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GRADES
9/10
EALR 1: The
student understands and applies the concepts and procedures of mathematics.
Component 1.1: Understand and
apply concepts and procedures from number sense.
1.1.1 Understand and apply scientific notation. W
Read and use scientific
and exponential notation. [MC, RL]
Identify a real-life situation
to match a particular number written in scientific or exponential notation
and justify the answer. [MC, RL]
Use scientific or exponential
notation to simplify a problem. [RL, MC]
Illustrate the meaning
of scientific notation using pictures, diagrams, or numbers. [CU]
Read and translate numbers
represented in scientific notation from calculators and other technology,
tables, and charts.
1.1.4 Apply understanding of direct and inverse proportion
to solve problems. W
Explain a method for determining
whether a real-world problem involves direct proportion or inverse proportion.
[SP, CU, MC]
Explain a method for solving
a real-world problem involving direct proportion. [CU, MC]
Explain a method for solving
a real-world problem involving inverse proportion. [CU, MC]
Solve problems using direct
or inverse models (e.g., similarity, age of car vs. worth). [SP, MC]
Explain, illustrate, or
describe examples of direct proportion. [CU]
Explain, illustrate, or
describe examples of inverse proportion. [CU]
Use direct or inverse proportion
to determine a number of objects or a measurement in a given situation.
Computation
1.1.6 Apply strategies to compute
fluently with rational numbers in all forms
including whole number exponents. W
Complete multi-step computations
using order of operations in situations involving combinations of rational
numbers including whole number exponents and square roots of square numbers.
[MC]
Calculate using order of
operations on all forms of rational numbers (e.g., (32+5)2-8, 22+ 32).
Use properties to reorder
and rearrange expressions to compute more efficiently. [ RL]
Estimation
1.1.8
Apply estimation strategies to determine the reasonableness of results
in situations involving multi-step computations with rational numbers including
whole number powers and square and cube roots. W
Identify when an approximation
is appropriate. [MC]
Explain situations involving
real numbers where estimates are sufficient and others for which exact value
is required. [CU]
Justify why an estimate
would be used rather than an exact answer in a given situation. [CU]
Describe various strategies
used during estimation involving integers, rational numbers. [CU]
Use estimation to predict
or to verify the reasonableness of calculated results. [RL]
Component
1.2: Understand and apply concepts and procedures from measurement.
Attributes, units, and systems
1.2.1 Analyze
how changes in one or two dimensions of an object affect perimeter, area,
surface area, and volume. W
Describe and compare the
impact that a change in one or more dimensions has on objects (e.g., how doubling
one dimension of a cube affects the surface area and volume). [CU, MC]
Describe how changes in
the dimensions of objects affect perimeter, area, and volume in real world
situations (e.g., how does the change in the diameter of an oil drum affect
the area and volume). [CU, MC]
Solve problems by deriving
the changes in two dimensions necessary to obtain a desired surface area and/or
volume (e.g., given a box with certain dimensions, make the volume of the
box y cubic units by changing two dimensions of the box). [SP]
Compare a given change
in one or two dimensions on the perimeter, area, surface areas, or volumes
of two objects.
Determine the change in
one dimension given a change in perimeter, area, volume, or surface area.
1.2.3 Understand how to convert units of measure within systems
(U.S. or metric). W
Understand how to convert
units of measure within U.S. or within metric systems to achieve an appropriate
level of precision.
Convert within a system
to a unit size appropriate to a given situation.
Convert to a larger unit
within a system while maintaining the same level of precision (e.g., represent
532 centimeters to 5.32 meters).
Convert to a smaller unit
within a system to increase the precision of a derived unit of measurement.
Procedures, precision, and
estimation
1.2.5 Apply
formulas to calculate measurements of right prisms or right circular cylinders.
W
Explain how to use a formula
for finding the volume of a prism or cylinder. [CU, MC]
Use a formula to find the
volume of a prism or cylinder. [RL, MC]
Use a formula to derive
a dimension of a right prism or right cylinder given other measures.
