Making the Transition: Instructional Shifts and CMP3...

Connected Mathematics Project 3 Making the Transition:

Instructional Shifts and CMP3 Implementation Essentials

Professional DevelopmentParticiPant Workbook

Connected Mathematics Project 3: Making the Transition: Instructional Shifts and CMP3 Implementation Essentials

© 2013 Pearson, Inc.2

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Pearson School achievement Servicesconnected Mathematics Project 3 Making the transition: instructional Shifts and cMP3 implementation EssentialsParticipant Workbook

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Table of Contents

agenda . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

outcomes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

Section 1: Program Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

Section 2: teaching with cMP3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

Section 3: assessment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

Section 4: Planning a cMP3 Unit to achieve the Focus, coherence, and rigor of the common core . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

Reflection and Closing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

references . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52



Agenda

Section

introduction

1. Program Structure

2. teaching with cMP3

3. assessment

4. Planning a cMP3 Unit to achieve the Focus, coherence, and rigor of the common core

Reflection and Closing



Outcomes

at the conclusion of this workshop, you will be able to

• implement CMP3 in alignment with the focus, coherence, and rigor of the Common Core;

• apply the CMP3 inquiry-based lesson structure to support learning through problem solving;

• utilize the formal and informal assessment components of CMP3 in an inquiry-based classroom; and

• demonstrate a deeper understanding of the mathematical content and practices of Unit 1 at your respective grade levels.



Introduction

The Factor Game

1.3 1.41.21.1

Playing the Factor Game is a fun way to practice finding factors of whole numbers. You may learn some interesting things about numbers that you didn’t know before.

1 2 3 4 5

6 7 8 9 10

11 12 13 14 15

16 17 18 19 20

21 22 23 24 25

26 27 28 29 30

1. Player A chooses a number on the game board and circles it.

2. Using a di�erent color, Player B circles all the proper factors of Player A’s number.

3. Player B circles a new number, and Player A circles all of the factors of the new number that are not already circled.

4. �e players take turns choosing numbers and circling factors.

5. If a player chooses a number with no uncircled factors, that player loses their current turn and scores no points.

6. �e game ends when there are no numbers left with uncircled factors.

7. Each player adds the numbers circled with his or her color. �e player with the greater total wins.

The Factor GameDirections

If you are Player A, what number would you choose as the first move? Why?

Playing the Factor Game: Finding Proper Factors1.1

Investigation 1 Building on Factors and Multiples 9

CMP14_SE06_U01_INV01.indd 9 03/01/13 11:27 PM



Introduction

(Pearson Education, inc. 2014c, 9–10)

1.3 1.41.21.1

Play the Factor Game several times with a partner.

A 1. How can you determine whether one number is a factor of another number?

2. If you know a factor of a number, can you find another factor? Explain.

3. Make a list of the factors of 18. Then make a list of the divisors of 18. Are the factors of a number also divisors of the number? Explain your reasoning.

B Give an example of a number that has many factors. Then give an example of a number that has few factors.

C How do you know when you have found all of the factors of a number?

Homework starts on page 17.

1.1Problem

Did you notice in the Factor Game that some numbers are better first moves than others? If you choose 22, you get 22 points and your opponent only gets 1 + 2 + 11 = 14 points. However, if you choose 18, you get 18 points and your opponent gets 1 + 2 + 3 + 6 + 9 = 21 points.

Playing to Win: Prime and Composite Numbers1.2

Prime Time10




Section 1: Program Structure

Driving Forces behind CMP3the following excerpt comes from Pearson Education, inc. (2014d, 1):

1. Engage students in active, personalized learning: CMP3 takes inquiry-based learning to the next level. new digital tools engage students while driving conceptual understanding, procedural skill, and real-world applications.

2. teach the common core, teach with greater ease: cMP3 aligns to the common core State Standards for Mathematics and prepares students for college and careers. Technology applications help you manage your classroom with fidelity, maximize instructional time, and capture needed student data.

3. Apply a research-proven instructional approach: CMP3 offers the most comprehensive research base of any middle school mathematics program.

Inquiry-Based LearningWhat is inquiry-based learning?

Types of Problem SolvingThe following excerpt appears in Van de Walle, Karp, and Bay-Williams’ Elementary and Middle School Mathematics: Teaching Developmentally (2013, 32):

in a classic publication on the types of teaching related to problem solving, Schroeder and Lester (1989) identified three types of approaches to problem solving:

1. Teaching for problem solving. This approach can be summarized as teaching a skill so that a student can later problem solve. teaching for problem solving often starts with learning the abstract concept and then moving to solving problems as a way to apply the learned skills. For example, students learn the algorithm for adding fractions and, once that is mastered, solve story problems that involve adding fractions. (this approach is used in many textbooks and is likely familiar to you.)

