Next Generation Standards and Objectives
for Mathematics in West Virginia Schools
Descriptive Analysis of Math 1 Objectives
Descriptive Analysis of the Objective – a narrative of what the child knows, understands and is able to do upon
mastery.
Relationships Between Quantities
Reason quantitatively and use units to solve problems. (Working with quantities and the relationships between them
provides grounding for work with expressions, equations, and functions.)
M.1HS.RBQ.1
use units as a way to understand problems and to guide
the solution of multi-step problems; choose and
interpret units consistently in formulas; choose and
interpret the scale and the origin in graphs and data
displays.
M.1HS.RBQ.2
define appropriate quantities for the purpose of
descriptive modeling.
M.1HS.RBQ.3
choose a level of accuracy appropriate to limitations on
measurement when reporting quantities.
Students understand how to use units as a method of interpreting problems
and to direct them to the solution of a problem. They are able to choose and
interpret units in formulas, in graphs, and in data displays. Students should
construct graphs using a variety of data sets. The following problem task can
be used: Your college savings fund has $1800 in it and you plan to spend $30 a
week. What would be an appropriate viewing window and scale to see the
remaining balance each week until the money is gone? Explain.
Students determine and interpret appropriate quantities when using descriptive
modeling. The following skill-based task can be used: How would you
measure the rate at which a swimming pool fills with water?
Students are able to determine the accuracy of values based on their limitations
in the context of the situation. Discuss misconceptions in resulting calculations
involving measurement, e.g., you cannot increase accuracy through calculation,
only through more accurate measurement. Discuss sensible levels of accuracy
when discussing the various distances or weights such as the weight of an
elephant, the distance to the moon, or the volume of a basketball. The
following problem task can be used: A rectangle is measured to be 3.4 cm x 5.2
cm. Why is it not accurate to say that the area of the rectangle is 17.68 cm 2?
Interpret the structure of expressions.
M.1HS.RBQ.4
interpret expressions that represent a quantity in terms
of its context.*
a. interpret parts of an expression, such as terms,
factors, and coefficients.
b. interpret complicated expressions by viewing
one or more of their parts as a single entity. For
example, interpret P(1+r)n as the product of P and a
factor not depending on P. (Limit to linear expressions
and to exponential expressions with integer exponents.)
Students will be able to identify the different parts of the expression and
explain the meaning of each within the context of a problem. They will
understand how to decompose expressions and make sense of the multiple
factors and terms. Design a game around identifying terms, bases, exponents,
coefficients, and factors. Use various formulas to discuss what the terms and
coefficients mean in terms of the real world context of the formula. The
following problem task can be used: Interpret the expression: 5 – 2 (x – y)2
Explain the possible output values.
Create equations that describe numbers or relationships.
M.1HS.RBQ.5
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. (Limit to linear and exponential
equations, and, in the case of exponential equations, limit to
situations requiring evaluation of exponential functions at integer
inputs.)
Students will be able to create linear and exponential equations and inequalities
in one variable and use them in a contextual situation to solve problems. The
following problem task can be used: Given that the following trapezoid has
area 54 cm2, set up an equation to find the length of the base, and solve the
equation.
M.1HS.RBQ.6
create equations in two or more variables to represent
relationships between quantities; graph equations on
coordinate axes with labels and scales. (Limit to linear and
exponential equations, and, in the case of exponential equations,
limit to situations requiring evaluation of exponential functions at
integer inputs.)
Students will be able to create equations in two or more variables to represent
relationships between quantities. Additionally, they will graph equations. Use
technology to explore a variety of linear and exponential graphs. The following
skill-based task can be used: Write and graph an equation that models the cost
of buying and running an air conditioner with a purchase price of $250 which
costs $0.38/hr. to run.
M.1HS.RBQ.7
represent constraints by equations or inequalities, and by
systems of equations and/or inequalities, and interpret
solutions as viable or non-viable options in a modeling
context. For example, represent inequalities describing
nutritional and cost constraints on combinations of
different foods. (Limit to linear equations and
inequalities.)
M.1HS.RBQ.8
rearrange formulas to highlight a quantity of interest,
using the same reasoning as in solving equations. For
example, rearrange Ohm’s law V = IR to highlight
resistance R. (Limit to formulas with a linear focus.)
Students will be able to write and to use a system of equations and/or
inequalities to solve a real world problem. They will recognize that the
equations and inequalities represent the constraints of the problem. The
following skill-based task can be used: A club is selling hats and jackets as a
fundraiser. Their budget is $1500 and they want to order at least 250 items.
They must buy at least as many hats as they buy jackets. Each hat costs $5 and
each jacket costs $8.
o Write a system of inequalities to represent the situation.
o Graph the inequalities.
o If the club buys 150 hats and 100 jackets, will the conditions be
satisfied?
o What is the maximum number of jackets they can buy and still meet
the conditions?
