patterns & functions workshop 4: in … & functions workshop 4: in practice ... if you...

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T h e M i s s i n g L i n k / W o r k s h o p 4 : P a t t e r n s & F u n c t i o n s — I n P r a c t i c e / T w o - H o u r A g e n d a

BEFORE WATCHING THIS PROGRAM ...

� Make sure to watch the first program on this topic(Workshop 3: Patterns &Functions — Discovery).

� Do the lessons from Workshop 3 on your own or with colleagues.

� If time permits, use one ormore of the lessons fromWorkshop 3 with at leastone of your classes.

� Have the following materi-als available to each viewer:

• Lesson 3: Buying T-Shirts from Workshop 3(page 57)

• Sample student work(page 64)

• Background materialson scoring guides(page 158)

• Background materialson creating good prob-lems or tasks (page 163).

PATTERNS & FUNCTIONSWORKSHOP 4: IN PRACTICEAgenda for Two-Hour Workshop

25 minutes Workshop Facilitator/Site LeaderReview and Discuss

Review Lesson 3: Buying T-Shirts from Workshop 3: Patterns & Functions— Discovery. Discuss the following questions:• How are slope and y-intercept important in the context of this problem?• Slope is a ratio sometimes stated as “the change in y over the change in

x.” What does this mean for the t-shirt problem?• The t-shirt problem is open-ended. What does that mean? What are the

benefits to giving students open-ended problems?

60 minutes Whole GroupView Workshop 4: Patterns & Functions — In Practice

While watching the program, consider the following focus questions:• On the Buying T-Shirts lesson, what would a “4” paper look like? • What would a “3” paper need to show? • How would a “3” differ from a “2”? A “2” from a “1”?

25 minutes Small Groups or with a PartnerDiscuss

Develop your own scoring criteria for the Buying T-Shirts lesson, using theabove questions. Review the sample student work (page 64) and use yourscoring guide to assess them. Discuss your scores. Focus on disagreements.How do your rankings compare with those of the Learner Teachers?

10 minutes Workshop Facilitator/Site LeaderHomework Assignment

• Review materials on creating a good problem or task (page 163).• Use the “Creating a Problem or Task” template and develop a new prob-

lem for the topic of Patterns & Functions. Or strengthen the problemsthat the Learner Teachers created (page 71).

• Review the Why This Topic Matters section (page 74).• Use your journal to reflect on the focus questions from this workshop.

What will you do now to continue deepening your content knowledgeabout Patterns & Functions? How will you continue to improve yourteaching methods? What is your personal action plan?

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T h e M i s s i n g L i n k / W o r k s h o p 4 : P a t t e r n s & F u n c t i o n s — I n P r a c t i c e / F o u r - H o u r A g e n d a

PATTERNS & FUNCTIONSWORKSHOP 4: IN PRACTICEAgenda for Four-Hour Workshop

15 minutes Workshop Facilitator/Site LeaderReview and Discuss

Review Lesson 3: Buying T-Shirts from Workshop 3. Discuss the followingquestions:• How are slope and y-intercept important in the context of this problem?• Slope is a ratio sometimes stated as “the change in y over the change in

x.” What does this mean for the t-shirt problem?• The t-shirt problem is open-ended. What does that mean? What are the

benefits to giving students open-ended problems?

15 minutes Whole GroupView Segment 1: Classroom Reflections

While watching Segment 1, consider the following focus questions:• Which representation (table, graph or equation) did your students tend

to prefer? How much of a stretch was it to get them to go from thetable or graph to using an equation?

• As some of the Learner Teachers observed, was it hard for you to standback and let the students explore the math on their own?

• Like the Learner Teachers, think through how one finds time to teachlessons like these — lessons that may be deeper and more complex thanthose you previously have been expected to teach. What can you giveup? How can you cover multiple concepts in the same lesson? Does itseem to you — as it did to the Learner Teachers — that this kind ofteaching actually could save you time? Why or why not?

• The Learner Teachers provided some tips that helped them with the les-son. Which, if any, seemed useful to you? Did you think of other tipsthat could be useful?

15 minutes Whole GroupDiscuss

Discuss Segment 1, considering the above questions.

30 minutes Whole Group View Segment 2: Task-Specific Scoring Guides

While watching Segment 2, consider the following focus questions:• On the Buying T-Shirts lesson, what would a “4” paper look like? • What would a “3” paper need to show? • How would a “3” differ from a “2”? A “2” from a “1”?

30 minutes Small Groups or with a Partner Read, Do and Discuss

Develop your own scoring criteria for the Buying T-Shirts lesson, using theabove questions. Review the sample student work (page 64) and use yourscoring guide to assess them. Discuss your scores. Focus on disagreements.

BEFORE WATCHING THIS PROGRAM ...