2. Teaching about problem solving. this second approach involves teaching students how to problem solve, which can include teaching the process (understand, design a strategy, implement, look back) or strategies for solving a problem. an example of a strategy is




“draw a picture,” in which students use a picture or diagram to help solve a problem. See “teaching about Problem Solving” in this chapter.

3. Teaching through problem solving. this approach generally means that students learn mathematics through real contexts, problems, situations, and models. the contexts and models allow students to build meaning for the concepts so that they can move to abstract concepts. teaching through problem solving might be described as upside down from teaching for problem solving—with the problem(s) presented at the beginning of a lesson and skills emerging from working with the problem(s). For example, in exploring the situation of combining 1/2 and 1/3 feet of ribbon to fi gure out how long the ribbon is, students would be led to discover the procedure for adding fractions.

Model CMP3 Lesson

1.3 1.41.21.1

Play the Factor Game several times with a partner.

A 1. How can you determine whether one number is a factor of another number?

2. If you know a factor of a number, can you find another factor? Explain.

3. Make a list of the factors of 18. Then make a list of the divisors of 18. Are the factors of a number also divisors of the number? Explain your reasoning.

B Give an example of a number that has many factors. Then give an example of a number that has few factors.

C How do you know when you have found all of the factors of a number?


1.1Problem

Did you notice in the Factor Game that some numbers are better first moves than others? If you choose 22, you get 22 points and your opponent only gets 1 + 2 + 11 = 14 points. However, if you choose 18, you get 18 points and your opponent gets 1 + 2 + 3 + 6 + 9 = 21 points.

Playing to Win: Prime and Composite Numbers1.2

Prime Time10





(Pearson Education, inc. 2014c, 10–11)

1.3 1.41.21.1

A 1. Make a table of all possible first moves (numbers from 1 to 30) in the Factor Game. For each move, list the proper factors of the number. Then record the scores for you and your opponent.

First Move

1

2

3

4

Lose a Turn

2

3

4

0

1

1

3

None

1

1

1, 2

Proper Factors My Score Opponent's Score

2. Describe an interesting pattern you see in your table.

B 1. What is the best first move? Why?

2. Which first move makes you lose your turn? Why?

C 1. List all first moves that allow your opponent to score only one point. These numbers are called prime numbers. Are all prime numbers good first moves? Explain.

2. List all first moves that allow your opponent to score more than one point. These numbers are called composite numbers. Are composite numbers good first moves? Explain.

D Find a number that has exactly

1. two factors. 2. three factors. 3. four factors. 4. six factors.

5. What is special about these numbers? Explain.

E Camila noticed that for some numbers her opponent’s score was equal to the number. For example, when her first move was 6, her opponent got 6 points. Are there any other numbers for which this is true?

F 1. For a first move of 24, Anna listed the proper factors as 1, 2, 4, and 6. Was she correct? Explain.

2. How can you be sure that you have listed all of the proper factors of a number?

1.2Problem


Investigation 1 Building on Factors and Multiples 11

cMP14_SE06_U01_inV01.indd 11 04/01/13 2:19 aM



Section 1: Program Structure6/10/13 Dash Web

media.pearsoncmg.com/curriculum/dash/hurix/CMP3_TP_G6_0530/course_base/resources/content_module_131/assets/A7A35307CAF5E8A43AD1E35AFB034… 1/1

(Pearson Education, inc. 2014c, 1)




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Scrambled Sentencesthe role of a student during the Launch phase:

the role of a student during the Explore phase:

The role of a student during the Summarize phase:

the role of the teacher during the Launch phase:

the role of the teacher during the Explore phase:

The role of the teacher during the Summarize phase:




CMP3: A Research-Proven Instructional ApproachSorting Aspects of CMP3 Supported by Research Findings

Summary of Research Findings Aspects of CMP3 Supported by These Research Findings

a “create a classroom learning context in which students can construct meaning. . .an important factor in teaching for meaning is connecting the new ideas and skills to students’ past knowledge and experience” (Grouws and cebulla 2000, 13–14).

b “attend to student meaning and student understanding in instruction. Students’ conceptions of the same idea will vary, as will their methods of solving problems and carrying out procedures. teachers should build on students’ intuitive notions and methods in designing and implementing instruction” (Grouws and cebulla 2000, 14).

c “Students can learn both concepts and skills by solving problems. . .Development of more sophisticated mathematical skills can also be approached by treating their development as a problem for students to solve. teachers can use students’ informal and intuitive knowledge in other areas to develop other useful procedures. instruction can begin with an example for which students intuitively know the answer. From there, students are allowed to explore and develop their own algorithm” (Grouws and cebulla 2000, 15–16).

