Grade 8 Overview
The Number System (NS)
 Know that there are numbers that are not rational, and
approximate them by rational numbers. ( Standards 1and 2)
Mathematical Practices (MP)
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.
Expressions and Equations (EE)
 Work with radicals and integer exponents. (Standards 1, 2, 3,
and 4)
 Understand the connections between proportional relationships,
lines, and linear equations. (Standards 5 and 6)
 Analyze and solve linear equations and pairs of simultaneous
linear equations. (Standards 7 and 8)
Functions (F)
 Define, evaluate, and compare functions. (Standards 1, 2 and 3)
 Use functions to model relationships between quantities.
(Standards 4 and 5)
Geometry (G)
 Understand congruence and similarity using physical models,
transparencies, or geometry software. (Standards 1, 2, 3, 4, and
5)
 Understand and apply the Pythagorean Theorem. (Standards 6,
7, and 8)
 Solve real-world and mathematical problems involving volume of
cylinders, cones and spheres. (Standard 9)
Statistics and Probability (SP)
 Investigate patterns of association in bivariate data. (Standards
1, 2, 3, and 4)
*Numbers following cluster heading indicate standards included in the cluster.
Tucson Unified School District
Mathematics Curriculum
Grade 8
Board Approved 03/27/2012
1
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 xcoordinate 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.
Tucson Unified School District
Mathematics Curriculum
Grade 8
Board Approved 03/27/2012
2
Standards for Mathematical Practice
Standards
Students are expected to:
Explanations and Examples
Mathematical Practices are
listed throughout the grade
level document in the 2nd
column to reflect the need to
connect the mathematical
practices to mathematical
content in instruction.
8.MP.1. Make sense of
problems and persevere in
solving them.
8.MP.2. Reason abstractly
and quantitatively.
8.MP.3. Construct viable
arguments and critique the
reasoning of others.
8.MP.4. Model with
mathematics.
8.MP.5. Use appropriate
tools strategically.
In grade 8, students solve real world problems through the application of algebraic and
geometric concepts. Students seek the meaning of a problem and look for efficient ways to
represent and solve it. They may check their thinking by asking themselves, “What is the most
efficient way to solve the problem?”, “Does this make sense?”, and “Can I solve the problem in a
different way?”
In grade 8, students represent a wide variety of real world contexts through the use of real
numbers and variables in mathematical expressions, equations, and inequalities. They examine
patterns in data and assess the degree of linearity of functions. Students contextualize to
understand the meaning of the number or variable as related to the problem and decontextualize
to manipulate symbolic representations by applying properties of operations.
In grade 8, students construct arguments using verbal or written explanations accompanied by
expressions, equations, inequalities, models, and graphs, tables, and other data displays (i.e.
box plots, dot plots, histograms, etc.). They further refine their mathematical communication
skills through mathematical discussions in which they critically evaluate their own thinking and
the thinking of other students. They pose questions like “How did you get that?”, “Why is that
true?” “Does that always work?” They explain their thinking to others and respond to others’
thinking.
In grade 8, students model problem situations symbolically, graphically, tabularly, and
contextually. Students form expressions, equations, or inequalities from real world contexts and
connect symbolic and graphical representations. Students solve systems of linear equations and
compare properties of functions provided in different forms. Students use scatterplots to
represent data and describe associations between variables. Students need many opportunities
to connect and explain the connections between the different representations. They should be
able to use all of these representations as appropriate to a problem context.
Students consider available tools (including estimation and technology) when solving a
mathematical problem and decide when certain tools might be helpful. For instance, students in
grade 8 may translate a set of data given in tabular form to a graphical representation to
compare it to another data set. Students might draw pictures, use applets, or write equations to
show the relationships between the angles created by a transversal.
Tucson Unified School District
Mathematics Curriculum
Grade 8
Board Approved 03/27/2012
3
Standards for Mathematical Practice
Standards
Students are expected to:
Explanations and Examples
Mathematical Practices are
listed throughout the grade
level document in the 2nd
column to reflect the need to
connect the mathematical
practices to mathematical
content in instruction.
8.MP.6. Attend to
precision.
8.MP.7. Look for and make
use of structure.
8.MP.8. Look for and
express regularity in
repeated reasoning.
In grade 8, students continue to refine their mathematical communication skills by using clear
and precise language in their discussions with others and in their own reasoning. Students use
appropriate terminology when referring to the number system, functions, geometric figures, and
data displays.
Students routinely seek patterns or structures to model and solve problems. In grade 8, students
apply properties to generate equivalent expressions and solve equations. Students examine
patterns in tables and graphs to generate equations and describe relationships. Additionally,
students experimentally verify the effects of transformations and describe them in terms of
congruence and similarity.
In grade 8, students use repeated reasoning to understand algorithms and make generalizations
about patterns. Students use iterative processes to determine more precise rational
approximations for irrational numbers. During multiple opportunities to solve and model
problems, they notice that the slope of a line and rate of change are the same value. Students
flexibly make connections between covariance, rates, and representations showing the
relationships between quantities.
Tucson Unified School District
Mathematics Curriculum
Grade 8
Board Approved 03/27/2012
4
Grade 8
The Number System (NS)
8.NS Know that there are numbers that are not rational, and approximate them by rational numbers. (Cluster 1- Standards 1
and 2)
Essential Concepts
Essential Questions
 The real numbers system contains both rational and irrational numbers.
The set of rational numbers contain subsets of numbers that build on
each other.
 Every rational number can be written as a ratio of two quantities
a 
 and as a decimal.
b 





What types of numbers are in the real number system?
How are the subsets of the rational numbers related to each other?
What is the difference between a rational and irrational number?
How do you estimate the value of an irrational number to help you
solve problems?
 Every real number has a decimal expansion; rational numbers have a
decimal expansion that will either terminate or repeat, where as
irrational numbers have a decimal expansion that will not terminate or
repeat.
 Square roots of perfect squares are rational numbers; where as square
roots of non-perfect squares are irrational numbers.
 Irrational numbers (such as  or 2 ) are estimated using truncated
decimal expansions, in order to be able to compare and place them on
a number line in order from least to greatest.
8.NS.1
Standard 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.
Mathematical
Practices
Examples & Explanations

Students can use graphic organizers to show the relationship between the subsets of the
real number system.
8.MP.2. Reason
abstractly and
quantitatively.
8.MP.6. Attend to
precision.
8.MP.7. Look for
and make use of
structure.
Connections: 8.EE.4;
8.EE.7b; 6-8.RST.4;
6-8.RST.7
Continued on next page
Tucson Unified School District
Mathematics Curriculum
Grade 8
Board Approved 03/27/2012
5

Students convert the fraction
2
to a decimal and determine if the number is rational or
3
0.66
2
 32.00  0.66 so this is rational because it repeats.
3

Examples & Explanations
irrational.
8.NS.2
Standard 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.
Connections: 8.G.7; 8.G.8; 68.RST.5;
ET08-S1C2-01
Mathematical
Practices
8. MP.2. Reason
abstractly and
quantitatively.
Students approximate square roots by iterative processes.

Examples:

8.MP.4. Model
with mathematics.
5 to the nearest hundredth.
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 iterative process with the tenths place value.
8.MP.7. Look for
and make use of
structure.
8.MP.8. Look for
and express
regularity in
repeated
reasoning.
Approximate the value of
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 
5 is between 2.23
and 2.24 since 2.232 is 4.9729 and 2.242 is 5.0176.
 

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.
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:
√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
Continued on next page
Tucson Unified School District
Mathematics Curriculum
Grade 8
Board Approved 03/27/2012
6
Additional Domain Information – The Number System (NS)
Key Vocabulary





Decimal expansion
Estimate
Fraction
Integer
Irrational number










Iterative process
Natural Number
Number line
Perfect Square
Ratio
Rational number
Real Number
Repeating decimal
Square (x2)
Square root



Subset
Terminating decimal
Whole Number
Example Resources

Books
 HOLT (4.6, 4.7)
 Math In Context (Revisiting Numbers)
 Looking for Pythagoras CMP Lesson 2
In this investigation, students will explore the relationship between the side lengths and areas of squares and use that relationship to
find the lengths of segments on dot grids.

Technology
 Khan academy
 My.hrw.com
 http://www.internet4classrooms.com/grade_level_help/number_rational_irrational_math_eighth_8th_grade.htm
Several links provide students with rational and irrational numbers and rational number subgroups. (Once link is opened, click
“Impatient?” to open.)
 http://www.quia.com/cm/110177.html
Students can sort numbers into rational and irrational numbers.
 http://tulyn.com/8th-grade-math/irrational-numbers
This website has videos on irrational numbers.

Exemplary Lessons
 http://www.lessonplanspage.com/mathcilaidentifyingandorderingrationalandirrationalnumbers8-htm/
An Excel spreadsheet is used in this rational and irrational number lesson. This lesson plan is part of a larger unit covering number
sense and operations. The students have demonstrated proficiency at addition, subtraction, multiplication, and division of integers,
decimals, and fractions. The objectives addressed in this lesson lead directly into the following objectives of identifying and ordering
rational and irrational numbers. Students will; Learn definitions of rational numbers, terminating decimals, and repeating decimals,
Express fractions as terminating or repeating decimals, Order rational numbers.

http://alex.state.al.us/lesson_view.php?id=24079
Students will participate in an activity that will help them obtain a feeling of how numbers are related to each other. They will determine
whether a number is rational or irrational. Students will also be able to order rational and irrational numbers on a number line. Students
will determine if a number is rational or irrational. Students will be able to order rational and irrational numbers on a number line.
Students will be able to locate square roots on a number line.

http://www.homeschoolmath.net/teaching/irrational_numbers.php Most children learn about Pi and square roots somewhere during the
middle school. They may hear said 'irrational number' and some even remember the phrase, but very few really understand what it
means. Well, irrational numbers are harder to understand than rational numbers, but I consider it worth the time and effort because
they have some fascinating properties. I just have to wonder in awe when I see the facts laid in front of me because it all sounds
Tucson Unified School District
Mathematics Curriculum
Grade 8
Board Approved 03/27/2012
7
unbelievable, yet proven true.
Assessment
All assessments for this domain must focus on assessing the degree, to which seventh grade students can, within the parameters specified in the 3
standards included in this domain,
 know that there are numbers that are not rational, and approximate them by rational numbers.
Both formative and summative assessments are vital components of effective mathematics curricula. Formative assessments, (e.g., pre-assessments,
daily checks for understanding, discussions of strategies students use to solve problems, etc.) assist in instructional planning and implementation;
summative assessments (e.g., unit assessments, quarterly benchmarks, etc.) inform learner growth related to important mathematics concepts. All
district-adopted resources contain multiple assessment tools and include online resources that can be used for the purposes delineated above.
PARCC also will provide two end of the year summative assessments. The first, a performance-based assessment, will focus on applying skills,
concepts, and understandings to solve multi-step problems requiring abstract reasoning, precision, perseverance, and strategic use of tools. The
performance measure will be administered as close to the end of the school year as possible. The second, an end of the year machine-scorable
summative assessment will be administered after approximately 90% of the school year. These assessments are to begin during the SY 2014-2015.
Common Student Misconceptions
Students fail to realize a ratio can be a fraction and all fractions are ratios: Explain to students that both ratios and fractions compare two
quantities. However a ratio can express a part-to-whole comparison or a part-to-part comparison, where as a fraction is always a part-to-whole
comparison.
Students believe repeating decimals are irrational: Show the student examples of fractions that are repeating decimals, such as 1/3. Because the
repeating decimal can be written as a fraction, by definition, makes it a rational number.
Students commonly believe that decimals with a pattern are rational. For example, students often think 0.010010001… is rational. This decimal is
great example to show students. They know what is coming next, however it does not repeat and therefore is irrational.
Students often misunderstand the relationship between rational numbers and its subgroups. Show examples of natural numbers, whole
numbers, and integers and how they relate to each other. The graphic organizer of nested subsets or a foldable is a powerful way to show the
relationships.
Students do not see the fraction bar as an operation: A fraction is a division problem and a division problem can be represented with a fraction.
Students do not see irrational numbers as numbers that do not repeat or terminate. Some students are surprised that the decimal representation
of Pi does not repeat. Some students believe that if only we keep looking at digits farther to the right, eventually a pattern will emerge. A few irrational
numbers are given special names (Pi and e), and much attention is given to square root of 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 number system. Using square roots of non perfect squares is a powerful way to explain this.
Tucson Unified School District
Mathematics Curriculum
Grade 8
Board Approved 03/27/2012
8
Grade 8
Expressions and Equations (EE) (3 Clusters)
8.EE Work with radicals and integer exponents(Cluster 1-Standards 1,2,3, and 4)
Essential Concepts
Essential Questions
 Properties of integer exponents are used to simplify and create
equivalent forms of numerical expressions.
 The inverse operation of squaring a number is finding the square root.
 The inverse operation of cubing a number is finding the cube root.
 Very large and very small numbers are represented using a single digit
times an integer power of 10 (scientific notation).
 Decimal form can be converted to scientific notation and vice-versa.
 Operations and rules for exponents are used to determine the value
and/or compare numbers in both decimal and scientific notation.
 Calculators and computers display scientific notation in different
formats.
8.EE.1
Standard 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.
Mathematical
Practices





Examples:

8.MP.5. Use
appropriate tools
strategically.

8.MP.7. Look for
and make use of
structure.

