STANDARDS FOR MATHEMATICS High School Algebra 1

exponential functions. For

example, identify percent
rate of change in
functions such as y =
(1.02)t, y = (0.97)t, y =
(1.01)12t, y = (1.2)t/10, and
classify them as
representing exponential
growth or decay.

Connection: 11-12.RST.7 Mathematical Examples & Explanations
Practices
HS.F-IF.9 Start F-IF.9 by focusing on linear and exponential functions. Include comparisons of two functions
HS.MP.6. Attend presented algebraically. Later in the year focus on expanding the types of functions to include linear,
HS.F-IF.9. Compare to precision. exponential, and quadratic. Extend work with quadratics to include the relationship between
properties of two functions coefficients and roots, and once roots are known, a quadratic equation can be factored.
each represented in a HS.MP.7. Look
different way (algebraically, for and make use Example:
graphically, numerically in of structure. • Given a graph of one quadratic function and an algebraic expression for another, say which
tables, or by verbal has the larger maximum.
descriptions). For example,
given a graph of one • Examine the functions below. Which function has the larger maximum? How do you know?
quadratic function and an
algebraic expression for f (x) = −2x2 − 8x + 20
another, say which has the
larger maximum.

Connections:
ETHS-S6C1-03;
ETHS-S6C2-03;
9-10.RST.7

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Additional Domain Information – Interpreting Functions (F-IF)

Key Vocabulary • Trigonometric Function • Maxima
• Period • Minima
• Average Rate of Change • Midline • End Behavior
• Function • Amplitude • Point-Slope Form
• Input/Domain • Sequence • Slope-Intercept Form
• Output/Range • Fibonacci Sequence • Standard Form
• Linear Function • Recursive • Vertex Form
• Piecewise Function • Arithmetic Sequence • x-intercept
• Quadratic Function • Geometric Sequence • y-intercept
• Exponential Function

Example Resources

• Books
¾ Developing Essential Understanding of Functions: Grades 9-12 by NCTM
¾ The X’s and Why’s of Algebra: Key Ideas and Common Misconceptions by Anne Collins & Linda Dacey

• Technology
Interpreting functions:
¾ http://www.brightstorm.com/math/algebra/graphs-and-functions/interpreting-graphs-problem-1/ 
Video on interpreting graphs for students in Algebra. Shows students how to look at what graphs mean and how to interpret slopes using
real world situations.
¾ http://www.khanacademy.org/ 
Conceptual videos: Linear, Quadratic, Exponential Functions Basic Linear Function, Recognizing Linear Functions, Linear Function
Graphs, Graphing a Quadratic Function, Applying Quadratic Functions 2, Quadratic Functions 1, Applying Quadratic Functions 3,
Applying Quadratic Functions 1, Quadratic Functions 2, Quadratic Functions 3, Exponential Growth Functions, Exponential Decay
Functions, See all videos from topic ck12.org Algebra 1 Examples, Graphing Exponential Functions
¾ http://www.schooltube.com/video/f44d3551932486f18d81/ 
Provides examples on how to build & analyze functions.
¾ www.classzone.com/ This is the site to access the book and extra resources online.
¾ http://illuminations.nctm.org/ This site has a bank of different lessons published by NCTM.
¾ http://www.hippocampus.org/ This site has online videos and interactive lessons for both teachers and students to use and can be
matched to a particular textbook.
¾ http://www.wolframalpha.com/ This site is a resource for looking up mathematical facts and also provides a place to solve and graph
equations.
¾ http://www.illustrativemathematics.org/ This is a webpage that has the new standards with sample classroom tasks linked to some of the
standards.

52 
 

• Example Lessons
¾ http://www.illustrativemathematics.org/standards/hs
Understand the concept of a function and use function notation.
Interpret functions that arise in applications in terms of the context. F-IF: Interpret functions that arise in applications in terms of the context.

Common Student Misconceptions

Students commonly have difficulty understanding the function notation Y = f(x). For example, f(2) = 5 is the function notation for the point (2, 5).
Students can have difficulty identifying the x and y values for a given point from the function notation.
Students often confuse the domain with the range.
Students commonly have difficulty identifying the zeros of a function. Students think that the zero of the function is when x = 0, as opposed to when
y = 0.
Students misunderstand how to write the interval where a function is increasing or decreasing. Often students write the answer in terms of y-
interval rather than in terms of the x-interval.
Students misunderstand how to identify the parameters (function transformation) to build new functions from existing functions. For example,
to shift the graph of y = x2 left by 6 units, they may write y = (x - 6)2 rather than y = (x + 6)2.
Students believe that the domain of a function is the same, regardless of the contextual problem that it is modeling. For example, the domain of
a quadratic includes negative values, but for a quadratic modeling the height of a falling object as a function of time t, the domain should be t ≥ 0.
Students often confuse the independent variable with the dependent variable.
Students often have difficulty identifying the quantities to be represented by variables, in modeling problems.

53 
 

Domain: Building Functions (2 Clusters)

Building Functions (F-BF) (Domain 2 – Cluster 1 – Standards 1 and 2)

Build a function that models a relationship between two quantities. (For F‐BF.1 and 2, linear, exponential and quadratic) 

Essential Concepts Essential Questions

• A function is a relationship between two quantities. • What data would you need to write a linear, basic quadratic, or basic
exponential function?
• The function representing a given situation may be a combination of
more than one standard function. • How do you translate a description of the relationship between two
quantities into an algebraic equation or inequality?
• Standard functions may be combined through arithmetic operations.
• Why are arithmetic sequences described by linear functions?
• Arithmetic and geometric sequences can be written both recursively and
with an explicit formula. • Why are geometric sequences described by exponential functions?

