Ratios
& Proportional Relationships 
The
Number System 
Expressions
& Equations 
Geometry 
Statistics
and Probability 
Mathematical
Practice 

Grade 6 
á
6.RP.A.1. Understand the concept of
a ratio and use ratio language to describe a ratio relationship between two
quantities. á
6.RP.A.2. Understand the concept of
a unit rate a/b associated with a ratio a:b with b 0, and use rate language in the context
of a ratio relationship. á
6.RP.A.3. Use ratio and rate
reasoning to solve realworld and mathematical problems, e.g., by reasoning
about tables of equivalent ratios, tape diagrams, double number line
diagrams, or equations. á
6.RP.A.3a. Make tables of
equivalent ratios relating quantities with wholenumber measurements, find missing values in the tables, and plot the pairs of
values on the coordinate plane. Use tables to compare ratios. á
6.RP.A.3b. Solve unit rate problems
including those involving unit pricing and constant speed. á
6.RP.A.3c. Find a percent of a
quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity);
solve problems involving finding the whole, given a part and the percent. á
6.RP.A.3d. Use ratio reasoning to
convert measurement units; manipulate and transform
units appropriately when multiplying or dividing quantities. 
á
6.NS.A.1. Interpret and compute
quotients of fractions, and solve word problems involving division of
fractions by fractions, e.g., by using visual fraction models and equations
to represent the problem. á
6. NS.B.2. Fluently divide
multidigit numbers using the standard algorithm. á
6. NS.B.3. Fluently add, subtract,
multiply, and divide multidigit decimals using the standard algorithm for
each operation. á
6. NS.B.4. Find the greatest common
factor of two whole numbers less than or equal to 100 and the least common
multiple of two whole numbers less than or equal to 12. Use the distributive
property to express a sum of two whole numbers 1Ð100 with a common
factor as a multiple of a sum of two whole numbers with no common factor. á
6.NS.C.5. Understand that positive
and negative numbers are used together to describe quantities having opposite
directions or values (e.g., temperature above/below zero, elevation
above/below sea level, credits/debits, positive/negative electric charge);
use positive and negative numbers to represent quantities in realworld
contexts, explaining the meaning of 0 in each situation. á
6.NS.C.6. Understand a rational
number as a point on the number line. Extend number line diagrams and
coordinate axes familiar from previous grades to represent points on the line
and in the plane with negative number coordinates. á
6.NS.C.6a. Recognize opposite signs
of numbers as indicating locations on opposite sides of 0 on the number line;
recognize that the opposite of the opposite of a number is the number itself,
e.g., Ð(Ð3) = 3, and that 0 is its own opposite. á
6.NS.C.6b. Understand signs of
numbers in ordered pairs as indicating locations in quadrants of the
coordinate plane; recognize that when two ordered pairs differ only by signs,
the locations of the points are related by reflections across one or both
axes. á
6.NS.C.6c. Find and position
integers and other rational numbers on a horizontal or vertical number line
diagram; find and position pairs of integers and other rational numbers on a
coordinate plane. á
6.NS.C.7. Understand ordering and
absolute value of rational numbers. á
6.NS.C.7a. Interpret statements of
inequality as statements about the relative position of two numbers on a
number line diagram. á
6.NS.C.7b. Write, interpret, and explain statements of order for rational
numbers in realworld contexts. á
6.NS.C.7c. Understand
the absolute value of a rational number as its distance from 0 on the number
line; interpret absolute value as magnitude for a positive or negative
quantity in a realworld situation. á
6.NS.C.7d. Distinguish comparisons
of absolute value from statements about order. á
6.NS.C.8. Solve realworld and
mathematical problems by graphing points in all four quadrants of the
coordinate plane. Include use of coordinates and absolute value to find
distances between points with the same first coordinate or the same second
coordinate. 
á 6.EE.A.1. Write and evaluate numerical expressions involving
wholenumber exponents. á 6.EE.A.2. Write, read, and evaluate expressions in which
letters stand for numbers. á 6.EE.A.2a. Write expressions that record operations with
numbers and with letters standing for numbers. á 6.EE.A.2b. Identify parts of an expression using mathematical