How do you use properties of integer exponents to simplify a
numerical expression?
Why would you want to simplify a numerical expression with integer
exponents?
What is the relationship between squares and square roots? Cube
and cube roots?
What is scientific notation?
Why would you want to use scientific notation to compare very large
or very small numbers?
How do you convert numbers between scientific notation and decimal
form or vice-versa?
How is scientific notation generated (shown) by technology?
Examples & Explanations
8.MP.2. Reason
abstractly and
quantitatively.
8.MP.6. Attend to
precision.

43
52
43
47

64
25
1
1
 43  7  4  4 

256
44

4-3
1 1 1
1 1
1
= 4-3 ´ 2 = 3 ´ 2 = ´ =
2
5
5 4 5 64 25 16, 000

323–5 = 3–3 = 1/33 = 1/27.
Tucson Unified School District
Mathematics Curriculum
Grade 8
Board Approved 03/27/2012
9
8.EE Work with radicals and integer exponents(Cluster 1-Standards 1,2,3, and 4)
8.EE.2
Standard 8.EE.2 Use
Mathematical
Examples & Explanations
square root and cube root
Examples:
Practices
2
symbols to represent
8.MP.2. Reason
 3  9 and 9   3
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.
Connections: 8.G.7; 8.G.8; 68.RST.4
abstractly and
quantitatively.
 13  1
1
and
    3  
 3
 3  27
2
Solve x  9
2
Solution: x  9
3

8.MP.5. Use
appropriate tools
strategically.

3
1

27
3
3
1
27

1
3
x2 = ± 9
x  3
8.MP.6. Attend to
precision.

8.MP.7. Look for
and make use of
structure.
Solve x  8
3
Solution: x  8
3
x3  3 8
3
x2
Know that √2 is irrational.
8.EE.3
Standard 8.EE.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108 and the
population of the world as
7109, and determine that the
world population is more than
20 times larger.
Mathematical
Practices
8.MP.2. Reason
abstractly and
quantitatively.
8.MP.5. Use
appropriate tools
strategically.
Examples & Explanations

Students have previously worked with powers of 10 and decimal placement in 5th grade. For
example in 5th grade they look at the patterns on placement of the decimal points when a
decimal is multiplied or divided by a positive power of ten.
o Example from 5th grade document page 7
“Students should be able to use the same type of reasoning as above to
explain why the following multiplication and division problem by powers of 10
make sense.

523  10 3  523,000 The place value of 523 is increased by 3
places.
8.MP.6. Attend to
precision.

5.223  102  522.3 The place value of 5.223 is increased by 2
places.

52.3  101  5.23 The place value of 52.3 is decreased by one
place.”
Continued on next page
Tucson Unified School District
Mathematics Curriculum
Grade 8
Board Approved 03/27/2012
10

8.EE.4
Standard 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.
Mathematical
Practices

8.MP.2. Reason
abstractly and
quantitatively.

8.MP.5. Use
appropriate tools
strategically.
Estimate the population of the United States as 3108 and the population of the world as 7109,
and determine that the world population is more than 20 times larger.
Examples & Explanations
Students can convert decimal forms to scientific notation and apply rules of exponents to
simplify expressions.
In working with calculators or spreadsheets, it is important that students recognize scientific
notation. Students should recognize that the output of 2.45E+23 is 2.45 x 10 23 and 3.5E-4 is 3.5
x 10-4. Students enter scientific notation using E or EE (scientific notation), * (multiplication),
and ^ (exponent) symbols.
8.MP.6. Attend to
precision.
Connections: 8.NS.1; 8.EE.1;
ET08-S6C1-03
8.EE Understand the connections between proportional relationships, lines, and linear equations (Cluster 2- Standards
5, 6, and 7)
Essential Concepts
Essential Questions
 A proportional relationship has a constant rate of change (or unit rate),
known as the slope.
 Equations for proportional relationships are linear equations of the form
y=mx, where m is the unit rate or slope.
 Linear equations when graphed are straight lines.
 Proportional relationship when graphed, are straight lines that goes
through the origin.
 Proportional relationships can be compared using graphs, tables, and
equations by analyzing the slopes (unit rates).
 Equations for linear relationship are of the form y=mx, where m is the
unit rate or slope and goes through the origin or y=mx+b for a line
intercepting the vertical axis at b.
 Proportional relationships are a special form of a linear relationship.
Tucson Unified School District
Mathematics Curriculum






How are proportional relationships represented on a graph, a table
and an equation?
How do you compare different proportional relationships represented
on graphs, tables and equations?
What is the difference between a proportional relationship and a linear
relationship on a graph? Equation?
How do you show that the slope is the same between any two distinct
points on a non- vertical line in the coordinate plane?
How can you determine if a linear equation has one solution, infinitely
many or no solutions?
What is the significance of the slope in a linear equation?
Continued on next page
Grade 8
Board Approved 03/27/2012
11
 The slope m is the same between any two distinct points on a nonvertical line in the coordinate plane. (This is shown using similar
triangles.)
 Linear equations in one variable have one solution, infinitely many
solutions or no solutions.
 Linear equations can be expanded and simplified using the distributive
property and combining like terms.
8.EE.5
Standard 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.
Connections: 8.F.2; 8.F.3; 68.RST.7; 6-8.WHST.2b;
SC08-S5C2-01; SC08-S5C205
Mathematical
Practices
8.MP.1. Make
sense of problems
and persevere in
solving them.
8.MP.2. Reason
abstractly and
quantitatively.
8.MP.3. Construct
viable arguments
and critique the
reasoning of
others.
Examples & Explanations

Using graphs of experiences that are familiar to students’ increases accessibility and supports
understanding and interpretation of proportional relationship. Students are expected to both
sketch and interpret graphs.
Example:
 Compare a distance-time graph to a distance-time equation to determine which of two
moving objects has greater speed.

Compare the scenarios to determine which represents a greater speed. Include a
description of each scenario including the unit rates in your explanation.
Scenario 1:
Scenario 2:
Scenario 1:
y = 50x
x is time2:in hours
Scenario
y is distance in
miles
8.MP.4. Model
with mathematics.
8.MP.5. Use
appropriate tools
strategically.
8.MP.6. Attend to
precision.
8.MP.7. Look for
and make use of
structure.
Continued on next page
Tucson Unified School District
Mathematics Curriculum
Grade 8
Board Approved 03/27/2012
12
8.MP.8. Look for
and express
regularity in
repeated
reasoning.
8.EE.6
Standard 8.EE.6
Use similar triangles to
explain why the slope m is
the same between any two
distinct points on a nonvertical 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.
Connections: 8.F.3; 8.G.4; 68.RST.3;
6-8.WHST.1b; ET08-S1C201; ET08-S6C1-03
Mathematical
Practices
8.MP.2. Reason
abstractly and
quantitatively.
8.MP.3. Construct
viable arguments
and critique the
reasoning of
others.
Examples & Explanations
Example:
 If you take two pairs of points on the same line, and then draw the corresponding triangles, the
triangles will be similar. Because the triangles are similar the ratio of side length must therefore
be the same, thus showing that slope between either pair of points is the same.

Explain why ACB is similar to DFE , and deduce that AB has the same slope as BE .
Express each line as an equation.
8.MP.4. Model
with mathematics.
8.MP.5. Use
appropriate tools
strategically.
8.MP.7. Look for
and make use of
structure.
8.MP.8. Look for
and express
regularity in
repeated
reasoning.
Tucson Unified School District
Mathematics Curriculum
Grade 8
Board Approved 03/27/2012
13
8.EE Understand the connections between proportional relationships, lines, and linear equations (Cluster 2- Standards
5, 6, and 7)
8.EE.7
Standard 8.EE.7
Mathematical
Examples & Explanations
Solve linear equations in one

As students transform linear equations in one variable into simpler forms, they discover the
Practices
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.
8.MP.2. Reason
abstractly and
quantitatively.
8.MP.5. Use
appropriate tools
strategically.
8.MP.6. Attend to
precision.
8.MP.7. Look for
and make use of
structure.
Connections: 8.F.3; 8.NS.1;
6-8.RST.3;
ET08-S1C3-01

equations can have one solution, infinitely many solutions, or no solutions.
When the equation has one solution, the variable has one value that makes the equation true
as in 12-4y=16. The only value for y that makes this equation true is -1.

When the equation has infinitely many solutions, the equation is true for all real numbers as in
7x + 14 = 7 (x+2). As this equation is simplified, the variable terms cancel leaving 14 = 14 or 0
= 0. Since the expressions are equivalent, the value for the two sides of the equation will be the
same regardless which real number is used for the substitution.

When an equation has no solutions it is also called an inconsistent equation. This is the case
when the two expressions are not equivalent as in 5x - 2 = 5(x+1). When simplifying this
equation, students will find that the solution appears to be two numbers that are not equal or -2
= 1. In this case, regardless which real number is used for the substitution, the equation is not
true and therefore has no solution.
Examples:
 Solve for x:
o  3( x  7)  4
o
o

3x  8  4x  8
3( x  1)  5  3x  2
Solve:
o 7( m  3)  7
o
1 2
3 1
 y  y
4 3
4 3
Tucson Unified School District
Mathematics Curriculum
Grade 8
Board Approved 03/27/2012
14
8.EE Analyze and solve linear equations and pairs of simultaneous linear equations (Cluster 3- Standard 8)
Essential Concepts
Essential Questions
 The solution to a system of linear equations in two variables is the point/
ordered pair on a graph where the two lines will intersect.
 The solution to a system of linear equations in two variables is the point/
ordered pair that satisfies both equations.
 System of linear questions can be solved algebraically to find the point
of intersection and then checked graphically.
8.EE.8
Standard 8.EE.8
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
inspection. For example,
3x + 2y = 5 and 3x + 2y =
6 have no solution
because 3x + 2y cannot
simultaneously be 5 and
6.
c. Solve real-world and
mathematical problems
leading to two linear
equations in two
variables. For example,
given coordinates for two
Mathematical
Practices
8.MP.1. Make
sense of problems
and persevere in
solving them.
8.MP.2. Reason
abstractly and
quantitatively.
8.MP.3. Construct
viable arguments
and critique the
reasoning of
others.
8.MP.4. Model
with mathematics.
8.MP.5. Use
appropriate tools
strategically.



How do you determine the solution to a system of linear equations on
a graph? Algebraically?
Do all system of linear equations have a solution? How do yow know?
What is an example of a real – world problem that can be solved using
a system of linear equations?
How do you determine if a given point is a solution to a system of
linear equations?
Examples & Explanations
A system of linear equations in two variables can be solved by graphing the two equations and
finding the point of intersection. Student may also discover, by graphing, that some systems will
never intersect because the lines are parallel. Therefore these systems will not have a solution.
A system of linear equations can also be solved algebraically using substitution. There are two
ways to solve using substitution.
1.) Given two equations equal to the same variable.
For example: C = 2x + 7 and C = 3x + 5
Set these two equations equal to each other and solve for the unknown (in this case x.)
2x+7 = 3x + 5
2.) Given two equations in which one equation is solved for a given variable.
For example: x = 7 + y and 3x + 2y = 6
Substitute 7+ y in for the x in 3x+2y = 6 that is 3(7+y) + 2y = 6 and solve for y.
Once you have solved for one variable, plug that answer into one of the original equations
to
find the missing coordinate in the ordered pair. Every solution to a system of linear
equations
should be given as an ordered pair with two variables.
To check whether a given a point/ ordered pair is a solution to a system of linear equations,
substitute the values in each equation and simplify to be sure the statements are true.
Example: Check whether (4, -3) is solution to the system x = 7 + y and 3x+2y = 6
3x  2y  6
x  7 y
8.MP.6. Attend to
precision.
That is
8.MP.7. Look for
and make use of
structure.
Tucson Unified School District
Mathematics Curriculum

4  7  (3)
44
True

15 
3(4)  2(3)  6
and
12  6  6
Therefore (4,-3) is a solution to the system!
66
True
Grade 8
Board Approved 03/27/2012
pairs of points, determine
whether the line through
the first pair of points
intersects the line through
the second pair.
8.MP.8. Look for
and express
regularity in
repeated
reasoning.
PLEASE NOTE! “The standard does say that students should solve systems algebraically,
although it doesn't specify a method, so I wouldn't say they have to know both substitution and
elimination, and a full-blown treatment comes later.” Bill McCallum (chief author of CCSS).
Connections: 6-8.RST.7;
ET08-S1C2-01;
ET08-S1C2-02
Continued on next page
Examples:

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.

Given coordinates for two pairs of points, determine whether the line through the first pair of
points intersects the line through the second pair.

