### Supporting Student Learning of Mathematics

```Study Group 2 – Algebra 1
Welcome Back!
Let’s spend some quality time discussing what we learned
from our Bridge to Practice exercises.
Part A From Bridge to Practice #1:
Practice Standards
Choose the Practice Standards students will have the opportunity to
use while solving these tasks we have focused on and find evidence
to support them.
Using the Assessment to Think About Instruction
In order for students to perform well on the CRA, what are the implications
for instruction?
•
•
What kinds of instructional tasks will need to be used in the
classroom?
What will teaching and learning look like and sound like in the
classroom?
Work all of the instructional task “Bike and Truck Task” and be prepared to
associated with it.
The CCSS for Mathematical Practice
1.
Make sense of problems and persevere in solving
them.
2.
Reason abstractly and quantitatively.
3.
Construct viable arguments and critique the reasoning
of others.
4.
Model with mathematics.
5.
Use appropriate tools strategically.
6.
Attend to precision.
7.
Look for and make use of structure.
8.
Look for and express regularity in repeated reasoning.
Common Core State Standards for Mathematics, 2010, NGA Center/CCSSO
3
1. Buddy Bags
For a student council fundraiser, Anna and Bobby have spent a total
of \$55.00 on supplies to create Buddy Bags. They plan to charge
\$2.00 per Buddy Bag sold.
Anna created the graph below from an equation to represent the
profit from the number of Buddy Bags sold.
a. Determine the equation Anna used to create the
graph if x represents the number of Buddy Bags
sold and y represents the profit in dollars. Use
mathematical reasoning to explain your
equation.
c. Anna says, “I connected the points to represent
the equation, but by connecting the points I am
not representing the context of the problem.”
Use mathematical reasoning to explain why she
is correct.
60
50
40
30
Profit in Dollars
b. Bobby claims that Anna’s graph is incorrect
because it does not show that they plan to
charge \$2.00 per Buddy Bag. Do you agree or
disagree with Bobby? Use mathematical
70
20
10
0
-10
0
10
20
30
40
50
-20
-30
-40
-50
-60
Number of Buddy Bags Sold
60
2. Disc Jockey Decisions
The student council has asked Dion to be the disc jockey for the Fall
Banquet. He has been asked to play instrumental music during the first
hour while the students are eating dinner. During the last 15 minutes of the
banquet the school choir will sing. For the remaining time, Dion will choose
popular songs to play.
a. Write an equation to determine the number of popular songs, p, that
Dion can choose if the songs Dion chooses have an average run time
of 3.5 minutes and the total time for the Fall Banquet is t minutes. Use
mathematical reasoning to justify that your equation is correct.
b. Use your equation from Part a to determine the number of popular
songs that Dion can choose if the banquet will be held from 6:00 –
10:00pm.
c. Dion decides to organize the music another way. He decides to play 50
popular songs. Write and solve an algebraic equation to determine the
average run time, r, of the 50 popular songs Dion can choose if the
average run time is represented in minutes by r. Use mathematical
reasoning to justify that your equation is correct.
Part B from Bridge to Practice #1:
Practice Standards
Choose the Practice Standards students will have the opportunity to use
while solving these tasks we have focused on and find evidence to support
them.
Using the Assessment to Think About Instruction
In order for students to perform well on the CRA, what are the
implications for instruction?
•
•
What kinds of instructional tasks will need to be used in the
classroom?
What will teaching and learning look like and sound like in the
classroom?
Work all of the instructional task “Bike and Truck Task” and be prepared to
associated with it.
Part C From Bridge to Practice #1:
Practice Standards
Choose the Practice Standards students will have the opportunity to use
while solving these tasks we have focused on and find evidence to support
them.
Using the Assessment to Think About Instruction
In order for students to perform well on the CRA, what are the implications
for instruction?
•
•
What kinds of instructional tasks will need to be used in the
classroom?
What will teaching and learning look like and sound like in the
classroom?
Work all of the instructional task “Bike and Truck Task” and be
prepared to talk about the task and the CCSSM Content and Practice
Standards associated with it.
Supporting Rigorous Mathematics
Teaching and Learning
Engaging In and Analyzing Teaching and
Tennessee Department of Education
High School Mathematics
Algebra 1
Rationale
By engaging in an instructional task,
teachers will have the opportunity to
consider the potential of the task and
engagement in the task for helping learners
develop the facility for expressing a
relationship between quantities in different
representational forms, and for making
connections between those forms.
Question to Consider…
What is the difference between the
Taken from TNCore’s FAQ Document:
Session Goals
Participants will:
• develop a shared understanding of teaching and
learning through an instructional task; and
• deepen content and pedagogical knowledge of
mathematics as it relates to the Common Core State
Standards (CCSS) for Mathematics.
