1. Study Group 2 – Algebra 1 Welcome Back!
Let’s spend some quality time discussing what we learned from our Bridge to Practice exercises.
Our Bridge to Practice through TNCore Training is linking our classroom instruction to the CCSS!

2. 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?
Complete the Instructional Task
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.
Go over Bridge to Practice Part A form last Study Group over Module 1
Part A: Show the next two slides to review the content AND practice standards for the selected CRA tasks.
Discuss the EVIDENCE that participants found for the standards they selected.
Remind participants that it would help to have a copy of the math practice standards from Module 1 handy (they are on the next slide for reference).
There is also an additional Powerpoint of the Math Practice Standards with visuals and clarifications called “CCSS Math Practices” as a reference, but there is not enough time in this session to go through each slide and discuss that Powerpoint. Teachers may want to keep it to refer to as they grow more familiar with the developing these practices in their students.

3. The CCSS for Mathematical Practice
Make sense of problems and persevere in solving them.
Reason abstractly and quantitatively.
Construct viable arguments and critique the reasoning of others.
Model with mathematics.
Use appropriate tools strategically.
Attend to precision.
Look for and make use of structure.
Look for and express regularity in repeated reasoning.
Use this slide as a quick reference to discuss the math practices relative to the two CRA problems.
Common Core State Standards for Mathematics, 2010, NGA Center/CCSSO

4. 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.
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.
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 reasoning to support your decision.
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.
(SAY): In what way does the prompt for the identified item elicit a student response that will demonstrate what s/he knows about specific content within the standard? (See content standards, slide 37.)
What is it about the prompt for the identified item that will require students to use standards for mathematical practice?
Make sense of problems and persevere in solving them.
Students have to move among multiple representations of the context given by the task.
Reason abstractly and quantitatively.
Construct viable arguments and critique the reasoning of others.
Students have to argue logically that Bobby is incorrect by noting the value determined as the slope of the graph.
Model with mathematics.
Use appropriate tools strategically.
Attend to precision.
Students accurately use the scale on the graph to determine the slope and label quantities correctly.
Look for and make use of structure.
Students note that the graph represents a linear function.
Look for and express regularity in repeated reasoning.

5. 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.
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.
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.
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.
(SAY): In what way does the prompt for the identified item elicit a student response that will demonstrate what s/he knows about specific content within the standard? (See content standards, slide 38.)
What is it about the prompt for the identified item that will require students to use standards for mathematical practice?
Make sense of problems and persevere in solving them.
Reason abstractly and quantitatively.
Construct viable arguments and critique the reasoning of others.
Explains reasoning in parts a and c; recognizes the need to account for the time allowed for popular songs and the nature of the relationship among the number of songs, the average run time of each song, and the total time allowed for popular songs in the explanation.
Model with mathematics.
Students will write an equation to model the situation. However, creating a model is actually also a content standard, in this instance.
Use appropriate tools strategically.
Attend to precision.
Accurately converts hours to minutes (or minutes to hours), determines the time allowed for popular songs and the average run time; labels all quantities as needed.
Look for and make use of structure.
Recognizes the “3.5 minutes” and the “r minutes” as a constant number of minutes per song; uses that fact to create linear equations in one or more variables.
Look for and express regularity in repeated reasoning.

6. 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?
Complete the Instructional Task
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.
Go over Bridge to Practice Part B from the last Study Group over Module 1
Lead a brief discussion over what kind of instructional tasks/lessons need to be used and what teaching and learning needs to look like to prepare students for these assessments.

7. 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?
Complete the Instructional Task
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.
Go over Bridge to Practice Part C from the last Study Group over Module 1
The remaining part of this session will be spent going through the following slides and analyzing the individual solutions each group member “brought to the table” for the instructional task.

8. Supporting Rigorous Mathematics Teaching and Learning
Engaging In and Analyzing Teaching and Learning through an Instructional Task
Tennessee Department of Education
High School Mathematics
Algebra 1
Overview of the Module:
Participants will consider what instruction that is aligned with the CCSS sounds like and looks like. We will engage in a lesson as adult learners. We will not pretend we are students or think about how students will respond. Instead we will engage in the lesson as adult learners. Our goal is to deepen our understanding of the standards and to make sense of the use of models when working with the concept.
No Prior Knowledge Necessary.
Materials:
Slides with note pages
Mathematics Common Core State Standards (CSSS) (the Standards for Mathematical Practice and the grade-level Standards for Mathematical Content)
Participant handouts (including Notes and Bridge to Practice #2)
Chart paper and markers
Graph paper
Rulers (optional)

9. 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.
(SAY) The CCSS include standards that focus on understanding of mathematical concepts AND the development of skills. We will engage in the lesson with the goal of deepening our understanding of concepts related to the task.

10. What is the difference between the following types of tasks?
Question to Consider…
What is the difference between the following types of tasks?
instructional task
assessment task
Ask participants if they have ever considered this question. Give them a minute to consider it and ask them to briefly share their thoughts.