Plant A and Plant B are on different watering schedules. This affects their rate of growth.
Compare the growth of the two plants to determine when their heights will be the same.
Let W = number of weeks
Let H = height of the plant after W weeks
W
0
1
2
3

Plant A
H
4
6
8
10
(0,4)
(1,6)
(2,8)
(3,10)
W
0
1
2
3
Plant B
H
2
6
10
14
(0,2)
(1,6)
(2,10)
(3,14)
Given each set of coordinates, graph their corresponding lines.
Solution:
Tucson Unified School District
Mathematics Curriculum
Grade 8
Board Approved 03/27/2012
16
Continued on next page

Write an equation that represent the growth rate of Plant A and Plant B.
Solution:
Plant A H = 2W + 4
Plant B H = 4W + 2

At which week will the plants have the same height?
Solution:
The plants have the same height after one week.
Plant A: H = 2W + 4
Plant B: H = 4W + 2
Plant A: H = 2(1) + 4
Plant B: H = 4(1) + 2
Plant A: H = 6
Plant B: H = 6
After one week, the height of Plant A and Plant B are both 6 inches.
Additional Domain Information – Expressions and Equations (EE)
Key Vocabulary (BOLD means most important)



Base
Cube
Cube Root



Exponent
Exponential form
Perfect Square



Power
Radical
Scientific notation


Square
Square root
Example Resources

Book

HOLT (Chapter 4.1-4.5, 11 and 12 and extension)
Tucson Unified School District
Mathematics Curriculum
Grade 8
Board Approved 03/27/2012
17



Math In Context (Revisiting Numbers, Algebra Rules!, Graphing Equations)
CMP2 (8th grade: Looking or Pythagoras inv 2, 7th grade: Moving Straight Ahead)
Algebra 1 book (8.1-8.4,3.1-3.4, chapter 4-5 and 6.1)

Technology
 http://nlvm.usu.edu/en/nav/category_g_3_t_2.html
Virtual manipulatives for exponents and equations under algebra.
 http://www.geogebra.org/cms/
Mathematics software for learning and teaching.
 http://www.ixl.com/math/grade-8
Practice skills for exponents and roots, scientific notation, single variable equations, and linear functions.
 http://www.mrbartonmaths.com/algebra.htm
Collecting Like Terms/Simplifying, Indices, Misconceptions,
Solving Equations, Straight Line Graphs, Substitution, Writing Expressions.
 https://eee.uci.edu/wiki/index.php/*Algebra_and_Functions
Common misconceptions in algebra.

Exemplary Lessons
 http://usablealgebra.landmark.edu/designing-evaluating-resources/sample-lesson-plan-exponents/
This exponent lesson plan provides goals, lessons and methods with online resources for students. In the lesson students will perform
operations using real numbers and variables; add, subtract, multiply, and divide signed numbers, write numbers in exponent form and
evaluate numerical expressions that contain exponents.

http://www.curriki.org/xwiki/bin/view/Coll_MSP2/AlgebraicThinkingABasicSkill
The resources highlighted here aim to reflect students' growing mathematical capacity over the span of the middle school years. The
activities and lessons, intended as supplementary materials, range from introduction to the fundamentals of algebra to work on linear
functions. Uniformly, they take into consideration the preference of the middle school student for concrete models, visual representations,
and interactive tasks. You will find resources on: working with algebraic expressions, solving equations, understanding graphs, moving
from patterns to rules to functions. Some are games, others are online simulations that can complement a lesson, and yet others are full-


blown lesson plans. We believe you will find tasks here that motivate your students to expand their basic skills in algebra.
http://www.beaconlearningcenter.com/lessons/1669.htm
This scientific notation lesson plan includes attached materials and extension lessons. It is hard to envision the distance to the Moon
without thinking about a very large number. Yet, a lunar dust particle is so small, several fit on the tip of a pinhead! Students explore the
extreme solving problems related to the Apollo space missions.
http://www.realworldmath.org/Real_World_Math/Lessons.html
Teachers choose plans by lesson type which uses Google Earth. Concept lessons will use Google Earth to present math topics, such as
rates or scientific notation in unique ways.
Project-Based Learning activities will include lessons that will require the collaborative efforts of students in pairs or groups. These
lessons may be of a longer duration and require additional outsource materials. Measurement lessons will make extensive use of the
Tucson Unified School District
Mathematics Curriculum
Grade 8
Board Approved 03/27/2012
18
ruler tool in Google Earth to accomplish problem solving activities. Exploratory lessons will follow non-traditional math topics such as
fractals, topology, or modern geometry.
Assessments
All assessments for this domain must focus on assessing the degree, to which seventh grade students can, within the parameters specified in the 3
standards included in this domain,
 work with radicals and integer exponents,
 demonstrate an understanding of the connections between proportional relationships, lines, and linear equations, and
 analyze and solve linear equations and pairs of simultaneous linear equations.
Both formative and summative assessments are vital components of effective mathematics curricula. Formative assessments, (e.g., pre-assessments,
daily checks for understanding, discussions of strategies students use to solve problems, etc.) assist in instructional planning and implementation;
summative assessments (e.g., unit assessments, quarterly benchmarks, etc.) inform learner growth related to important mathematics concepts. All
district-adopted resources contain multiple assessment tools and include online resources that can be used for the purposes delineated above.
PARCC also will provide two end of the year summative assessments. The first, a performance-based assessment, will focus on applying skills,
concepts, and understandings to solve multi-step problems requiring abstract reasoning, precision, perseverance, and strategic use of tools. The
performance measure will be administered as close to the end of the school year as possible. The second, an end of the year machine-scorable
summative assessment will be administered after approximately 90% of the school year. These assessments are to begin during the SY 2014-2015.
Common Student Misconceptions
Students commonly think that an exponent means to multiply the base times the exponent. For example, students think that 32 = 3x2 as opposed
3
to 3x3 or 23 to be 2 x 3 instead of (2)(2)(2). Students need to understand the fact that, x , means x x x, means x times x times x, not 3x or 3 times x.
2
Students mistakenly apply the sign of the exponent to the base, that is students think x -n = -xn . For example, students believe 3
1
2
as 3 or -9 instead of 2 132 or 19. Patterns are a great way to show theprogression from positive exponents to negative exponents.
3
to be the same
 a  symbol. For example,
Students do not understand that square root problems have two possible answers and this should be denoted using

25 , there are two possible answers, 5 or -5. So when solving a problem such as x 2  25 students
 square root of both sides leaving x   25 . By using the  they are showing they understand that there are two possible answers, one
should take the
positive and one negative.


Students commonly confuse the product
with exponents,
 of powers property and the power of a power property. When multiplying variables
2
3 is equivalent to x6 as
students misinterpret the operation of the
variables
as
the
operation
of
the
exponents.
For
example,
students
believe
that
x
⋅
x


if students are asked to find the square root of 25,
opposed to x5.
Tucson Unified School District
Mathematics Curriculum
Grade 8
Board Approved 03/27/2012
19
Students believe that in scientific notation, when a number contains one nonzero digit and a positive exponent, the number of zeros equals
the exponent. For example, 2,300,000 – 2.3 x 105. This pattern may incorrectly be applied to scientific notation values with negative values or with more
than one nonzero digit.
Students commonly believe that the E on a calculator display means an error as opposed to an exponent in scientific notation.
Students believe that only the letters x and y can be used for variables: Explain to students the definition of a variable. Tell them that, while any
variable can be used, some are more common than others.
Students think that variables always have one specific value. Students solve one equation for x and find that x= 2. Then they think that x will always
be 2. Students need to understand that variables can change value based on a given problem. For example, in x + 5 =10, x =5 but in x + 3 = 4 now x =1.
Another example; p + q = 12; p and q have multiple values based what number type is being utilized.
Students believe the variable is always on the left side of the equation: Students think that you always need a variable = a constant as a solution.
Providing several examples with the variable on the left and right side of the equation will help students to recognize that it does not matter which side the
variable is on.
Students commonly combine unlike terms such as 5x+ 2y. Student may answer the question as 7xy. Here they do not understand the concept of like
terms and the order of operations. Another common error is to say that 4x+6 = 10x. Again this stems from not understanding that the multiplication
needs to be done before the addition.
When students are simplifying expression with the distributive property, students commonly don’t multiply and show x x  x . For example,
given the problem 2x(5x+3) student commonly give the answer 10x +6x as opposed to 10x 2 + 6x.
2
As students begin to build and work with expressions containing more than two operations, students tend to set aside the order of
operations. For example, having a student simplify an expression like 8 + 4(2x - 5) + 3x can bring to light several misconceptions.
Do the students

immediately add the 8 and 4 before distributing the 4? Do they only multiply the 4 and the 2x and not distribute the 4 to both terms in the parenthesis? Do
they collect all like terms?
Students commonly forget that the negative sign in front of a variable is really a coefficient of -1. For example, –x is really -1x.
Students may think that the notation 4x means forty-something. For example, when evaluating 3x + 4 if x = 5, substitution does not result in the
expression meaning 35 +4, students need to understand that 3x means x multiplied by 3 or x+x+x. Using parentheses when substituting in the value can
prevent some of this confusion. For example, 3(5) + 4 = 15+4 =19.
Students forget to combine like numbers to write the equation in slope intercept form: Equations are not always in the slope intercept form,
y=mx+b. For example, 4x +5x + 9 +1 = y can be written as 9x +10 = y.
Students confuse one-variable and two-variable equations. Students should be able to plug an ordered pair into two-variable equations to see if the
equation is true, or to compare it to another equation to see if they are both equal.
Tucson Unified School District
Mathematics Curriculum
Grade 8
Board Approved 03/27/2012
20
Functions (F) (2 Clusters)
8.F Define, evaluate, and compare functions (Cluster 1- Standards 1, 2, and 3)
Essential Concepts
Essential Questions
 A function is a rule that assigns each input exactly one output.
 A graph of an equation is also the graph of that function consisting of
inputs and the corresponding outputs.
 Functions can be represented algebraically, graphically, numerically in
tables or by verbal descriptions.
 Linear functions are represented by the equation y=mx+b and a straight
line on a graph.
8.F.1
Standard 8.F.1
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.
(Function notation is not
required in Grade 8.)
Mathematical
Practices
8.MP.2. Reason
abstractly and
quantitatively.
8.MP.6. Attend to
precision.
Connection: SC08-S5C2-05




Given a table or graph, how do determine if it represents a function?
How do you compare properties of functions that are represented in
different ways?
How do you determine if an equation represents linear function?
Give an example of a function that is not a linear function; explain why
it is a function but not a linear function.
Examples & Explanations
For example, the rule that takes x as input and gives x2+5x+4 as output is a function. Using y to
stand for the output we can represent this function with the equation y = x2+5x+4, and the graph of
the equation is the graph of the function. Students are not yet expected use function notation such
as
f(x) = x2+5x+4.
Example: Determine which if the following tables represent a function and explain why.
A
B
Input
Output
Input
Output
0
1
0
0
1
2
1
2
2
2
1
3
3
4
4
5
Solution: A represents a function because for each input there is exactly one output.
B doe NOT represent a function because the input 1 has two outputs (2 and 3).
Explanation:
A vertical line test can be preformed to determine whether a graph represents a function. By
definition a function, each x value (input) of a function can have only one y value (output). If a
vertical line is drawn for an x value then that line can only hit the graph at one point (that is one
output).
Continued on next page
Tucson Unified School District
Mathematics Curriculum
Grade 8
Board Approved 03/27/2012
21
Example:
Determine if the graph represents a function:
A
B
C
Solution:
A & B are functions because the vertical line only hits the graph at one point no matter where you
draw the line
C is not a function because the vertical line hits the graph in two points.
Continued on next page
Tucson Unified School District
Mathematics Curriculum
Grade 8
Board Approved 03/27/2012
22
8.F Define, evaluate, and compare functions (Cluster 1- Standards 1, 2, and 3)
8.F.2
Standard 8.F.2
Mathematical
Examples & Explanations
Compare properties of two
Examples:
Practices
functions each represented in
a different way (algebraically,
graphically, numerically in
tables, or by verbal
descriptions). For example,
given a linear function
represented by a table of
values and a linear function
represented by an algebraic
expression, determine which
function has the greater rate
of change.
Connections: 8.EE.5; 8.F.2;
6-8.RST.7;
6-8.WHST.1b; ET08-S1C3-01
8.MP.1. Make
sense of problems
and persevere in
solving them.

Given a linear function represented by a table of values and a linear function represented by an
algebraic expression, determine which function has the greater rate of change.

Compare the two linear functions listed below and determine which equation represents a
greater rate of change.
8.MP.2. Reason
abstractly and
quantitatively.
Function 1:
Function 2:
8.MP.3. Construct
viable arguments
and critique the
reasoning of
others.
The function
whose input x and
output y are
related by
y = 3x + 7
8.MP.4. Model
with mathematics.
8.MP.5. Use
appropriate tools
strategically.
8.MP.6. Attend to
precision.
8.MP.7. Look for
and make use of
structure.
8.MP.8. Look for
and express
regularity in
repeated
reasoning.

Compare the two linear functions listed below and determine which has a negative slope.
Function 1: Gift Card
Samantha starts with $20 on a gift card for the book store. She spends $3.50 per week to buy a
magazine. Let y be the amount remaining as a function of the number of weeks, x.
x
0
1
2
3
4
Continued on next page
Tucson Unified School District
Mathematics Curriculum
y
20
16.50
13.00
9.50
6.00
Grade 8
Board Approved 03/27/2012
23
Function 2:
The school bookstore rents graphing calculators for $5 per month. It also collects a non-refundable
fee of $10.00 for the school year. Write the rule for the total cost (c) of renting a calculator as a
function of the number of months (m).
Solution:
Function 1 is an example of a function whose graph has negative slope. Samantha starts with $20
and spends money each week. The amount of money left on the gift card decreases each week.
The graph has a negative slope of -3.5, which is the amount the gift card balance decreases with
Samantha’s weekly magazine purchase. Function 2 is an example of a function whose graph has
positive slope. Students pay a yearly nonrefundable fee for renting the calculator and pay $5 for
each month they rent the calculator. This function has a positive slope of 5 which is the amount of
the monthly rental fee. An equation for Example 2 could be c = 5m + 10.
8.F.3
Standard 8.F.3
Interpret the equation y = mx
+ b as defining a linear
function, whose graph is a
straight line; give examples of
functions that are not linear.
For example, the function A =
s2 giving the area of a square
as a function of its side length
is not linear because its graph
contains the points (1,1), (2,4)
and (3,9), which are not on a
straight line.
Connections: 8.EE.5; 8.EE.7a
; 6-8.WHST.1b; ET08-S6C103
Mathematical
Practices
8.MP.2. Reason
abstractly and
quantitatively.
8.MP.4. Model
with mathematics.
8.MP.5. Use
appropriate tools
strategically.
Examples & Explanations
Example:
 The function A = s2 giving the area of a square as a function of its side length is not linear
because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line.