Students will be able to solve literal equations for a specific variable. The
following problem task can be used: Paul just arrived in England and heard the
temperature in degrees Celsius. He remembers that C = (5/9)(F – 32). How
will Paul find the temperature in Fahrenheit?
Linear and Exponential Relationships
Represent and solve equations and inequalities graphically.
M.1HS.LER.1
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). (Focus on linear and exponential equations and be able to
adapt and apply that learning to other types of equations in future
courses.)
M.1HS.LER.2
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); find the
solutions approximately, e.g., using technology to graph
the functions, make tables of values or find successive
approximations. Include cases where f(x) and/or g(x)
are linear, polynomial, rational, absolute value,
exponential and logarithmic functions.* (Focus on cases
where f(x) and g(x) are linear or exponential.)
M.1HS.LER.3
graph the solutions to a linear inequality in two variables
as a half-plane (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 half-planes.
Understand that all solutions to an equation in two variables are contained on
the graph of that equation. Create a matching game in which students match
equations, graphs of equations, and solutions. The following skill-based tasks
can be used: Given a graph of the equation , find three solutions that will
satisfy the equation: 3x + y = 9. Given a graph representing the growth of a
savings account over time with a given rate of return, determine the value of
the account after 3 years, 5 years, 10 years, 12 years and 6 months.
Students understand and can explain why the intersection of y = f(x) and y =
g(x) is the solution of f(x) = g(x) for any combination of linear, polynomial,
rational, absolute value, exponential, and logarithmic functions.
They find the solution(s) by using technology to graph the functions, by using
tables of values, or by using successive approximations. The following skillbased task can be used: Use technology to graph and compare a beginning
salary of $25 per day increased by $5 each day and a beginning salary of $0.01
per day, which doubles each day. When are the salaries equal? How do you
know? The following problem task can be used: Explain why a company has
to sell 100 soccer balls before they will make a profit. The cost of producing a
soccer ball is modeled by C = 10x + 1000. The retail price of a soccer ball is
$20.
Students are able to graph the solutions to a linear inequality in two variables as
a half-plane. Additionally, they can graph the solution set to a system of linear
inequalities in two variables as the intersection of corresponding half-planes.
Use technology to model examples of intersections of inequalities. The
following problem task can be used: Graph the system of linear inequalities
below and determine if (3, 2) is a solution to the system.
Understand the concept of a function and use function notation. (Students should experience a variety of types of
situations modeled by functions. Detailed analysis of any particular class of function at this stage is not advised. Students
should apply these concepts throughout their future mathematics courses. Draw examples from linear and exponential
functions. In M.1HS.LER.6, draw connection to M.1HS.LER.5, which requires students to write arithmetic and geometric
sequences. Emphasize arithmetic and geometric sequences as examples of linear and exponential functions.)
M.1HS.LER.4
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).
Students will use the definition of a function to determine whether a
relationship is a function. Given the function f(x), students should be able to
identify x as an element of the domain, the input, and f(x) is an element in the
range, the output. Know that the graph of the function, f, is the graph of the
equation y=f(x). The following problem task can be used: Write a story that
would generate a relation that is a function. Write a story that would generate a
relation that is not a function.
M.1HS.LER.5
use function notation, evaluate functions for inputs in
their domains and interpret statements that use function
notation in terms of a context.
M.1HS.LER.6
recognize that sequences are functions, sometimes
defined recursively, whose domain is a subset of the
integers. For example, the Fibonacci sequence is defined
recursively by f(0) = f(1) = 1, f(n+1) = f(n)+ f(n-1) for
n ≥ 1.
Students will understand that when a relation is determined to be a function,
they should use f(x) notation. Additionally, they should evaluate functions for
inputs in their domain and interpret statements that use function notation in
terms of the context in which the functions are used. The following problem
task can be used: Find a function from science, economics, or sports. Write it
in function notation and explain its meaning at several points in the domain.
Students recognize that sequences, sometimes defined recursively, are functions
whose domain is a subset of the set of integers. Have students generate
recursive sequences from contexts and define them in recursive notation. The
following skill-based task can be used: Write a recursive formula in function
notation for the sequence generated by adding 3 to each successive term when
beginning with 7.
Interpret functions that arise in applications in terms of a context.
M.1HS.LER.7
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.
Key features include: intercepts; intervals where the
function is increasing, decreasing, positive, or negative;
relative maximums and minimums; symmetries; end
behavior; and periodicity. (Focus on linear and
exponential functions.)
When given a linear or exponential function, students will be able to identify
key features in graphs and tables. Given the key features of a function, students
will sketch the graph. The following skill-based task can be used: Identify the
intervals where the function is increasing and decreasing.