• A recursive formula for a sequence describes how to determine the next • How could you translate a recursive formula for a sequence into an
term from the previous term(s). explicit formula? Vice versa?

• An explicit formula for a sequence describes how to determine any term
in the sequence.

• Arithmetic sequences can be described by linear functions.

• Geometric sequences can be described by exponential functions.

• Sequences model situations in which the domain is a set of integers.

HS.F-BF.1

HS.F-BF.1 Mathematical Examples & Explanations
Practices
Write a function that Start by limiting F-BF.1a, 1b to linear and exponential functions. Later in the year extend this work
describes a relationship HS.MP.1. Make to focus on situations that exhibit a quadratic relationship.
between two quantities. sense of problems
and persevere in Students will analyze a given problem to determine the function expressed by identifying patterns
Connections: solving them. in the function’s rate of change. They will specify intervals of increase, decrease, constancy, and, if
ETHS-S6C1-03; possible, relate them to the function’s description in words or graphically. Students may use
ETHS-S6C2-03 HS.MP.2. Reason graphing calculators or programs, spreadsheets, or computer algebra systems to model functions.
abstractly and
a. Determine an explicit quantitatively. Examples:
expression, a recursive • You buy a $10,000 car with an annual interest rate of 6 percent compounded annually and
process, or steps for HS.MP.4. Model make monthly payments of $250. Express the amount remaining to be paid off as a
calculation from a with mathematics. function of the number of months, using a recursion equation.
context.

Connections: HS.MP.5. Use • A cup of coffee is initially at a temperature of 93º F. The difference between its temperature
ETHS-S6C1-03; appropriate tools and the room temperature of 68º F decreases by 9% each minute. Write a function
ETHS-S6C2-03; strategically. describing the temperature of the coffee as a function of time.
9-10.RST.7; 11-12.RST.7
HS.MP.6. Attend (Continued on next page)

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b. Combine standard to precision. Examples & Explanations
function types using
arithmetic operations. For HS.MP.7. Look for For F-BF.2 limit to linear and exponential functions. Connect arithmetic sequences to linear
example, build a function and make use of functions and geometric sequences to exponential functions.
that models the structure.
temperature of a cooling An explicit rule for the nth term of a sequence gives an as an expression in the term’s position n; a
body by adding a HS.MP.8. Look for recursive rule gives the first term of a sequence, and a recursive equation relates an to the
constant function to a and express preceding term(s). Both methods of presenting a sequence describe an as a function of n.
decaying exponential, regularity in
and relate these functions repeated Examples:
to the model. reasoning. • Generate the 5th-11th terms of a sequence if A1= 2 and A(n+1) = (An )2 − 1

Connections: Mathematical • Use the formula: An = A1 + d(n - 1) where d is the common difference to generate a
ETHS-S6C1-03; Practices sequence whose first three terms are: -7, -4, and -1.
ETHS-S6C2-03
HS.MP.4. Model • There are 2,500 fish in a pond. Each year the population decreases by 25 percent, but
HS.F-BF.2 with mathematics. 1,000 fish are added to the pond at the end of the year. Find the population in five
years. Also, find the long-term population.
HS.F-BF.2 HS.MP.5. Use
appropriate tools • Given the formula An= 2n - 1, find the 17th term of the sequence. What is the 9th term in
Write arithmetic and strategically. the sequence 3, 5, 7, 9, ?
geometric sequences both
recursively and with an HS.MP.8. Look for
explicit formula, use them to and express
model situations, and regularity in
translate between the two repeated
forms. reasoning.

• Given a1 = 4 and an = an-1 + 3, write the explicit formula.

55 
 

Building Functions (F-FB) (Domain 2 – Cluster 2 – Standards 3 and 4)

Build new functions from existing functions. (Linear, exponential, quadratic and absolute value; for F‐BF.4a, linear only) 

Essential Concepts Essential Questions

• f(x) + k will translate the graph of the function f(x) up or down by k units. • Create a graph and explain what transformation(s) were done on the
parent function to create that graph.
• k f(x) will expand or contract the graph of the function f(x) vertically by a
factor of k. If k<0 the graph will reflect across the x-axis. • What are the transformations that can be done to a graph and how
can they be represented algebraically?
• f(kx) will expand or contract the graph of the function f(x) horizontally by
a factor of k. If k<0 the graph will reflect across the y-axis. • How do you determine if a graph is odd, even, or neither?
• Why are the two descriptions of an even function equivalent?
• f(x + k) will translate the graph of the function f(x) left or right by k units. • Why are the two descriptions of an odd function equivalent?
• How do you determine if two functions are inverses of one another?
• If f(-x) = f(x) then the function is even, therefore its graph is symmetrical • Given a function, how do you find its inverse?
across the y-axis.

• If f(-x) = - f(x) then the function is odd, therefore its graph is symmetrical
across the origin.

• Two functions f and g are inverses of one another if for all values of x in
the domain of f, f(x)=y and g(y)=x.

• Not all functions have an inverse.