Determine which of the functions listed below are linear and which are not linear and explain
your reasoning.
o
o
o
o
y = -2x2 + 3
y = 2x
A = πr2
y = 0.25 + 0.5(x – 2)
non linear
linear
non linear
linear
8.MP.6. Attend to
precision.
8.MP.7. Look for
and make use of
structure.
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8.F Use functions to model relationships between quantities (Cluster 2- Standards 4 and 5)
Essential Concepts
Essential Questions
 Linear functions are functions that have a constant rate of change
(slope) and an initial value.
 The initial value of a linear function is the place where the line will
intersect the vertical axis or the y-intercept.
 Linear functions are represented as verbal descriptions, tables, graphs
and equations that are all related by the same rate of change (slope)
and initial value.
 Real world functional relationships between two quantities can be
represented using verbal descriptions and graphs.
8.F.4
Standard 8.F.4
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.
Connections: 8.EE.5; 8.SP2;
8.SP.3;
ET08-S1C2-01; SC08-S5C201;
SC08-S1C3-02
Mathematical
Practices
8.MP.1. Make
sense of problems
and persevere in
solving them.
8.MP.2. Reason
abstractly and
quantitatively.
8.MP.3. Construct
viable arguments
and critique the
reasoning of
others.
8.MP.4. Model
with mathematics.





What are the attributes of a function that tell you it is a linear function?
How do you find the rate of change and initial value of a linear function
from the verbal description, table, graph or equation?
How do write an equation for a linear function given a table? Graph?
Verbal description?
How do you determine if an equation, a table and a graph are related?
Why would you want to represent a real world functional relationship
with a graph?
Examples & Explanations
Examples:
 The table below shows the cost of renting a car. The company charges $45 a day for the
car as well as charging a one-time $25 fee for the car’s navigation system (GPS).Write an
expression for the cost in dollars, c, as a function of the number of days, d.
Students might write the equation c = 45d + 25 using the verbal description or by first
making a table.
Days (d)
1
2
3
4
Cost (c) in dollars
70
115
160
205
Students should recognize that the rate of change is 45 (the cost of renting the car) and that initial
cost (the first day charge) also includes paying for the navigation system. Classroom discussion
about one time fees vs. recurrent fees will help students model contextual situations.
8.MP.5. Use
appropriate tools
strategically.
Continued on next page
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8.MP.6. Attend to
precision.
8.MP.7. Look for
and make use of
structure.
8.MP.8. Look for
and express
regularity in
repeated
reasoning.
8.F.5
Standard 8.F.5
Describe qualitatively the
functional relationship
between two quantities by
analyzing a graph (e.g.,
where the function is
increasing or decreasing,
linear or nonlinear). Sketch a
graph that exhibits the
qualitative features of a
function that has been
described verbally.
Connections: 6-8.WHST.2a-f;
ET08-S1C2-01; SC08-S5C205
Mathematical
Practices
8.MP.2. Reason
abstractly and
quantitatively.
8.MP.3. Construct
viable arguments
and critique the
reasoning of
others.

When scuba divers come back to the surface of the water, they need to be careful not to
ascend too quickly. Divers should not come to the surface more quickly than a rate of 0.75 ft
per second. If the divers start at a depth of 100 feet, the equation d = 0.75t – 100 shows the
relationship between the time of the ascent in seconds (t) and the distance from the surface in
feet (d).
o Will they be at the surface in 5 minutes? How long will it take the divers to surface from
their dive?
Make a table of values showing several times and the corresponding distance of the divers from
the surface. Explain what your table shows. How do the values in the table relate to your equation?
Examples & Explanations
Example:

The graph below shows a student’s trip to school. This student walks to his friend’s house
and, together, they ride a bus to school. The bus stops once before arriving at school.
Describe how each part A-E of the graph relates to the story.
8.MP.4. Model
with mathematics.
8.MP.5. Use
appropriate tools
strategically.
8.MP.6. Attend to
precision.
8.MP.7. Look for
and make use of
structure.
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Additional Domain Information – Functions (F)
Key Vocabulary (BOLD means most important)




Domain
Function
Input
Linear function







Nonlinear function
Output
Point Slope Form
Range
Slope Intercept Form
Slope/Rate of Change
Vertical line test


X Intercept
Y Intercept
Example Resources

Books




HOLT (Chapter 3.3-3.5,13.4)
Math In Context (Ups and Downs, Algebra Rules!, Patterns and Figures)
CMP2 (7th grade: Moving Straight Ahead)
Algebra 1 book (1.6-1.7, 4.2-4.3,5.2)

Technology
 http://www.jonathanwray.com/resources.html#algebra
Provides several useful interactive sites to use with students on functions.
 http://www.geogebra.org/cms/
Create graphs or show examples of functions.
 http://www.learningwave.com/lwonline/algebra_section2/slope3.html
This site explains how to find slope, gives simple examples, and provides problem solving examples.
 http://www.nctm.org/standards/content.aspx?id=26883
Student can modify a rate of change and learn how modifications in that rate affect the linear graph displaying accumulation with real life
scenarios.

Exemplary Lessons
 http://www.curriki.org/xwiki/bin/view/Coll_MSP2/AlgebraicThinkingABasicSkill
After giving several links on background information for teachers, this resource guide provides several links to exemplary resources
under the subtitles: Working with algebraic expressions, Solving equations, Understanding graphs, From patterns to relations to
functions.
 http://www.realworldmath.org/Real_World_Math/Lunar_Sports.html
This lesson combines math and science on a field trip that takes place on the moon. Students will analyze the effects of low gravity
environments on physical activity, measure and record various types of physical activity, and write a function and apply a factor of 6 to
data measurements.
 http://www.shodor.org/interactivate/lessons/ImpossibleGraphs/
This lesson is devoted to impossible graphs. Users of the module can learn to distinguish between possible and impossible graphs of
functions, and to learn why some graphs are impossible. These activities together give a brief lesson that can be completed in as little as
30 minutes class-time, depending on how many teams need to share their ideas. The discovery process takes about 15 minutes, and
each presentation about 5 minutes.
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

http://www.shodor.org/interactivate/lessons/GraphsAndFunctions/
This lesson is designed to introduce students to graphing functions. These activities can be done individually or in teams of as many as
four students. Allow for 2-3 hours of class time for the entire lesson if all portions are done in class.
Alignment 1: 8.F.4 from http://www.illustrativemathematics.org/standards/k8
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.
Assessments
All assessments for this domain must focus on assessing the degree, to which seventh grade students can, within the parameters specified in the 3
standards included in this domain,
 define, evaluate, and compare functions and
 use functions to model relationships between quantities.
Both formative and summative assessments are vital components of effective mathematics curricula. Formative assessments, (e.g., pre-assessments,
daily checks for understanding, discussions of strategies students use to solve problems, etc.) assist in instructional planning and implementation;
summative assessments (e.g., unit assessments, quarterly benchmarks, etc.) inform learner growth related to important mathematics concepts. All
district-adopted resources contain multiple assessment tools and include online resources that can be used for the purposes delineated above.
PARCC also will provide two end of the year summative assessments. The first, a performance-based assessment, will focus on applying skills,
concepts, and understandings to solve multi-step problems requiring abstract reasoning, precision, perseverance, and strategic use of tools. The
performance measure will be administered as close to the end of the school year as possible. The second, an end of the year machine-scorable
summative assessment will be administered after approximately 90% of the school year. These assessments are to begin during the SY 2014-2015.
Common Student Misconceptions:
Students commonly mislabel a table as a function, because they mix up the inputs and outputs. For example, students think that when they see
two of the same outputs, that the table cannot be a function. This is not true. A table is not a function if it has the same input with two different outputs.
Students often view a graph as a literal picture of a situation (iconic representation). When asked what a graph like the one pictured above might
represent, a student might answer that it shows someone traveling up a hill, down the other side, due east along a flat road, and then down a hill.
Conversely, if asked to show graphically a runner’s trip around an oval track, a student might draw a picture of the track (an oval).
Students may misunderstand what the graph represents in context. For example, a line moving up or down on a graph does not necessarily mean
that a person is moving up or down. Students need to make sure they are reading the graph in terms of the labels on the x and y axis. For example a flat
line on a distance vs. time graph means that the person has stopped moving, whereas on a speed vs. time graph a straight line indicates they are not
speeding up or slowing down; they are staying at a constant speed.
Students reverse coordinates when representing the slope. Students should have several opportunities to remember that the formula for slope is rise
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over run. Students have difficulty with slope because they are taught to graph points left or right and then up or down and the slope is backwards. One
way to help students with this misconception is to draw your steps above the line instead of below the line, showing up or down first then left or right.
Students commonly think that vertical lines are linear functions. This is because they are told that if it is a straight line, then it is linear function. But
a vertical line will not pass the vertical line test to show that it is a function.
Some students will mix up x- and y-axes on the coordinate plane, or mix up the ordered pairs. When emphasizing that the first value is plotted on
the horizontal axes (usually x, with positive to the right) and the second is the vertical axis (usually called y, with positive up), point out that this is merely
a convention. It could have been otherwise, but it is very useful for people to agree on a standard customary practice.
Some students do not pay attention to the scale on a graph, assuming that the scale units are always “one”. When making axes for a graph,
some students do not use equal intervals to create the scale.
Students often confuse a recursive rule with an explicit formula for a function. For example, after identifying that a linear function shows an
increase of 2 in the values of the output for every change of 1 in the input, some students will represent the table with the equation y = x + 2, instead of
realizing that this means y = 2x + b. When tables are constructed with increasing consecutive integers for input values, then the distinction between the
recursive and explicit formulas is about whether you are reasoning vertically or horizontally in the table. Both types of reasoning—and both types of
formulas—are important for developing proficiency with functions.
Students mistakenly believe that linear functions (with a constant rate of change) are the only type of functions. Students need to be exposed to
other functions, such as quadratics and exponential functions.
Students commonly do not recognize a constant rate of change when entries in a table are absent. When input values are not increasing
consecutive integers (e.g., when the input values are decreasing, when some integers are skipped, or when some input values are not integers), some
students have more difficulty identifying the pattern and calculating the slope. It is important that all students have experience with such tables, to be sure
that they do not over generalize from the easier examples.
Students frequently attempt to “solve” expressions. Many students add “ = 0 ” to an expression they were asked to simplify, so that they can solve it.
Students need to understand the difference between an expression and an equation.
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Geometry (G) (3 Clusters)
8.G Understand congruence and similarity using physical models, transparencies, or geometry software. (Cluster 1Standards 1, 2, 3, 4, and 5)
Essential Concepts
Essential Questions
 Translating a point, line, line segment or angle does not change any
attributes of that object, it will just move the object to a new location.
 When a point is reflected across a line that reflected point stays the
same distance from the line of reflection as the original point.
 When a line segment or angle is rotated, reflected or translated, the
length of that line segment and measure of the angle will not change.
 A sequence of rotations, reflections, and/or translations to a twodimensional figure will create a congruent two-dimensional figure.
 An image is the figure created by doing a transformation on the preimage (or original object).
 A dilation of a two-dimensional figure will create an image that is a
similar figure to the original by a multiplicative relationship.
 A translated, reflected or rotated two-dimensional figure will create an
image that is a congruent figure to the original.
 Similar figures are produced by doing a sequence of rotations,
reflections, translations AND dilations. The sequence of transformation
must include dilation in order to produce a similar figure.
 Similar figures are figures that have the same angles and proportional
side lengths.
 Parallel lines cut by a transversal will create pairs of angles that are
either congruent or supplementary.
 The relationships between the angles made by parallel lines cut by a
transversal can be used to informally prove that the interior angles of a
triangle will add up to 180o.
8.G.1
Standard 8.G.1
Verify experimentally the
properties of rotations,
reflections, and translations:
a. Lines are taken to lines,
and line segments to line
segments of the same
length.
b. Angles are taken to
angles of the same
Mathematical
Practices
8.MP.4. Model
with mathematics.
8.MP.5. Use
appropriate tools
strategically.







What effect do translating, rotating and reflecting have on a point?
Line? Line segment? Angle?
Why does doing a sequence of rotations, reflections, and translations
create congruent figures?
Why does a dilation create a similar figure as opposed to a congruent
figure?
What effect does a dilation, translation, rotation and reflection have on
a two-dimensional figure on the coordinate plane?
How do you determine if two figures are similar?
How do determine if a pair of angles created by parallel lines cut by a
transversal will be congruent or supplementary?
Describe how parallel lines cut by a transversal can be used to prove
that the interior angles of a triangle will add up to 180o.
Examples & Explanations
Students need multiple opportunities to explore the transformation of figures so that they can
appreciate that points stay the same distance apart and lines stay at the same angle after they
have been rotated, reflected, and/or translated.
Students are not expected to work formally with properties of dilations until high school.
8.MP.6. Attend to
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measure.
Parallel lines are taken to
parallel lines.
precision.
8.MP.7. Look for
and make use of
structure.
8.MP.8. Look for
and express
regularity in
repeated
reasoning.
8.G.2
Standard 8.G.2
Understand that a twodimensional figure is
congruent to another if the
second can be obtained from
the first by a sequence of
rotations, reflections, and
translations; given two
congruent figures, describe a
sequence that exhibits the
congruence between them.
Connections: 6-8.WHST.2b,f;
ET08-S6C1-03
Mathematical
Practices
8.MP.2. Reason
abstractly and
quantitatively.
Examples & Explanations
Examples:
 Is Figure A congruent to Figure A’? Explain how you know.
8.MP.4. Model
with mathematics.
8.MP.6. Attend to
precision.
8.MP.7. Look for
and make use of
structure.