M.1HS.LER.8
relate the domain of a function to its graph and where
applicable, to the quantitative relationship it describes.
For example, if the function h(n) gives the number of
person-hours it takes to assemble n engines in a factory,
then the positive integers would be an appropriate
domain for the function. (Focus on linear and
exponential functions.)
M.1HS.LER.9
calculate and interpret the average rate of change of a
function (presented symbolically or as a table) over a
specified interval. Estimate the rate of change from a
graph. (Focus on linear functions and intervals for
exponential functions whose domain is a subset of the
integers. Mathematics II and III will address other
function types).
When students are given the graph of a function, they will determine the
practical domain of the function as it relates to the numerical relationship it
describes. Find examples of functions with limited domains from other
curricular areas (science, physical education, social studies, consumer science).
The following skill-based task can be used: You are hoping to make a profit on
the school play and have determined the function describing the profit to be
f(t) = 8t – 2654 where t is the number of tickets sold. What is a reasonable
domain for this function? Explain.
Calculate the average rate of change over a specified interval of a linear or
exponential function. Estimate the average rate of change over a specified
interval of a linear or exponential function from the function’s graph.
Interpret, in context, the average rate of change of a linear or exponential
function over a specified interval. The following skill-based task can be used:
Use the following table to find the average rate of change of g over the
intervals [-2, -1] and [0,2]:
x
g(x)
-2
2
-1
-1
0
-4
2
-10
Analyze functions using different representations. (Focus on linear and exponential functions. Include comparisons of
two functions presented algebraically. For example, compare the growth of two linear functions, or two exponential functions
such as y=3n and y=100·2n.)
M.1HS.LER.10
graph functions expressed symbolically and show key
features of the graph, by hand in simple cases and using
technology for more complicated cases.
a. graph linear and quadratic functions and
show intercepts, maxima, and minima.
b. graph exponential and logarithmic functions,
showing intercepts and end behavior and
trigonometric functions, showing period,
midline and amplitude.
M.1HS.LER.11
compare properties of two functions each represented
in a different way (algebraically, graphically, numerically
in tables, or by verbal descriptions). For example, given
a graph of one quadratic function and an algebraic
expression for another, say which has the larger
maximum.
Graph linear functions expressed symbolically and identify intercepts. Graph
exponential functions expressed symbolically and show intercepts and end
behavior.
The student will compare the key features of two functions represented in
different ways. For example, compare the end behavior of a function which is
represented graphically and a function that is represented symbolically. The
following problem task can be used: Examine the functions below. Which
function has the larger maximum? How do you know?
f ( x )  2x 2  8x  20
Build a function that models a relationship between two quantities. (Limit to linear and exponential functions.)
M.1HS.LER.12
write a function that describes a relationship between
two quantities.
a. Determine an explicit expression, a recursive
process, or steps for calculation from a
context.
b. Combine standard function types using
arithmetic operations.
For example, build a function that models the
temperature of a cooling body by adding a constant
function to a decaying exponential, and relate these
functions to the model.
Students are able to write a function, combine standard function types, and
determine an explicit expression, a recursive process, or calculation steps.
Match functions expressed using different representations that have the same
properties. The following problem task can be used: A cup of coffee is initially
at a temperature of 93º F. The difference between its temperature and the
room temperature of 68º F decreases by 9% each minute. Write a function
describing the temperature of the coffee as a function of time. Create a graphic
organizer to highlight your understanding of functions and their properties by
comparing two functions using at least two different representations.
M.1HS.LER.13
write arithmetic and geometric sequences both
recursively and with an explicit formula, use them to
model situations, and translate between the two forms.
(Connect arithmetic sequences to linear functions and
geometric sequences to exponential functions.)
Students will be able to write arithmetic sequences and geometric sequences
recursively and explicitly, use recursive and explicit forms to model a situation,
and translate between the two forms. They will understand that linear functions
are the explicit form of recursively-defined arithmetic sequences and
exponential functions are the explicit form of recursively-defined geometric
sequences. Match sequences expressed recursively with those expressed
explicitly. The following skill-based task can be used: Write two formulas that
model the pattern 3, 9, 27, 81…
Build new functions from existing functions. (Focus on vertical translations of graphs of linear and exponential functions.
Relate the vertical translation of a linear function to its y-intercept. While applying other transformations to a linear graph is
appropriate at this level, it may be difficult for students to identify or distinguish between the effects of the other
transformations included in this standard.)
M.1HS.LER.14
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
(both positive and negative); find the value of k given
the graphs. Experiment with cases and illustrate an
explanation of the effects on the graph using
technology. Include recognizing even and odd functions
from their graphs and algebraic expressions for them.