HS.F-BF.3

HS.F-BF.3 Mathematical Examples & Explanations
Practices
Identify the effect on the Start by focusing on vertical translations of graphs of linear and exponential functions. Relate the
graph of replacing f(x) by f(x) HS.MP.4. Model vertical translation of a linear function to its y-intercept. While applying other transformations to a
+ k, k f(x), f(kx), and f(x + k) with linear graph is appropriate at this level, it may be difficult for students to identify or distinguish
for specific values of k (both mathematics. between the effects of the other transformations included in this standard. Later in the year, extend
positive and negative); find this work to quadratic functions, and consider including absolute value functions.
the value of k given the HS.MP.5. Use
graphs. Experiment with appropriate tools Students will apply transformations to functions and recognize functions as even and odd. Students
cases and illustrate an strategically. may use graphing calculators or programs, spreadsheets, or computer algebra systems to graph
explanation of the effects on functions.
the graph using technology. HS.MP.7. Look
for and make use Examples:
Include recognizing even and of structure. • Compare the graphs of f(x)=3x with those of g(x)=3x+2 and h(x)=3x -1 to see that parallel
odd functions from their lines have the same slope AND to explore the effect of the transformation of the function,
graphs and algebraic f(x)=3x such that g(x)=f(x)+2 and h(x)=f(x) – 1.
expressions for them.

Connections: • Explore the relationship between f(x)=3x, g(x)= 5x, and h(x) = 1 x with a calculator to
ETHS-S6C2-03; 2
11-12.WHST.2e
develop a relationship between the coefficient on x and the slope.

(Continued on next page)

56 
 

• Describe the effect of varying the parameters a, h, and k on the shape and position of the

graph f(x) = ab(x + h) + k, orally or in written format. What effect do values between 0 and 1
have? What effect do negative values have?

• Is f(x) = x3 - 3x2 + 2x + 1 even, odd, or neither? Explain your answer orally or in written
format.

• Compare the shape and position of the graphs of f (x) = x2 and g(x) = 2 x2 , and explain

the differences in terms of the algebraic expressions for the functions.

• Describe the effect of varying the parameters a, h, and k have on the shape and position of
the graph of f(x) = a(x-h)2 + k.

HS.F-BF.4 Mathematical Examples & Explanations
HS.F-BF.4 Practices
For F-BF.4a, focus on linear functions but consider simple situations where the domain of the
Find inverse functions. HS.MP.2. function must be restricted in order for the inverse to exist, such as f(x) = x2, x>0. This work will be
Reason extended in Algebra II to include simple rational, simple radical and simple exponential functions.
Connection: abstractly and
ETHS-S6C2-03 quantitatively. Students may use graphing calculators or programs, spreadsheets, or computer algebra systems to
model functions.
a. Solve an equation of the HS.MP.4. Model
form f(x) = c for a simple with Examples:
function f that has an mathematics. • For the function h(x) = 3(x – 2), defined on the domain of all real numbers, find the inverse
inverse and write an function if it exists or explain why it doesn’t exist.
expression for the HS.MP.5. Use (Continued on next page)
inverse.
57 
 

For example, f(x) =2x3 or appropriate tools • Graph h(x) and h-1(x) and explain how they relate to each other graphically for a linear
f(x) = (x+1)/(x-1) for x ≠ 1. strategically. function.

HS.MP.7. Look • Consider simple situations where the domain of the function must be restricted in order for
for and make use the inverse to exist, such as f(x) = x2, x>0.
of structure.

Additional Domain Information – Building Functions (F-BF)

Key Vocabulary

• Domain • Expand/contract • Translate
• Range • Linear function • Recursive formula
• Rate of change • Exponential function • Explicit formula
• Transformation • Quadratic function • Even/odd function
• Inverse

Example Resources

• Books
¾ SpringBoard by College Board [activity 2.2, 4.2]

¾ Algebra I by McDougal and Littell [8.2, 8.5, 8.6]
¾ Developing Essential Understanding of Functions: Grades 9-12 by NCTM

• Technology
¾ http://www.khanacademy.org/#algebra-functions provides an extensive list of function video lectures
¾ http://www.purplemath.com/modules/index.htm provides resources about teaching functions
¾ http://a4a.learnport.org/page/comparing-functions provides multiple interactive applet investigations: introduction to recursive notation, etc.
¾ http://www.padowan.dk/graph/ A free downloadable graph generator
¾ www.classzone.com/ This is the site to access the book and extra resources online.
¾ http://illuminations.nctm.org/ This site has a bank of different lessons published by NCTM.
¾ http://www.hippocampus.org/ This site has online videos and interactive lessons for both teachers and students to use and can be matched
to a particular textbook.
¾ http://www.wolframalpha.com/ This site is a resource for looking up mathematical facts and also provides a place to solve and graph
equations.
¾ http://www.illustrativemathematics.org/ This is a webpage that has the new standards with sample classroom tasks linked to some of the
standards.

• Example Lessons
¾ http://www.regentsprep.org/regents/math/algebra/AE7/PennyLabSheet.pdf Exponential growth and decay: Penny Activity
¾ http://illuminations.nctm.org/LessonDetail.aspx?ID=U142 Population of Trout Pond 4 lessons: recursion, numerical analysis, graphical
analysis, and symbolic analysis.

58 
 

Common Student Misconceptions

Students have difficulty identifying that a variable squared with a coefficient does not mean to also square the coefficient. For example, 3x2 is
misrepresented as 9x2.

Students have difficulty applying rules of exponents when the exponent is rational. For example, adding/subtracting rational exponents that do not
have like denominators.

Students confuse power functions and exponential functions. For example, students think that n2 is an exponential function because it contains an
exponent.

Domain: Linear, Quadratic and Exponential Models (2 Clusters)

Linear, Quadratic and Exponential Models (F-LE) (Domain 3 – Cluster 1 – Standards 1, 2 and 3)

Construct and compare linear, quadratic, and exponential models and solve problems.