Use formulas to describe
and compare the surface areas and volumes of two or more right prisms and/or
right cylinders. [RL]
Use formulas to obtain
measurements needed to describe a right cylinder or right prism.
1.2.6 Understand
and apply strategies to obtain reasonable measurements at an appropriate level
of precision. W
Identify situations in
which approximate measurements are sufficient.
Estimate a reasonable measurement
at an appropriate level of precision. [MC]
Estimate quantities using
derived units of measure (e.g., distance or time using miles per hour, cost
using unit cost). [MC]
Estimate derived units
of measure (e.g., miles per hour, people/year, grams/cubic centimeters). [MC]
Apply a process that can
be used to find a reasonable estimate for the volume of prisms, pyramids,
cylinders, and cones.
Estimate volume and surface
area for right cylinders and right prisms.
Component
1.3: Understand and apply concepts and procedures from geometric sense.
Properties and relationships
1.3.1 Understand the relationship
among characteristics of one-dimensional, two-dimensional, and three-dimensional
figures. W
Identify and label one-
and two-dimensional characteristics (rays, lines, end points, line segments,
vertices, and angles) in three-dimensional figures. [CU]
Match or draw three-dimensional
objects from different perspectives using the same properties and relationships
(e.g., match to the correct net, draw the top view). [RL]
Draw and label with names
and symbols nets of right prisms and right cylinders. [RL, CU]
Describe everyday objects
in terms of their geometric characteristics. [CU]
Describe or classify various
shapes based on their characteristics.
Make and test conjectures
about two-dimensional and three-dimensional shapes and their individual attributes
and relationships using physical, symbolic, and technological models (e.g.,
diagonal of a rectangle or prism is the longest interior segment; what figures
make up cross-sections of a given three-dimensional shape). [SP, RL, CU, MC]
1.3.2 Apply understanding of geometric properties and
relationships. W
Use geometric properties
and relationships to describe, compare, and draw two-dimensional and three-dimensional
shapes and figures.
Construct geometric figures
using a variety of tools and technologies (e.g., angle bisectors, perpendicular
bisectors, triangles given specific characteristics). [MC]
Draw a plane shape and
justify the answer given a set of characteristics. [RL, CU]
Use the properties of two-dimensional
and three-dimensional shapes to solve mathematical problems (e.g., find the
width of a river based on similar triangles; given a set of parallel lines,
a transversal, and an angle, find the other angles). [SP, RL, CU, MC]
Compare two-dimensional
and three-dimensional shapes according to characteristics including faces,
edges, and vertices, using actual and virtual modeling. [RL, CU]
Use technology to generate
two and three dimensional models of geometric figures with given geometric
characteristics (e.g., generate a two-dimensional animation using pentagons
with fixed coordinates for one edge). [RL, SP]
Create a three-dimensional
scale drawing with particular geometric characteristics. [SP, CU, MC]
Locations and transformations
1.3.3 Apply understanding of geometric properties and
location of points to figures. W
Use coordinates to describe
or identify the location of objects on coordinate grids.
Describe geometric characteristics
of two-dimensional objects using coordinates on a grid. [MC]
Describe the location of
points that satisfy given conditions (e.g., the set of points equidistant
from a given point; a point equidistant from a given set of points). [CU]
Represent situations on
a coordinate grid or describe the location of points that satisfy given conditions
(e.g., locate a gas station to be equidistant from given cities; locate a
staking point to maximize the grazing area of a tethered goat). [MC, SP, RL]
Use tools and technology
to draw objects on a coordinate grid based on given conditions. [CU]
Identify, interpret, and
use the meaning of slope of a line as a rate of change using physical, symbolic,
and technological models. [SP, RL, MC]
1.3.4 Apply understanding of multiple transformations.
W
Apply multiple transformations
to create congruent and similar figures in any or all of the four quadrants.
Use multiple transformations
(combinations of translations, reflections, or rotations) to draw an image.
