Welcome to Day Four

Agenda Number Talk Linear Function Video Cases Break Hiring Problem Lunch Connecting Number to Algebra Break Scientific Notation, Patterns, Algebra Reflection

Number Talk Is 2/3 closer to 0, ½, or 1?

Solving the Task

Analyzing the Task Individually, consider the structure of the task. How many minutes have lapsed when there are 25 dots? Consider the question for yourself then discuss with a partner.

Working on the Task

Sharing the Task Individual Think/Work Time Small Group Discuss Whole Group Sharing

Preparing to View Video Norms for viewing teacher video. Video Transcriptions

Video 1: “Working Backwards”

Video Notes Make notes of important and/or significant aspects of the video. Preparing for 2 nd Video Video Transcripts

Video Discussion Consider these questions individually, then discuss with a partner. 1.What evidences of student thinking did you observe? 2.What was an important point you noticed in the video? 3.Where there any evidences of the SMP’s?

Possible Solutions

Math Notes What SMP’s did you find yourself engaging in with the task? What was mathematically significant for you as you interacted with the task, task strategies and videos? What are the important mathematical ideas you want to remember about this session?

Algebraic Habits of Mind Building Rules to Represent Functions  Organizing information  Predicting patterns  Chunking the information  Describing a rule  Different representations  Describing change  Justifying a rule

Habit 2 Abstracting from Computation  Computational shortcuts  Calculating without computing  Generalizing beyond examples  Equivalent expressions  Symbolic expressions  Justifying shortcuts

Habit 3 Doing/Undoing  Input from output  Working backward (Driscoll, 2006)

Algebraic Habits of Mind/SMP’s How do these “algebraic habits of mind” compare with the SMP’s? Have you used any of these in the activities this week? Discuss with a partner Keep these in mind as we continue to work during the remaining COMP activities

BREAK

Hiring Problem Mission: Your group of three to four individuals is in charge of hiring workers to help clean, paint, and move furniture in a school district named Prince Matney. Workers must complete the job within 4 days. Conditions: 1. You can hire from each company only once. You must use all the workers that are provided by the company. This is non-negotiable. 2. You have a Worksite Supervisor who can supervise at most 12 workers per day. Assume that each hired worker works the same amount of time and produces the same amount of work per hour. 3. You need at least 14 workers for moving furniture, 14 workers for painting, and 14 workers for cleaning within the deadline.

LUNCH

Connecting Number to Algebra

Arrays – Grade 2

Multi-Digit Multiplication–Grade 4

Abstracting Multiplication

Traditional Algorithm Does this make sense to most of our students?

Algebra Tiles Area model representations Consider: Add: Multiply:

Multiply Binomials - Algebra

How do you visualize 3 x 5? Which of these is/are most accurate? How would you sequence the sharing of these models?

1.Anticipating (e.g., Fernandez & Yoshida, 2004; Schoenfeld, 1998) 2.Monitoring (e.g., Hodge & Cobb, 2003; Nelson, 2001; Shifter, 2001) 3.Selecting (e.g., Lampert, 2001; Stigler & Hiebert, 1999) 4.Sequencing (e.g., Schoenfeld, 1998) 5.Connecting (e.g., Ball, 2001; Brendehur & Frykholm, 2000) The Five Practices (Peg Smith)

Problem

Samantha

Eric

Discussion Questions Would you want Samantha or Eric to present her/his solution first to the class? Why? Is it necessary or even important for both of them to present their solutions?

BREAK

SCIENTIFIC NOTATION, PATTERNS, AND ALGEBRA Helping Students to Make Connections

FOLD A PIECE OF PAPER “Fold it in half”

Analysis What’s going on in this problem? Why is it so difficult to fold the paper more than about 7 times? Can we represent the problem mathematically, using a table? Are there patterns going on that we might summarize?

Myth Busters

String Representation Let’s see if we can locate some points on a line represented by a piece of string …

Richter Scale A rise of each number on the Scale represents an earthquake 10 times more powerful than the previous. Typical for California: 2 on the scale San Francisco, 1906: 7.1 on the scale How much more powerful was the 1906 quake than a “typical” daily earthquake in the area?

How Many “Times” More? If something costs $80 today but only cost $5 a couple decades ago, how many times more expensive is it today than it was then?

How Many “Times” More? So, it might make sense that to figure out how many more times powerful a Richter Scale earthquake of 7.0 is than a 2.0, we would:

Multiplying How might we help students to sort out this problem?

Making Connections Problems typical later in an algebra context:

Consider Addition Consider: But what about:

Extend the Connection Again, in an algebra context: Simplify: What about:

As if that weren’t enough … Where else have our students seen situations where multiplication and division can be done most easily but addition and subtraction require that something be made the “same” first?

Graphing Calculators Converting to and from Scientific Notation Performing Operations What is ENG? How is it similar to or different from SCI mode?

Powers of Ten Video

Curriculum Connections Review the CCSSM Algebra Standards for your grade level. How do the activities today align with the algebra expectations at your grade level? How will you use the activities next year with students?

Time of Reflection Take a few moments to reflect on SMP’s connected to the content tasks we did today. -- Name of the task and related SMP’s -- Evidence for the chosen SMP’s -- Jot down how you contributed to our shared community of professionals and what mathematical and/or pedagogical knowledge you are taking away from today.