Essential Concepts Essential Questions

• Linear functions grow by equal differences over equal intervals. • How do you determine if a given situation is modeled by a linear or
exponential function?
• Exponential functions grow by equal factors over equal intervals.
• How do you construct an exponential function given a graph? Table?
• Linear functions have an additive recursive pattern; exponential Description of a relationship?
functions have a multiplicative recursive pattern.
• How do you know an exponential growth model will eventually exceed
• Linear and exponential functions can be constructed given a graph, a in quantity any linear or quadratic growth model?
description of a relationship, or a set of input-output pairs (which may be
given in a table).

• An exponential growth model will eventually exceed in quantity any
linear or quadratic growth model.

HS.F-LE.1

HS.F-LE.1 Mathematical Examples & Explanations

Distinguish between Practices Students may use graphing calculators or programs, spreadsheets, or computer algebra systems
situations that can be to model and compare linear and exponential functions.
modeled with linear functions HS.MP.3.
and with exponential Construct viable Examples:
functions. arguments and • A cell phone company has three plans. Graph the equation for each plan, and analyze the
critique the change as the number of minutes used increases. When is it beneficial to enroll in Plan 1?
Connections: reasoning of Plan 2? Plan 3?
ETHS-S6C2-03; others. 1. $59.95/month for 700 minutes and $0.25 for each additional minute,
SSHS-S5C5-03 2. $39.95/month for 400 minutes and $0.15 for each additional minute, and
HS.MP.4. Model 3. $89.95/month for 1,400 minutes and $0.05 for each additional minute.
a. Prove that linear with mathematics. (Continued on next page)

59 
 

functions grow by equal HS.MP.5. Use • A computer store sells about 200 computers at the price of $1,000 per computer. For each
differences over equal appropriate tools $50 increase in price, about ten fewer computers are sold. How much should the computer
intervals, and that strategically. store charge per computer in order to maximize their profit?
exponential functions
grow by equal factors HS.MP.7. Look for Students can investigate functions and graphs modeling different situations involving simple and
over equal intervals. and make use of compound interest. Students can compare interest rates with different periods of compounding
structure. (monthly, daily) and compare them with the corresponding annual percentage rate. Spreadsheets
Connection: and applets can be used to explore and model different interest rates and loan terms.
11-12.WHST.1a-1e HS.MP.8. Look for
and express • A couple wants to buy a house in five years. They need to save a down payment of $8,000.
b. Recognize situations in regularity in They deposit $1,000 in a bank account earning 3.25% interest, compounded quarterly.
which one quantity repeated How much will they need to save each month in order to meet their goal?
changes at a constant reasoning.
rate per unit interval • Sketch and analyze the graphs of the following two situations. What information can you
relative to another. conclude about the types of growth each type of interest has?
o Lee borrows $9,000 from his mother to buy a car. His mom charges him 5% interest a
Connection: 11-12.RST.4 year, but she does not compound the interest.
o Lee borrows $9,000 from a bank to buy a car. The bank charges 5% interest
c. Recognize situations in compounded annually.
which a quantity grows or
decays by a constant
percent rate per unit
interval relative to
another.

Connections:
ETHS-S6C1-03;
ETHS-S6C2-03; 11-12.RST.4

HS.F-LE.2 Mathematical Examples & Explanations
Practices
HS.F-LE.2 In constructing linear functions in F-LE.2, draw on and consolidate previous work in Grade 8 on
HS.MP.4. Model finding equations for lines and linear functions (8.EE.6, 8.F.4). In 8th grade students should do work
Construct linear and with with identifying slope and unit rates for linear functions given two points, a table or a graph.
exponential functions, mathematics.
including arithmetic and Students may use graphing calculators or programs, spreadsheets, or computer algebra systems to
geometric sequences, given a HS.MP.8. Look construct linear and exponential functions.
graph, a description of a for and express
relationship, or two input- regularity in (Continued on next page)
output pairs (include reading
these from a table).

60 
 

Connections: repeated Examples:
ETHS-S6C1-03; reasoning. • Determine an exponential function of the form f(x) = abx using data points from the table.
ETHS-S6C2-03; Graph the function and identify the key characteristics of the graph.
11-12.RST.4; SSHS-S5C5-03
x f(x)
01
13
3 27

HS.F-LE.3 Mathematic • Sara’s starting salary is $32,500. Each year she receives a $700 raise. Write a sequence in
explicit form to describe the situation.
HS.F-LE.3 al Practices
• Solve the equation 2x = 300.
Observe using graphs and HS.MP.2. Possible solution using a graphing calculator: enter y = 2x and y = 300 into a graphing
tables that a quantity Reason calculator and find where the graphs intersect, by viewing the table to see where the
increasing exponentially abstractly and function values are about the same.
eventually exceeds a quantity quantitatively.
increasing linearly, Examples & Explanations
quadratically, or (more
generally) as a polynomial Start F-LE.3 by limiting to comparisons between linear and exponential models. Later in the year
function. compare linear and exponential growth to quadratic growth.

Example:
• Contrast the growth of the functions f(x)=3x, f(x)=3x and f(x) = x2 + 3.

61 
 

Linear, Quadratic and Exponential Models (F-LE) (Domain 3 – Cluster 2 – Standard 5)

Interpret expressions for functions in terms of the situation they model. (Linear and exponential of the form f(x)=bx+k)

Essential Concepts Essential Questions
• A given situation will set parameters for any linear or exponential
• Create an example of a linear situation and give the function that can
function that models the situation. be used to model the situation. What does each part of the function
represent in the context of the problem? What are the parameters for
HS.F-LE.5 this function?