[RL]
Use dilation (expansion
or contraction) of a given shape to form a similar shape. [RL, CU]
Determine the final coordinates
of a point after a series of transformations. [RL, CU]
Examine figures to determine
rotational symmetry about the center of the shape. [RL, MC]
Define a set of transformations
that would map one onto the other given two similar shapes. [SP, RL]
Create a design with or
without technology using a combination of two or more transformations with
one or two two-dimensional figures. [SP, RL]
Use technology to create
two- and three-dimensional animations using combinations of transformations.
[MC, SP, RL]
Component
1.4: Understand and apply concepts and procedures from probability and
statistics.
Probability
1.4.1 Understand the concept of conditional probability.
W
Compare the probabilities
of dependent and independent events. [CU, MC]
Determine and justify whether
the outcome of a first event affects the probability of a later event (e.g.,
drawing cards from a deck with or without replacement). [CU]
Explain the difference
between dependent and independent events. [CU]
Explain and give examples
of compound events. [CU]
1.4.2 Apply understanding of dependent and
independent events to calculate probabilities. W
Determine probabilities
of dependent and independent events. [SP]
Generate the outcomes and
probability of multiple independent and dependent events using a model or
procedure (e.g., tree diagram, area model, counting procedures).
Generate the outcomes and
probability of events using a counting procedure (e.g., the number of license
plates that can be made with three letters and three numbers; winning the
lottery). [MC]
Explain the relationship
between theoretical probability and empirical frequency of dependent events
using simulations with and without technology. [CU]
Create a simple game based
on independent probabilities wherein all players have an equal probability
of winning. [MC, SP]
Create a simple game based
on compound probabilities. [MC, SP]
Determine the sample space
for independent or dependent events.
Statistics
1.4.3 Apply appropriate methods and technology to collect
data or evaluate methods used by others for a given research questions. W
Identify sources of bias
in data collection questions, samples, and/or methods and describe how such
bias can be controlled. [RL, CU]
Evaluate methods and technology
used to investigate a research question. [CU, MC]
Collect data using appropriate
methods.
Use technology appropriately
to collect data. [RL, MC]
Identify appropriate data
collection methods that might impact the accuracy of the results of a given
situation. [RL, CU]
Determine whether the sample
for a given study was from a representative sample.
Determine whether the methods
of data collection used were appropriate for a given question or population.
[RL]
1.4.4 Understand and apply techniques to find the equation
for a reasonable linear model. W
Determine the equation
for a reasonable line to describe a set of bivariate data. [RL, MC]
Determine the equation
of a line that fits the data displayed on a scatter plot. [SP, RL]
Use technology to determine
the line of best fit for a set of data. [MC]
Match an equation with
a set of data. [MC]
Match an equation with
a graphic display. [MC]
Create a graph based on
the equation for the line.
1.4.5 Analyze a linear model to judge its appropriateness
for a data set. W
Determine whether a straight
line is an appropriate way to describe a trend in a set of bivariate data.
[MC, RL]
Determine whether the underlying
model for a set of data is linear. [RL, MC]
Decide and explain whether
it is appropriate to extend a given data set following a line of best fit.
[RL, MC]
Determine whether a linear
prediction from a given set of data is appropriate for the data and support
the decision with data. [MC].
Determine whether an equation
for a line is appropriate for a given set of data and support the judgment
with data. [RL, MC]
Use technology to generate
data to fit a linear model. [SP, MC]
1.4.6 Apply understanding of statistics to
make, analyze, or evaluate a statistical argument. W
Identify trends in a set
of data in order to make a prediction based on the information. [CU, MC]
Justify a prediction or
an inference based on a set of data. [CU, MC]
State possible factors
that may influence a trend but not be reflected in the data (e.g., population
growth of deer vs. availability of natural resources or hunting permits).