terms (sum, term, product, factor, quotient, coefficient); view one or more
parts of an expression as a single entity. á 6.EE.A.2c. Evaluate expressions at specific values of their
variables. Include expressions that arise from formulas used in realworld
problems. Perform arithmetic operations, including those involving
wholenumber exponents, in the conventional order when there are no
parentheses to specify a particular order (Order of Operations). á 6.EE.A.3. Apply the properties of operations to generate
equivalent expressions. á 6.EE.A.4. Identify when two expressions are equivalent (i.e.,
when the two expressions name the same number regardless of which value is
substituted into them). á 6.EE.B.5. Understand solving an equation or inequality as a
process of answering a question: which values from a specified set, if any,
make the equation or inequality true? Use substitution to determine whether a
given number in a specified set makes an equation or inequality true. á 6.EE.B.6. Use variables to represent numbers and write
expressions when solving a realworld or mathematical problem; understand that a variable can represent an unknown
number, or, depending on the purpose at hand, any number in a specified set. á 6.EE.B.7. Solve realworld and mathematical problems by
writing and solving equations of the form x + p = q and px = q for cases in which p, q and x are all
nonnegative rational numbers. á 6.EE.B.8. Write an inequality of the form x > c or x < c
to represent a constraint or condition in a realworld or mathematical
problem. Recognize that inequalities of the form x > c or x < c have
infinitely many solutions; represent solutions of such inequalities on number
line diagrams. á 6.EE.C.9. Use variables to represent two quantities in a
realworld problem that change in relationship to one another; write an
equation to express one quantity, thought of as the dependent variable, in
terms of the other quantity, thought of as the independent variable. Analyze
the relationship between the dependent and independent variables using graphs
and tables, and relate these to the equation. For example, in a problem
involving motion at constant speed, list and graph ordered pairs of distances
and times, and write the equation d = 65t to represent the relationship
between distance and time 
á 6.G.A.2. Find the volume of a right rectangular prism with
fractional edge lengths by packing it with unit cubes of the appropriate unit
fraction edge lengths, and show that the volume is the same as would be found
by multiplying the edge lengths of the prism. Apply the formulas V = l w h
and V = b h to find volumes of right rectangular prisms with fractional edge
lengths in the context of solving realworld and mathematical problems. 
á 6.SP.A.1. Recognize a statistical question as one that
anticipates variability in the data related to the question and accounts for
it in the answers. á 6.SP.A.2. Understand that a set of data collected to answer a
statistical question has a distribution which can be
described by its center, spread, and overall shape. á 6.SP.A.3. Recognize that a measure of center for a numerical
data set summarizes all of its values with a single number, while a measure
of variation describes how its values vary with a single number. á 6.SP.B.4. Display numerical data in plots on a number line,
including dot plots, histograms, and box plots. á 6.SP.B.5. Summarize numerical data sets in relation to their context,
such as by: á 6.SP.B.5a. Reporting the number of observations. á 6.SP.B.5b. Describing the nature of the attribute under
investigation, including how it was measured and its units of measurement. á 6.SP.B.5c. Giving quantitative measures of center (median
and/or mean) and variability (interquartile range and/or mean absolute
deviation), as well as describing any overall pattern and any striking
deviations from the overall pattern with reference to the context in which
the data were gathered. á 6.SP.B.5d. Relating the choice of
measures of center and variability to the shape of the data distribution and
the context in which the data were gathered. 
á MP.1. Make sense of problems and persevere in solving them. á MP.2. Reason abstractly and quantitatively. á MP.3. Construct viable arguments and critique the reasoning of
others. á MP.4. Model with mathematics. á MP.5. Use appropriate tools strategically. á MP.6. Attend to precision. á MP.7. Look for and make use of structure. á MP.8. Look for and express regularity in repeated reasoning. 
Grade 7 