• Create an example of an exponential situation and give the function
that can be used to model the situation. What does each part of the
function represent in the context of the problem? What are the
parameters for this function?

HS.F-LE.5 Mathematical Examples & Explanations
Practices Limit exponential functions to those of the form f(x) = bx + k.
Interpret the parameters in a HS.MP.2.
linear or exponential function Reason Students may use graphing calculators or programs, spreadsheets, or computer algebra systems to
in terms of a context. abstractly and model and interpret parameters in linear, quadratic or exponential functions.
quantitatively.
Connections: Examples:
ETHS-S6C1-03; HS.MP.4. • The total cost for a plumber who charges $50 for a house call and $85 per hour would be
ETHS-S6C2-03; Model with expressed as the function y = 85x + 50. If the rate were raised to $90 per hour, how would
SSHS-S5C5-03; mathematics. the function change?
11-12.WHST.2e
• The equation y = 8,000(1.04)x models the rising population of a city with 8,000 residents when
the annual growth rate is 4%.
o What would be the effect on the equation if the city’s population were 12,000 instead
of 8,000?
o What would happen to the population over 25 years if the growth rate were 6%
instead of 4%?

• A function of the form f(n) = P(1 + r)n is used to model the amount of money in a savings
account that earns 5% interest, compounded annually, where n is the number of years since
the initial deposit. What is the value of r? What is the meaning of the constant P in terms of
the savings account? Explain either orally or in written format.

62 
 

Additional Domain Information – Linear, Quadratic and Exponential Models (F-LE)

Key Vocabulary

• Linear function • Explicit formula • Interest rate
• Exponential function • Even/odd function • Growth/decay rate
• Quadratic function • Principal

Example Resources

• Books
¾ Algebra I by McDougal and Littell [9.8, 11.3]

¾ SpringBoard Algebra I by CollegeBoard [2.2]
¾ Developing Essential Understanding of Functions: Grades 9-12 by NCTM

• Technology
¾ http://www.ixl.com/math/algebra-1/write-linear-quadratic-and-exponential-functions Activities in creating functions based on data tables.
¾ http://a4a.learnport.org/page/comparing-functions Apply a linear and exponential model to a small data set in order to explore the
differences in growth pattern, depending on which model is selected.
¾ http://www.padowan.dk/graph/ Free downloadable graph generator.
¾ eltrevoogmath.weebly.com/uploads/6/6/8/2/6682813/9-8.pdf This PDF is a note-taking guide for 9.8 McDougal Littell.
¾ www.classzone.com/ This is the site to access the book and extra resources online.
¾ http://illuminations.nctm.org/ This site has a bank of different lessons published by NCTM
¾ http://www.hippocampus.org/ This site has online videos and interactive lessons for both teachers and students to use and can be
matched to a particular textbook.
¾ http://www.wolframalpha.com/ This site is a resource for looking up mathematical facts and also provides a place to solve and graph
equations.
¾ http://www.illustrativemathematics.org/ This is a webpage that has the new standards with sample classroom tasks linked to some of the
standards.

• Example Lessons
¾ http://www.nsa.gov/academia/_files/collected_learning/high_school/algebra/matchstick_math.pdf Use matchsticks and straws to create
models of the three types of functions and analyze them.
¾ http://www.yummymath.com/2011/harry-potter/ Using the income and expense data from the Harry Potter films, create a function and
predict the income of a hypothetical ninth movie.
¾ Technology Time SpringBoard [2.2 pg 80] Given 6 different types of functions, determine the domain and range of the functions.
¾ Investigation 3 Thinking with Mathematical Models, Connected Math 2 [pg 47] Inverse Variation

63 
 

Common Student Misconceptions

Students tend to draw all graphs of functions as linear. For example, they may graph x2 as 2x.
Students misunderstand how to identify the parameters (function transformation) to build new functions from existing functions. For example,
to shift the graph of y = x2 left by 6 units, they may write y = (x - 6)2 rather than y = (x + 6)2.
Students often confuse the independent variable with the dependent variable.
Students often have difficulty identifying the quantities to be represented by variables, in modeling problems,.

Assessment

Both formative and summative assessments are vital components of effective mathematics curricula. Formative assessments (e.g., pre-assessments,
daily checks for understanding, discussions of strategies students use to solve problems, etc.) assist in instructional planning and implementation;
summative assessments (e.g., unit assessments, quarterly benchmarks, etc.) inform learner growth related to important mathematics concepts. All
district-adopted resources contain multiple assessment tools and include online resources that can be used for the purposes delineated above.
PARCC also will provide two end of the year summative assessments. The first, a performance-based assessment, will focus on applying skills,
concepts, and understandings to solve multi-step problems requiring abstract reasoning, precision, perseverance, and strategic use of tools. The
performance measure will be administered as close to the end of the school year as possible. The second, an end of the year machine-scorable
summative assessment, will be administered after approximately 90% of the school year. These assessments are to begin during the SY 2014-2015.

64 
 

High School Algebra 1

Conceptual Category: Statistics and Probability (1 Domain, 3 Clusters)

Domain: Interpreting Categorical and Quantitative Data (3 Clusters)

Interpreting Categorical and Quantitative Data (S-ID) (Domain 1 – Cluster 1 – Standards 1, 2 and 3)

Summarize, represent, and interpret data on a single count or measurement variable. 

