GRADE LEVEL CONTENT AREA: 10th-11th Grade ELL Mathematics
Theme: Data Analysis, Family of Functions, Exponential and Quadratic Functions
Essential Question: How are Linear, Exponential and Quadratic functions related and how do are they modeled in data sets?
Skills and Content
Standards
Key Vocabulary
Resources
Assessments





please see online test generator




Essentials
21st Century Skills
Important to Know and Do




Conveying ideas graphically to create data displays.
Reading and comprehending with accuracy information about data sets.
Sometimes it is helpful to organize numerical data into intervals.
Frequency tables and histograms display numerical data organized into
intervals.
S.ID.1 Represent data with plots on the real number line (dot plots, histograms, and box plots).
N.Q.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.
N.Q.2 Define appropriate quantities for the purpose of descriptive
modeling.
N.Q.3 Choose a level of accuracy appropriate to limitations on
measurement when reporting quantities.
frequency
frequency table
histogram
uniform
symmetric
skewed
cumulative frequency table
data
Pearson Algebra 1
Foundations,
Chap 12 Sec 2
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From Pearson Exam View and Teacher Resources Standardized Test Practice, Quizzes, Chapter Tests and Performance Tasks
Identifying similarities and differences and interpreting/analyzing which measure is most useful for the situation.
Communicating with peers about data displays.
Different measures can be used to interpret and compare data sets.
Three measures of central tendency of a set of data are mean, median, and mode.
S.ID.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.
S.ID.3 Interpret differences in shape, center, and spread in the context
of the data sets, accounting for possible effects of extreme data points
(outliers).
N.Q.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.
N.Q.2 Define appropriate quantities for the purpose of descriptive
modeling.
N.Q.3 Choose a level of accuracy appropriate to limitations on
measurement when reporting quantities.
measure of central tendency
outlier
mean
median
mode
measure of dispersion
range of a set of data
between, most, middle
order
standard deviation
variance
Chap 12 Sec 3
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Summarizing information given from a data set and comparing it to other data sets
Conveying ideas graphically to create data displays.
Separating data into subsets is a useful way to summarize and compare data sets.
A box-and-whisker plot displays the maximum, mininmum, and quartiles of a data set.
S.ID.1 Represent data with plots on the real number line (dot plots, histograms, and box plots).
S.ID.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.
N.Q.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.
N.Q.2 Define appropriate quantities for the purpose of descriptive
modeling.
N.Q.3 Choose a level of accuracy appropriate to limitations on
measurement when reporting quantities.
quartile
interquartile range
box-and-whisker plot
percentile
percentile rank
minimum
maximum
Chap 12 Sec 4
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Classifying various types of data and surveys.
When collecting data, it is important for the results to accurately represent
the situation.
Surveys can use random, systematic, or stratified sampling methods.
S.IC.3 Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each.
quantitative
qualitative
univariate
bivariate
population
sample
bias
survey
Chap 12 Sec 5
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Conveying ideas about the relationships between equations, graphs and tables.
Test predictions about their graphs.
Collaborative work to communicate verbally about learning.
Family of Functions (come back to)
A.REI.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).
F.BF.3 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.
transformations
shifts
reflections
stretches
shrinks
Wiki resources for Graphing Functions Supplement with transformations material

Classifying polynomials.
Identify similarities and differences between expressions and functions.
Monomials can be used to form larger expressions called polynomials.
Polynomials can be added and subtracted.
A.APR.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.
monomial
degree of a monomial
polynomial
standard form of a polynomial
degree of a polynomial
binomial
trinomial
Chapter 8 Section 1 Calculator