á 7.EE.A.2. Understand that rewriting an expression in different
forms in a problem context can shed light on the problem and how the
quantities in it are related. á 7.EE.B.3. Solve multistep reallife and mathematical problems
posed with positive and negative rational numbers in any form (whole numbers,
fractions, and decimals), using tools strategically. Apply properties of
operations to calculate with numbers in any form; convert between forms as
appropriate; and assess the reasonableness of answers using mental
computation and estimation strategies.

á 7.G.A.3. Describe the twodimensional figures that result from
slicing threedimensional figures, as in plane sections of right rectangular
prisms and right rectangular pyramids. á 7.G.B.4. Know the formulas for the area and circumference of a
circle and use them to solve problems; give an informal derivation of the
relationship between the circumference and area of a circle.

á 7.SP.A.2. Use data from a random sample to draw inferences
about a population with an unknown characteristic of interest. Generate
multiple samples (or simulated samples) of the same size to gauge the
variation in estimates or predictions. á 7.SP.B.3. Informally assess the degree of visual overlap of
two numerical data distributions with similar variabilities,
measuring the difference between the centers by expressing it as a multiple
of a measure of variability. á 7.SP.B.4. Use measures of center and measures of variability
for numerical data from random samples to draw informal comparative
inferences about two populations. á 7.SP.C.5. Understand that the probability of a chance event is
a number between 0 and 1 that expresses the likelihood of the event
occurring. Larger numbers indicate greater likelihood. A probability near 0
indicates an unlikely event, a probability around 1/2 indicates an event that
is neither unlikely nor likely, and a probability near 1 indicates a likely
event.


Grade 8 

á 8.EE.A.1. Know and apply the properties of integer exponents
to generate equivalent numerical expressions. á 8.EE.A.2. Use square root and cube root symbols to represent
solutions to equations of the form x² = p and x³ = p, where p is a
positive rational number. Evaluate square roots of small perfect squares and
cube roots of small perfect cubes. Know that is irrational. á 8.EE.A.3. Use numbers expressed in the form of a single digit
times a wholenumber power of 10 to estimate very large or very small
quantities, and to express how many times as much one is than the other. á 8.EE.A.4. Perform operations with numbers expressed in
scientific notation, including problems where both decimal and scientific
notation are used. Use scientific notation and choose units of appropriate
size for measurements of very large or very small quantities (e.g., use
millimeters per year for seafloor spreading). Interpret scientific notation
that has been generated by technology. á 8.EE.B.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
distancetime graph to a distancetime equation to determine which of two
moving objects has greater speed. á 8.EE.B.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.
8.EE.C.7. Solve linear equations in one variable. á 8.EE.C.7a. Give examples of linear equations in one variable
with one solution, infinitely many solutions, or no solutions. Show which of
these possibilities is the case by successively transforming the given
equation into simpler forms, until an equivalent equation of the form x = a,
a = a, or a = b results (where a and b are different numbers). á 8.EE.C.7b. Solve linear equations with rational number
coefficients, including equations whose solutions require expanding
expressions using the distributive property and collecting like terms. á 8.EE.C.8. Analyze and solve pairs of simultaneous linear
equations. á 8.EE.C.8a. Understand that solutions to a system of two linear
equations in two variables correspond to points of intersection of their
graphs, because points of intersection satisfy both equations simultaneously. á 8.EE.C.8b. Solve systems of two linear equations in two
variables algebraically, and estimate solutions by graphing the equations.
Solve simple cases by inspection. For example, 3x + 2y = 5 and 3x + 2y = 6
have no solution because 3x + 2y cannot simultaneously be 5 and 6. á 8.EE.C.8c. Solve realworld 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. 