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 (both positive and negative); find the value of k
given the graphs. Experiment with cases and illustrate an explanation of the
effects on the graph using technology. Include recognizing even and odd
functions from their graphs and algebraic expressions for them. The following
problem task can be used: Compare and contrast the graph of any function,
f(x), and the graph of f(x) + k.
Construct and compare linear, quadratic, and exponential models and solve problems.
M.1HS.LER.15
distinguish between situations that can be modeled with
linear functions and with exponential functions.
a. prove that linear functions grow by equal
differences over equal intervals; exponential
functions grow by equal factors over equal
intervals.
b. recognize situations in which one quantity
changes at a constant rate per unit interval
relative to another.
c. recognize situations in which a quantity
grows or decays by a constant percent rate per
unit interval relative to another.
M.1HS.LER.16
construct linear and exponential functions, including
arithmetic and geometric sequences, given a graph, a
description of a relationship or two input-output pairs
(include reading these from a table).
M.1HS.LER.17
observe using graphs and tables that a quantity
increasing exponentially eventually exceeds a quantity
increasing linearly, quadratically, or (more generally) as a
polynomial function. (Limit to comparisons between
exponential and linear models.)
Given a contextual situation, students will be able to describe whether the
situation in question has a linear pattern of change or an exponential pattern of
change. They will be able to show that linear functions change and how
exponential functions change. Model and explore several linear and
exponential functions, analyzing what action results in a change. The following
problem task can be used: A computer store sells about 200 computers at the
price of $1,000 per computer. For each $50 increase in price, about ten fewer
computers are sold. How much should the computer store charge per
computer in order to maximize their profit?
Students will create linear and exponential functions given the following
situations:
- arithmetic and geometric sequences
- a graph
- a description of a relationship
- two points, which can be read from a table
The following skill-based task can be used: Sara’s starting salary is $32,500.
Each year she receives a $700 raise. Write a sequence in explicit form to
describe the situation.
Through observation of graphs and tables, students develop the understanding
that a quantity increasing exponentially eventually exceeds a quantity increasing
linearly. The following skill-based task can be used: Which increases faster,
f(x) = 3x or g(x) = 3x? Justify your answer.
Interpret expressions for functions in terms of the situation they model. (Limit exponential functions to those of the
form f(x) = bx + k.)
M.1HS.LER.18
interpret the parameters in a linear or exponential
function in terms of a context.
Based on the context of a situation, students will be able to explain the meaning
of the coefficients, factors, exponents, and/or intercepts in a linear or
exponential function. Use applets or graphing technology to explore the effect
of changing the slope and y-intercept of a linear equation. Use applets or
graphing technology to explore the effect of changing the base value and
constant of an exponential function. The following problem task can be used:
Annie is picking apples with her sister. The number of apples in her basket is
described by n = 22t + 12, where t is the number of minutes Annie spends
picking apples. What do the numbers 22 and 12 tell you about Annie’s apple
picking?
Reasoning with Equations
Understand solving equations as a process of reasoning and explain the reasoning. (Students should focus on and
master M1.RWE.1 for linear equations and be able to extend and apply their reasoning to other types of equations in future
courses. Students will solve exponential equations with logarithms in Mathematics III.)
M.1HS.RWE.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.
Assuming an equation has a solution, construct a convincing argument that
justifies each step in the solution process. Justifications may include the
associative, commutative, and division properties, combining like terms,
multiplication by 1, etc. Solve simple rational and radical equations in one
variable and provide examples of how extraneous solutions arise. Understand,
apply, and explain the results of using inverse operations. Justify the steps in
solving equations by applying and explaining the properties of equality, inverse,
and identity. Use the names of the properties and common sense explanations
to explain the steps in solving an equation. The following example problem
task can be used: When Sally picks any number between 1 and 20, doubles it,
adds 6, divides by 2 and subtracts 3, she always gets the number she started
with. Why? Evaluate and use algebraic evidence to support your conclusion.
Solve equations and inequalities in one variable. (Extend earlier work with solving linear equations to solving linear
inequalities in one variable and to solving literal equations that are linear in the variable being solved for. Include simple
exponential equations that rely only on application of the laws of exponents, such as 5x = 125 or 2x = 1/16.)
M.1HS.RWE.2
solve linear equations and inequalities in one variable,
including equations with coefficients represented by
letters.
Write equations in equivalent forms to solve problems. Analyze and solve
literal equations for a specified variable. Understand and apply the properties
of inequalities. Verify that a given number or variable is a solution to the
equation or inequality. Interpret the solution of an inequality in real terms.