11. Taken from TNCore’s FAQ Document:
Allow participants to read about the two types of tasks. Discuss a summary of the differences.
(ASK): What type of tasks are the CRAs we worked in our first study group? (Assessment tasks; however, they could be modified to be more open-ended to allow for more solutions paths and discussion)
(ASK): What type of task is The Bike and Truck Task from the Bridge to Practice? (Instructional task)

12. 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)
Directions:
Read the session goals.

13. 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)
(SAY) We will engage in a task, and then step out and reflect on our engagement in the task. We will consider how our learning was supported, and which standards we had opportunities to think about and use when figuring out the solution path.
We will engage with the task for the sake of our thinking and learning about the mathematics.
Facilitator Information:
If participants start to describe how their students would do the task or how their students think about the mathematics, remind them that for now we are focusing on our thinking and understanding of the task, and the underlying mathematics. Our goal in this module is to deepen our understanding of the standards and to make sense of the use of the mathematical practices when working with the concept(s).

14. 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?
(SAY) Take a look at the focus cluster standards (on pages 11–12) in the handout. They are also on slides for reference, but you don’t need to select which standards this task addresses now because the Bridge to Practice will be over aligning the content standards with this particular task.

15. Bike and Truck Task
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.
Distance from start of road (in feet)
Time (in seconds)
(SAY): You already solved this task privately and now we are going to look at HOW to facilitate an instructional task by analyzing our solutions in small group and then whole group discussion.
Directions:
Circulate while participants are working in small groups and ask assessing and advancing questions, based on where the participants are in the problem solving process.
Possible assessing and advancing questions are in the notes on the next slide.

16. The Structures and Routines of a Lesson
MONITOR: Teacher selects examples for the Share, Discuss, and Analyze Phase based on:
Different solution paths to the
same task
Different representations
Errors
Misconceptions
Set Up of the Task
The Explore Phase/Private Work Time
Generate Solutions
The Explore Phase/Small Group Problem Solving
Generate and Compare Solutions
Assess and Advance Student Learning
SHARE: Students explain their methods, repeat others’ ideas, put ideas into their own words, add on to ideas and ask for 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.
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
This slide is a model of how the Structures and Routines of a lesson should unfold as teachers facilitate an Instructional Task.
(SAY) This is how we will engage together. Structures and routines are patterned ways of working that help students know what to expect.
When we engage in lessons, we first set up the task. This usually takes a few minutes. Be careful not to provide students with too much scaffolding so they can develop their own ideas about how to solve it.
Then, you will have approximately 5 minutes of private time to solve the task independently. It is VERY important to give students this time prior to breaking up into groups so they can process the problem for themselves.
(For the purpose of our training, we completed the private think time as our Bridge to Practice)
Next, you will work in small groups for about 15 minutes. While you are working, I will circulate asking assessing and advancing questions. During this time, I will be looking for a variety of solution paths to have shared with the whole group. I may be asking you to write your method on chart paper.
Finally, we will engage in a group discussion of the different solution paths and make connections between the paths to arrive at the essential understandings of the standards related to this task.

17. Solve the Task (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)
Work with others at your table. Compare your solution 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.
Directions:
Read through the directions on the slide.
Focus participants on considering the relationship among the data and any equations they may write.
In question 4 you may need to ask what the meaning of average rate of change is, and then ask participants to re-focus on the data in the graph.

18. 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
asking questions about the ideas shared.
The goal is to understand the mathematics and to make connections among the various solution paths.
Directions:
Use the information on the slide to describe the Share, Discuss, and Analyze Phase of the lesson.

19. Bike and Truck Task
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.
Distance from start of road (in feet)
Time (in seconds)
Facilitator Note:
This slide is here for you to reference during the group discussion.
Directions:
Circulate while participants are working in small groups and ask assessing and advancing questions, based on where the participants are in the problem solving process.
Possible assessing and advancing questions are in the notes on the next slide.

20. Bike and Truck Task
Label the graphs appropriately with B(t) and K(t). Explain how you made your decision.
Describe the movement of the truck. Explain how you used the values of B(t) and K(t) to make decisions about your description.
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.
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.
Facilitator Note:
This slide is here for you to reference during the group discussion.