(This will be completed as the Bridge to Practice)
Overview of Activities
Participants will:
• engage in a lesson; and
• reflect on learning in relationship to the CCSS.
(This will be completed as the Bridge to Practice #2)
Looking Over the Standards
• Briefly look over the focus cluster standards.
• We will return to the standards at the end of the
lesson and consider:
 What focus cluster standards were addressed in
the lesson?
 What gets “counted” as learning?
Distance from start of road (in feet)
A bicycle traveling at a steady rate and a truck are
moving along a road in the same direction. The graph
below shows their positions as a function of time. Let
B(t) represent the bicycle’s distance and K(t) represent
the truck’s distance.
Time (in seconds)
The Structures and Routines of a Lesson
The Explore Phase/Private Work Time
Generate Solutions
The Explore Phase/Small Group Problem
Solving
1. Generate and Compare Solutions
2. Assess and Advance Student Learning
Share, Discuss, and Analyze Phase of the Lesson
1. Share and Model
2. Compare Solutions
3. Focus the Discussion on
Key Mathematical Ideas
4. Engage in a Quick Write
MONITOR: Teacher selects
examples for the Share,
Discuss, and Analyze Phase
based on:
• Different solution paths to the
• Different representations
• Errors
• Misconceptions
SHARE: Students explain their
methods, repeat others’ ideas,
put ideas into their own words,
clarification.
REPEAT THE CYCLE FOR EACH
SOLUTION PATH
COMPARE: Students discuss
similarities and difference
between solution paths.
FOCUS: Discuss the meaning
of mathematical ideas in each
representation
REFLECT: By engaging
students in a quick write or a
discussion of the process.
(Private Think Time and Small Group Time)
• Work privately on the Bike and Truck Task.
(This should have been completed as the Bridge to
Practice prior to this session)
paths. If everyone used the same method to solve the
task, see if you can come up with a different way.
• Consider the information that can be determined about
the two vehicles.
Expectations for Group Discussion
• Solution paths will be shared.
• Listen with the goals of:
– putting the ideas into your own words;
– adding on to the ideas of others;
– making connections between solution paths; and
• The goal is to understand the mathematics and to make
connections among the various solution paths.
Distance from start of road (in feet)
A bicycle traveling at a steady rate and a truck are
moving along a road in the same direction. The graph
below shows their positions as a function of time. Let
B(t) represent the bicycle’s distance and K(t) represent
the truck’s distance.
Time (in seconds)
1. Label the graphs appropriately with B(t) and K(t).
2. Describe the movement of the truck. Explain how
you used the values of B(t) and K(t) to make
3. Which vehicle was first to reach 300 feet from the
start of the road? How can you use the domain
and/or range to determine which vehicle was the first
to reach 300 feet? Explain your reasoning in words.
4. Jack claims that the average rate of change for both
the bicycle and the truck was the same in the first 17
seconds of travel. Explain why you agree or disagree
with Jack.
(Whole Group Discussion)
• How did you describe the movement of the truck,
as opposed to that of the bike? What information
from the graph did you use to make those
decisions?
• In what ways did you use the information you
determined about the two vehicles to determine
which vehicle was first to reach 300 feet from the
• When, if ever, is the average rate of change the
same for the two vehicles?
Reflecting on Our Learning
• Which of the supports listed will
EL students benefit from during
instruction?
Connections between Representations
Pictures
Manipulative
Models
Written
Symbols
Real-world
Situations
Oral
Language
Adapted from Lesh, Post, & Behr, 1987
Five Different Representations of a Function
Language
Context
Table
Graph
Van De Walle, 2004, p. 440
Equation
The CCSS for Mathematical Content
CCSS Conceptual Category – Algebra
Creating Equations*
(A–CED)
Create equations that describe numbers or relationships.
A-CED.A.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.A.2 Create equations in two or more variables to represent
relationships between quantities; graph equations on coordinate
axes with labels and scales.
A-CED.A.3 Represent constraints by equations or inequalities, and by systems
of equations and/or inequalities, and interpret solutions as viable or
nonviable options in a modeling context. For example, represent
inequalities describing nutritional and cost constraints on
combinations of different foods.
A-CED.A.4 Rearrange formulas to highlight a quantity of interest, using the
same reasoning as in solving equations. For example, rearrange
Ohm’s law V = IR to highlight resistance R.
*Mathematical Modeling is a Standard for Mathematical Practice (MP4) and a Conceptual Category, and specific
modeling standards appear throughout the high school standards indicated with a star (★). Where an entire domain is
with aOF
star,
each standard in that domain is a modeling standard.