Describe the sequence of transformations that results in the transformation of Figure A to
Figure A’.
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8.G Understand congruence and similarity using physical models, transparencies, or geometry software. (Cluster 1Standards 1, 2, 3, 4, and 5)
8.G.3
Standard 8.G.3
Mathematical
Examples & Explanations
Describe the effect of
A dilation is a transformation that moves each point along a ray emanating from a fixed center, and
Practices
dilations, translations,
rotations, and reflections on
two-dimensional figures using
coordinates.
Connections: 6-8.WHST.2b,f;
ET08-S6C1-03
8.MP.3. Construct
viable arguments
and critique the
reasoning of
others.
multiplies distances from the center by a common scale factor. In dilated figures, the dilated figure
is similar to its pre-image.
Translation: A translation is a transformation of an object that moves the object so that every point
of the object moves in the same direction as well as the same distance. In a translation, the
translated object is congruent to its pre-image.
8.MP.4. Model
with mathematics.
8.MP.5. Use
appropriate tools
strategically.
Example:
has been translated 7 units to the right and 3 units up. To get from A (1,5) to A’ (8,8), move A 7
units to the right (from x = 1 to x = 8) and 3 units up (from y = 5 to y = 8). Points B + C also move in
the same direction (7 units to the right and 3 units up).
8.MP.6. Attend to
precision.
8.MP.7. Look for
and make use of
structure.
Reflection: A reflection is a transformation that flips an object across a line of reflection (in a
coordinate grid the line of reflection may be the x or y axis). In a reflection, the reflected object is
congruent to its pre-image.
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When an object is reflected across the y axis, the reflected x coordinate is the opposite of the preimage x coordinate.
Rotation: A rotated figure is a figure that has been turned about a fixed point. This is called the
center of rotation. A figure can be rotated up to 360˚. Rotated figures are congruent to their preimage figures.
Consider when DEF is rotated 180˚ clockwise about the origin. The coordinates of DEF are
D(2,5), E(2,1), and F(8,1). When rotated 180˚, D' E ' F ' has new coordinates D’(-2,-5), E’(-2,-1)
and F’(-8,-1). Each
coordinate is the opposite of its pre-image.
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8.G Understand congruence and similarity using physical models, transparencies, or geometry software. (Cluster 1Standards 1, 2, 3, 4, and 5)
8.G.4
Standard 8.G.4
Mathematical
Examples & Explanations
Understand that a twoExamples:
Practices
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.
8.MP.2. Reason
abstractly and
quantitatively.
Connections: 8.EE.6; 68.WHST.2b,f;
ET08-S6C1-03; ET08-S1C101
8.MP.6. Attend to
precision.

Is Figure A similar to Figure A’? Explain how you know.

Describe the sequence of transformations that results in the transformation of Figure A to
Figure A’.
8.MP.4. Model
with mathematics.
8.MP.5. Use
appropriate tools
strategically.
8.MP.7. Look for
and make use of
structure.
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8.G Understand congruence and similarity using physical models, transparencies, or geometry software. (Cluster 1Standards 1, 2, 3, 4, and 5)
8.G.5
Standard 8.G.5
Mathematical
Explanations & Examples
Use informal arguments to
Angle relationships that can be explored include but are not limited to:
Practices
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.
Connections: 6-8.WHST.2b,f;
6-8.WHST.1b; ET08-S6C103; ET08-S1C1-01; ET08S1C3-03
8.MP.3. Construct
viable arguments
and critique the
reasoning of
others.
8.MP.4. Model
with mathematics.
8.MP.5. Use
appropriate tools
strategically.


Same-side (consecutive) interior and same-side (consecutive) exterior angles are
supplementary.
Corresponding, Alternate interior angles and alternate exterior angles.
Examples:
 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.

Students can informally prove relationships with transversals.
Show that m 3 + m 4 + m 5 = 180˚ if l and m are parallel lines and t1 & t2 are
transversals.
8.MP.6. Attend to
precision.
1 +  2 + 3 = 180˚. Angle 1 and Angle 5 are congruent because they are
corresponding angles ( 5  1 ). 1 can be substituted for 5 .
8.MP.7. Look for
and make use of
structure.
4  2 : because alternate interior angles are congruent.
 4 can be substituted for  2
Therefore m 3 + m  4 + m 5 = 180˚
Students can informally conclude that the sum of a triangle is 180º (the angle-sum theorem) by
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applying their understanding of lines and alternate interior angles. In the figure below, line x is
parallel to line yz:
Angle a is 35º because it alternates with the angle inside the triangle that measures 35º. Angle c is
80º because it alternates with the angle inside the triangle that measures 80º. Because lines have
a measure of 180º, and angles a + b + c form a straight line, then angle b must be 65 º (180 – 35 +
80 = 65). Therefore, the sum of the angles of the triangle are 35º + 65 º + 80 º
Examples:
 Write and solve an equation to find the measure of angle x.

Write and solve an equation to find the measure of angle x.
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8.G Understand and apply the Pythagorean Theorem (Cluster 2- Standards 6, 7, and 8)
Essential Concepts
Essential Questions
 Pythagorean theorem states that for a right triangle the sum of the
square of the two legs is equal to the square of the hypotenuse. (a2+b2
= c2)
 The converse of the Pythagorean theorem states that if the sum of the
squares of the smaller sides in a triangle equals the square of the third
side, then the triangle must be a right triangle.
 If a triangle is a right triangle, Pythagorean theorem can be used to find
a missing side length or hypotenuse.
 Real world problems in both two and three dimensions that involve right
triangles can be solved using Pythagorean theorem.
 The distance between two points on a coordinate plane can be found by
drawing the vertical and horizontal lines from the points to create a right
triangle and then applying the Pythagorean theorem.
8.G.6
Standard 8.G.6
Explain a proof of the
Pythagorean Theorem and its
converse.
Connections: 6-8.WHST.2a-f;
ET08-S1C2-01






What is a proof of the Pythagorean theorem?
What is a proof of the converse of the Pythagorean theorem?
How do you use Pythagorean theorem to find the missing side length
or hypotenuse of a right triangle?
What is an example of a real world problem that can be solved using
Pythagorean theorem? Describe how you use Pythagorean theorem
to solve the problem.
What is an example of a real world problem that can be solved using
the converse of Pythagorean theorem? Describe how you use
Pythagorean theorem to solve the problem.
How do you use Pythagorean theorem to find the distance between
two points on a coordinate plane?
Mathematical
Practices
Examples & Explanations
8.MP.3. Construct
viable arguments
and critique the
reasoning of
others.
Pythagorean Theorem: Students should verify, using a model, that the sum of the squares of the
legs is equal to the square of the hypotenuse in a right triangle.
8.MP.4. Model
with mathematics.
8.MP.6. Attend to
precision.
8.MP.7. Look for
and make use of
structure.
Image from: myastrologybook.com
Converse of Pythagorean Theorem: Students should also understand that if the sum of the squares
of the 2 smaller legs of a triangle is equal to the square of the third leg, then the triangle is a right
triangle
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8.G Understand and apply the Pythagorean Theorem (Cluster 2- Standards 6, 7, and 8)
8.G.7
Standard 8.G.7
Mathematical
Examples & Explanations
Apply the Pythagorean
Through authentic experiences and exploration, students should use the Pythagorean Theorem to
Practices
Theorem to determine
unknown side lengths in right
triangles in real-world and
mathematical problems in two
and three dimensions.
Connections: 8.NS.2; ET08S2C2-01
8.MP.1. Make
sense of problems
and persevere in
solving them.
solve problems. Problems can include working in both two and three dimensions. Students should
be familiar with the common Pythagorean triplets.
8.MP.2. Reason
abstractly and
quantitatively.
8.MP.4. Model
with mathematics.
8.MP.5. Use
appropriate tools
strategically.
Image from: akhnatonsjournal.org
8.MP.6. Attend to
precision.
8.MP.7. Look for
and make use of
structure.
8.G.8
Standard 8.G.8
Apply the Pythagorean
Theorem to find the distance
between two points in a
coordinate system.
Connections: 8.NS.2; ET08S6C1-03
Mathematical
Practices
8.MP.1. Make
sense of problems
and persevere in
solving them.
Examples & Explanations
Example:

Students will create a right triangle from the two points given (as shown in the diagram
below) and then use the Pythagorean Theorem to find the distance between the two
given points.
8.MP.2. Reason
abstractly and
quantitatively.
8.MP.4. Model
with mathematics.
Tucson Unified School District
Mathematics Curriculum
Grade 8
Board Approved 03/27/2012
38
8.MP.5. Use
appropriate tools
strategically.
8.MP.6. Attend to
precision.
8.MP.7. Look for
and make use of
structure.
8.G Solve real-world and mathematical problems involving volume of cylinders, cones, and spheres (Cluster 3Standard 9)
Essential Concepts
Essential Questions
2
 The volume of a cylinder is the area of the base multiplied the height
 Why is the volume of a cylinder V  r  h ? What does each part of
2
the formula represent in terms of the cylinder?
(circle x height) that is V  r  h .

 The volume of the cone is 1/3 the volume of a cylinder.
1 2
r  h
r  h or V 

3
3

4 3
 The volume of a sphere is V  r
3
2
V


How can you prove that the volume of a cone is 1/3 of the volume of a
cylinder with the same radius and height?
What are examples ofreal world problems that are solved by finding
the volume of a three-dimensional figure such as a cone, cylinder or
sphere?
 The formulas for finding the volume of three-dimensional figures are
used to solve real world problems that involve filling three-dimensional
figures.
8.G.9

Standard 8.G.9
Know the formulas for the
volumes of cones, cylinders,
and spheres and use them to
solve real-world and
mathematical problems.
Connections: 6-8.RST.3; 68.RST.7;
ET08-S2C2-01; ET08-S1C401
Mathematical
Practices
8.MP.1. Make
sense of problems
and persevere in
solving them.
8.MP.2. Reason
abstractly and
quantitatively.
8.MP.3. Construct
viable arguments
Tucson Unified School District
Mathematics Curriculum
Examples & Explanations
Volume of a cylinder: V  r  h
2
1 2
r 2  h
Volume of a Cone: V  r  h or V 
3
3

4 3
Volume of a sphere is V  r
3

Continued on next page

Grade 8
Board Approved 03/27/2012
39
and critique the
reasoning of
others.
Example:

James wanted to plant pansies in his new planter. He wondered how much potting soil
he should buy to fill it. Use the measurements in the diagram below to determine the
planter’s volume.
8.MP.4. Model
with mathematics.
8.MP.5. Use
appropriate tools
strategically.
8.MP.6. Attend to
precision.
8.MP.7. Look for
and make use of
structure.
8.MP.8. Look for
and express
regularity in
repeated
reasoning.
Additional Domain Information – Geometry (G)
Key Vocabulary (BOLD means most important)





Adjacent angles
Alternate interior angles
Cone
Congruent
Corresponding angles





Cylinder
Dilation
Exterior angles
Hypotenuse
Interior angles





Parallel Lines
Perpendicular lines
Prism
Pythagorean
Theorem
Reflection





Right angle
Rotation
Similar
Skew lines
Sphere






Transformation
Translation
Transversal
Triangle Sum
Theorem
Vertical angles
Volume
Example Resources

Book




HOLT (Chapter 7, 8.5-8.6,8.9,4.8, 5.5-5.6)
Math In Context (Patterns and Symbols, It’s All the Same, Looking at Angle)
CMP2 (8th grade: Looking or Pythagoras, Kaleidoscopes, Hubcaps and Mirrors, 7 th grade: Filling and Wrapping)
Algebra 1 book (11.4)
Tucson Unified School District
Mathematics Curriculum
Grade 8
Board Approved 03/27/2012
40


