Study Group 2 – Algebra 1

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? 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.

The CCSS for Mathematical Practice Common Core State Standards for Mathematics, 2010, NGA Center/CCSSO • 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.

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.

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.

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.

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.

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

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 following types of tasks? instructional task assessment task

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?

Bike and Truck Task Distance from start of road (in feet) Time (in seconds) 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.

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

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.

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.

Bike and Truck Task Distance from start of road (in feet) Time (in seconds) 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.

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.

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?

Reflecting on Our Learning What supported your learning? Which of the supports listed will EL students benefit from during instruction?

Pictures Manipulative Models Written Symbols Real-world Situations Oral Language Linking to Research/LiteratureConnections between Representations Adapted from Lesh, Post, & Behr, 1987

Language Context Table Graph Equation Five Different Representations of a Function Van De Walle, 2004, p. 440

The CCSS for Mathematical ContentCCSS Conceptual Category – Algebra *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 ContentCCSS Conceptual Category – Algebra Common Core State Standards, 2010, p. 65, NGA Center/CCSSO

The CCSS for Mathematical ContentCCSS Conceptual Category – Functions 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 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.

Research Connection: Findings by Tharp and Gallimore Tharp & Gallimore, 1991 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.

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.

Study Group 2 – Algebra 1