1OH #

Quarter Three Concept LessonProfessional Development“Algebra and Functions”

Los Angeles Unified School DistrictElementary Mathematics

2OH #

Outcomes for the Day

Understand how to help students developconcepts of functional relationships

Engage in the lesson as learners to betterunderstand the implementation of theQuarter 3 Concept Lesson

Understand how to use talk moves andthoughtful questioning to promote useful andproductive classroom discussion

3OH #

Content Outcomes

Learn how to help students understand that multiplerepresentations can be used to represent functionalrelationships

Understand how to determine the rigor of a task Understand how rigorous tasks can make

connections between several Big Ideas, Concepts,and Skills

4OH #

Warm-Up Activity:“Guess My Rule”

Find either an ordered pair or equation onyour tableExample of an Ordered Pair: (2, 3)

Example of an Equation: y = 2x + 3

If you are an ordered pair line up on one sideof the room. If you are an equation, line upon the other.

5OH #

Warm-Up Activity:“Guess My Rule”

If you have an equation, find a matchingordered pair. If you have an ordered pair,find a matching equation.

Introduce yourself. Share how you knowthat you match.

Example:2x + 12 = y would match (2, 16)

6OH #

Debriefing the Activity

What were some strategies that youused to find your match?

What do you have to know and be ableto do in order to participate in anactivity such as this?

How would an activity like this addressthe needs of our diverse learners?

7OH #

Reading: “Function Concepts andRepresentations” Part 1

Read Exploring Functions from Van deWalle, pages 436-440(Stop at “Developing Function Concepts in the Classroom”)

Choose one of the three contexts fromyour card

Develop a skit, visual, or demonstrationthat communicates your context throughthe lens of your assigned representation

Be prepared to share with the wholegroup

8OH #

Guiding Questions

Consider the following questions as youdevelop your presentations: How does this representation illustrate

functional relationships? What is important to remember when

representing functions with this model? What connections can you make between

this model and other models?

9OH #

Reading: “Function Concepts andRepresentations” Part 2a

Read “Developing Function Conceptsin the Classroom” from van de Walle,pages 440-441

Discuss at your tables responses tothe questions on handout # 2

10OH #

Reading: “Function Concepts andRepresentations” Part 2b

Under which circumstances would one ofthese representations be more appropriatethan another?

What questions will you ask to addressstudents who do not see a particularrepresentation, considering the diverselearners in our classrooms (ELs, SELs,GATE, and students with special needs)?

What questions will you ask to makeconnections between each representation?

11OH #

Activity:“Piles of Tiles”

Pile 1 Pile 2 Pile 3 Pile 4?

?

Look at the piles of tiles below. Draw or use your tiles toshow how you would build the next pile.

Adapted from Lessons forAlgebraic Thinking Grades 3-5,pages 197-221 by Wickett,Kharas, and Burns

12OH #

Identifying Rigor

Consider the “Piles of Tiles” task.

Circle bulleted statements from the“Categories of Mathematical Tasks”handout that describe what this taskdoes.

Doing Mathematics Tasks•Requires complex and non-algorithmic thinking (i.e., there isnot a predictable, well-rehearsedapproach or pathway explicitlysuggested by the task, taskinstructions, or a worked-outexample).•Requires students to explore andto understand the nature ofmathematical concepts, processes,or relationships.•Demands self-monitoring or self-regulation of one’s own cognitiveprocesses.•Requires students to accessrelevant knowledge andexperiences and make appropriateuse of them in working through thetask.•Requires students to analyze thetask and actively examine taskconstraints that may limit possiblesolution strategies and solutions.•Requires considerable cognitiveeffort and may involve some levelof anxiety for the student due tothe unpredictable nature of thesolution process required.

Procedures WithConnections Tasks•Focus students’ attention on theuse of procedures for the purposeof developing deeper levels ofunderstanding of mathematicalconcepts and ideas.•Suggest pathways to follow(explicitly or implicitly) that arebroad general procedures thathave close connections tounderlying conceptual ideas asopposed to narrow algorithms thatare opaque with respect tounderlying concepts.•Usually are represented in multipleways (e.g., visual diagrams,manipulatives, symbols, problemsituations). Making connectionsamong multiple representationshelps to develop meaning.•Require some degree of cognitiveeffort. Although generalprocedures may be followed, theycannot be followed mindlessly.Students need to engage with theconceptual ideas that underlie theprocedures in order to successfullycomplete the task and developunderstanding.