Technology
 Geogebra.org
An interactive geometry intended for teachers and students. Constructions can be made with points, vectors, segments, lines polygons,
conic sections. Teachers and students can use GeoGebra to make conjectures and prove geometric theorems.
 http://www.mathwarehouse.com/solid-geometry/cylinder/formula-volume-of-cylinder.php#
This site examines the properties of right circular cylinder with examples and practice problems
 http://www.calculatoredge.com/enggcalc/volume.html
Online volume calculator
 http://www.brightstorm.com/math/geometry/
Free online geometry video lessons to help students with the formulas, terms and theorems related to triangles, polygons, circles, and
other geometric shapes to improve their math problem solving skills. Titles include: Geometry Building Blocks; Reasoning, Diagonals,
Angles, Parallel Lines; Constructions; Triangles; Transformations; Pythagorean Theorem; Volume; and Similarity
Exemplary Lessons
 http://illuminations.nctm.org/LessonDetail.aspx?id=L791
In this lesson, students will use formulas they have explored for the volume of a cylinder and convert them into the same volume for
rectangular prisms while trying to minimize the surface area. Various real world cylindrical objects will be measured and converted into a
prism to hold the same volume. As an extension, students may design and create a rectangular prism container according to their
dimensions to compare and contrast with the cylinder.
 http://www.pbs.org/teachers/connect/resources/4397/preview/
This lesson on Pythagorean theorem helps students understand the statement of the Pythagorean theorem, apply the Pythagorean
theorem to solve for unknown sides of right triangles, and prove the Pythagorean theorem using the method developed by President
James Garfield.
 http://www.pbs.org/teachers/connect/resources/4356/preview/
This lesson on transformations has students describe three types of symmetry, categorize symmetric figures based on type of symmetry,
and create figures using different type of symmetry
 http://www.misterteacher.com/abc.html
This lesson on transformations has students predict and describe positions and orientations of two-dimensional shapes after
transformations such as reflections, rotations, and translations. It also includes Advanced Transformations where students predict and
describe the results of translations, reflections, and rotations of objects in the coordinate plane.
 http://www.lessonplanspage.com/mathparallellinescutbytransversalproperties910-htm/
This lesson on parallel lines helps students understand and apply the angle theorems related to parallel lines to find given the
measurement angles, provides practice the geometric proof technique to discover learning by comparing angle measurements and
types of angles
Assessments
All assessments for this domain must focus on assessing the degree, to which seventh grade students can, within the parameters specified in the 3
standards included in this domain,
Tucson Unified School District
Mathematics Curriculum
Grade 8
Board Approved 03/27/2012
41



demonstrate an understanding of congruence and similarity using physical models, transparencies, or geometry software,
demonstrate an understanding of and ability to apply the Pythagorean Theorem, and
solve real-world and mathematical problems involving volume of cylinders, cones and spheres.
Both formative and summative assessments are vital components of effective mathematics curricula. Formative assessments, (e.g., pre-assessments,
daily checks for understanding, discussions of strategies students use to solve problems, etc.) assist in instructional planning and implementation;
summative assessments (e.g., unit assessments, quarterly benchmarks, etc.) inform learner growth related to important mathematics concepts. All
district-adopted resources contain multiple assessment tools and include online resources that can be used for the purposes delineated above.
PARCC also will provide two end of the year summative assessments. The first, a performance-based assessment, will focus on applying skills,
concepts, and understandings to solve multi-step problems requiring abstract reasoning, precision, perseverance, and strategic use of tools. The
performance measure will be administered as close to the end of the school year as possible. The second, an end of the year machine-scorable
summative assessment will be administered after approximately 90% of the school year. These assessments are to begin during the SY 2014-2015.
Common Student Misconceptions
Students tend to believe that similar figures will always face the same directions. For example, in the past students have been exposed to similar
figures where the image has not been rotated at all.
It is important the students be exposed to similar figures that are created by a
sequence of transformations, including rotations, reflections, translations and dilations. For example,
Students commonly think that the angles in similar figures are different. Students tend to think that because the side length get bigger or smaller,
so do the angles. Students need opportunities to measure and compare angles in figures created by dilations.
Students do not identify the variables as different legs on a right triangle: There is no connection made between the formula and the definition.
Rather than saying a2 + b2 = c2, it is more useful for them to say (leg)2 + (leg)2 = (hypotenuse)2.
Students commonly don not understand that the squared terms are the squares drawn off of each side of the right triangle. For example,
a2 + b2 = c2 is in reference to the three squares that you draw off the legs and hypotenuse of the right triangle. It is not the sum of the lengths of the legs
that will equal the length of the hypotenuse; it is the area of the squares off the legs that will sum to equal the area of the square off the hypotenuse.
Students believe that the Pythagorean Theorem applies to all triangles. It is important to emphasize that Pythagorean Theorem includes the
conditional statement that if you have a right triangle, then a2+b2 = c2. Students need opportunities to try Pythagorean theorem on all different types of
triangles to determine which types of triangle the theorem holds true for.
Students believe that volume is a number that results from substituting other numbers into a formula rather than a measure related to the
Tucson Unified School District
Mathematics Curriculum
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Board Approved 03/27/2012
42
amount of space occupied. It is important to provide students opportunities for hands on experiences where students fill three dimensional objects.
Students misinterpret coordinates. Students must understand that the x-axis is horizontal and the y-axis is vertical. Students must understand
directions for positive and negative coordinates. Students need to understand that (x, y) is not the same as (y, x).
Students assume the origin is always the center of rotation. The center of rotation should be carefully notated. Students should be exposed to
multiple examples of rotations where the center of rotation is not at the origin.
Students believe “x” and “y” are the only variables in ordered pairs. Use other variables, changing the variables of the axes accordingly.
Tucson Unified School District
Mathematics Curriculum
Grade 8
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43
Statistics and Probability (SP) (1 Cluster)
8.SP Investigate patterns of association in bivariate data (Cluster 1- Standards 1, 2, 3, and 4)
Essential Concepts
Essential Questions
 Data that is collected using two variables is called bivariate data.
 Scatterplots and two-way frequency tables are used to show patterns of
association and relationships between bivariate categorical data.
 Scatterplots can suggest a linear association/relationships.
 If a scatterplot suggests a linear relationship, then a line of best fit can
be drawn and a linear equation can be created to model the relationship
between the bivariate data.
 An equation of a line of best fit can be used to interpret and solve
problems in the context of bivariate measurement data.
8.SP.1
Standard 8.SP.1
Construct and interpret
scatterplots 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.
Connections: 6-8.WHST.2b,f;
ET08-S1C3-01; ET08-S1C302; ET08-S6C1-03;
SS08-S4C1-01;SS08-S4C203;
SS08-S4C1-05; SC08-S1C302;
SC08-S1C3-03
Mathematical
Practices
8.MP.2. Reason
abstractly and
quantitatively.
8.MP.4. Model
with mathematics.
8.MP.5. Use
appropriate tools
strategically.
8.MP.6. Attend to
precision.
8.MP.7. Look for
and make use of
structure.




What are two ways to represent and/or display bivariate data and why
would you use one as opposed to the other?
What information will each of the two representations tell us about the
bivariate data?
How do you decide if a line of best fit is a good model to determine the
relationship of bivariate data?
What can a line of best fit tell us about bivariate data?
Examples & Explanations
Students create scatterplots and examine relationships between variables. They analyze
scatterplots to determine positive and negative associations, the degree of association, and type of
association. Students examine outliers to determine if data points are valid or represent a recording
or measurement error. Students can use tools such as those at the National Center for Educational
Statistics to create a graph or generate data sets.
(http://nces.ed.gov/nceskids/createagraph/default.aspx)
Examples:
 Data for 10 students’ Math and Science scores are provided in the chart. Describe the
association between the Math and Science scores.
Student
1
2
3
4
5
6
7
8
9
10
Math
64
50
85
34
56
24
72
63
42
93
Science
68
70
83
33
60
27
74
63
40
96
Continued on next page
Tucson Unified School District
Mathematics Curriculum
Grade 8
Board Approved 03/27/2012
44

Data for 10 students’ Math scores and the distance they live from school are provided
in the table below. Describe the association between the Math scores and the distance
they live from school.
Student
Math score
Dist from
school (miles)

1
64
2
50
3
85
4
34
5
56
6
24
7
72
8
63
9
42
10
93
0.5
1.8
1
2.3
3.4
0.2
2.5
1.6
0.8
2.5
Data from a local fast food restaurant is provided showing the number of staff
members and the average time for filling an order are provided in the table below.
Describe the association between the number of staff and the average time for filling
an order.
Number of staff
Average time to fill order (seconds)

3
180
4
138
5
120
6
108
7
96
8
84
The chart below lists the life expectancy in years for people in the United States every
five years from 1970 to 2005. What would you expect the life expectancy of a person in
the United States to be in 2010, 2015, and 2020 based upon this data? Explain how
you determined your values.
Life Expectancy
(in years)
Date
Tucson Unified School District
Mathematics Curriculum
70.8
72.6
73.7
74.7
75.4
75.8
76.8
77.4
1970
1975
1980
1985
1990
1995
2000
2005
Grade 8
Board Approved 03/27/2012
45
8.SP Investigate patterns of association in bivariate data (Cluster 1- Standards 1, 2, 3, and 4)
8.SP.2
Standard 8.SP.2
Mathematical
Examples & Explanations
Know that straight lines are
Examples:
Practices
widely used to model
relationships between two
quantitative variables. For
scatterplots that suggest a
linear association, informally
fit a straight line, and
informally assess the model
fit by judging the closeness of
the data points to the line.
Connections: 8.EE.5; 8.F.3;
ET08-S1C3-01; ET08-S6C103;
SS08-S4C1-05;

The capacity of the fuel tank in a car is 13.5 gallons. The table below shows the
number of miles traveled and how many gallons of gas are left in the tank. Describe
the relationship between the variables. If the data is linear, determine a line of best fit.
Do you think the line represents a good fit for the data set? Why or why not? What is
the average fuel efficiency of the car in miles per gallon?
Miles Traveled 0
75
120
160
250
300
Gallons Used
0
2.3
4.5
5.7
9.7
10.7
8.MP.2. Reason
abstractly and
quantitatively.
8.MP.4. Model
with mathematics.
8.MP.5. Use
appropriate tools
strategically.
8.MP.6. Attend to
precision.
8.MP.7. Look for
and make use of
structure.
8.SP.3
Standard 8.SP.3
Use the equation of a linear
model to solve problems in
the context of bivariate
measurement data,
interpreting the slope and
intercept. For example, in a
linear model for a biology
experiment, interpret a slope
of 1.5 cm/hr as meaning that
an additional hour of sunlight
each day is associated with
an additional 1.5 cm in
mature plant height.
Connections: 8.EE.5; 8.F.3;
8.F.4;ET08-S1C3-03; ET08S2C2-01;
Mathematical
Practices
8.MP.2. Reason
abstractly and
quantitatively.
8.MP.4. Model
with mathematics.
Examples & Explanations
Examples:
 In a linear model for a biology experiment, interpret a slope of 1.5 cm/hr as meaning that an
additional hour of sunlight each day is associated with an additional 1.5 cm in mature plant
height.

Given data from students’ math scores and absences, make a scatterplot.
8.MP.5. Use
appropriate tools
strategically.
8.MP.6. Attend to
precision.
8.MP.7. Look for
Continued on next page
Tucson Unified School District
Mathematics Curriculum
Grade 8
Board Approved 03/27/2012
46
and make use of
structure.

Draw a line of best fit, paying attention to the closeness of the data points on either side of the
line.

From the line of best fit, determine an approximate linear equation that models the given data
(about y   25 x  95 )
3

Students should recognize that 95 represents the y intercept and 
25 represents the slope of
3
the line.

Students can use this linear model to solve problems. For example, through substitution, they
can use the equation to determine that a student with 4 absences should expect to receive a
math score of about 62. They can then compare this value to their line.
Tucson Unified School District
Mathematics Curriculum
Grade 8
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47
8.SP Investigate patterns of association in bivariate data (Cluster 1- Standards 1, 2, 3, and 4)
8.SP.4
Standard 8.SP.4
Mathematical
Examples & Explanations
Understand that patterns of
Example:
Practices
association can also be seen
in bivariate categorical data
by displaying frequencies and
relative frequencies in a twoway table. Construct and
interpret a two-way table
summarizing data on two
categorical variables
collected from the same
subjects. Use relative
frequencies calculated for
rows or columns to describe
possible association between
the two variables. For
example, collect data from
students in your class on
whether or not they have a
curfew on school nights and
whether or not they have
assigned chores at home. Is
there evidence that those
who have a curfew also tend
to have chores?
8.MP.2. Reason
abstractly and
quantitatively.

Collect data from students in your class on whether or not they have a curfew on school nights
and whether or not they have assigned chores at home. Is there evidence that those who have
a curfew also tend to have chores?

The table illustrates the results when 100 students were asked the survey questions: Do you
have a curfew? and Do you have assigned chores? Is there evidence that those who have a
curfew also tend to have chores?
8.MP.3. Construct
viable arguments
and critique the
reasoning of
others.
8.MP.4. Model
with mathematics.
8.MP.5. Use
appropriate tools
strategically.
8.MP.6. Attend to
precision.
8.MP.7. Look for
and make use of
structure.
Connections: 6-8.WHST.2b,f;
ET08-S1C1-01; ET08-S1C302; ET08-S1C3-03;
SS08-S4C2-03; SS08-S4C105;
SC08-S1C3-02
Solution: Of the students who answered that they had a curfew, 40 had chores and 10 did not. Of
the students who answered they did not have a curfew, 10 had chores and 40 did not. From this
sample, there appears to be a positive correlation between having a curfew and having chores.
Additional Domain Information – Statistics and Probability (SP)
Key Vocabulary (BOLD means most important)



Bivariate Data
Clustering
Frequency



Line of best fit
Linear association
Linear model



Negative association
Nonlinear association.
Outliers
Tucson Unified School District
Mathematics Curriculum



Positive association
Relative Frequency
Scatterplot


Slope
Two-way Table
Grade 8
Board Approved 03/27/2012
48
Example Resources

Book






HOLT (9.7, 12.7)
Math in Context (Insights into Data)
CMP2( Samples and Population Investigation 4)
Algebra 1 (5.6)
Statistical Questions from the Classroom – NCTM
This book presents eleven short discussions of some of the most frequently asked questions about statistics, questions that are
consistently raised by statistics students and by classroom teachers alike. The authors offer teachers of statistics some quick insight and
support in understanding these issues and explaining these ideas to their own students. Examples and visual representations of the
ideas are included.
Statistical Misconceptions – Schuyler W Huck
This brief, inexpensive book is designed to help readers identify and then discard the major statistical misconceptions that have infiltrated
the way they think about data.