[MC, CU, RL]
Use statistics to support
different points of view. [RL]
Analyze a set of statistics
to develop a logical point of view. [RL. CU, MC]
Justify or refute claims
and supporting arguments based on data. [CU, MC]
Determine whether statistics
have been used or misused to support a point of view or argument and support
the evaluation with data. [RL]
Component
1.5: Understand and apply concepts and procedures from algebraic sense.
Patterns, functions,
and other relations
1.5.1 Apply processes that use repeated addition
(linear) or repeated multiplication (exponential). W
Recognize, extend, or create
a pattern or sequence between sets of numbers and/or linear patterns. [RL,
CU, MC]
Identify, extend, or create
a geometric or arithmetic sequence or pattern. [RL, CU]
Translate among equivalent
numerical, graphical, and algebraic forms of a linear function. RL, MC]
Make predictions based
on a pattern or sequence.
1.5.2 Analyze a pattern, table, graph, or model
involving repeated addition (linear) or repeated multiplication (exponential)
model to write an equation or rule. W
Find the equation of a
line in a variety of ways (e.g., from a table, graph, slope-intercept, point-slope,
two points). [RL, MC]
Generate and use rules
for a pattern to make predictions about future events (e.g., population growth,
future sales, growth of corn stalks, future value of savings account). [SP,
RL, MC]
Identify or write an equation
or rule to describe a pattern, sequence, and/or a linear function. [RL, CU,
MC]
Write an equation for a
line given a set of information (e.g., two points, point-slope, etc.). [CU,
MC]
Write a recursive definition
of a geometric pattern (e.g., Start and New = Old * Number). [CU, MC]
Represent systems of equations
and inequalities graphically. [RL, MC]
Write a story that represents
a given linear equation or expression. [CU, MC]
Write an expression, equation,
or inequality with two variables representing a linear model of a real-world
problem. [CU, MC]
Symbols and
representations
1.5.4 Apply understanding of equations, tables,
or graphs to represent situations involving relationships that can be written
as repeated addition (linear) or repeated multiplication (exponential). W
Represent variable quantities
through expressions, equations, inequalities, graphs, and tables to represent
linear situations involving whole number powers and square and cube roots.
[CU, MC]
Identify and use variable
quantities to read and write expressions and equations to represent situations
that can be described using repeated addition (e.g., models that are linear
in nature). [CU, MC]
Identify and use variable
quantities to read and write expressions and equations to represent situations
that can be described using repeated multiplication (e.g., models that are
exponential such as savings accounts and early stages of population growth).
[CU, MC]
Recognize and write equations
in recursive form for additive models (e.g., starting value, New=Old + some
number). [CU, MC]
Recognize and write equations
in recursive form for additive models (e.g., starting value, New=Old some
number). [CU, MC]
Select an expression or
equation to represent a given real world situation. [MC]
Evaluating
and solving
1.5.5 Apply procedures to simplify expressions.
W
Simplify expressions and
evaluate formulas involving exponents.
Justify a simplification
of an expression involving exponents. [RL, CU]
Use multiple mathematical
strategies and properties to simplify expressions.
1.5.6 Apply procedures to solve equations and systems
of equations. W
Rearrange formulas to solve
for a particular variable (e.g., given
, solve for h). [MC, CU]
Solve real-world situations
involving linear relationships and verify that the solution makes sense in
relation to the problem. [SP, RL, CU, MC]
Find the solution to a
system of linear equations using tables, graphs, and symbols. [CU, MC]
Interpret solutions of
systems of equations. [CU, MC]
Solve multi-step equations.
[SP, RL]
Use systems of equations
to analyze and solve real-life problems. [SP, CU, MC]
Determine when two linear
options yield the same outcome (e.g., given two different investment or profit
options, determine when both options will yield the same result).
Use systems of equations
to determine the most advantageous outcome given a situation (e.g., given
two investment options, determine under what conditions each will yield the
best result.). [MC, SP]
EALR 2: The student uses mathematics to define
and solve problems.
Component 2.1: Investigate situations.