Procedures WithoutConnections Tasks•Are algorithmic. Use ofthe procedure is eitherspecifically called for orits use is evident basedon prior instruction,experience, or placementof the task.•Require limited cognitivedemand for successfulcompletion. There islittle ambiguity aboutwhat needs to be doneand how to do it.•Have no connection tothe concepts or meaningthat underlie theprocedure being used.•Are focused onproducing correctanswers rather thandeveloping mathematicalunderstanding.•Require no explanations,or explanations thatfocus solely ondescribing the procedurethat was used.

Memorization Tasks•Involves eitherproducing previouslylearned facts, rules,formulae, or definitionsOR committing facts,rules, formulae, ordefinitions to memory.•Cannot be solved usingprocedures because aprocedure does notexist or because thetime frame in which thetask is being completedis too short to use aprocedure.•Are not ambiguous –such tasks involve exactreproduction ofpreviously seen materialand what is to bereproduced is clearlyand directly stated.•Have no connection tothe concepts ormeaning that underliethe facts, rules,formulae, or definitionsbeing learned orreproduced.

Higher Level Cognitive DemandsLower-Level Cognitive Demands

Identifying Rigor: Task Analysis Guide

© University of Pittsburgh OH # 13

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Rigor in Tasks

Based on what we’ve circled here,how might you characterize therigor of this task?

How might tasks such as theseaddress the needs of diverselearners (Els, SELs, GATE, andstudents with special needs)?

15OH #

Rigor in Tasks

When might you use tasks such asthese in your classroom?

What might make it problematic for youto use tasks such as these? What mightbe some possible solutions?

Thinking Through a Lesson Protocol:Classroom Discourse and Asking Questions4th Grade

Los Angeles Unified School District

Elementary Mathematics

OH #1

Outcomes

Understand how to use the TTLP as a planningtool for maintaining rigor and teachingconceptually

Understand how to use talk moves andthoughtful questioning to promote useful andproductive classroom discussion for diverselearners: ELs, SELs, GATE students, andstudents with disabilities

Develop open-ended questions to promotecritical thinking and mathematical reasoning

OH #2

Thinking Through a Lesson Protocol

Highlight those elements of the TTLP that thefacilitator considered when planning thelesson.

How did the facilitator’s consideration and useof these elements maintain the rigor fordiverse learners?

OH #3

Connecting to the Big Idea, Concepts and Skills for Quarter 3

Algebraic ReasoningProblem situations can be represented as algebraic

expressions and equations, as variables, and ascharts and graphs.

Algebraicexpressions are usedto represent problem

situations.

Functions can beexpressed with

words, symbols,tables, and graphs.

•Use variables.

•Use and interpret formulas.

•Understand the functional relationshipwithin equations such as y = 3x.

•Use coordinate grids.

•Graph ordered pairs and lines.

•Find the distance between two points ona coordinate grid.

OH #4

Research Quotes

Read and discuss the 2 quotes in your group.

Think about how our concept lessons addressthe ideas of “dynamic places” and “highquality talk.”

Choose one person to record and share keypoints.

OH #5

Quote #1

“Today’s mathematics classroomsshould be dynamic places where children

are involved and engaged in theirown learning. This can be achieved through

activities that promote higher levelthinking, cooperative problem solving,

and communication.”

~Sullivan and Lilburn, 2002

OH #6

Quote #2

“Our goal is not to increase the amountof talk in our classrooms, but to

increase the amount of high quality talk in ourclassrooms – the mathematically

productive talk.”

~Chapin, O’Connor and Anderson, 2003

OH #7

Final Thoughts

“When students become familiar withour inventory of phrases and expressions, they

usually know what we expect of them. Although werarely stop to think about our most common

conversational prompts, they are among the mostimportant instructional tools. It matters what you say

and how you say it. The tools include strategies –what we call talk moves.”

~Chapin, O’Connor and Anderson, 2003

OH #8

Classroom Discussions

ConceptualUnderstanding

ProblemSolving

ProceduralSkills

OH #9

BalancedCurriculum

The 5 Talk Moves of Classroom Discussions

Choose one talk move from the envelope at yourtable.

Form new groups by talk move.

Take 15 minutes of independent reading time.

Discuss reading as a group.

Create a commercial, jingle, visual, or dramatizationto advertise your talk move.

OH #10

Making Connections

How will using this talkmove meet the needs of your diverse

students (SEL, EL, Special Ed,GATE)?

OH #11

A Question to Consider

In thinking about your practice, whatpercentage of the things that you say

are questions?

OH #12

Asking Questions

What percentage of the things thatteachers say are questions?

A. 35%

B. 10%

C. 60%

OH #13

Sullivan and Lilburn, 2002

Asking Questions

What is the average of 5, 6, 9, 3, and 7?

The average of five numbers is 6. Whatmight the numbers be?

OH #14

Open-Ended Questions

“Requires students to think more deeply

and to give a response that involves more than

recalling a fact or reproducing a skill.”

