Unit 2: Relations and Functions
Timeframe: 10-13 days
Number of lessons in this unit: 9
Learning Outcomes
Common Core Learning Standards addressed in this Unit: F-IF.1, F-IF.2, F-IF.5, F-BF.1c(+), F-BF.3, F-BF.4a, F-BF.4b(+)
NYS 2005 Algebra 2 Core Curriculum standards addressed in this Unit: A2.A.37, A2.A.38, A2.A.39, A2.A.40, A2.A.41,
A2.A.42, A2.A.43, A2.A.44, A2.A.45, A2.A.46, A2.A.51, A2.A.52
Standards for Mathematical Practices addressed in this Unit:
1. Make sense of problems and persevere in solving them.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
8. Look for and express regularity in repeated reasoning.
Enduring Understandings for this Unit:
A function is a relation in which each input value in its domain is
related to exactly one output value in its range. While the
relationship between a function’s inputs and outputs can be
represented graphically, not all graphs are produced by functions.
A function and equation differ in that a function describes the way
that input and output (e.g. x and y) are related, whereas an
equation states that two expressions are equal (and often involves
solving for the value of a variable).
Essential Questions for this Unit:
- How are relations and functions related?
- When are functions “one-to-one,” “onto,” both, or
neither?
- What are the different ways to represent relations and
functions?
- Under what circumstances do functions have inverses?
- How can the domain and range of a function be
determined?
Content of this Unit:
- Relations
- Functions
- Domain and range
- Function notation
- Composition of functions
- One-to-one and onto
- Inverses of functions
- Transformations of functions
Skills of this Unit:
- Represent and interpret functions relations using tables, graphs, coordinates, and
maps.
- Define relations and functions and distinguish between the two from their
graphs.
- Assess when a relation is a function.
- Understand domain, range, and the relationship between the two as they pertain
to functions.
- Determine domain and range of a function using its equation or graph.
- Represent functions in function notation and use function notation to evaluate
functions.
- Determine the composition of two functions.
- Explain when a function is one-to-one and/or onto.
- Predict and describe the effect of a transformation on a function’s graph.
- Determine and interpret the inverse of a function.
- Understand the relationship between a function and its inverse.
Key Vocabulary & Language of this Unit:
Relation, function, domain, range, composition, one-to-one, onto,
inverse, transformation, undefined (review)
Assessments
Resources used in this Unit:
- JMAP.org Resources for each Standard
- TI Smartview
- Regents Reference table
- Prentice Hall Algebra 2/Trigonometry
- Holt Algebra 2 Resource Page
- Graphical representations of geometric connections
- SmartBoard software to demonstrate functions
Formative Assessments:
- Functions math sprint with trade-and-grade
- Mini dry erase boards
- Daily exit slips
- Pair, group, and class discussions
- Homework assignments
- Writing prompts/journaling
- Quizzes
- Peer- and self-assessments
- Results and observations
- Warm up collection and review
- Summaries
- Questioning
Summative Assessments, including Performance Tasks:
- Student presentations
- Unit Performance Task
- Unit Exam
Instructional Pathway
Learning Activities & Teaching Strategies Used in This Unit
AD = agree/disagree: statement given related to functions for students to agree/disagree with and defend
answers
CAP = create a problem: students create a problem from a given scenario
C = explicit connection to another math course or other discipline
CS = card sort: sort cards into groups with a common theme related to functions
DA = discovery activity
DT = discussion topic: for any grouping structure
E = extension topic/problem
EA = error analysis: teacher models a common mistake and students determine where the mistake was
made, or teacher presents alternative approaches, and students determine which, if any, are wrong
GO = graphic organizer: use of Frayer model for definition
OEQ = open-ended question: question that has many possible correct answers
Grouping Structures
(I) = individual
(P) = with a partner
(G) = in a student group
(C) = whole class
MR = multiple representations
MS = math sprint: quick timed assessment for building fluency and reviewing
PA = peer assessment with a partner
RWC = real world connection
SP = student presentation
T = use of technology: calculators, SmartBoard function generator
TPS = think-pair-share
WB = whiteboard: students do/correct their work on mini whiteboards and share with a partner, group, or
class
WP = writing prompt
Standards Aim
Lesson Content
-
A2.A.37
A2.A.38
1. What are relations and
functions?