á 8.G.A.1. Verify experimentally the properties of rotations,
reflections, and translations: á 8.G.A.1a. Lines are taken to lines, and line segments to line
segments of the same length. á 8.G.A.1b. Angles are taken to angles of the same measure. á 8.G.A.1c. Parallel lines are taken to parallel lines. á 8.G.A.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. á 8.G.A.3. Describe the effect of dilations, translations,
rotations, and reflections on twodimensional figures using coordinates. á 8.G.A.4. Understand that a twodimensional 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
twodimensional figures, describe a sequence that exhibits the similarity
between them. á 8.G.A.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 angleangle criterion for
similarity of triangles. á 8.G.B.6. Explain a proof of the Pythagorean Theorem
and its converse. á 8.G.B.7. Apply the Pythagorean Theorem
to determine unknown side lengths in right triangles in realworld and
mathematical problems in two and three dimensions. á 8.G.B.8. Apply the Pythagorean Theorem
to find the distance between two points in a coordinate system. á 8.G.C.9. Know the formulas for the volumes of cones,
cylinders, and spheres and use them to solve realworld and mathematical
problems. 


Number
and Quantity 
Algebra 
Functions 
Geometry 
Statistics
and Probability 







Grade 9 
á HSNRN.A.1. Explain how the definition of the meaning of
rational exponents follows from extending the properties of integer exponents
to those values, allowing for a notation for radicals in terms of rational
exponents. á HSNRN.A.2. Rewrite expressions involving radicals and
rational exponents using the properties of exponents. á HSNRN.B.3. Explain why the sum or product of two rational
numbers is rational; that the sum of a rational number and an irrational
number is irrational; and that the product of a nonzero rational number and
an irrational number is irrational. 
á
HSASSE.A.1. Interpret expressions
that represent a quantity in terms of its context. á
HSASSE.A.1a. Interpret parts of an
expression, such as terms, factors, and coefficients. á
HSASSE.A.1b. Interpret complicated
expressions by viewing one or more of their parts as a single entity. á HSASSE.A.2. Use the structure of an expression to identify
ways to rewrite it. á
HSASSE.B.3. Choose and produce an
equivalent form of an expression to reveal and explain properties of the
quantity represented by the expression. á
HSASSE.B.3a. Factor a quadratic
expression to reveal the zeros of the function it defines. á HSASSE.B.3b. Complete the square in a quadratic expression to
reveal the maximum or minimum value of the function it defines. 
á
HSFIF.A.1. Understand that a
function from one set (called the domain) to another set (called the range)
assigns to each element of the domain exactly one element of the range. If f
is a function and x is an element of its domain, then f(x)
denotes the output of f corresponding to the input x. The graph of f is the
graph of the equation y = f(x). á
HSFIF.A.2. Use function notation,
evaluate functions for inputs in their domains, and interpret statements that
use function notation in terms of a context. á
HSFIF.B.4. For a function that
models a relationship between two quantities, interpret key features of
graphs and tables in terms of the quantities, and sketch graphs showing key
features given a verbal description of the relationship. á
HSFIF.B.5. Relate the domain of a
function to its graph and, where applicable, to the quantitative relationship
it describes á
HSFIF.C.7. Graph functions
expressed symbolically and show key features of the graph, by hand in simple
cases and using technology for more complicated cases. á HSFIF.C.7a. Graph linear and quadratic functions and show intercepts, maxima, and minima. HSFIF.C.8. Write a function defined by an expression in
different but equivalent forms to reveal and explain different properties of
the function. á HSFIF.C.8a. Use the process of factoring and completing the
square in a quadratic function to show zeros, extreme values, and symmetry of
the graph, and interpret these in terms of a context. 
á
HSGCO.A.1. Know precise
definitions of angle, circle, perpendicular line, parallel line, and line
segment, based on the undefined notions of point, line, distance along a
line, and distance around a circular arc. á
HSGCO.A.2. Represent
transformations in the plane using, e.g., transparencies and geometry software;
describe transformations as functions that take points in the plane as inputs
and give other points as outputs. Compare transformations that preserve
distance and angle to those that do not (e.g., translation versus horizontal
stretch). á
HSGCO.A.3. Given
a rectangle, parallelogram, trapezoid, or regular polygon, describe the
rotations and reflections that carry it onto itself. á
HSGCO.A.4. Develop definitions of
rotations, reflections, and translations in terms of angles, circles,
perpendicular lines, parallel lines, and line segments. á HSGCO.A.5. Given a geometric figure and a rotation,
reflection, or translation, draw the transformed figure using, e.g., graph
paper, tracing paper, or geometry software. Specify a sequence of
transformations that will carry a given figure onto another. 
á
HSSID.A.1. Represent data with
plots on the real number line (dot plots, histograms, and box plots). á
HSSID.A.2. Use statistics
appropriate to the shape of the data distribution to compare center (median,
mean) and spread (interquartile range, standard deviation) of two or more
different data sets. á
HSSID.B.5. Summarize categorical
data for two categories in twoway frequency tables. Interpret relative
frequencies in the context of the data (including joint, marginal and conditional
relative frequencies). Recognize possible associations and trends in the
data. á
HSSID.B.6. Represent data on two
quantitative variables on a scatter plot and describe how the variables are
related. á
HSSID.B.6a. Fit a function to the
data; use functions fitted to data to solve problems in the context of the
data. Use given functions or choose a function suggested by the context.
Emphasize linear, quadratic, and exponential models. á
HSSID.B.6b. Informally assess the
fit of a model function by plotting and analyzing residuals. á HSSID.B.6c. Fit a linear function
for scatter plots that suggest a linear association. á
HSSID.C.7. Interpret the slope
(rate of change) and the intercept (constant term) of a linear fit in the
context of the data. á HSSID.C.8. Compute (using technology) and interpret the
correlation coefficient of a linear fit. 