21. Discuss the Task (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 start of the road?
When, if ever, is the average rate of change the same for the two vehicles?
Directions:
The following sequence of mathematical concepts and questions can be used to guide your facilitation of the Share, Discuss, and Analyze Phase.
Facilitator Questions and Possible Responses:
Connecting features of the graph to rate of change
Do either of these functions ever have a negative rate of change? What would a negative rate of change look like? What would it mean in context?
What do the points of intersection represent on this graph? What is true about the bike and truck’s speed at and near these points?
A rate of change describes how one variable quantity changes with respect to another. In other words, a rate of change describes the co-variation between two variables.
Connecting rate of change to context
What two variables are being related in this problem? What does the rate of change tell you about the relationship between these variables?
Does the bike or the truck cover more distance for each one second increase in time over the course of the race? Is this true over every interval of the race?
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.
Calculating and comparing rates of change
Does the bike or the truck have a greater average rate of change over the 300 feet indicated in the graph? How do you know?
Do either of these functions have a constant rate of change? How can you determine this from the graph?
Over which intervals is bike’s average rate of change greater than truck’s? Is the bike in the lead over these intervals? Explain.

22. Reflecting on Our Learning
What supported your learning?
Which of the supports listed will EL students benefit from during instruction?
Directions:
Chart responses.
Probing Facilitator Questions and Possible Responses:
What supported your learning?
Private think time
Small group problem solving time
The whole group discussion
The design of the task—It was open and we were able to come up with our own ways of thinking about the problem.
The Turn and Talk time—Because sometimes a question was hard to think about and I needed the time to think.
Did the talk help anyone else? What about the talk? Did anyone else find it helpful to hear others repeat ideas and to add on to other’s ideas?
Which of these supports would be helpful to English learners?
All of the things that supported our learning are good for all learners.
ELs will especially benefit from the structures built in that allow them to talk to their peers. Multiple representations are also helpful.
Cognates would be helpful if any apply to the problem. Cognates are the use of English words that have a word in Spanish that sounds similar. ELs might also benefit from hearing the problem before other students hear it. Teaching the vocabulary words in advance of the problem or having picture available would also be helpful.

23. Linking to Research/Literature Connections between Representations
Pictures
Written
Symbols
Manipulative
Models
Real-world
Situations
Oral
Language
(Say) Many of you moved between representations when solving and discussing the solution paths to the task.
Research has shown that some of the better problem solvers are those who, when they are struggling to figure out a problem, have the resources or know how to use different representations in order to solve a problem. If this is true, then what are the implications for instruction in our classrooms?
Adapted from Lesh, Post, & Behr, 1987

24. Five Different Representations of a Function
Language
Table
Context
Graph
Equation
(SAY) This diagram varies slightly from the one shown on the earlier slide. This one includes equations and graphs, which are more appropriate for middle school and high school students.
Both diagrams are included so that we can consider how students can benefit from looking at multiple representations of function relationships at all levels of mathematics learning.
Van De Walle, 2004, p. 440

25. 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.
Slides are only here for reference as part of the Focus Clusters that this task MIGHT address. Participants will decide which standards align to the task for their Bridge to Practice #2
*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

26. 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.
Slides are only here for reference as part of the Focus Clusters that this task MIGHT address. Participants will decide which standards align to the task for their Bridge to Practice #2
Common Core State Standards, 2010, p. 65, NGA Center/CCSSO

27. 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 half-plane (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.
Slides are only here for reference as part of the Focus Clusters that this task MIGHT address. Participants will decide which standards align to the task for their Bridge to Practice #2
Common Core State Standards, 2010, p. 65, NGA Center/CCSSO

28. 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 marked with a star, each standard in that domain is a modeling standard.
Slides are only here for reference as part of the Focus Clusters that this task MIGHT address. Participants will decide which standards align to the task for their Bridge to Practice #2
Probing Facilitator Questions and Possible Responses:.
What key features of the graph did we interpret?
We interpreted rate of change and intervals where the function is increasing or decreasing.
We used the y-intercept as we interpreted the graphs.
How did we address F-IF.B.5?
When we determined which vehicle reached 300 feet first, we had to realize that the function with the “smaller” domain was the one that reached 300 feet first.
This standard doesn’t talk about range. But similarly, we realized when calculating the average rate of change, one function has a more restricted range than the other.
The CCSS Key Shift of Rigor indicates that conceptual understanding, procedural skill and fluency, and application should be pursued with “equal intensity.” How does this cluster of standards support this Key Shift?
These standards ask students to interpret, relate, and calculate. The verbs imply all three components of the key shift.
Common Core State Standards, 2010, p. 69, NGA Center/CCSSO

29. Bridge to Practice #2: Time to Reflect on Our Learning
1. Using the Bike and Truck Task:
a. Choose the Content Standards from pages of the handout that this task addresses and find evidence to support them.
Choose the Practice Standards students will have the opportunity to use while solving this task and find evidence to support them.
Using the quotes on the next page, write a few sentences to summarize what Tharp and Gallimore are saying about the learning process.
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.
Bridge to Practice #2: Make sure participants have a copy of the Bridge to Practice #2 handout that slides explain.

30. 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.
Bridge to Practice #2: 2) Read the Tharp and Gallimore Quotes. Write a few sentences to summarize what Tharp and Gallimore are saying about the learning process.
Tharp & Gallimore, 1991

31. 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.
Bridge to Practice #2: 3) 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.