2013 UNIVERSITY
PITTSBURGH
Common Core State Standards, 2010, p. 65, NGA Center/CCSSO
The CCSS for Mathematical Content
CCSS Conceptual Category – Algebra
Reasoning with Equations and Inequalities
(A–REI)
Solve equations and inequalities in one variable.
A-REI.B.3
Solve linear equations and inequalities in one variable,
including equations with coefficients represented by letters.
A-REI.B.4
Solve quadratic equations in one variable.
A-REI.B.4a Use the method of completing the square to transform any
quadratic equation in x into an equation of the form (x – p)2 = q
that has the same solutions. Derive the quadratic formula from
this form.
A-REI.B.4b 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. Recognize when the quadratic formula gives
complex solutions and write them as a ± bi for real numbers a
and b.
Common Core State Standards, 2010, p. 65, NGA Center/CCSSO
The CCSS for Mathematical Content
CCSS Conceptual Category – Algebra
Reasoning with Equations and Inequalities
(A–REI)
Represent and solve equations and inequalities graphically.
A-REI.D.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).
A-REI.D.11
Explain why the x-coordinates of the points where the graphs of the
equations y = f(x) and y = g(x) intersect are the solutions of the
equation f(x) = g(x); find the solutions approximately, e.g., using
technology to graph the functions, make tables of values, or find
successive approximations. Include cases where f(x) and/or g(x) are
linear, polynomial, rational, absolute value, exponential, and
logarithmic functions.★
A-REI.D.12
Graph the solutions to a linear inequality in two variables as a halfplane (excluding the boundary in the case of a strict inequality), and
graph the solution set to a system of linear inequalities in two
variables as the intersection of the corresponding half-planes.
★Mathematical Modeling is a Standard for Mathematical Practice (MP4) and a Conceptual Category, and specific
modeling standards appear throughout the high school standards indicated with a star (★). Where an entire domain is
marked with a star, each standard in that domain is a modeling standard.
Common Core State Standards, 2010, p. 65, NGA Center/CCSSO
The CCSS for Mathematical Content
CCSS Conceptual Category – Functions
Interpreting Functions
(F–IF)
Interpret functions that arise in applications in terms of the context.
F-IF.B.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.★
F-IF.B.5
Relate the domain of a function to its graph and, where applicable, to the
quantitative relationship it describes. For example, if the function h(n) gives the
number of person-hours it takes to assemble n engines in a factory, then the
positive integers would be an appropriate domain for the function.★
F-IF.B.6
Calculate and interpret the average rate of change of a function (presented
symbolically or as a table) over a specified interval. Estimate the rate of
change from a graph.★
★Mathematical Modeling is a Standard for Mathematical Practice (MP4) and a Conceptual Category, and specific
modeling standards appear throughout the high school standards indicated with a star (★). Where an entire domain is
UNIVERSITY
OF a
PITTSBURGH
with
star, each
standard in that domain is a modeling standard.
Common Core State Standards, 2010, p. 69, NGA Center/CCSSO
Bridge to Practice #2:
Time to Reflect on Our Learning
1. Using the Bike and Truck Task:
a. Choose the Content Standards from pages 11-12 of the handout that this
b. Choose the Practice Standards students will have the opportunity to use
while solving this task and find evidence to support them.
2. Using the quotes on the next page, write a few sentences to
summarize what Tharp and Gallimore are saying about the learning
process.
3. Read the given Essential Understandings. Explain why I need to
know this level of detail about rate of change in order to determine if
a student understands the concept behind rate of change.
Research Connection: Findings by
Tharp and Gallimore
• For teaching to have occurred - Teachers must “be aware of
the students’ ever-changing relationships to the subject
matter.”
• They [teachers] can assist because, while the learning
process is alive and unfolding, they see and feel the
student's progression through the zone, as well as the
stumbles and errors that call for support.
• For the development of thinking skills—the [students’] ability to
form, express, and exchange ideas in speech and writing—the
critical form of assisting learners is dialogue -- the
questioning and sharing of ideas and knowledge that
happen in conversation.
Tharp & Gallimore, 1991
Underlying Mathematical Ideas Related to
the Lesson (Essential Understandings)
• The language of change and rate of change (increasing,
decreasing, constant, relative maximum or minimum) can
be used to describe how two quantities vary together over
a range of possible values.
• A rate of change describes how one variable quantity
changes with respect to another – in other words, a rate of
change describes the covariation between two variables
(NCTM, EU 2b).
• The average rate of change is the change in the
dependent variable over a specified interval in the
domain. Linear functions are the only family of functions
for which the average rate of change is the same on every
interval in the domain.