� Make sure to watch the first program on this topic(Workshop 3: Patterns &Functions — Discovery).

� Do the lessons fromWorkshop 3 on your ownor with colleagues.

� If time permits, use one ormore of the lessons fromWorkshop 3 with at leastone of your classes.

� Have the following materi-als available to each viewer:

• Lesson 3: Buying T-Shirts from Workshop 3 (page 57)

• Sample student work(page 64)

• Background materialson scoring guides(page 158)

• Background materialson creating good prob-lems or tasks (page 163).

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T h e M i s s i n g L i n k / W o r k s h o p 4 : P a t t e r n s & F u n c t i o n s — I n P r a c t i c e / F o u r - H o u r A g e n d a


Discuss rankings. Focus on disagreements. What evidence does the student give that he or she understands thecore concepts of Patterns & Functions? How do your rankings compare with those of the Learner Teachers?

30 minutes Whole GroupView Segment 3: Designing a New Problem

While watching Segment 3, consider the following focus questions:• What makes a good problem or task?• What sources have you used to create good problems for your students?• What have you learned so far about creating problem stems (Workshop 2)?

30 minutes Small Groups or with a Partner Read, Do and Discuss

Review material on creating good problems or tasks (page 163). Create your own problem. Or strengthen the prob-lems that the Learner Teachers developed (page 71).


Discuss the problems that the Learner Teachers began to create, considering the following questions: • What are the strengths and weaknesses of each of the new problems?• Are the expected math skills appropriate for middle school?• Is each problem description clear? • Is each problem engaging?• Do you agree with the Learner Teachers’ criteria for a “4” score?• How would you strengthen these problems?• Do they cover the essential math concepts about Patterns & Functions? It’s not enough to create an engaging

lesson. Your lesson also needs to allow students to do rigorous math.

25 minutes Whole GroupFinal Reflections

View the last segment of Workshop 4. Share your own final reflections. Use these focus questions as a startingpoint: • Can all students do this kind of work?• Do performance tasks help engage some students who traditionally have been hard to reach?• What adjustments do you have to make in your teaching style and the physical layout of your classrooms?• Like Nicolle (one of the Learner Teachers), are you finding Patterns & Functions everywhere in your daily life —

and then sharing these examples with your students? • What’s your action plan for using this learning and building on it?

10 minutes Workshop Facilitator/Site LeaderHomework Assignment

• Continue developing a new problem or strengthen the problems that the Learner Teachers created. Share withyour colleagues and revise, based on the feedback.• Review the Why This Topic Matters section (page 74). Add your own suggestions to the ideas for making this

topic more relevant. • Use your journal to reflect on the focus questions from this workshop. What will you do now to continue deep-

ening your content knowledge about Patterns & Functions? How will you continue to improve your teachingmethods? What is your personal action plan?

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T h e M i s s i n g L i n k / P a t t e r n s & F u n c t i o n s / A d d i t i o n a l N o t e s

PATTERNS & FUNCTIONSADDITIONAL NOTESRecommendations• Remember, in this program Learner Teachers are working on a more sophisticated scoring guide than in

Workshop 2: Proportionality & Similar Figures — In Practice. The first step is to do the kind of basic sorting andranking discussed in Workshop 2 — mainly to get a good picture in your mind of work that meets standards andwork that does not meet standards. This workshop takes the process one step farther so that you’re using studentwork to develop more task-specific guides.

• Make sure that you clarify for students that the problem calls for generating a linear graph.• Observe how the Little-by-Little sample task connects math to literature.• Sometimes (as in the Little-by-Little task) it does not matter on which axis the dependent and independent vari-

ables are plotted. A “dependent variable” means “the thing you want to find out.” An “independent variable”means “the thing you already know” (and can be used to help you find out the dependent variable). So, if youwere the crow in the fable, you could say, “I need the water level to be at 10 inches in the pitcher to get mydrink. How many rocks will it take?” In this case the independent variable (x-axis) should be “water level ininches” and the dependent variable (y-axis) should be “number of rocks.” But you could also say, “I have 100rocks to drop into the pitcher. How much will it raise the water level if I drop in 50? 80? 100?” In this case, theindependent variable is “number of rocks,” and the dependent variable is “water level in inches.”

Focus: TIMSSContinuing the discussion from Workshop 2, spend some more time discussing how the findings from the Third Inter-national Mathematics and Science Study (TIMSS) are impacting your classroom practice and professional development.How is TIMSS influencing how you teach? Are you doing less lecturing and having your students do more of the workthemselves? Use Shannon’s observations between Segments 2 and 3 as a starting point for your discussions.

The Missing Link Web site connects you to all the background materials on the TIMSS research, as well as to examplesof how other districts are using the findings. Go to www.learner.org/channel/workshops/missinglink.