Technology
 http://nces.ed.gov/nceskids/createagraph/default.aspx
Allows students to create their own graph in a bar, line, area, pie, and XY chart.
 http://cnx.org/content/m10949/latest/
Measures of central tendency, variability, and spread summarize a single variable by providing important information about its
distribution. Often, more than one variable is collected on each individual. This site is a teacher resource for bivariate data including
examples and vocabulary. Prerequisites are noted.
 http://www.northcanton.sparcc.org/~technology/excel/files/scatterplots.html
As the user enters ordered pairs, the spreadsheet graphs a scatterplot, creates a line of best fit, and displays its equation (in linear or
polynomial form).
 http://cwx.prenhall.com/bookbind/pubbooks/esm_sincich_pracstats_2/chapter2/custom1/deluxe-content.html
This site explains how to make several types of tables, charts and graphs using Excel. A computer lab would be needed to make this a
class project.

Exemplary Lessons
 http://www.pbs.org/teachers/connect/resources/4103/preview/
This science-based lesson on data and scatterplots has students construct and evaluate graphs of the current sunspot cycle. This lesson
plan includes a worksheet and extra resources.
 http://www.shodor.org/interactivate/lessons/UnivariateBivariateData/
This lesson is designed to introduce students to the differentiation between univariate and bivariate data. Students will gain experience
determining what types of graphs and measures are appropriate for each type of data. This lesson is designed for students who are
familiar with graphs and measures related to univariate data, even if they don't know the vocabulary term.
 http://ims.ode.state.oh.us/ODE/IMS/Lessons/Web_Content/CMA_LP_S05_BA_L09_I01_01.pdf. In this lesson, students create and
Tucson Unified School District
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Grade 8
Board Approved 03/27/2012
49

interpret various types of graphs from data gathered in class. In the analysis of the graphs, students classify data as univariate or
bivariate and as quantitative or qualitative. Students also interpret frequency distributions based on spread, symmetry, skewness,
clusters and outliers.
http://illuminations.nctm.org/LessonDetail.aspx?ID=L298. Exploring Linear Data - Students model linear data in a variety of settings that
range from car repair costs to sports to medicine. Students work to construct scatter- plots, interpret data points and trends, and
investigate the notion of line of best fit.
Assessments
All assessments for this domain must focus on assessing the degree, to which seventh grade students can, within the parameters specified in the 3
standards included in this domain,
 investigate patterns of association in bivariate data.