Essential Concepts Essential Questions

• Sets of data can be represented on number lines via dot plots, • For a given data set, which measure of center or variability best
histograms, and box plots, in order to look at and compare the overall describes the data and why?
shape of the data, measures of center and spread.
• How can extreme data points affect the shape, measures of center
• Extreme data points (outliers) can affect the shape, measures of center, and spread of a data set?
and spread of a given data set.
• What types of data would you want to display on a number line and
• The measure of center or variability that best interprets a data set will why?
depend upon the shape of the data distribution and context of data
collection.

HS.S-ID.1

HS.S-ID.1 Mathematical Examples & Explanations
Practices
Represent data with plots on In grades 6 – 8, students describe center and spread in a data distribution. Here they choose a
the real number line (dot HS.MP.4. Model summary statistic appropriate to the characteristics of the data distribution, such as the shape of the
plots, histograms, and box with distribution or the existence of extreme data points.
plots). mathematics.
A statistical process is a problem-solving process consisting of four steps:
Connections: HS.MP.5. Use 1. formulating a question that can be answered by data;
SCHS-S1C1-04; appropriate tools 2. designing and implementing a plan that collects appropriate data;
SCHS-S1C2-03; strategically. 3. analyzing the data by graphical and/or numerical methods;
SCHS-S1C2-05; 4. and interpreting the analysis in the context of the original question.
SCHS-S1C4-02;
SCHS-S2C1-04; Example:
ETHS-S6C2-03;
SSHS-S1C1-04; • What measure of center or variability would best represent the data distribution for the
9-10.RST.7 height of basketball players on this team? Why?

(Continued on next page)

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Are there any extreme data points that may skew the data?

Basketball Team – Height of Players in inches for 2010-2011 Season
75, 73, 76, 78, 79, 78, 79, 81, 80, 82, 81, 84, 82, 84, 80, 84

HS.S-ID.2 Mathematical Examples & Explanations

HS.S-ID.2 Practices Students may use spreadsheets, graphing calculators and statistical software for calculations,
summaries, and comparisons of data sets.
Use statistics appropriate to HS.MP.2.
the shape of the data Reason Examples:
distribution to compare center abstractly and • The two data sets below depict the housing prices sold in the King River area and Toby
(median, mean) and spread quantitatively. Ranch areas of Pinal County, Arizona. Based on the prices below, which price range can be
(interquartile range, standard expected for a home purchased in Toby Ranch? In the King River area? In Pinal County?
deviation) of two or more HS.MP.3. o King River area {1.2 million, 242000, 265500, 140000, 281000, 265000, 211000}
different data sets. Construct viable o Toby Ranch homes {5million, 154000, 250000, 250000, 200000, 160000, 190000}
arguments and
Connections: critique the • Given a set of test scores: 99, 96, 94, 93, 90, 88, 86, 77, 70, 68, find the mean, median and
SCHS-S1C3-06; reasoning of standard deviation. Explain how the values vary about the mean and median. What
ETHS-S6C2-03; others. information does this give the teacher?
SSHS-S1C1-01
HS.MP.4. Model
with
mathematics.HS.
MP.5. Use
appropriate tools
strategically.

HS.MP.7. Look
for and make use
of structure.

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HS.S-ID.3 Mathematical Examples & Explanations

HS.S-ID.3 Practices Students may use spreadsheets, graphing calculators and statistical software to statistically identify
outliers and analyze data sets with and without outliers as appropriate.
Interpret differences in shape, HS.MP.2.
center, and spread in the Reason Comparing two data sets using a histogram. Not only can the shape of the distribution be observed,
context of the data sets, abstractly and but so can the distribution's location and spread. Figure 16 shows how a mean has increased -- a
accounting for possible quantitatively. transition from the distribution shown at the left (blue) to the one shown on the right (green). Figure
effects of extreme data points 17 shows a different method of comparing distributions. The original data set (shown in green) has
(outliers). HS.MP.3. greater variability than the later data set (the blue histogram superimposed over the original data
Construct viable set).
arguments and
critique the
reasoning of
others.

HS.MP.4. Model
with
mathematics.

HS.MP.5. Use
appropriate tools
strategically.

HS.MP.7. Look
for and make use
of structure.

Figure 16: Histogram of two data sets, one with increased mean

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Figure 17: Histogram of two data sets, one with increased variability

From: http://illuminae.info/matec/index.php?option=com_content&view=article&id=12:quality-tools-
and-spc-charts&catid=9

Interpreting Categorical and Quantitative Data (S-ID) (Domain 1 – Cluster 2 – Standards 5 and 6)

Summarize, represent, and interpret data on two categorical and quantitative variables. (Linear focus, discuss general principles) 

Essential Concepts Essential Questions

• Two-way frequency tables can be used to interpret joint, marginal and • How are two-way frequency tables used to interpret joint, marginal and
conditional relative frequencies of categorical data. conditional relative frequencies of categorical data?

• Two-way frequency tables and scatter plots of categorical data can be • How do you use a two-way frequency table or scatter plot to identify

used to identify possible associations and trends in the data. associations or trends in a data set?

• Scatter plots of data sets can be used to identify the type of function that • Why would you want to identify trends or associations in a data set?

best represents the shape of the data (linear, quadratic or exponential). • Why would you want to informally assess and identify a type of function

• Residuals (lines of regressions) are drawn on scatter plots in order to to fit a data set?

informally assess the fit of a function to a data set.

• If a scatter plot has a linear association, then a line of best fit can be
drawn to interpret the data set.