~Sullivan and Lilburn, 2002

Review Asking Questions from the renewedMIG in the Appendix, page 20.

Compare open-ended and closed questions

OH #15

How to Create Good Questions

Working Backward

Step 1: Identify a topic.

Step 2: Think of a closedquestion and write downthe answer.

Step 3: Make up aquestion that includes (oraddresses) the answer.

Adapting a StandardQuestion

Step 1: Identify a topic.

Step 2: Think of astandard question.

Step 3: Adapt it to makeit a good (open) question.

OH #16

Thinking Through the Lesson Protocol:Content into Practice

How can the TTLP become an integral part ofyour practice?

Which talk move will you choose to focus onusing in Quarter 3?

OH #17

Thinking Through a Lesson Protocol:Classroom Discourse and Asking Questions5th Grade

Los Angeles Unified School District

Elementary Mathematics

OH #1

Outcomes

Understand how to use the TTLP as a planningtool for maintaining rigor and teachingconceptually

Understand how to use talk moves andthoughtful questioning to promote useful andproductive classroom discussion for diverselearners

Develop open-ended questions to promotecritical thinking and mathematical reasoning

OH #2

Thinking Through a Lesson Protocol

Highlight those elements of the TTLP that thefacilitator considered when planning thelesson.

How did the facilitator’s consideration and useof these elements maintain the rigor fordiverse learners?

OH #3

Connecting to the Big Idea, Concepts and Skills for Quarter 3

Algebraic ReasoningEquations, expressions, and variables

are mathematical models used torepresent real situations.

Linear relationships are presented in multipleways.

•Write and evaluate simple algebraicexpressions using one variable.

•Use the distributive property in equationsand expressions with variables.

•Identify and graph ordered pairs in the fourquadrants.

•Graph ordered pairs of integers on a gridbased on a linear equations

OH #4

Research Quotes

Read and discuss the 2 quotes in your group.

Think about how our concept lessons addressthe ideas of “dynamic places” and “highquality talk.”

Choose one person to record and share keypoints.

OH #5

Quote #1

“Today’s mathematics classroomsshould be dynamic places where children

are involved and engaged in theirown learning. This can be achieved through

activities that promote higher levelthinking, cooperative problem solving,

and communication.”

~Sullivan and Lilburn, 2002

OH #6

Quote #2

“Our goal is not to increase the amountof talk in our classrooms, but to

increase the amount of high quality talk in ourclassrooms – the mathematically

productive talk.”

~Chapin, O’Connor and Anderson, 2003

OH #7

Final Thoughts

“When students become familiar withour inventory of phrases and expressions, they

usually know what we expect of them. Although werarely stop to think about our most common

conversational prompts, they are among the mostimportant instructional tools. It matters what you say

and how you say it. The tools include strategies –what we call talk moves.”

~Chapin, O’Connor and Anderson, 2003

OH #8

Classroom Discussions

ConceptualUnderstanding

ProblemSolving

ProceduralSkills

OH #9

BalancedCurriculum

The 5 Talk Moves of Classroom Discussions

Choose one talk move from the envelope at yourtable.

Form new groups by talk move.

Take 15 minutes of independent reading time.

Discuss the reading as a group.

Create a commercial, jingle, visual, or dramatizationto advertise your talk move.

OH #10

Making Connections

How will using this talkmove meet the needs of your diverse

students (SEL, EL, Special Ed,GATE)?

OH #11

A Question to Consider

In thinking about your practice, whatpercentage of the things that you say

are questions?

OH #12

Asking Questions

What percentage of the things thatteachers say are questions?

A. 35%

B. 10%

C. 60%

OH #13

Sullivan and Lilburn, 2002

Asking Questions

What is the average of 5, 6, 9, 3, and 7?

The average of five numbers is 6. Whatmight the numbers be?

OH #14

Open-Ended Questions

“Requires students to think more deeply

and to give a response that involves more than

recalling a fact or reproducing a skill.”

~Sullivan and Lilburn, 2002

Review Asking Questions from the renewedMIG in the Appendix, page 20.

Compare open-ended and closed questions

OH #15

How to Create Good Questions

Working Backward

Step 1: Identify a topic.

Step 2: Think of a closedquestion and write downthe answer.

Step 3: Make up aquestion that includes (oraddresses) the answer.

Adapting a StandardQuestion

Step 1: Identify a topic.

Step 2: Think of astandard question.

Step 3: Adapt it to makeit a good (open) question.

OH #16

Thinking Through the Lesson Protocol:Content into Practice

How can the TTLP become an integral part ofyour practice?

Which talk move will you choose to focus onusing in Quarter 3?

OH #17