Grade 10 













Grade 9 
á
HSNQ.A.1. Use units as a way to
understand problems and to guide the solution of multistep problems; choose
and interpret units consistently in formulas; choose and interpret the scale
and the origin in graphs and data displays. á
HSNQ.A.2. Define appropriate
quantities for the purpose of descriptive modeling. á
HSNQ.A.3. Choose a level of
accuracy appropriate to limitations on measurement when reporting quantities. 
á HSAAPR.A.1. Understand that polynomials form a system
analogous to the integers, namely, they are closed under the operations of
addition, subtraction, and multiplication; add, subtract, and multiply
polynomials. á
HSAAPR.B.2. Know and apply the
Remainder Theorem: For a polynomial p(x) and a
number a, the remainder on division by x Ð a is p(a), so p(a) = 0 if
and only if (x Ð a) is a factor of p(x). á
HSAAPR.B.3. Identify zeros of
polynomials when suitable factorizations are available, and use the zeros to
construct a rough graph of the function defined by the polynomial. á
HSAAPR.C.4. Prove polynomial
identities and use them to describe numerical relationships. á HSAAPR.C.5. (+) Know and apply the Binomial Theorem for the
expansion of (x + y)n in powers of x and y for a
positive integer n, where x and y are any numbers, with coefficients
determined for example by PascalÕs Triangle. 

á
HSGSRT.C. Define trigonometric
ratios and solve problems involving right triangles á
HSGSRT.C.6. Understand that by
similarity, side ratios in right triangles are properties of the angles in
the triangle, leading to definitions of trigonometric ratios for acute angles. á
HSGSRT.C.7. Explain and use the
relationship between the sine and cosine of complementary angles. á
HSGSRT.C.8. Use trigonometric
ratios and the Pythagorean Theorem to solve right
triangles in applied problems. á HSGSRT.D.9. (+) Derive the formula A = ½ ab sin© for the area of a triangle by drawing an
auxiliary line from a vertex perpendicular to the opposite side. 
á HSSIC.A. Understand and evaluate random processes underlying
statistical experiments á HSSIC.A.1. Understand that statistics is a process for making
inferences about population parameters based on a random sample from that
population. 