Equations can represent real world and mathematical problems. Include
equations and inequalities that arise when comparing the values of two
different functions, such as one describing linear growth and one describing
exponential growth. Solve for specified variables using common formulas used
in science, economics, or other disciplines. Examine and prove why dividing or
multiplying by a negative reverses the inequality sign. Use applications from a
variety of disciplines to motivate solving linear equations and inequalities. The
following problem task can be used: The perimeter of a rectangle is given by P
= 2W + 2L. Solve for W and restate in words the meaning of this new formula
in terms of the meaning of the other variables.
Solve equations and inequalities in one variable. (Extend earlier work with solving linear equations to solving linear
inequalities in one variable and to solving literal equations that are linear in the variable being solved for. Include simple
exponential equations that rely only on application of the laws of exponents, such as 5x = 125 or 2x = 1/16.)
M.1HS.RWE.3
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.
Solve systems of equations using the elimination method (sometimes called
linear combinations). Solve a system of equations by substitution (solving for
one variable in the first equation and substituting it into the second equation).
Explain the use of the multiplication property of equality to solve a system of
equations. Explain why the sum of two equations is justifiable in the solving of
a system of equations (property of equality). Relate the process of linear
combinations with the process of substitution for solving a system of linear
equations. Use modeling with a balance scale to make the point that multiples
of an equation are equivalent for purposes of elimination. Have students start
with a system of equations and use the addition and multiplication properties of
equality to create a new and equivalent system of equations then verify that the
two systems have the same solution.
M.1HS.RWE.4
solve systems of linear equations exactly and
approximately (e.g., with graphs), focusing on pairs of
linear equations in two variables.
Solve a system of equations exactly (with algebra) and approximately (with
graphs). Test a solution to the system in both original equations (both
graphically and algebraically). Analyze a system of equations using slope to
predict if the system has one, infinitely many, or no solutions. Solve contextual
problems using systems of equations. Have students generate scenarios which
might yield one, many, or no solutions. Have students create systems of
equations to model various contextual situations such as cell phone plan costs.
Use graphing calculators to estimate solutions to systems. Approximate the
solution to a system of equations graphically and then verify the solution
algebraically. The following problem task can be used: The high school is
putting on the musical Footloose. The auditorium has 300 seats. Student tickets
are $3 and adult tickets are $5. The royalty for the musical is $1300. What
combination of student and adult tickets do you need to fill the house and pay
the royalty? How could you change the price of tickets so more students can
go?
Descriptive Statistics
Summarize, represent, and interpret data on a single count or measurement variable. (In grades 6 – 8, students
describe center and spread in a data distribution. Here they choose a summary statistic appropriate to the characteristics of the
data distribution, such as the shape of the distribution or the existence of extreme data points.)
M.1HS.DST.1
represent data with plots on the real number line (dot
plots, histograms, and box plots).
M.1HS.DST.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.
M.1HS.DST.3
interpret differences in shape, center and spread in the
context of the data sets, accounting for possible effects
of extreme data points (outliers).
Construct dot plots, histograms and box plots for data on a real number line.
Describe and give a simple interpretation of a graphical representation of data.
Determine which type of data plot would be most appropriate for a specific
situation. Gather or provide data and have students plot each type of graph.
Analyze the strengths and weaknesses inherent in each type of plot by
comparing different plots of the same data. Have students collect their own
data and choose a graph to represent it. Students may use spreadsheets,
graphing calculators and statistical software for calculations, summaries, and
comparisons of data sets. The following problem task can be used: On the
midterm math exam, students had the following scores: 95, 45, 37, 82, 90, 100,
91, 78, 67, 84, 85, 85, 82, 91, 92, 93, 92, 76, 84, 100, 59, 92, 77, 68, and 88.
What are the strengths and weaknesses of presenting this data in a certain type
of plot for the students in the class, the parents, or the school board?
Describe a distribution using center and spread. Use the correct measure of
center and spread to describe a distribution that is symmetric or skewed.
Identify outliers (extreme data points) and their effects on data sets. Compare
two or more different data sets using the center and spread of each. Given two
sets of data or two graphs, identify similarities and differences in shape, center,
and spread. Compare data sets and be able to summarize the similarities and
differences between the shape and measures of centers and spreads of the data
sets. Discuss what it means when related data sets have different centers or
spreads in relation to the context of the data set. The following problem task
can be used: Plot data based on populations of European countries. Plot data
based on populations of Asian countries. Compare and discuss differences in
center and spread.
Interpret differences in different data sets in context. Interpret differences due
to possible effects of outliers. Given two data sets or two graphs, identify
similarities and differences in shape, center, and spread. Students may use
spreadsheets, graphing calculators and statistical software to statistically identify
outliers and analyze data sets with and without outliers as appropriate. Use
technology to manipulate plots of data sets to explore how changing data
affects the measures of center and spread. State the effects of any existing
outliers. Use data from multiple sources to interpret differences in shape,
center, and spread. Use data that includes outliers and explore what happens
when outliers are removed. Discuss the effect of outliers on measures of
center and spread and the effect on the shape. The following problem task can
be used: Find two similar data sets A and B (use textbook or internet
resources). What changes would need to be made to data set A to make it look
like the graph of set B?