Example: The following are the times (in seconds) of the
Olympics in the given years. Using this information, is it reasonable to
believe that the women will run as fast as the men in this event? Justify
your answer using this data:
Year Mens Womens
Year Mens Womens
1948 10.3 11.9
1976 10.06 11.08
1952 10.4 11.5
1980 10.25 11.06
1956 10.5 11.5
1984 9.99 10.97
1960 10.2 11.0
1988 9.92 10.54
1964 10 11.4
1992 9.96 10.82
1968 9.95 11.0
1996 9.84 10.94
1972 10.14 11.07
2000 9.87 10.75
2.1.1 Analyze a situation
to define a problem. W
Use strategies to become
informed about the situation (e.g., listing information; examine the table
for patterns; create a scatter plot to look for patterns; asking questions).
Summarize the problem (e.g.,
there are Olympic winning times over the past 50 years; both mens and womens
times are decreasing; will there come a time when women run faster than men).
Determine whether enough
information is given to find a solution (e.g., list what is needed to be found;
extend the pattern to see if womens times will be less).
Determine whether information
is missing or extraneous (e.g., compare the list of known things to the list
of needed things to see if there are things that are not needed).
Define the problem (e.g.,
if the pattern continues in the same fashion, will women run faster than men
and, if so, when will that occur).
Component 2.2: Apply strategies
to construct solutions.
2.2.1 Apply strategies,
concepts, and procedures to devise a plan to solve the problem. W
Organize relevant information
from multiple sources (e.g., create a list of known and unknown information;
create a scatter plot of mens and womens times vs. time on the same coordinate
axis to analyze the patterns).
Select and apply appropriate
mathematical tools to devise a strategy in a situation (e.g., if the data,
in either tabular or graphical form, suggest a linear relationship, plan to
find a linear equation for each set of data; solve those equations simultaneously
[or use technology to find the intersection of the two lines] to answer the
question). If the data pattern suggests a non-linear model, plan to project
what the pattern is and extend that pattern.
2.2.2 Apply mathematical
tools to solve the problem. W
Implement the plan devised
to solve the problem (e.g., solve the set of simultaneous equations to arrive
at a time where the two times are the same).
Use mathematics to solve
the problem (e.g., use algebra to write equations for the two linear models,
solve the system of equations using either symbols or technology).
Identify when an approach
is unproductive and modify or try a new approach (e.g., if the result does
not make sense in the context, return to the plan to see if something has
gone wrong and adjust accordingly).
Check the solution to see
if it works (e.g., the solution may be a partial year [i.e., 2003.6]; decide
how to deal with this and also if the year is reasonable [i.e., 1925 does
not make sense given the context]).
EALR 3: The student uses
mathematical reasoning.
Component
3.1: Analyze information.
3.1.1. Synthesize information
from multiple sources in order to answer questions. W
Use the properties of two-dimensional
and three-dimensional figures to solve mathematical problems (e.g., find the
width of a river based on similar triangles; given a set of parallel lines,
a transversal, and an angle, find the other angles).
Component
3.2: Make predictions, inferences, conjectures, and draw conclusions.
3.2.1 Apply skill of conjecturing and analyze conjectures
by formulating a proof or constructing a counter example. W
Make and test conjectures
about two-dimensional and three-dimensional figures and their individual attributes
and relationships using physical, symbolic, and technological models (e.g.,
diagonal of a rectangle or prism is the longest interior segment; what figures
make up cross-sections of a given three-dimensional shape). (1.3.1)
3.2.2 Analyze information to draw conclusions and
support them using inductive and deductive reasoning. W
Compare and describe the
volume of cylinders, cones, and prisms when an attribute is changed (e.g.,
the area of the base, the height of solid). (1.2.4)
Draw a plane shape of a
given set of characteristics and justify the answer. (1.3.2)
Identify trends in a set
of data in order to make a prediction based on the information. (1.4.6)
Use statistics to support
different points of view. (1.4.6)
3.2.3 Analyze procedures to determine
appropriateness of claims and arguments. W
Examine claims and supporting
arguments based on data and make needed revisions. (1.4.6)
