Download DC-CAS: PERFORMANCE LEVEL DESCRIPTORS

DC-CAS: PERFORMANCE LEVEL DESCRIPTORS
Mathematics Grade 10
The DC-CAS is a standards-based assessment. Based on performance, each student is classified as performing
at one of four performance levels: advanced, proficient, basic, or below basic. The descriptions below provide a
brief summary of typical performance for each level. The skills identified in each descriptor represent, but are
not all-inclusive of, the skills a student is able to demonstrate at each performance level.
Students in the Below Basic level are able to:
• simplify expressions involving positive and negative exponents
• identify and delineate the difference between a rational and an irrational number
• use the properties of exponents to rewrite radical numbers under 100 as rational numbers with rational
exponents
• choose and interpret the scale and the origin in graphs and data displays
• identify parts of an algebraic expression or equation (term, factor, coefficient, etc.)
• solve linear equations and inequalities in one variable
• solve a system of linear equations approximately by graphing
• write linear equations and inequalities in one variable to represent a context in a word problem
• demonstrate the understanding that polynomials are closed under addition
• represent linear equations with integer coefficients in one and two variables graphically with and
without technology
• understand that a function assigns to each element of the domain exactly one element of the range
• understand and use function notation
• estimate the rate of change of a function for a specified interval from a graph
• identify a function and a non-function
• evaluate functions for inputs in their domains
• graph linear functions and show intercepts
• construct a linear function given a graph, a description of a relationship, or two input-output pairs
• understand that the x-coordinates 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)
• know 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
• given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections
that carry it onto itself
• develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular
lines, parallel lines, and line segments
• given a geometric figure and a rotation, reflection, or translation, identify the transformed figure
• specify a sequence of translations, reflections, and rotations that will carry a given figure onto another
• use geometric shapes, their measures, and their properties to describe objects
• identify the shapes of two-dimensional cross-sections of spheres and rectangular prisms
• represent data with plots on the real-number line
• describe a data set in terms of center and spread
• summarize categorical data for two categories in two-way frequency tables
Students in the Basic level are able to:
• use the properties of exponents to rewrite expressions involving rational exponents
• rewrite radicals as rational numbers with rational exponents and vice versa
• choose and interpret units consistently in formulas with a familiar context
• know that rational numbers are closed under addition and multiplication
• complete measurement conversions with simple units
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DC-CAS: PERFORMANCE LEVEL DESCRIPTORS
Mathematics Grade 10
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perform operations with rational numbers and common irrational numbers
interpret parts of an expression (term, factor, coefficient, etc.) in terms of its context
interpret complicated expressions by viewing one or more of their parts as a single entity
graph the solutions to a linear inequality in two variables as a half-plane
graph equations on a coordinate plane
solve a system of linear equations algebraically
understand that the graph of an equation in two variables is the set of all its solutions plotted on the
coordinate plane, often forming a curve (which could be a line)
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
graphically find the approximate solutions of the equation f(x) = g(x) when equations y = f(x) and y =
g(x) intersect in linear functions
add, subtract, and multiply single variable polynomials with a degree of 2 or less
add and subtract simple rational expressions
represent constraints by equations
represent quadratic equations with integer coefficients in one and two variables graphically
calculate the average rate of change of a function over a specified interval from a table
relate the domain of a function to its graph
recognize that sequences are functions
interpret statements that use function notation in terms of a context
identify the intercepts of linear functions shown graphically
add and subtract linear functions
recognize situations in which one quantity changes at a constant rate per unit interval relative to
another
find the inverse for a simple linear function f that has an inverse
observe that a quantity increasing exponentially eventually exceeds a quantity increasing linearly
describe transformations as functions that take points on the coordinate plane as inputs and give other
points as outputs
understand that the dilation of a line segment is longer or shorter in the ratio given by the scale factor
understand that a dilation takes a line not passing through the center of the dilation to a parallel line and
leaves a line passing through the center unchanged
use the definition of congruence in terms of rigid motions to decide if two figures are congruent
understand and be able to show that triangles are congruent if and only if their corresponding angles
and corresponding sides are congruent
understand and interpret basic proofs about lines, angles, triangles, and parallelograms
given a geometric figure and a rotation, reflection, or translation construct the transformed figure
use geometric descriptions of rigid motions to transform figures
use the definition of similarity in terms of similarity transformations to decide if two figures are similar
identify relationships among inscribed angles, radii, and chords
know the trigonometric ratios
give informal arguments for the formulas for the circumference and area of a circle
use the volume formula for cylinders to solve problems
identify the shapes of two-dimensional cross-sections of three-dimensional objects
use coordinates to compute perimeters of polygons and areas of triangles and rectangles
use statistics to compare the center and spread of two or more different data sets
identify differences in the center and spread of two comparable data sets
summarize categorical data for two categories in two-way frequency tables
represent data on two quantitative variables on a scatter plot
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DC-CAS: PERFORMANCE LEVEL DESCRIPTORS
Mathematics Grade 10
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interpret the slope of a linear model in the context of the data
fit a linear function for a scatter plot that suggests a linear association
In addition to meeting all “Basic” requirements, students at the Proficient level are able to:
• choose a level of accuracy appropriate to limitations on measurement
• 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
• extend rules of operations with integer exponents to operations involving rational exponents
• use units as a way to guide the solution of multi-step problems in familiar contexts
• create the scale and origin of a graph or data display
• rewrite rational expressions in radical form and vice versa
• use the structure of an expression to identify ways to rewrite it
• factor a quadratic expression to reveal the zeros or the maximum or minimum values of the function it
defines
• use the quadratic formula for solving a quadratic equation
• graph the solution set to a system of linear inequalities in two variables as the intersection of the
corresponding half-planes
• solve a simple system consisting of a linear equation and a quadratic equation in two variables
