PROBLEM SOLVING VIA FUNCTIONS TASKS
Day 2
REFLECTING ON FUNCTIONAL THINKING
Go to mscsummercourses2013.wikispaces.com
and then to the Problem Solving via Functions Tasks
page.
Respond to the discussion prompt “Reflections
on Day 1” on the bottom of the page.
Math & Science Collaborative at the Allegheny Intermediate Unit
CONNECTING TO THE STANDARDS
 How do you see the Learning Progressions helping you or applying in your
classroom?
 Use an example to illustrate this
FUNCTIONAL THINKING
Based on your reading assignment:
How has your thinking about functions and functional
thinking changed? Why? What new insights have you
gained?
CUBES IN A LINE TASK
How many faces (face units) are there when 2 cubes are put together
sharing a face? 10 cubes? 100 cubes? t cubes?
MATHEMATICAL TASK QUESTIONS
Predict how a student might solve this
problem using:
• a visual, closed method
• a recursive method
• a table
MATHEMATICAL TASK QUESTIONS
Look over how you solved this problem. Why did it
make sense to you to solve it this way? How is this
similar/different than how you solved the Tiling the
Patio task on Day 1?
What are some of the ways students might solve it?
How might they use the cubes to generate the
number of faces for any number of cubes? What
misconceptions might they bring?
What might a teacher need to do to prepare to use
this task with students?
VIDEO SEGMENT FOCUS QUESTIONS
What moments or interchanges appear to be
interesting/important mathematically?
What about them makes this so?
For additional resources for this task, see the article
Developing Algebraic Reasoning Through Generalization
in your binder
Language
Context
Table
Graph
Equation
Van de Walle, 2004
Five Different Representations of a Function
REPRESENTATIONS AND SMP
Where do you see evidence for each of
the 5 representations of function?
Which SMPs are best illuminated by the
Cubes in a Line Task? Provide evidence.
LINEAR VS. EXPONENTIAL FUNCTIONS
Linear
Rate of change is constant
Exponential
Cube faces
1
6
2
10
3
14
Recursive: next= now + rate of change
f(t+1)= f(t)+m; m is the rate of change
Closed form: f(x)=mx + b, m = rate of
change (slope), b= y-intercept (constant)
Arithmetic sequences can be thought of
linear functions whose domains are pos.
integers
1, 4, 7, 10, 13,….
LINEAR AND EXPONENTIAL MODELS
Construct and compare linear and exponential models to solve problems
 Distinguish between situations that can be modeled with linear
functions and with exponential functions (F-LE1a)
 Linear and exponential functions should receive the bulk of attention
PA MODEL CURRICULUM- ALGEBRA
The PK-12 PA Common Core Standards for Mathematics stress both procedural skills and conceptual
understanding to ensure students are learning and applying the critical information they need to succeed at
higher levels. The introduction at each grade level articulates a small number of critical mathematical areas that
should be the focus for that grade. The Standards emphasize applying mathematical ways of thinking to real
world issues and challenges.
Students will be able to independently use their learning to:
1.
Make sense of and persevere in solving complex and novel mathematical problems.
2.
Use effective mathematical reasoning to construct viable arguments and critique the reasoning of others.
3.
Communicate precisely when making mathematical statements and express answers with a degree of
precision appropriate for the context of the problem/situation.
4.
Apply mathematical knowledge to analyze and model situations/relationships using multiple
representations and appropriate tools in order to make decisions, solve problems, and draw conclusions.
5.
Make use of structure and repeated reasoning to gain a mathematical perspective and formulate
generalized problem solving strategies.
Algebra 1
At this level it is expected that students will formalize and expand on Algebraic concepts established in previous
coursework. Students will deepen and extend their understanding of linear and exponential relationships by
contrasting them with each other and by applying linear models to data that exhibit a linear trend. Students will
engage in methods for analyzing and using functions. Students will fluently move between multiple
representations of functions including but not limited to linear, exponential, and quadratics.
Algebra 1 Modules
The modules should equate to a full year of instruction:

Module 1: Relationships Between Quantities and Reasoning with Equations

Module 2: Linear and Exponential Relationships

Module 3: Descriptive Statistics

Module 4: Equations and Expressions

Module 5: Quadratic Functions and Modeling
LINEAR AND EXPONENTIAL RELATIONSHIPS
 By the end of eighth grade students have learned to solve linear equations
in one variable and have applied graphical and algebraic methods to analyze
and solve systems of linear equations in two variables. This unit builds on
these earlier experiences by asking students to analyze and explain the
process of solving an equation.
 Students develop fluency writing, interpreting, and translating between
various forms of linear equations and inequalities, and using them to solve
problems.
 They master the solution of linear equations and apply related
solution techniques and the laws of exponents to the creation and
solution of simple exponential equations.
 All of this work is grounded on understanding quantities and on
relationships between them.
GROWTH TASK
(MATHEMATICS IN CONTEXT, ROODHARDT, ET AL. 1998)
The plants’ heights (in millimeters) over several weeks (t) appear in the
tables in the figure below. h(t) and g(t) are functions that represent the
height of the plants’ over time, respectively.
GROWTH TASK
(MATHEMATICS IN CONTEXT, ROODHARDT, ET AL. 1998)
LINEAR VS. EXPONENTIAL FUNCTIONS
Linear
Exponential
Rate of change is constant
Rate of change is not constant, but it changes
Rate of change Increases/decreases over time
When the input increases by 1unit, the
output is multiplied by a constant factor
Recursive: next= now + rate of change
f(t+1)= f(t)+m; m is the rate of change
Recursive: next =now X positive constant
g(t+1)= g(t) X b; b is not the rate of change
of the sequence but it does tell us how the
sequence is changing
Closed form: f(x)=mx + b,
constants
Closed form: g(t)=aXbt, where b is the “base”
that determines the rate of change; a is the
“starting point” or y-intercept of the graph
m & b are
Arithmetic sequences can be thought of
linear functions whose domains are pos.
integers
1, 4, 7, 10, 13,….
Geometric sequences can be thought of as
exponential functions whose domains are
pos. integers
10, 20, 40, 80, ….
QUALITATIVE GRAPHING:
A Context for Exploring Functional
Relationships
IMPORTANCE OF QUALITATIVE GRAPHS
Rather than beginning with tasks that require students
to plot points on a scaled Cartesian coordinate
system, students should first be introduced to
qualitative graphs and asked to view them globally.
This approach utilizes students’ everyday knowledge
of real-world events and provides a basis for
interpreting graphs of functions.
Leinhardt, Zaslavsky, and Stein , 1990, p.28
IMPORTANCE OF QUALITATIVE GRAPHS
Introducing the function concept by using graphs or
pictorial representations highlights the importance of
graphs in linking representations of functions. There is a
natural progression from qualitative graphs to quantitative
graphs to tables to equations and students are more
comfortable working with the function concept when it is
introduced in this progression.
Van Dyke, 2003, p. 126
KEISHA’S BICYCLE RIDE TASK
Solve the task
Share your story with a partner
Identify the key elements that you would be
looking for in a student-generated story
Identify the misconceptions you might expect to
surface as students work on this task
REFLECTION
What is functional thinking? What new insights
have you gained? How has your thinking changed?
FUNCTIONAL THINKING IN YOUR
CLASSROOM
Plan task/activity to develop/extend
students’ functional thinking. This will be
submitted (electronically) for inclusion on
the course wikipage
Focus on one content standard and one
standard for math practice
Resources are found at
www.mscsummercourses2013.wikispaces.com
MA T H & S C I E N C E C O L L A B O R A T I V E A T
T H E A L L E G H E N Y I N T E R ME D I A T E U N I T
HOMEWORK
Complete the task/activity you started working on
this afternoon.
Read the Case of Robert Carter found in your
binder, day 2. Complete the form Supporting Students’
Capacity to Engage in the Standards for Mathematical Practice The Case of Robert Carter to
MA T H & S C I E N C E C O L L A B O R A T I V E A T
T H E A L L E G H E N Y I N T E R ME D I A T E U N I T
guide your reading.