Summarize, represents, and interpret data on two categorical and quantitative variables. (Students take a more
sophisticated look at using a linear function to model the relationship between two numerical variables. In addition to fitting a
line to data, students assess how well the model fits by analyzing residuals.)
M.1HS.DST.4
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.
M.1HS.DST.5
represent data on two quantitative variables on a scatter
plot and describe how the variables are related.
a. 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 and
exponential models.
b. informally assess the fit of a function by
plotting and analyzing residuals. (Focus should
be on situations for which linear models are
appropriate.)
c. fit a linear function for scatter plots that
suggest a linear association.
Create a two-way table from two categorical variables and read values from
two-way tables. Interpret joint, marginal, and relative frequencies in context.
Recognize associations and trends in data from a two-way table. Use
contextual situations to have students create a two-way frequency table
showing the relationship between two categorical variables such as height and
weight or blood pressure and incidence of heart disease. Use technology to
create two-way tables. Compare various tables and discuss frequencies that are
evident. The following problem task can be used: Collect data that compares
populations of countries with square miles. What trends emerge when we
compare living in geographically large countries with those that are highly
populated?
Create a scatter plot from two quantitative variables. Describe the form,
strength, and direction of the relationship. Categorize the data as linear or not.
Use algebraic methods and technology to fit a linear function to the data. Use
the function to predict values. Explain the meaning of the slope and yintercept in context. Categorize data as exponential. Use algebraic methods
and technology to fit an exponential function to the data. Use the function to
predict values. Explain the meaning of the growth rate and y-intercept in
context. Calculate a residual. Create and analyze a residual plot. Create a
scatter plot of bivariate data and estimate a linear or exponential function that
fits the data and use this function to solve problems in the context of the data.
Find residuals using technology and analyze their meaning. Fit a linear function
(trend line) to a scatter plot with and without technology. Students may use
spreadsheets, graphing calculators, and statistical software to represent data,
describe how the variables are related, fit functions to data, perform
regressions, and calculate residuals. The following problem task can be used:
Collect data on forearm length and height in a class. Plot the data and estimate
a linear function for the data. Compare and discuss different student
representations of the data and equations they discover. Could the equation(s)
be used to estimate the height for any person with a known forearm length?
Why or why not?
Interpret linear models.
M.1HS.DST.6
interpret the slope (rate of change) and the intercept
(constant term) of a linear model in the context of the
data. (Build on students’ work with linear relationships
in eighth grade and introduce the correlation coefficient.
The focus here is on the computation and interpretation
of the correlation coefficient as a measure of how well
the data fit the relationship.)
M.1HS.DST.7
compute (using technology) and interpret the correlation
coefficient of a linear fit.
M.1HS.DST.8
distinguish between correlation and causation. (The
important distinction between a statistical relationship
and a cause-and-effect relationship arises here.)
Explain the meaning of the slope and y-intercept in context. Find and graph
data sets from the Internet and discuss the meaning of their slopes and
intercepts in context. Students may use spreadsheets or graphing calculators to
create representations of data sets and create linear models. The following
problem task can be used: Create a poster of bivariate data with a linear
relationship. Describe for the class the meaning of the data, including the
meaning of the slope and intercept in the context of the data.
Use a calculator or computer to find the correlation coefficient for a linear
association. Interpret the meaning of the value in the context of the data.
Determine whether the correlation coefficient shows a weak positive, strong
positive, weak negative, strong negative, or no correlation. Have students enter
data into graphing technology, calculate the regression equation, and interpret
what the correlation coefficient is telling about the data. The following
problem task can be used: Hypothesize the correlation between two sets of
seemingly related data. Gather data to support or refute your hypothesis.
Explain the difference between correlation and causation. The focus here is on
the computation and interpretation of the correlation coefficient as a measure
of how well the data fit the relationship. Explain the distinction between a
statistical relationship and a cause and effect relationship. Understand and
explain that a strong correlation does not mean causation. Discuss data that
has correlation but no causation (height vs. foot length). Discuss data that has
correlation and causation (number of M&Ms in a cup vs. the weight of the
cup). The following problem task can be used: Find media artifacts that make
claims of causation and evaluate them.
Congruence, Proof, and Constructions
Experiment with transformations in the plane.
M.1HS.CPC.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.
M.1HS.CPC.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).
M.1HS.CPC.3
given a rectangle, parallelogram, trapezoid or regular
polygon, describe the rotations and reflections that carry
it onto itself.
