume.
Domain: Statistics and Probability
8SP (Quarter 4)
Cluster: Investigate patterns of association in bivariate data
Construct and interpret scatter plots for bivariate measurement data to investigate patterns of
association between two quantities. Describe patterns such as clustering, outliers, positive or negative association,
linear association, and nonlinear association
Standard: 8SP1
Mastery, Patterns of Reasoning:
Example:
Conceptual:
 Understand clustering patterns of positive or
negative association, linear association, and
nonlinear association
 Know what outliers are
Procedural:
 Collect, record, and construct a set of bivariate data
using a scatter plot
 Interpret patterns on a scatter plot such as
clustering, outliers, and positive, negative or not
association
 What is an outlier?
Representational:
 Graphically represent the values of a bivariate data
set with a scatter plot
 Construct a scatter plot and describe any association
you observe with the values below
Critical Background Knowledge:
 Do the point plotted below have a positive, negative,
or not association?
Height
70 in
72 in
61 in
62 in
68 in
Hand span
10 in
9.5 in
8 in
9.5 in
9 in
Bridge to previous instruction:
Conceptual:
5.G.2, 5.OA.3
 Understand graphing of linear values and points
 Understand the meaning of linear and nonlinear
Procedural:
5.OA.3, 6.NS.8
 Graph points on a coordinate system
Representational:
7.RP.2, 7.EE.4
 Represent linear relationships graphically
Common misconceptions:
o Students sometimes confuse the x- and y-coordinates as well as the x- and y-axis
o Students often confuse vertical and horizontal change in slope