Grade 10 
á
HSNQ.A.1. Use
units as a way to understand problems and to guide the solution of multistep
problems; choose and interpret units consistently in formulas; choose and
interpret the scale and the origin in graphs and data displays. á
HSNQ.A.2.
Define appropriate quantities for the purpose of descriptive modeling. á
HSNQ.A.3.
Choose a level of accuracy appropriate to limitations on measurement when
reporting quantities. 












Grade 9 
á HSNVM.A.1. (+) Recognize vector quantities as having both magnitude
and direction. Represent vector quantities by directed line segments, and use
appropriate symbols for vectors and their magnitudes (e.g., v, v, v,
v). á HSNVM.A.2. (+) Find the components of a vector by subtracting
the coordinates of an initial point from the coordinates of a terminal point. á HSNVM.B.4. (+) Add and subtract vectors. á HSNVM.B.4a. Add vectors endtoend, componentwise, and by
the parallelogram rule. Understand that the magnitude of a sum of two vectors
is typically not the sum of the magnitudes. á HSNVM.B.4b. Given two vectors in magnitude and direction
form, determine the magnitude and direction of their sum. á HSNVM.B.4c. Understand vector subtraction v Ð w as v +
(Ðw), where Ðw is the additive inverse of w, with the same magnitude
as w and pointing in the opposite direction. Represent vector subtraction
graphically by connecting the tips in the appropriate order, and perform
vector subtraction componentwise. á HSNVM.B.5. (+) Multiply a vector by a scalar. á HSNVM.B.5a. Represent scalar multiplication graphically by
scaling vectors and possibly reversing their direction; perform scalar
multiplication componentwise, e.g., as c(v?, v?) =
(cv?, cv?). HSNVM.B.5b. Compute the magnitude of a scalar multiple cv using cv = cv. Compute the direction of cv knowing that when cv ? 0, the direction of cv is either along v (for c > 0) or against v (for c < 0). 
á HSACED.A.1. Create equations and inequalities in one variable
and use them to solve problems. Include equations arising from linear and
quadratic functions, and simple rational and exponential functions. á HSACED.A.2. Create equations in two or more variables to
represent relationships between quantities; graph equations on coordinate
axes with labels and scales. á HSACED.A.3. Represent constraints by equations or
inequalities, and by systems of equations and/or inequalities, and interpret
solutions as viable or nonviable options in a modeling context. á
HSACED.A.4. Rearrange formulas to
highlight a quantity of interest, using the same reasoning as in solving
equations.


á
HSGGPE.B.4. Use coordinates to
prove simple geometric theorems algebraically. á
HSGGPE.B.5. Prove the slope
criteria for parallel and perpendicular lines and use them to solve geometric
problems (e.g., find the equation of a line parallel or perpendicular to a
given line that passes through a given point). á HSGGPE.B.6. Find the point on a directed line segment between
two given points that divide the segment in a given ratio. 
á
HSSCP.A.1. Describe events as
subsets of a sample space (the set of outcomes) using characteristics (or
categories) of the outcomes, or as unions, intersections, or complements of
other events (Òor,Ó Òand,Ó ÒnotÓ). 

Grade 10 



á 
á
HSSCP.A.1.
Describe events as subsets of a sample space (the set of outcomes) using
characteristics (or categories) of the outcomes, or as unions, intersections,
or complements of other events (Òor,Ó Òand,Ó ÒnotÓ). á
HSSCP.A.2.
Understand that two events A and B are independent if the probability of A
and B occurring together is the product of their probabilities, and use this
characterization to determine if they are independent. á
HSSCP.A.3.
Understand the conditional probability of A given B as P(A
and B)/P(B), and interpret independence of A and B as saying that the
conditional probability of A given B is the same as the probability of A, and
the conditional probability of B given A is the same as the probability of B. á
HSSCP.A.5.
Recognize and explain the concepts of conditional probability and
independence in everyday language and everyday situations. á
HSSCP.B.6.
Find the conditional probability of A given B as the fraction of BÕs outcomes
that also belong to A and interpret the answer in terms of the model. á
HSSCP.B.7.
Apply the Addition Rule, P(A or B) = P(A) + P(B)
Ð P(A and B), and interpret the answer in terms of the model. 