graphically
• use equations and inequalities in one variable to solve problems
• create equations in two or more variables to represent relationships between quantities
• add, subtract, and multiply polynomials with multiple variables
• understand polynomials are closed under subtraction and multiplication
• rewrite simple rational expressions in different forms
• multiply and divide rational expressions
• find the approximate solutions of the equation f(x) = g(x) when equations y = f(x) and y = g(x) intersect
in linear, polynomial, rational, and absolute functions
• represent constraints by inequalities
• manipulate equations and formulas to highlight a quantity of interest
• use the method of completing the square to transform any quadratic equation in x into an equation of
the form (x – p)2 = q that has the same solutions
• sketch graphs of linear functions showing key features given a verbal description of the relationship
• solve quadratic equations with integer roots by inspection, square roots, and factoring
• interpret the average rate of change of a function over a specified interval
• for a function that models a relationship between two quantities, interpret key features of graphs and
tables in terms of the quantities
• relate the domain of a function to the quantitative relationship it describes
• graph quadratic functions and show intercepts, maxima, and minima
• graph square root and cube root functions
• determine an explicit expression
• multiply and divide linear functions
• compare properties of two functions each represented in a different way
• interpret the parameters in an exponential function in terms of a context
• construct exponential functions, including arithmetic and geometric sequences, given a graph, a
description of a relationship, or two input-output pairs
• prove that linear functions grow by equal differences over equal intervals
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DC-CAS: PERFORMANCE LEVEL DESCRIPTORS
Mathematics Grade 10
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recognize situations in which a quantity grows or decays by a constant rate per unit interval relative to
another
find the inverse for a simple quadratic function f that has an inverse
identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values
of k; find the value of k given the graphs
compare transformations that preserve distance and angle to those that do not (e.g., translation versus
horizontal stretch)
specify a sequence of transformations that will carry a given figure onto another
prove theorems about lines, angles, triangles, and parallelograms
use geometric descriptions of rigid motions to predict the effect of a given rigid motion on a given
figure
use the properties of similarity transformations to establish the AA criterion for two triangles to be
similar
use congruence and similarity criteria for triangles to solve problems
use the relationship between the sine and cosine of complementary angles
use the Pythagorean theorem to solve right triangles in applied problems
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
explain how the criteria for triangle congruence follow from the definition of congruence in terms of
rigid motion
describe relationships among inscribed angles, radii, and chords
understand that the length of the arc intercepted by an angle is proportional to the radius
define the radian measure of the angle intercepting an arc as the constant of proportionality
give an informal argument for the formula for the volume of a cylinder
use formulas to find the volumes of pyramids, cones, and spheres
identify three-dimensional objects generated by rotations of circles, rectangles, or triangles
apply concepts of density based on area and volume in modeling situations
use coordinates to prove simple geometric theorems algebraically
prove the slope criteria for parallel and perpendicular lines
find the point on a directed line segment between two given points that partitions the segment in a
given ratio
interpret effects of extreme data points on the shape, center, and spread of two comparable data sets
interpret relative frequencies in the context of data in two-way frequency tables and recognize possible
associations and trends in the data
describe how the variables in a scatter plot are related
fit a function to the data
use functions fitted to data to solve problems in the context of the data
interpret the slope and the intercept of a linear model in the context of the data
compute the correlation coefficient of a linear fit
distinguish between correlation and causation
In addition to meeting all “Proficient” requirements, students at the Advanced level are able to:
• explain how the meaning of rational exponents follows from extending the rules of operations with
integer exponents to operations involving rational exponents
• rewrite expressions involving radicals and rational exponents using the properties of exponents
• use units to solve multi-step problems with unfamiliar contexts
• define appropriate quantities for the purpose of descriptive modeling
• use the properties of exponents to transform expressions for exponential functions
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DC-CAS: PERFORMANCE LEVEL DESCRIPTORS
Mathematics Grade 10
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understand that polynomials are closed under subtraction and multiplication
solve a simple system consisting of a linear equation and a quadratic equation in two variables
algebraically
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
construct a viable argument to justify a solution method when solving a simple equation
interpret solutions of equations and inequalities as viable or non-viable options in a modeling context
represent constraints by a system of equations or inequalities
use the method of completing the square to derive the quadratic formula
recognize when the quadratic formula gives complex solutions and write them as a ± bi for real
numbers a and b
sketch a graph showing the key features of a given verbal description of a relationship
explain why the x-coordinates 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)
graph piecewise-defined functions, including step functions and absolute value functions
use the properties of exponents to interpret expressions for exponential functions and to review and
explain different properties of the function
determine a recursive process or steps for calculation from a context
interpret the parameters in an exponential function in terms of the context
prove that exponential functions grow by equal factors over equal intervals
solve an equation of the form f(x) = c for a simple function f that has an inverse and find the inverse
graph functions both by hand and by using technology
prove that all circles are similar
use congruence and similarity criteria to prove relationships in geometric figures
explain the relationship between the sine and cosine of complementary angles
use similarity transformations to explain that corresponding angles of similar triangles are congruent
and that corresponding side lengths are proportional
use trigonometric ratios to solve right triangles in applied problems
derive the formula for the area of a sector of a circle
give an informal argument for the formulas for the volume of a pyramid, cone, and sphere
use formulas to find the volume of three-dimensional figures made of prisms, pyramids, cylinders, and
spheres to solve problems
identify three-dimensional objects generated by rotations of two-dimensional objects
apply geometric methods to solve design problems
use the slope criteria for parallel and perpendicular lines to solve geometric problems
interpret relative frequencies in the context of a data set
visually assess the fit of a function to data in a scatter plot by plotting and analyzing residuals
interpret the correlation coefficient of a linear fit
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