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HS.S-ID.5 Mathematical Examples & Explanations

HS.S-ID.5 Practices Students may use spreadsheets, graphing calculators, and statistical software to create frequency
tables and determine associations or trends in the data.
Summarize categorical data HS.MP.1. Make
for two categories in two-way sense of Examples:
frequency tables. Interpret problems and
relative frequencies in the persevere in • Two-way Frequency Table
context of the data (including solving them. A two-way frequency table is shown below displaying the relationship between age and
joint, marginal, and baldness. We took a sample of 100 male subjects, and determined who is or is not bald. We
conditional relative HS.MP.2. also recorded the age of the male subjects by categories.
frequencies). Recognize Reason
possible associations and abstractly and Bald Two-way Frequency Table Total
trends in the data. quantitatively. Age
No 46
Connections: HS.MP.3. Yes Younger than 45 45 or older 54
ETHS-S1C2-01; Construct viable Total 35 11 100
ETHS-S6C2-03; arguments and 24 30
11-12.RST.9; critique the 59 41
11-12.WHST.1a-1b; reasoning of
11-12.WHST.1e others. The total row and total column entries in the table above report the marginal frequencies, while
entries in the body of the table are the joint frequencies.

HS.MP.4. Model • Two-way Relative Frequency Table
with The relative frequencies in the body of the table are called conditional relative frequencies.
mathematics.

HS.MP.5. Use Two-way Relative Frequency Table
appropriate tools
strategically. Bald Age Total

HS.MP.8. Look No Younger than 45 45 or older
for and express Yes
regularity in Total 0.35 0.11 0.46
repeated
reasoning. 0.24 0.30 0.54

0.59 0.41 1.00

HS.S-ID.6 Mathematic Examples & Explanations

HS.S-ID.6 al Practices Students take a more sophisticated look at using a linear function to model the relationship between

Represent data on two HS.MP.2. two numerical variables. In addition to fitting a line to data, students assess how well the model fits by
quantitative variables on a Reason
scatter plot, and describe how analyzing residuals. (Continued on next page)

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the variables are related. abstractly and S.ID.6b should be focused on linear models, but may be used to preview quadratic functions for later
quantitatively. in the year.
Connections:
SCHS-S1C2-05; HS.MP.3. This concept can be explained through the use of technology to model the idea and allow students to
SCHS-S1C3-01; Construct explore this standard.
ETHS-S1C2-01; viable
ETHS-S1C3-01; arguments and The residual in a regression model is the difference between the observed and the predicted y for
ETHS-S6C2-03 critique the some x (where y is the dependent variable and x is the independent variable). So if we have a model
reasoning of
a. Fit a function to the data; others. ( ) ( )y = ax + b and a data point x i, y i , the residual for this point is ri = y i − ax i + b .
use functions fitted to
data to solve problems in HS.MP.4. Students may use spreadsheets, graphing calculators, and statistical software to represent data,
the context of the data. Model with describe how the variables are related, fit functions to data, perform regressions, and calculate
Use given functions or mathematics. residuals.
choose a function
suggested by the context. HS.MP.5. Use Example:
Emphasize linear, appropriate
quadratic, and tools • Measure the wrist and neck size of each person in your class and make a scatter plot. Find
exponential models. strategically. the least squares regression line. Calculate and interpret the correlation coefficient for this
linear regression model. Graph the residuals and evaluate the fit of the linear equation. Use
Connection: 11-12.RST.7 HS.MP.7. Look the line of best fit to predict the wrist size for a person not in your class.
for and make
b. Informally assess the fit use of
of a function by plotting structure.
and analyzing residuals.
HS.MP.8. Look
Connections: 11-12.RST.7; for and express
11-12.WHST.1b-1c regularity in
repeated
c. Fit a linear function for a reasoning.
scatter plot that suggests
a linear association.

Connection: 11-12.RST.7

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Interpreting Categorical and Quantitative Data (S-ID) (Domain 1 – Cluster 3 – Standards 7, 8 and 9)

Interpret linear models 

Essential Concepts Essential Questions

• If a scatter plot has a linear association, then a linear model can be • How do you interpret the meaning of a slope in a linear model in
drawn and used to identify and interpret the meaning of the slope context?
(constant rate of change) and the intercept (constant term) between the
data sets. • What is the meaning of an intercept in terms of a linear model for a
given data set?
• Technology is used to compute and interpret the correlation coefficient
(the slope) of a linear model. • How are the slope and correlation coefficient related?

• A correlation does not necessarily mean there is causation. • How do you use technology to compute the correlation coefficient?

HS.S-ID.7 • What is the difference between a correlation and causation?

• Give an example of a relationship that has a correlation but is not
causation and explain why.

HS.S-ID.7 Mathematical Examples & Explanations

Interpret the slope (rate of Practices Build on students’ work with linear relationships in eighth grade and introduce the correlation
change) and the intercept coefficient.
(constant term) of a linear HS.MP.1. Make
model in the context of the sense of Students may use spreadsheets or graphing calculators to create representations of data sets and
data. problems and create linear models.
persevere in
Connections: solving them. Example:
SCHS-S5C2-01;
ETHS-S1C2-01; HS.MP.2. • Lisa lights a candle and records its height in inches every hour. The results recorded as
ETHS-S6C2-03; Reason (time, height) are (0, 20), (1, 18.3), (2, 16.6), (3, 14.9), (4, 13.2), (5, 11.5), (7, 8.1), (9, 4.7),
9-10.RST.4; 9-10.RST.7; abstractly and and (10, 3). Express the candle’s height (h) as a function of time (t) and state the meaning
9-10.WHST.2f quantitatively. of the slope and the intercept in terms of the burning candle.