Grade 9 

á
MP.1. Make sense of problems and
persevere in solving them. á
MP.2. Reason abstractly and
quantitatively. á
MP.3. Construct viable arguments
and critique the reasoning of others. á
MP.4. Model with mathematics. á
MP.5. Use appropriate tools
strategically. á
MP.6. Attend to precision. á
MP.7. Look for and make use of
structure. MP.8. Look for and express regularity in repeated reasoning. 

á
HSGMG.A.1. Use geometric shapes,
their measures and their properties to describe objects (e.g., modeling a
tree trunk or a human torso as a cylinder). á
HSGMG.A.2. Apply concepts of
density based on area and volume in modeling situations (e.g., persons per
square mile, BTUs per cubic foot). á HSGMG.A.3. Apply geometric methods to solve design problems
(e.g., designing an object or structure to satisfy constraints or minimize
cost; working with typographic grid systems based on ratios). 


Grade 10 

á
MP.1. Make
sense of problems and persevere in solving them. á MP.2. Reason abstractly and quantitatively á MP.3. Construct viable arguments and critique the reasoning of
others á
MP.4. Model
with mathematics. á
MP.5. Use
appropriate tools strategically. á
MP.6. Attend
to precision. á
MP.7. Look for
and make use of structure. á
MP.8. Look for
and express regularity in repeated reasoning. 

á
HSGMG.A.1.
Use geometric shapes, their measures and their properties to describe objects
(e.g., modeling a tree trunk or a human torso as a cylinder). á
HSGMG.A.2.
Apply concepts of density based on area and volume in modeling situations
(e.g., persons per square mile, BTUs per cubic foot). á
HSGMG.A.3.
Apply geometric methods to solve design problems (e.g., designing an object
or structure to satisfy constraints or minimize cost; working with
typographic grid systems based on ratios). 









Grade 9 



á
MP.1. Make sense of problems and
persevere in solving them. á
MP.3. Construct viable arguments
and critique the reasoning of others. á
MP.4. Model with mathematics. á
MP.5. Use appropriate tools
strategically. MP.6. Attend to precision. 


Grade 10 



á
MP.1. Make
sense of problems and persevere in solving them. á
MP.3.
Construct viable arguments and critique the reasoning of others. á
MP.4. Model
with mathematics. á
MP.5. Use
appropriate tools strategically. á
MP.6. Attend
to precision. 