Teach and DiscussBy far, the best way to strengthen your practice is to teach these lessons in your own classrooms, just as theLearner Teachers did. If you can’t teach the lessons during the eight-week series, teach them sometime during theschool year. Then reflect on your experience, preferably with a group of colleagues.

Analyze work from your own students. • How did your students’ work samples compare with the samples from this program?• How would you clarify your Launch or strengthen your Summarize to help your students understand your math-

ematical expectations?• Does the process of deciding which scoring guide to use help you clarify exactly what math you expect your stu-

dents to learn in each lesson?

Share insights and advice with others in The Missing Link network: www.learner.org/channel/workshops/missinglink — select the “Teacher Talk” section.

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T h e M i s s i n g L i n k / P a t t e r n s & F u n c t i o n s / S a m p l e S t u d e n t W o r k

Student Work Sample A, page 1: Buying T-Shirts

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Student Work Sample A, page 2: Buying T-Shirts

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Student Work Sample B: Buying T-Shirts

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Student Work Sample C: Buying T-Shirts


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Student Work Sample D, page 1: Buying T-Shirts


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Student Work Sample D, page 2: Buying T-Shirts


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Student Work Sample E: Buying T-Shirts

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T h e M i s s i n g L i n k / P a t t e r n s & F u n c t i o n s / S a m p l e T a s k s

Sample Task Grade Level: 8

BABY-SITTING BUSINESS PLAN

Mathematical Concepts Mathematical Skills

• Linear functions • Graphing linear functions• Solving systems • Evaluating algebraic expressions

• Writing equations

Problem Statement or PromptYou have been baby sitting for a few months and have noticed that, when people hire you for one hour,it’s not worth the time and energy to take the job. Develop two pricing plans — one that you can usewhen you are hired for a short period of time and another that you can use when you are hired for a longperiod of time. Compare the two plans mathematically. Determine which plan would be best if you wereto baby-sit for one hour, four hours, six hours and 12 hours. Is there ever a time when the plans wouldpay the same amount?

Task Criteria or Problem Guidelines

• Create table of solutions.• Compare both plans.• Show graph of both equations.• Write equations.

Time Frame for Task Completion

• 60 minutes

Scoring Guide: Baby-Sitting Business Plan — “4”

• States two pricing plans clearly in words.• Creates comparative graph showing both plans and the amount charged for zero to 12 hours.• Represents each plan with an equation.• Shows table of solutions for amounts earned for both plans for zero to 12 hours.• Identifies clearly benefits of both plans.

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Sample Task Grade Level: 7–8

CANDY SALES

Mathematical Concepts Mathematical Skills• Linear functions • Setting up equations• Problem-solving and reasoning • Constructing a table

• Graphing

Problem Statement or Prompt

Your school is going to sell candy for $1 per bar. Company A will ship only 5,000 bars at a time. It willcharge 40 cents per bar and a $150 delivery charge for each 5,000 bars. Company B will ship any amountof bars for a flat rate of $90 and charge 50 cents per bar.

How many candy bars from each company would you need to sell to raise $4,000 for a senior trip? Write aproposal to the principal and the company, explaining your choice. Back up your choice with mathemati-cal support.


• Explain table comparing both companies (not necessary for solution).• Complete graph and table (not necessary for solution).• Write proposal to the company and the principal.• Evaluate process for the profit.• Mathematically compare the two companies.


• 3 class periods

Scoring Guide: Candy Sales — “4”

• Shows complete understanding of the task.• Includes information for the solution process.• Creates proposals that demonstrate understanding of profit.• Demonstrates a solution process that is easy to follow.

This task is very rich and can be solved in a variety of ways. It is not necessary, though, for students to usegraphs for the solution.

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Sample Task Grade Level: 7–8

LITTLE-BY-LITTLE DOES THE TRICK

Mathematical Concepts Mathematical Skills

• Linear function as a constant • Collecting/recording data on table, linear graphrate of change • Interpreting/predicting from table or graph

• Identify and apply dependent and independent variables

• Identify the slope and y-intercept and express them in an equation

Problem Statement or Prompt

• Why does Little-by-Little do the trick as in the fable, “The Crow and the Pitcher”?


• Follow lab procedures.• Record data on a table and linear graph.• Define dependent and independent variables.• Interpret the meaning of the y-intercept.• Explain the slope as a rate of change in an equation.


• 2–3 days in a block, 4 in a nonblock

Scoring Guide: Little-By-Little Does the Trick — “4”

• Correctly identifies dependent variable, independent variable and y-intercept.• Correctly states the equation.• Identifies the slope as a rate of change.• Constructs a table and linear graph.

patterns & functions workshop 4: in … & functions workshop 4: in practice ... if you...

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