Both formative and summative assessments are vital components of effective mathematics curricula. Formative assessments, (e.g., pre-assessments,
daily checks for understanding, discussions of strategies students use to solve problems, etc.) assist in instructional planning and implementation;
summative assessments (e.g., unit assessments, quarterly benchmarks, etc.) inform learner growth related to important mathematics concepts. All
district-adopted resources contain multiple assessment tools and include online resources that can be used for the purposes delineated above.
PARCC also will provide two end of the year summative assessments. The first, a performance-based assessment, will focus on applying skills,
concepts, and understandings to solve multi-step problems requiring abstract reasoning, precision, perseverance, and strategic use of tools. The
performance measure will be administered as close to the end of the school year as possible. The second, an end of the year machine-scorable
summative assessment will be administered after approximately 90% of the school year. These assessments are to begin during the SY 2014-2015.
Common Student Misconceptions
Students commonly try to connect all the points plotted on a scatter plot. This comes from their experience with linear functions and creating line
graphs. Be sure to explain that these are discrete points and may not have data in between, therefore they cannot be connected the same way certain
linear functions can be.
Students believe that correlation between two variables automatically implies that one causes the other. Students should recognize bivariate
data as two separate sets of data that can be represented in one display. They need to understand that correlation does not mean causation.
Students may believe bivariate data is only displayed in scatter plots. Cluster 8.SP.4 provides the opportunity to display bivariate, categorical data
in a table.
Students assume the line of best fit must connect all data in a singular line. Because students are informally drawing lines of best fit, the lines will
vary slightly. To obtain the exact line of best fit, students would use technology to find the line of regression.
Tucson Unified School District
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50
Students tend to try to find the line that has the most points on it. When students are trying to find lines of best fit, they need understand that you
are also looking for the line with the closest number of points above and below the line as well as the number of points on the line.
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COMPARATIVE ANALYSIS: GRADE 8 MATHEMATICS
PLANNING FOR CONTENT SHIFTS
2010 STANDARD
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.
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.
2008 PO
M08-S1C1-02 Classify real numbers as rational or
irrational.
8.EE.1 Know and apply the properties of
integer exponents to generate equivalent
numerical expressions. For example, 323–
5 = 3–3 = 1/ 3 = 1/
3
27.
M08-S1C2-05 Simplify numerical expressions using
the order of operations that include grouping symbols,
square roots, cube roots, absolute values, and
positive exponents. (positive exponents only)
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.
8.EE.3 Use numbers expressed in the
form of a single digit times an integer
PLAN
M08-S1C1-03 Model the relationship between the
subsets of the real number system.
M06-S1C3-01 Use benchmarks as meaningful points
of comparison for rational numbers.
M07-S1C3-01 Estimate and apply benchmark
numbers for rational number and common irrational
numbers.
M07-S1C3-03 Estimate square roots of numbers less
than 1000 by locating them between two consecutive
whole numbers.
M08-S1C3-02 Estimate the location of rational and
common irrational numbers on a number line.
MHS-S1C2-01 Solve word problems involving
absolute value, powers, roots, and scientific notation.
MHS-S1C2-03 Calculate powers and roots of rational
and irrational numbers.
M06-S1C1-06 Express the inverse relationships
between exponents and roots for perfect squares and
cubes.
M08-S1C1-02 Classify real numbers as rational or
irrational.
M07-S1C2-04 Represent and interpret numbers
using scientific notation (positive exponents only).
Tucson Unified School District
Mathematics Curriculum
Grade 8
Board Approved 03/27/2012
52
2010 STANDARD
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108 and the population
of the world as 7109, and determine that
the world population is more than 20 times
larger.
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.
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.
2008 PO
M08-S1C2-04 Convert standard notation to scientific
notation and vice versa (include positive and negative
exponents).
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
M08-S3C3-04 Translate between different
representations of linear equations using symbols,
graphs, tables, or written descriptions.
M08-S3C4-01 Interpret the relationship between a
linear equation and its graph, identifying and
PLAN
M08-S1C2-04 Convert standard notation to scientific
notation and vice versa (include positive and negative
exponents).
MHS-S1C2-04 Compute using scientific notation.
M08-S3C2-01 Sketch and interpret a graph that
models a given context; describe a context that is
modeled by a given graph.
M08-S3C2-05 Demonstrate that proportional
relationships are linear using equations, graphs, or
tables.
M08-S3C3-04 Translate between different
representations of linear equations using symbols,
graphs, tables, or written descriptions.
M08-S3C4-01 Interpret the relationship between a
linear equation and its graph, identifying and
computing slope and intercepts. (missing comparison
of 2 different graphs and/or equations)
M08-S5C2-12 Make, validate, and justify conclusions
and generalizations about linear relationships.
Tucson Unified School District
Mathematics Curriculum
Grade 8
Board Approved 03/27/2012
53
2010 STANDARD
equation y = mx + b for a line intercepting
the vertical axis at b.
2008 PO
computing slope and intercepts. (does not specifically
address similar triangles)
M08-S5C2-12 Make, validate, and justify conclusions
and generalizations about linear relationships.
PLAN
8.EE.7 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.
8.EE.8 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.
8.EE.8
b. Solve systems of two linear equations
in two variables algebraically, and
estimate solutions by graphing the
M07-S3C3-05 Create and solve two-step equations
that can be solved using inverse operations with
rational numbers.
M08-S3C3-01 Write or identify algebraic expressions,
equations, or inequalities that represent a situation.
M07-S3C3-03 Solve multi-step equations using
inverse properties with rational numbers.
M07-S3C3-05 Create and solve two-step equations
that can be solved using inverse operations with
rational numbers.
M08-S3C3-03 Analyze situations, simplify, and solve
problems involving linear equations and inequalities
using the properties of the real number system.
MHS-S3C2-05 Recognize and solve problems that
can be modeled using a system of two equations in
two variables.
M08-S1C3-01 Make estimates appropriate to a given
situation.
MHS-S3C3-07 Solve systems of two linear equations
in two variables.
Tucson Unified School District
Mathematics Curriculum
Grade 8
Board Approved 03/27/2012
54
2010 STANDARD
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.
c. 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.
8.F.1 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. (Function notation is
not required in Grade 8.)
8.F.2 Compare properties of two functions
each represented in a different way
(algebraically, graphically, numerically in
tables, or by verbal descriptions). For
example, given a linear function
represented by a table of values and a
linear function represented by an algebraic
expression, determine which function has
the greater rate of change.
8.F.3 Interpret the equation y = mx + b as
defining a linear function, whose graph is a
straight line; give examples of functions
that are not linear. For example, the
function A = s2 giving the area of a square
as a function of its side length is not linear
because its graph contains the points (1,1),
(2,4) and (3,9), which are not on a straight
line.
8.F.4 Construct a function to model a
linear relationship between two quantities.
2008 PO
MHS-S4C3-07 Determine the solution to a system of
linear equations in two variables from the graphs of
the equations.
PLAN
MHS-S3C2-05 Recognize and solve problems that
can be modeled using a system of two equations in
two variables.
MHS-S3C3-07 Solve systems of two linear equations
in two variables.
M08-S3C2-02 Determine if a relationship represented
by a graph or table is a function.
*
M08-S3C2-04 Identify functions as linear or nonlinear
and contrast distinguishing properties of functions
using equations, graphs, or tables.
M08-S5C2-12 Make, validate, and justify conclusions
and generalizations about linear relationships.
M07-S3C4-01 Use graphs and tables to model and
analyze change. (analysis of change)
Tucson Unified School District
Mathematics Curriculum
Grade 8
Board Approved 03/27/2012
55
2010 STANDARD
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.
8.F.5 Describe qualitatively the functional
relationship between two quantities by
analyzing a graph (e.g., where the function
is increasing or decreasing, linear or
nonlinear). Sketch a graph that exhibits the
qualitative features of a function that has
been described verbally.
8.G.1 Verify experimentally the properties
of rotations, reflections, and translations:
2008 PO
M08-S3C1-01 Recognize, describe, create, and
analyze numerical and geometric sequences using
tables, graphs, words, or symbols; make conjectures
about these sequences.
M08-S3C2-03 Write the rule for a simple function
using algebraic notation. (writing the rule)
M08-S3C2-04 Identify functions as linear or nonlinear
and contrast distinguishing properties of functions
using equations, graphs, or tables. (properties of
functions)
M08-S3C3-04 Translate between different
representations of linear equations using symbols,
graphs, tables, or written descriptions. (different forms
– equation/graphs)
M08-S3C4-01 Interpret the relationship between a
linear equation and its graph, identifying and
computing slope and intercepts. (slope and rate of
change)
M08-S5C2-12 Make, validate, and justify conclusions
and generalizations about linear relationships.
M08-S3C2-01 Sketch and interpret a graph that
models a given context; describe a context that is
modeled by a given graph.
M08-S3C2-04 Identify functions as linear or nonlinear
and contrast distinguishing properties of functions
using equations, graphs, or tables.
PLAN
M07-S4C2-01 Model the result of a double
transformation (translations or reflections) of a 2dimensional figure on a coordinate plane using all four
quadrants.
M08-S4C2-01 Model the result of rotations in
multiples of 45 degrees of a 2-dimensional figure
about the origin.
Tucson Unified School District
Mathematics Curriculum
Grade 8
Board Approved 03/27/2012
56
2010 STANDARD
a.
b.
c.
Lines are taken to lines, and line
segments to line segments of the
same length.
Angles are taken to angles of the
same measure.
2008 PO
M08-S4C2-02 Describe the transformations that
create a given tessellation.
Aligned P.O.s do not specifically address line
segments, angles, parallel lines. Rotations, reflections
and translations of figures are specifically addressed.
PLAN
Parallel lines are taken to parallel
lines.
8.G.2 Understand that a two-dimensional
figure is congruent to another if the second
can be obtained from the first by a
sequence of rotations, reflections, and
translations; given two congruent figures,
describe a sequence that exhibits the
congruence between them.
M07-S4C1-05 Identify corresponding parts of
congruent figures.
M07-S4C2-01 Model the result of a double
transformation (translations or reflections) of a 2dimensional figure on a coordinate plane using all four
quadrants.
M08-S4C2-01 Model the result of rotations in
multiples of 45 degrees of a 2-dimensional figure
about the origin.
M08-S4C2-02 Describe the transformations that
create a given tessellation.
M06-S4C2-01 Identify a simple translation or
reflection and model its effect on a 2-dimensional
figure on a coordinate plane using all four quadrants.
8.G.3 Describe the effect of dilations,
translations, rotations, and reflections on
two-dimensional figures using coordinates.
M06-S4C2-01 Identify a simple translation or
reflection and model its effect on a 2-dimensional
figure on a coordinate plane using all four quadrants.
M07-S4C2-01 Model the result of a double
Tucson Unified School District
Mathematics Curriculum
Grade 8
Board Approved 03/27/2012
57
2010 STANDARD
2008 PO
transformation (translations or reflections) of a 2dimensional figure on a coordinate plane using all four
quadrants.
M07-S4C2-01 Model the result of rotations in
multiples of 45 degrees of a 2-dimensional figure
about the origin.
MHS-S4C2-01 Determine whether a transformation
of a 2-dimensional figure on a coordinate plane
represents a translation, reflection, rotation, or dilation
and whether congruence is preserved.
MHS-S4C2-02 Determine the new coordinates of a
point when a single transformation is performed on a
2-dimensional figure.
MHS-S4C2-03 Sketch and describe the properties of
a 2-dimensional figure that is the result of two or more
transformations.
MHS-S4C2-04 Determine the effects of a single
transformation on linear or area measurements of a 2dimensional figure
8.G.4 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.
M07-S4C2-01 Model the result of a double
transformation (translations or reflections) of a 2dimensional figure on a coordinate plane using all four
quadrants.
M08-S4C1-03 Use proportional reasoning to
determine congruence and similarity of triangles.
M08-S4C2-01 Model the result of rotations in
multiples of 45 degrees of a 2-dimensional figure
about the origin.
M08-S4C4-02 Solve geometric problems using ratios
and proportions.
PLAN
MHS-S4C2-01 Determine whether a transformation
of a 2-dimensional figure on a coordinate plane
represents a translation, reflection, rotation, or dilation
and whether congruence is preserved.
MHS-S4C2-02 Determine the new coordinates of a
point when a single transformation is performed on a
2-dimensional figure.
Tucson Unified School District
Mathematics Curriculum
Grade 8
Board Approved 03/27/2012
58
2010 STANDARD
2008 PO
MHS-S4C2-03 Sketch and describe the properties of
a 2-dimensional figure that is the result of two or more
transformations.
MHS-S4C2-04 Determine the effects of a single
transformation on linear or area measurements of a 2dimensional figure.
8.G.5 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.
M05-S4C1-02 Solve problems by understanding and
applying the property that the sum of the interior
angles of a triangle is 180˚.
M07-S4C1-02 Analyze and determine relationships
between angles created by parallel lines cut by a
transversal.
M08-S4C1-03 Use proportional reasoning to
determine congruence and similarity of triangles.
8.G.6 Explain a proof of the Pythagorean
Theorem and its converse.
M08-S5C2-11 Identify simple valid arguments using
if… then statements.
M08-S5C2-13 Verify the Pythagorean Theorem using
a valid argument. (does not address converse)
M08-S4C1-04 Use the Pythagorean Theorem to
solve problems (does not address 3D).
8.G.7 Apply the Pythagorean Theorem to
determine unknown side lengths in right
triangles in real-world and mathematical
problems in two and three dimensions.
8.G.8 Apply the Pythagorean Theorem to
find the distance between two points in a
coordinate system.
8.G.9 Know the formulas for the volumes
of cones, cylinders, and spheres and use
them to solve real-world and mathematical
problems.
PLAN
M08-S4C3-02 Use the Pythagorean Theorem to find
the distance between two points in the coordinate
plane.
M08-S5C2-13 Verify the Pythagorean Theorem using
a valid argument.
M08-S4C4-03 Calculate the surface area and volume
of rectangular prisms, right triangular prisms, and
cylinders.
MHS-S4C4-05 Calculate the surface area and
volume of 3-dimensional figures and solve for missing
measures.
Tucson Unified School District
Mathematics Curriculum
Grade 8
Board Approved 03/27/2012
59
2010 STANDARD
8.SP.1 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.
2008 PO
M07-S2C1-01 Solve problems by selecting,
constructing, and interpreting displays of data
including multi-line graphs and scatterplots.
M07-S2C1-02 Interpret trends in a data set, estimate
values for missing data, and predict values for points
beyond the range of the data set.
M07-S2C1-03 Identify outliers and determine their
effect on mean, median, mode, and range.
M08-S2C1-01 Solve problems by selecting,
constructing, interpreting, and calculating with
displays of data, including box and whisker plots and
scatterplots.
8.SP.2 Know that straight lines are widely
used to model relationships between two
quantitative variables. For scatter plots that
suggest a linear association, informally fit a
straight line, and informally assess the
model fit by judging the closeness of the
data points to the line.
8.SP.3 Use the equation of a linear model
to solve problems in the context of bivariate
measurement data, interpreting the slope
and intercept. For example, in a linear
model for a biology experiment, interpret a
slope of 1.5 cm/hr as meaning that an
additional hour of sunlight each day is
associated with an additional 1.5 cm in
mature plant height.
M08-S2C1-01 Solve problems by selecting,
constructing, interpreting, and calculating with
displays of data, including box and whisker plots and
scatterplots.
PLAN
M08-S3C3-01 Write or identify algebraic expressions,
equations, or inequalities that represent a situation.
M08-S3C3-04 Translate between different
representations of linear equations using symbols,
graphs, tables, or written descriptions.
M08-S3C4-01 Interpret the relationship between a
linear equation and its graph, identifying and
computing slope and intercepts.
M08-S5C2-12 Make, validate, and justify conclusions
and generalizations about linear relationships.
MHS-S2C1-03 Display data, including paired data, as
lists, tables, matrices, and plots with or without
technology; make predictions and observations about
patterns or departures from patterns.
Tucson Unified School District
Mathematics Curriculum
Grade 8
Board Approved 03/27/2012
60
2010 STANDARD
8.SP.4 Understand that patterns of
association can also be seen in bivariate
categorical data by displaying frequencies
and relative frequencies in a two-way table.
Construct and interpret a two-way table
summarizing data on two categorical
variables collected from the same subjects.
Use relative frequencies calculated for
rows or columns to describe possible
association between the two variables. For
example, collect data from students in your
class on whether or not they have a curfew
on school nights and whether or not they
have assigned chores at home. Is there
evidence that those who have a curfew
also tend to have chores?
2008 PO
M08-S2C1-02 Make inferences by comparing the
same summary statistic for two or more data sets.
8.MP.1 Make sense of problems and
persevere in solving them.
M08-S5C2-01 Analyze a problem situation to
determine the question(s) to be answered.
PLAN
MHS-S2C1-08 Design simple experiments or
investigations and collect data to answer questions.
M08-S5C2-02 Analyze and compare mathematical
strategies for efficient problem solving; select and use
one or more strategies to solve a problem.
M08-S5C2-03 Identify relevant, missing, and
extraneous information related to the solution to a
problem.
M08-S5C2-04 Represent a problem situation using
multiple representations, describe the process used to
solve the problem, and verify the reasonableness of
the solution.
M08-S5C2-05 Apply a previously used problemsolving strategy in a new context.
M08-S5C2-06 Communicate the answer(s) to the
question(s) in a problem using appropriate
representations, including symbols and informal and
formal mathematical language.
M08-S5C2-07 Isolate and organize mathematical
Tucson Unified School District
Mathematics Curriculum
Grade 8
Board Approved 03/27/2012
61
2010 STANDARD
8.MP.2 Reason abstractly and
quantitatively.
8.MP.3 Construct viable arguments and
critique the reasoning of others.
8.MP.4 Model with mathematics.
2008 PO
information taken from symbols, diagrams, and
graphs to make inferences, draw conclusions, and
justify reasoning.
M08-S5C2-08 Describe when to use proportional
reasoning to solve a problem.
M08-S5C2-09 Make and test conjectures based on
information collected from explorations and
experiments.
M08-S5C2-07 Isolate and organize mathematical
information taken from symbols, diagrams, and
graphs to make inferences, draw conclusions, and
justify reasoning.
M08-S5C2-08 Describe when to use proportional
reasoning to solve a problem.
PLAN
M08-S5C2-04 Represent a problem situation using
multiple representations, describe the process used to
solve the problem, and verify the reasonableness of
the solution.
M08-S5C2-07 Isolate and organize mathematical
information taken from symbols, diagrams, and
graphs to make inferences, draw conclusions, and
justify reasoning.
M08-S5C2-09 Make and test conjectures based on
information collected from explorations and
experiments.
M08-S5C2-02 Analyze and compare mathematical
strategies for efficient problem solving; select and use
one or more strategies to solve a problem.
M08-S5C2-04 Represent a problem situation using
multiple representations, describe the process used to
solve the problem, and verify the reasonableness of
the solution.
M08-S5C2-05 Apply a previously used problemsolving strategy in a new context.
M08-S5C2-06 Communicate the answer(s) to the
question(s) in a problem using appropriate
representations, including symbols and informal and
Tucson Unified School District
Mathematics Curriculum
Grade 8
Board Approved 03/27/2012
62
2010 STANDARD
8.MP.5 Use appropriate tools strategically.
8.MP.6 Attend to precision.
8.MP.7 Look for and make use of
structure.
8.MP.8 Look for and express regularity in
repeated reasoning.
MOVEMENT
MOVED TO GRADE 6
2008 PO
formal mathematical language.
M08-S5C2-07 Isolate and organize mathematical
information taken from symbols, diagrams, and
graphs to make inferences, draw conclusions, and
justify reasoning.
M08-S5C2-08 Describe when to use proportional
reasoning to solve a problem.
M08-S5C2-02 Analyze and compare mathematical
strategies for efficient problem solving; select and use
one or more strategies to solve a problem.
M08-S5C2-09 Make and test conjectures based on
information collected from explorations and
experiments.
M08-S5C2-06 Communicate the answer(s) to the
question(s) in a problem using appropriate
representations, including symbols and informal and
formal mathematical language.
PLAN
M08-S5C2-07 Isolate and organize mathematical
information taken from symbols, diagrams, and
graphs to make inferences, draw conclusions, and
justify reasoning.
M08-S5C2-09 Make and test conjectures based on
information collected from explorations and
experiments.
M08-S5C2-04 Represent a problem situation using
multiple representations, describe the process used to
solve the problem, and verify the reasonableness of
the solution.
2008 PO
M08-S1C1-01 Compare and order real numbers
PLAN
including very large and small integers, and decimals
and fractions close to zero.
MOVED TO GRADE 7
M08-S1C1-04 Model and solve problems
Tucson Unified School District
Mathematics Curriculum
Grade 8
Board Approved 03/27/2012
63
involving absolute values.
MOVED TO GRADE 6
M08-S1C2-01 Solve problems with factors,
multiples, divisibility or remainders, prime numbers,
and composite numbers.
MOVED TO GRADE 5
M08-S1C2-02 Describe the effect of multiplying and
dividing a rational number by
 a number less than zero,
 a number between zero and one,
 one, and
 a number greater than one.
MOVED TO GRADE 7
M08-S1C2-03 Solve problems involving
percent increase, percent decrease, and
simple interest rates.
MOVED TO GRADE 7
M08-S2C1-03 Describe how summary statistics
relate to the shape of the distribution.
M08-S2C1-04 Determine whether information is
represented effectively and appropriately given a
graph or a set of data by identifying sources of bias
and compare and contrast the effectiveness of
different representations of data.
M08-S2C1-05 Evaluate the design of an
experiment.
REMOVED
MOVED TO GRADE 7
MOVED TO GRADE 7
MOVED TO GRADE 7
MOVED TO GRADE 7
REMOVED
M08-S2C2-01 Determine theoretical and
experimental conditional probabilities in
compound probability experiments.
M08-S2C2-02 Interpret probabilities within
a given context and compare the outcome
of an experiment to predictions made prior
to performing the experiment.
M08-S2C2-03 Use all possible outcomes (sample
space) to determine the probability of dependent and
independent events.
M08-S2C3-01 Represent, analyze, and solve
counting problems with or without ordering and
Tucson Unified School District
Mathematics Curriculum
Grade 8
Board Approved 03/27/2012
64
repetitions.
REMOVED
REMOVED
MOVED TO GRADE 6
MOVED TO GRADE 6
REDISTRIBUTED TO GRADE 6 AND 7
MOVED TO HIGH SCHOOL
MOVED TO GRADE 7
REMOVED
MOVED TO HIGH SCHOOL
REMOVED
MOVED TO HIGH SCHOOL
REMOVED
M08-S2C3-02 Solve counting problems and
represent counting principles algebraically including
factorial notation.
M08-S2C4-01 Use directed graphs to solve
problems.
M08-S3C3-02 Evaluate an expression containing
variables by substituting rational numbers for the
variables.
M08-S3C3-05 Graph an inequality on a number line.
M08-S3C4-02 Solve problems involving
simple rates.
M08-S4C1-01 Identify the attributes of
circles: radius, diameter, chords, tangents,
secants, inscribed angles, central angles,
intercepted arcs, circumference, and area.
M08-S4C1-02 Predict results of combining,
subdividing, and changing shapes of plane
figures and solids.
M08-S4C2-03 Identify lines of symmetry in
plane figures or classify types of symmetries
of 2-dimensional figures.
M08-S4C3-01 Make and test a conjecture
about how to find the midpoint between any
two points in the coordinate plane.
M08-S4C4-01 Solve problems involving
conversions within the same measurement
system.
M08-S5C1-01 Create an algorithm to solve
problems involving indirect measurements,
using proportional reasoning, dimensional
analysis, and the concepts of density and
rate.
M08-S5C2-10 Solve logic problems
Tucson Unified School District
Mathematics Curriculum
Grade 8
Board Approved 03/27/2012
65
involving multiple variables, conditional
statements, conjectures, and negation using
words, charts, and pictures.
Tucson Unified School District
Mathematics Curriculum
Grade 8
Board Approved 03/27/2012
66