Algebra 
Geometry 





Grade 9 
á
HSAREI.A.1. Explain each step in
solving a simple equation as following from the equality of numbers asserted
at the previous step, starting from the assumption that the original equation
has a solution. Construct a viable argument to justify a solution method. á
HSAREI.A.2. Solve simple rational
and radical equations in one variable, and give examples showing how
extraneous solutions may arise. á
HSAREI.B.3. Solve linear equations
and inequalities in one variable, including equations with coefficients
represented by letters. á
HSAREI.B.4. Solve quadratic
equations in one variable. á
HSAREI.B.4a. Use the method of
completing the square to transform any quadratic equation in x into an
equation of the form (x Ð p)² = q that
has the same solutions. Derive the quadratic formula from this form. á
HSAREI.B.4b. Solve quadratic
equations by inspection (e.g., for x² = 49), taking square roots,
completing the square, the quadratic formula and factoring, as appropriate to
the initial form of the equation. Recognize when the quadratic formula gives
complex solutions and write them as a ± bi for real numbers a and b.HSAREI.C.5. Prove that, given a system of two
equations in two variables, replacing one equation by the sum of that
equation and a multiple of the other produces a system with the same
solutions. á
HSAREI.C.6. Solve systems of
linear equations exactly and approximately (e.g., with graphs), focusing on
pairs of linear equations in two variables. á
HSAREI.C.7. Solve a simple system
consisting of a linear equation and a quadratic equation in two variables
algebraically and graphically. For example, find the points of intersection
between the line y = Ð3x and the circle x² + y² = 3. á
HSAREI.C.8. (+) Represent a system
of linear equations as a single matrix equation in a vector variable. á
HSAREI.D.10. Understand that the
graph of an equation in two variables is the set of all its solutions plotted
in the coordinate plane, often forming a curve . á
HSAREI.D.11. Explain why the
xcoordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the
equation f(x) = g(x); find the solutions approximately, values, or find
successive approximations. Include cases where f(x)
and/or g(x) are linear, polynomial,. Rational, absolute value, exponential,
and logarithmic functions. á HSAREI.D.12. Graph the solutions to a linear inequality in two
variables as a halfplane (excluding the boundary in the case of a strict
inequality), and graph the solution set to a system of linear inequalities in
two variables as the intersection of the corresponding halfplanes. 
á
HSGGMD.A.1. Give an informal
argument for the formulas for the circumference of a circle, area of a
circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, CavalieriÕs principle, and informal limit arguments. á
HSGGMD.A.2. (+) Given
an informal argument using CavalieriÕs principle
for the formulas for the volume of a sphere and other solid figures. á
HSGGMD.A.3. Use volume formulas
for cylinders, pyramids, cones and spheres to solve problems. á HSGGMD.B.4. Identify crosssectional shapes of slices of
threedimensional objects, and identify threedimensional objects generated
by rotations of twodimensional objects. 

Grade 10 
á
HSAREI.A.1.
Explain each step in solving a simple equation as following from the equality
of numbers asserted at the previous step, starting from the assumption that
the original equation has a solution. Construct a viable argument to justify
a solution method. á
HSAREI.A.2.
Solve simple rational and radical equations in one variable, and give
examples showing how extraneous solutions may arise. á
HSAREI.B.3.
Solve linear equations and inequalities in one variable, including equations
with coefficients represented by letters. á
HSAREI.C.5.
Prove that, given a system of two equations in two variables, replacing one
equation by the sum of that equation and a multiple of the other produces a
system with the same solutions. á
HSAREI.C.6.
Solve systems of linear equations exactly and approximately (e.g., with
graphs), focusing on pairs of linear equations in two variables. á
HSAREI.C.8.
(+) Represent a system of linear equations as a single matrix equation in a
vector variable. á
HSAREI.C.9.
(+) Find the inverse of a matrix if it exists and use it to solve systems of
linear equations (using technology for matrices of dimension 3 × 3 or
greater). á
HSAREI.D.10.
Understand that the graph of an equation in two variables is the set of all
its solutions plotted in the coordinate plane, often forming a curve (which
could be a line). á
HSAREI.D.12.
Graph the solutions to a linear inequality in two variables as a halfplane
(excluding the boundary in the case of a strict inequality), and graph the
solution set to a system of linear inequalities in two variables as the
intersection of the corresponding halfplanes. 
á
HSGGMD.A.1.
Give an informal argument for the formulas for the circumference of a circle,
area of a circle, volume of a cylinder, pyramid, and cone. Use dissection
arguments, CavalieriÕs principle, and informal
limit arguments. á HSGGMD.A.2. (+) Given an informal
argument using CavalieriÕs principle for the
formulas for the volume of a sphere and other solid figures. á HSGGMD.A.3. Use volume formulas for cylinders, pyramids,
cones and spheres to solve problems. á
HSGGMD.B.4.
Identify crosssectional shapes of slices of threedimensional objects, and
identify threedimensional objects generated by rotations of twodimensional
objects. 