HS.MP.4. Model Solution:
with h = -1.7t + 20
mathematics. Slope: The candle’s height decreases by 1.7 inches for each hour it is burning.
Intercept: Before the candle begins to burn, its height is 20 inches.
HS.MP.5. Use
appropriate tools
strategically.

HS.MP.6. Attend
to precision.

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HS.S-ID.8 Mathematical Examples & Explanations
HS.S-ID.8 Practices
The focus here is on the computation and interpretation of the correlation coefficient as a measure
Compute (using technology) HS.MP.4. Model of how well the data fit the relationship.
and interpret the correlation with
coefficient of a linear fit. mathematics. 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
Connections: HS.MP.5. Use residuals and correlation coefficients.
ETHS-S1C2-01; appropriate tools
ETHS-S6C2-03; strategically. Example:
11-12.RST.5; • Collect height, shoe-size, and wrist circumference data for each student. Determine the
11-12.WHST.2e HS.MP.8. Look best way to display the data. Answer the following questions: Is there a correlation between
for and express any two of the three indicators? Is there a correlation between all three indicators? What
HS.S-ID.9 regularity in patterns and trends are apparent in the data? What inferences can be made from the data?
HS.S-ID.9 repeated
reasoning.
Distinguish between
correlation and causation. Mathematical Examples & Explanations
Practices
Connection: 9-10.RST.9 The important distinction between a statistical relationship and a cause-and-effect relationship
HS.MP.3. arises in S-ID.9.
Construct viable
arguments and Some data leads observers to believe that there is a cause and effect relationship when a strong
critique the relationship is observed. Students should be careful not to assume that correlation implies
reasoning of causation. The determination that one thing causes another requires a controlled randomized
others. experiment.

HS.MP.4. Model Example:
with
mathematics. • Diane did a study for a health class about the effects of a student’s end-of-year math test
scores on height. Based on a graph of her data, she found that there was a direct
HS.MP.6. Attend relationship between students’ math scores and height. She concluded “doing well on your
to precision. end-of-year math tests makes you tall.” Is this conclusion justified? Explain any flaws in
Diane’s reasoning.

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Additional Domain Information – Interpreting Categorical and Quantitative Data (S-ID)

Key Vocabulary

• Outliers • Standard deviation • Line of regression • Line of best fit
• Quartile intervals • Interval (residual) • Distribution
• Frequency • Scatter plot • Correlation
• Two-way frequency tables • Causation • Data distribution
• Correlation coefficient

Example Resources

• Books
¾ Textbooks (Pending)
¾ Focus in High School Mathematics: Reasoning and Sense Making in Statistics and Probability, National Council of Teachers of Mathematics
Publication

• Technology
¾ http://www.shodor.org/interactivate/lessons/LinearRegressionCorrelation/ Within a complete lesson with categorical and quantitative data
concepts, with vocabulary, there is a useful tool for scatter plots where the student is able to plot the data, find the best line of fit, and analyze
residuals.
¾ http://nlvm.usu.edu/en/nav/topic_t_5.html This page contains a variety of data analysis formats to present, analyze and predict data.
¾ www.classzone.com/ This is the site to access the book and extra resources online.
¾ http://illuminations.nctm.org/ This site has a bank of different lessons published by NCTM.
¾ http://www.hippocampus.org/ This site has online videos and interactive lessons for both teachers and students to use and can be matched
to a particular textbook.
¾ http://www.wolframalpha.com/ This site is a resource for looking up mathematical facts and also provides a place to solve and graph
¾ www.Illustrativemathematics.org – This is a webpage that has the new standards with sample classroom tasks linked to some of the
standards.

• Example Lessons

¾ www.malamaaina.org/files/mathematics/lesson7.pdf This lesson is a series of activities designed to use previous data gathered at a

higher level with each succeeding activity.
¾ http://www.indiana.edu/~iucme/mathmodeling/lessons.htm - A series of 40 problem-based activities to develop data-gathering techniques

from 7th – 12th grade.

Common Student Misconceptions

Students confuse the measures of center and how an outlier can have different impacts on the mean and median. For example, students think
that including an outlier changes the median, when in fact it changes the mean.

Students tend to have difficulty with the distinction between experimental and predicted values.

Students sometimes have difficulty connecting lines to functions. For example, they may have difficulty in predicting that a steeper line will change
more quickly than a gradual one.

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Students have more difficulty with negative correlations than with positive correlations.
Students may have difficulty identifying when a group of data represent a linear or nonlinear function.
Students have difficulty placing lines of best fit. For example, they may try to connect all of the points on a scatter plot, or may not draw the line
through the middle of the data.
Students may believe that a correlation implies causation.

Assessment

Both formative and summative assessments are vital components of effective mathematics curricula. Formative assessments, (e.g., pre-assessments,
daily checks for understanding, discussions of strategies students use to solve problems, etc.) assist in instructional planning and implementation;
summative assessments (e.g., unit assessments, quarterly benchmarks, etc.) inform learner growth related to important mathematics concepts. All
district-adopted resources contain multiple assessment tools and include online resources that can be used for the purposes delineated above.
PARCC also will provide two end of the year summative assessments. The first, a performance-based assessment, will focus on applying skills,
concepts, and understandings to solve multi-step problems requiring abstract reasoning, precision, perseverance, and strategic use of tools. The
performance measure will be administered as close to the end of the school year as possible. The second, an end of the year machine-scorable
summative assessment, will be administered after approximately 90% of the school year. These assessments are to begin during the SY 2014-2015.

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