M.1HS.CPC.4
develop definitions of rotations, reflections and
translations in terms of angles, circles, perpendicular
lines, parallel lines and line segments.
M.1HS.CPC.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.
Understand and use definitions of angles, circles, perpendicular lines, parallel
lines, and line segments based on the undefined term of a point, a line, the
distance along a line, and the length of an arc. Use precise definitions to
identify and model an angle, circle, perpendicular line, parallel line, and line
segment. Demonstrate mathematical notation for each term. Have students
write their own understanding of a given term. Give students formal and
informal definitions of each term and compare them. Develop precise
definitions through use of examples and non-examples. Discuss the
importance of having precise definitions. Identify real-life examples of each
term in the student’s environment, using definitions.
Use various technologies such as transparencies, geometry software, interactive
whiteboards, and digital visual presenters to represent and compare rigid and
size transformations of figures in a coordinate plane. Comparing
transformations that preserve distance and angle to those that do not.
Describe and compare function transformations on a set of points as inputs to
produce another set of points as outputs, to include translations and horizontal
and vertical stretching. Understand that a function has one output for every
input whether the input is a number or a point in the plane. Use M.C. Escher
pictures to compare and contrast rigid and non-rigid transformations. The
following problem task can be used: If a transformation preserves distances,
what other information would you need to know to determine an output for
the point (1, 0)?
Describe the rotations and reflections of a rectangle, parallelogram, trapezoid,
or regular polygon that maps each figure onto itself. Describe and identify lines
and points of symmetry. Provide sets of polygons for students to manipulate.
Use mirrors or a reflective device to help students see lines of symmetry. The
following problem task can be used: Given any number between 0 and 180,
can you find a polygon that has that rotational symmetry? Explain.
Using previous comparisons and descriptions of transformations, develop and
understand the meaning of rotations, reflections, and translations based on
angles, circles, perpendicular lines, parallel lines, and line segments. Draw
rotations, reflections, and translations. Use geometry software to model
rotations, reflections, and translations. Given a polygon and its transformation,
identify the angle of rotation or the distance of translation.
Transform a geometric figure given a rotation, reflection, or translation using
graph paper, tracing paper, or geometric software. Create sequences of
transformations that map a geometric figure on to itself and another geometric
figure. Perform rotations, reflections, and translations using a variety of
methods. Understand that the composition of transformations is not
commutative. Have students use a variety of tools to explore and perform
simple, multi-step, and composite rotations, reflections, and translations.
Given a transformation, work backwards to discover the sequence that led to
the transformation. Prove that every rotation is a composition of two
reflections.
Understand congruence in terms of rigid motions. (Rigid motions are at the foundation of the definition of congruence.
Students reason from the basic properties of rigid motions (that they preserve distance and angle), which are assumed without
proof. Rigid motions and their assumed properties can be used to establish the usual triangle congruence criteria, which can
then be used to prove other theorems.)
M.1HS.CPC.6
use geometric descriptions of rigid motions to transform
figures and to predict the effect of a given rigid motion
on a given figure; given two figures, use the definition of
congruence in terms of rigid motions to decide if they
are congruent.
Knowing that rigid transformations preserve size and shape or distance and
angle, use this fact to connect the idea of congruency and develop the
definition of congruent. Transform figures using geometric descriptions of
rigid motions. Predict the effect of rotating, reflecting, or translating a given
figure. Justify the congruence of two figures using properties of rigid motions.
Use graph paper, tracing paper, physical models and geometry software to
verify predictions regarding rigid motion and congruence. Use frieze patterns
and Escher art to explore congruency in transformations.
M.1HS.CPC.7
use the definition of congruence in terms of rigid
motions to show that two triangles are congruent if and
only if corresponding pairs of sides and corresponding
pairs of angles are congruent.
M.1HS.CPC.8
explain how the criteria for triangle congruence (ASA,
SAS, and SSS) follow from the definition of congruence
in terms of rigid motions.
Identify corresponding parts of two triangles. Show that two triangles are
congruent if and only if corresponding pairs of side and corresponding pairs of
angles are congruent (CPCTC). Match pairs of cardboard congruent triangles
and justify congruence. Measure angles and side lengths of triangles resulting
from rigid transformations using a variety of technology and paper based
methods (e.g. patty paper).
Use the definition of congruence, based on rigid motion, to develop and
explain the minimum conditions necessary for triangle congruence (ASA, SSS,
and SAS). Understand, explain, and demonstrate why ASA, SAS, or SSS are
sufficient to show congruence. Understand, explain, and demonstrate why SSA
and AAA are not sufficient to show congruence. Explore the minimum
conditions necessary to show triangles are congruent using technology,
reflective devices, patty paper, spaghetti, or grid paper. Establish triangle
congruence criteria using properties of rigid motion. Demonstrate visually why
some conditions like SSA or AAA are not sufficient to show congruence.