2. When is a relation a
function, and when is it
not?
-
-
Reference to a function as a “machine”
and “black box” (review)
Examples of relations
Examples of functions
Mapping representation
Vocabulary for graphical word wall:
relation, functions
Mapping representations
Graphical representations
Coordinate representations
Vertical Line Test (review)
Assess whether any given
representation of a relation also
represents a function or not
Activities & Strategies
- RWC: Vending machine analogy; when you
press M3, you always get a Snickers bar! If
you get something else when you press M3,
the vending machine is not predictable or is
broken (i.e. not functioning!).
- T(C): SmartBoard Function Machine, let
students pick x values and the function
machine outputs y values, and students work
to find the “rule.”
- DA(G): Provide graphs, sets of points,
tables, and mappings to students, with each
identified as a function or non-function, and
have students come up with “rules” for how
to determine if a given representation is a
function or not.
- GO(I): Frayer model for the word
“function” (include definition, illustration,
examples, and non-examples)
F-IF.1
F-IF.5
A2.A.51
3. How do we determine
the domain and range
of a function from its
graph?
-
F-IF.1
F-IF.5
A2.A.39
4. How do we determine
the domain and range
of a function from its
equation?
-
-
F-IF.2
A2.A.40
A2.A.41
5. How can function
notation be used to
evaluate functions?
-
Definitions of domain and range
Examples of domains/ranges that are
intervals and all real numbers
Given a graph of a function, determine
the domain and the range
Real world scenarios and constraints on
domains/ranges
Vocabulary for graphical word wall:
domain, range
Values that make an expression
undefined (0 denominator) or not real
(negative radicand)
Use structure and form to determine
range (such as absolute value and
quadratics)
Include examples with open/closed
endpoints
What is function notation, why does it
exist, what does it show us that “y =”
doesn’t
Convert functions in y = form (or other
form) into function notation with
correct variables
Evaluate functions using function
notation for given values in the domain
Include negative numbers that get
squared (review) or that generate nonreal/undefined values (what does this
-
-
-
-
-
DT(P): Are all relations functions? Are all
functions relations? Justify your answers.
MS(I): As opener/review from previous
day: determine function or non-function,
given several graphs, tables, sets of
coordinates, and element maps.
CAP(P): students develop graphs modeled
from real world scenarios that they come up
with, then trade with other pairs to determine
domain and range.
AD(G): All functions have domains and
ranges.
MR: Represent domains and ranges in set
builder notation and interval notation.
WB(C): Students determine domains and
ranges on their boards and display to teacher
and the class.
E: Find domain and range from equations of
non-functions (e.g. basic circles)
TPS: How are parentheses used differently
in the following? f(x), 4(x – 3), (6, -5)
OEQ(P): Come up with two different
functions for which f(–3) = 15; get creative!
WP(I): One student represented a function
with y = 15x and another students
represented the same function with f(x) =
15x. Are both the same? Is one
representation more useful than another?
Explain your thinking.
F-BF.1c
A2.A.42
6. How are compositions
of functions
determined, and why
are they useful?
-
-
A2.A.43
7. How and when are
functions classified as
one-to-one or onto?
-
F-BF.4a
F-BF.4b
A2.A.44
A2.A.45
8. What is a function’s
inverse, and how is it
found?
-
-
then mean for that input?)
Vocabulary for graphical word wall:
function notation
The composition symbol, ᵒ, and the
order in which functions are composed
Correct usage of the composition
symbol as it relates to function notation,
e.g. f ᵒ g = f(g(x))
Perform compositions of two functions;
include an example where the two
functions are inverses (set the stage for
lesson 10)
Vocabulary for graphical word wall:
composition
Definitions of one-to-one and onto
Graphical and mapping representations
of one-to-one and onto functions
Vocabulary for graphical word wall:
one-to-one, onto
Define what an inverse is; use mapping
representation
Find the inverse of a function
Verify that two functions are or are not
inverses by using compositions
Show that the domain and range of a
function and its inverse are reversed;
use this idea to draw the graph of a
function’s inverse given the graph (or
input/output pairs) of the function
Vocabulary for graphical word wall:
-
-
-
-
-
EA(C): 1. Compose two functions in the
wrong order, 2. Compose two functions in
the correct order, but replace only one x in
the function rather than both x’s, e.g. f(x) =
x2 + 4x, g(x) = x – 7, f ᵒ g (x) = (x – 7)2 + 4x
C(geometry): Teacher writes R90 ᵒ rx-axis on
board and asks how the use of the ᵒ symbol
here is related to compositions of functions.