Using cues, questions and advanced organizers to understand how to use the Distributive Property and finding the GCF for factoring polynomials.
Listening actively
A monomial can be multiplied by a polynomial using the Distributive Property.
Factoring a polynomial reverses the multiplication process
The first step when factoring a monomial from a polynomial is finding the greatest common factor of the terms of the polynomial.
A.APR.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
A.SSE.1 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.
factoring
greatest common factor
Chapter 8 Section 2
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Using multiple sources to find information such as models, algebra and tables to find the product of two binomials.
There are several ways to find the product of two binomials, including models, algebra, and tables.
The properties of real numbers can be used to multiply two binomials.
A.APR.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.
dstributive property
Chapter 8 Section 3
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Summarizing the special rules.
Communicating the special rules to another student.
There are special rules for simplifying the square of a binomial or the product of a sum and a difference.
A.SSE.2 Use the structure of an expression to identify ways to rewrite it.
square of a binomial
the product of a sum and difference
Chapter 8 Section 4
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Generating and testing predictions.
Determining accuracy.
Some trinomials of the form x2 + bx + c can be factored into equivalent forms that are the product of two binomials.
The signs and factors of the coefficients of the trinomial indicate how the trinomial can be factored.
A.SSE.2 Use the structure of an expression to identify ways to rewrite it.
trinomial
Chapter 8 Section 5
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Generating and testing predictions.
Determining accuracy.
Some trinomials of the form ax2 + bx + c can be factored out before the remaining polynomial is factored.
Sometimes the greatest common factor of the polynomial should be factored out before the remaining polynomial is factored.
A.SSE.2 Use the structure of an expression to identify ways to rewrite it.
greatest common factor
Chapter 8 Section 6
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Determining the relationship to special case binomials and using prior knowledge to solve a problem.
Some trinomials such as squares of binomials or differences of two squares, can be factored by reversing the rule for multiplying special-case binomials.
A.SSE.2 Use the structure of an expression to identify ways to rewrite it. For example, see x4– y4 as (x2)2– (y2)2, thus recognizing it as a
difference of squares that can be factored as (x2– y2)(x2+ y2).
perfect-square trinomial
difference of two squares
Chapter 8 Section 7
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Generating and testing predictions and attending to accuracy.
Some polynomials of a degree greater than 2 can be factored.
If a polynomial has 4 or more terms, it may be possible to group the terms and factor binomials from the groups.
A.SSE.2 Use the structure of an expression to identify ways to rewrite it.
factor by grouping
Chapter 8 Section 8
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Convey ideas about the relationships between graphs and equations.
Use multiple representations to analyze characteristics of a graph.
The family of quadratic functions models certain situations where the rate of change is not constant.
Quadratic functions are graphed by a symmetric curve with a highest or lowest point corresponding to a maximum or minimum value.
A.REI.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).

F.IF.7 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.
quadratic function
standard form of a quadratic function
quadratic parent function
parabola
axis of symmetry
vertex
minimum
maximum
vertex form
Chapter 9
Section 1
Vertex form graphing McDougal Littel Algebra 1 p. 669 and Algebra 2 Section 4.2
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Use multiple representations to emphasize different characteristics of a function.
In the quadratic function y= ax2+ bx + c,
the value of b translates the position of the axis of symmetry.
The axis of symmetry for the graph of the quadratic function
y= ax2+ bx + c is x= -b
2a
The x-coordinate of the vertex of the graph is x= -b
2a
F.IF.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. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.
quadratic equations in polynomial form
intercept form
axis of symmetry
vertex
Chapter 9 Section 2
Intercept form McDougal Littel Alg 1 p. 641 Alg 2 4.2
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Define the problem and its variables to generate equations.
Use cues and questions to determine an appropriate domain.
Quadratic equations can be solved by a variety of methods, including graphing and finding square roots.
In many cases the negative solutions of a quadratic equation will not be a reasonable solution to the original problem.
F.IF.9 Compare properties of two functions each represented in a
different way (algebraically, graphically, numerically in tables, or by
verbal descriptions).
A.REI.4 Solve quadratic equations in one variable.
A.CED.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.
A.CED.2 Create equations in two or more variables to represent
relationships between quantities; graph equations on coordinate axes
with labels and scales.
A.CED.3 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
F.IF.8 Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.
a. 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.
quadratic equation
standard form of a quadratic equation
root of an equation
zero of a function
Chapter 9 Section 3
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Develop a strategy for solving equations through collaborative team work.
Some quadratic equations can be solved by using the Zero-Product Property.
Sometimes it is useful to write a quadratic equation in standard form before solving.
A.REI.4 Solve quadratic equations in one variable.
b. 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.
Zero-Product Property
Chapter 9 Section 4
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Define the problem and use advanced organizer to complete the square.
Any quadratic can be solved by first writing it in he form m2= n
A.REI.4 Solve quadratic equations in one variable.
b. 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.
completing the square
Chapter 9 Section 5
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Define the problem and use advanced organizer to use quadratic formula.
Any quadratic equation can be solved using the quadratic formula.
The discriminant of a quadratic equation can be used to determine the number of solutions an equation has.
A.REI.4 Solve quadratic equations in one variable.
b. 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.
quadratic formula
discriminant
Chapter 9 Section 6