Make geometric constructions. (Build on prior student experience with simple constructions. Emphasize the ability to
formalize and defend how these constructions result in the desired objects. Some of these constructions are closely related to
previous standards and can be introduced in conjunction with them.)
M.1HS.CPC.9
make formal geometric constructions with a variety of
tools and methods (compass and straightedge, string,
reflective devices, paper folding, dynamic geometric
software, etc.). Copying a segment; copying an angle;
bisecting a segment; bisecting an angle; constructing
perpendicular lines, including the perpendicular bisector
of a line segment; and constructing a line parallel to a
given line through a point not on the line.
M.1HS.CPC.10
construct an equilateral triangle, a square and a regular
hexagon inscribed in a circle.
Perform the constructions listed in the standard using a variety of tools and
methods including compass, straightedge, string, reflective devices, paper
folding, and dynamic geometric software. Explain why these constructions
result in the desired objects. Have students explore how to make a variety of
constructions using different tools. Ask students to justify how they know
their method results in the desired construction. Discuss the underlying
principles that different tools rely on to produce the desired constructions.
Given two quadrilaterals that are reflections of each other, find the line of that
reflection.
Construct an equilateral triangle, square, and regular hexagon so that each
vertex of the shape is on the circle. Allow students to explore possible
methods for constructing equilateral triangles, squares, and hexagons and
methods for constructing each inscribed in a circle. The following problem
task can be used: Find two ways to construct a hexagon inscribed in a circle.
Connecting Algebra and Geometry through Coordinates
Use coordinates to prove simple geometric theorems algebraically. (Reasoning with triangles in this unit is limited to
right triangles; e.g., derive the equation for a line through two points using similar right triangles.)
M.1HS.CAG.3 provides practice with the distance formula and its connection with the Pythagorean theorem.
M.1HS.CAG.1
use coordinates to prove simple geometric theorems
algebraically. For example, prove or disprove that a
figure defined by four given points in the coordinate
plane is a rectangle; prove or disprove that the point (1,
√3) lies on the circle centered at the origin and
containing the point (0, 2).
M.1HS.CAG.2
prove the slope criteria for parallel and perpendicular
lines; 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).
(Relate work on parallel lines to work on M.1HS.RWE.3
involving systems of equations having no solution or
infinitely many solutions.)
Use coordinates to prove simple geometric theorems algebraically, focusing on
lines, segments, and angles. Prove that points in a plane determine defined
geometric figures. Explore properties of geometric figures plotted on a
coordinate axes system using graphing technology and dynamic software.
Generalize coordinates of geometric figures using variables for one or more of
the vertices. Derive the equation for a line through two points using similar
right triangles. Take a picture or find a picture which includes a polygon.
Overlay the picture on a coordinate plane (manually or electronically).
Determine the coordinates of the vertices. Classify the polygon. Use the
coordinates to justify the classification. The following problem task can be
used: Take a picture or find a picture which includes a polygon. Overlay the
picture on a coordinate plane (manually or electronically). Determine the
coordinates of the vertices. Classify the polygon. Use the coordinates to justify
the classification.
Prove that the slopes of parallel lines are equal. Prove that the product of the
slopes of perpendicular lines is -1. Use slope criteria for parallel and
perpendicular lines to solve geometric problems. Write the equation of a line
parallel or perpendicular to a given line, passing through a given point. Allow
students to explore and make conjectures about relationships between lines and
segments using a variety of methods. Discuss the role of algebra in providing a
precise means of representing a visual image. Relate work on parallel lines to
systems of equations having no solution or infinitely many solutions. Verify
that the distance between two parallel lines is constant.
M.1HS.CAG.3
use coordinates to compute perimeters of polygons and
areas of triangles and rectangles, e.g., using the distance
formula. (Provides practice with the distance formula
and its connection with the Pythagorean theorem.)
Use the distance formula to compute perimeters of polygons and areas of
triangles and rectangles. Graph polygons using coordinates and determine side
lengths and perimeters of the polygons. Calculate areas of triangles and
rectangles. Given a triangle, use slopes to verify that the length and height are
perpendicular and find the area. Explore perimeter and area of a variety of
polygons, including convex, concave, and irregularly shaped polygons. Find the
area and perimeter of a real-world shape using a coordinate grid and Google
Earth. Select a shape (your yard, a parking lot, the school, etc.). Use the tool
menu to overlay a grid. Use coordinates to find the perimeter and area of the
shape you’ve selected. Determine the scale factor of the picture as related to
the actual real-life view. Then find the actual perimeter and area.