CS(G): Students categorize cards with
mixed graphs and mapping representations
into the following categories: non-functions,
one-to-one functions, onto functions, one-toone and onto functions, functions that are
neither one-to-one nor onto
PA(P): Students work individually to
calculate the inverse of a function given an
equation and draw the inverse of a function
given the function’s graph; they then work in
pairs to trade and grade each other’s work
SP(P): Students present to the class about
their findings related to finding the inverse;
follow-up questions to the pairs from the
teacher will include questions related to
whether the inverse is a function, one-to-one,
onto, etc.
inverse, invertible
-
-
F-BF.3
A2.A.46
9. How do we transform
functions and
relations?
-
Use the function f(x) = x2 and write
expressions for: f(x) + 5, f(x + 5),
5f(x), and f(-x)
Based on this, have students predict the
effects of these “shifts” on the graphs of
the original function, f(x).
-
-
C(geometry): What transformation is
created when x and y values are reversed?
(reflection over y = x!) What does this say
about the graphical relationship between a
function and its inverse?
E: Given the graph of a function’s inverse,
determine the equation of the original
function.
T: Use calculators to explore different
shifts, including f(x ± a), f(x) ± a, f(-x), and
af(x), with different values for a
DA(G): By exploring and analyzing many
different examples, students come up with
verbal and graphical descriptions for how
different transformations to a function effect
the graph of f(x)
C(geometry): What transformations are
occurring? (Translations, dilations, and
reflections!)
10. Review Class
11. Unit Exam
Differentiation strategies used in this unit & modifications embedded within this unit to provide access for all learners
-
Use of projected calculator emulator (TI Smartview).
Use of graphic organizers to enhance learning of key vocabulary.
Provide multiple representations of functions and relations throughout unit.
Extension problems provide additional challenge to students.
Focus on supporting students with proper use and interpretation of symbols related to functions.
-
Use of open-ended questions provides students at varying levels with entry points.
Think-alouds used with certain students and groups, as needed.
Students are provided with independent think time prior to answering questions in any grouping setting.
Explicit connections of this unit’s content to other courses help to build relevance.
Choice in student presentation format and delivery.
Development of Academic & Personal Behaviors and 21st Century Skills
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-
Many pairing and grouping exercises help students to build collaboration skills.
Students maintain portfolios and contribute work from this unit to it; students are provided with opportunities to revise their work, after
receiving feedback from peers and teacher (including the unit performance task).
Through the use of strategic grouping, driven by formative assessment data, students interact with pairs, groups, and the whole class on a
routinely changing basis; tasks and classroom activities are developed to promote independence (e.g. the use of “Ask Three Before Me” and
related strategies), effective collaboration, and leadership (group leaders are rotated).
Student presentations provide students with the opportunity to present and defend their work in front of their peers, and take questions.
Instructional Shifts
Instructional Shift: Focus
Where in this unit is there evidence of focusing
deeply on the concepts that are prioritized in
the standards?
Instructional Shift: Coherence
How does this unit build upon knowledge of
prior years, and how does it support future
coursework?
Instructional Shift: Rigor
Where is there evidence of rigor in this unit?
The sequence of lessons in this unit provide
students with the opportunity to develop an in
depth understanding of functions,
representations of functions, and the use of
functions.
These lessons build upon the skills developed in
8th grade and algebra with regard to introduction
to functions and connect to the transformations
unit in geometry. This unit helps to prepare
students for courses after algebra 2 in which
Several rigorous activities are planned for this
unit, including providing students with
opportunities to create their own problems and
questions, agree/disagree and justify their ideas
with partners and the class, and engage with
function notation is widely utilized.
open-ended questions and extension questions.
Multiple discovery activities provide students
with the opportunity to investigate certain
mathematical phenomena, analyze what they
see, and develop mathematical rules or
techniques with their peers.