Determine relationship between positive, negative, and zero exponents through cues, questioning and advanced organizers.
Extend the use of exponents to include zero and negative exponents.

base
exponent
power
Chap 7 Sec 1
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Determining relationships of size/amount accurately.
Powers of 10 can be used to more easily write and compare very large
or very small numbers.
Scientific notation is a shorthand way to write numbers using powers of 10.

scientific notaton
power of 10
standard notation
Chap 7 Sec 2
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Defining the problem and its variables and using properties to simplify them.
A property of exponents can be used to multiply powers with the same base.

simplified form
Chap 7 Sec 3
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Defining the problem and its variables and using properties to simplify them
Properties of exponents can be used to simplify a power raised to a power or
a product raised to a power.

power
product
area
Chap 7 Sec 4
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Defining the problem and its variables and using properties to simplify them
Properties of exponents can be used to divide powers with the same base.


Chap 7 Sec 5

Determine relationships between equivalent expressions
Rational Exponents (need other resources)
N.RN.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.

N.RN.2 Rewrite expressions involving radicals and rational exponents using the properties of exponents.
simplifying square roots
simplifying non-square roots (cubed and fourth roots and variables)
rational exponents
Other resources needed

Convey ideas about the relatiohsip between graphs and equations.

Identify similarities and differences between sequences and functions.
Some functions model an initial amount that is repeatedly multiplied by the
same positive number. In the rules for these functions, the independent variable
is an exponent.
A.REI.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).
F.IF.3 Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the Fibonacci sequence is defined recursively byf(0) = f(1) = 1, f(n+1) = f(n) + f(n-1) for n > 1.
F.IF.7 Graph functions expressed symbolically and show key features
of the graph, by hand in simple cases and using technology for more
complicated cases.★
e. Graph exponential and logarithmic functions, showing intercepts
and end behavior, and trigonometric functions, showing period,
midline, and amplitude.
F.BF.2 Write arithmetic and geometric sequences both recursively and
with an explicit formula, use them to model situations, and translate
between the two forms.★
A.CED.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.
A.CED.2 Create equations in two or more variables to represent
relationships between quantities; graph equations on coordinate axes
with labels and scales.
exponential function
real number
domain
range
rule
geometric sequences
Chap 7 Sec 6

Use multiple representations to analyze characteristics of graphs.

Use multiple representations to emphasize different characteristics of a function.
An exponential growth function can model growth or decay of an initial amount
F.IF.9 Compare properties of two functions each represented in a
different way (algebraically, graphically, numerically in tables, or by
verbal descriptions).
F.LE.1 Distinguish between situations that can be modeled with linear functions and with exponential functions.
c. Recognize situations in which a quantity grows or decays by a
constant percent rate per unit interval relative to another.
A.CED.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.
exponential growth
growth factor
compound interest
exponential decay
decay factor
Chapter 7 Section 7
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Create and analyze information from multiple data displays.

Test predictions about their graphs.

Identify similarities and differences of various residual plots.
Linear, quadratic, or exponential functions can be used to model various sets of data.
Graphing and testing data can show which type of function best models the data.
Regression of linear, quadratic, and exponential functions can be modeled by various sets of data and interpreted.
A.REI.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).
F.LE.1 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; and that 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.
S.ID.6 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.
F.LE.3 Observe using graphs and tables that a quantity increasing
exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.
linear regression
quadratic regression
exponential regression
residual plot
Chapter 9 Section 7
McDougal Littel Alg1 p. 692-93; need other resources for regression