D “Giving students both an opportunity to discover and invent new knowledge and an opportunity to practice what they have learned improves student achievement. . .The findings from a number of research studies show that when students discover mathematical ideas and invent mathematical procedures, they have a stronger conceptual understanding of connections between mathematical ideas” (Grouws and cebulla 2000, 17).

E “Findings from american studies clearly demonstrate two important principles that are associated with the development of students’ deep conceptual understanding of mathematics. First, student achievement and understanding are significantly improved when teachers are aware of how students construct knowledge, are familiar with the intuitive solution methods that students use when they solve problems, and utilize this knowledge when planning and conducting instruction in mathematics. . .Second, structuring instruction around carefully chosen problems, allowing students to interact when solving these problems, and then providing opportunities for them to share their solution methods result in increased achievement on problem-solving measures. . .research has also demonstrated that when students have opportunities to develop their own solution methods, they are better able to apply mathematical knowledge in new problem situations” (Grouws and cebulla 2000, 19).




Summary of Research Findings Aspects of CMP3 Supported by These Research Findings

F “Using small groups of students to work on activities, problems and assignments can increase student mathematics achievement. . .considerable research evidence within mathematics education indicates that using small groups of various types for different classroom tasks has positive effects on student learning” (Grouws and cebulla 2000, 21).

G “Whole-class discussion following individual and group work improves student achievement. . .Research suggests that whole-class discussion can be effective when it is used for sharing and explaining the variety of solutions by which individual students have solved problems. it allows students to see the many ways of examining a situation and the variety of appropriate and acceptable solutions” (Grouws and cebulla 2000, 23).

Aspects of CMP3 Supported by Research Findings 1. During all three phases of a cMP3 lesson, teachers are able to informally assess student

thinking by observing student responses and solution methods.

2. cMP3 lessons employ a teaching through problem solving philosophy that pushes students to develop conceptual fluency prior to procedural fluency.

3. cMP3 student texts provide examples of student work to provide students with alternate strategies that provide multiple representations of a problem’s solution method.

4. In CMP3 inquiry-based lessons, students are actively engaged in doing mathematics and constructing knowledge.

5. in cMP3 lessons, students explore problems with multiple entry points and use their own prior knowledge and skill set to be “doers” of math.

6. Each lesson has an overarching focus question that provides students with a question to consider while working through the problem knowing that they should be able to respond to the question during the Summarize phase.

7. in the Launch phase, the teacher helps students make connections to prior knowledge and find the “edge” that peaks their curiosity, without over anticipating potential student struggles and “giving away” too much or alleviating the challenge.

8. Students work in cooperative learning groups (pairs and/or small groups) to explore problems during the Explore phase of the lesson.

9. in the Explore phase, students work in cooperative learning groups to delve into a rich contextual problem to analyze and, eventually, generalize a concept or skill.

10. In the Explore phase, the teacher uses assessing and advancing questions to support struggling students with care toward following the students’ thinking rather than the teacher’s pathway of reasoning.




11. in the Explore phase, students struggle with problems and over time develop and learn to rely on their own problem-solving skill set.

12. In the Summarize phase, the teacher uses talk moves and higher-order questioning techniques to guide students to arrive at their own generalizations of a concept or a skill.

13. In the Summarize phase, the teacher sequences the presentation of the solution methods of multiple student groups so that the class can compare and contrast the multiple pathways (representations) for solving the same problem.

14. in common core Mathematical Practices at the end of each investigation, students write reflections on how they used the mathematical practices in their own problem solving.

15. In Mathematical Reflections at the end of each Investigation, Students write summaries to explain and describe their learning.

16. Unit Projects provide opportunities for students to demonstrate a broader understanding of the connections between ideas of the unit and across learning progressions.

17. Application-Connections-Extensions (ACE) homework problems provide a chance for students to show procedural knowledge, the ability to apply concepts, and conceptual understanding.

(Pearson Education, inc. 2014b)




Revisit the Section 1 Big Questions • What is CMP3’s instructional philosophy?

• What is CMP3’s lesson structure?



Section 2: Teaching with CMP3

Standards for Mathematical PracticeAround the Room and Back Again: Mathematical Practices in My Classroom

Standards for Mathematical Practice

What should I see in my classroom when students meet this standard?

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.

(nGa center and ccSSo 2010, 6–8)




Using the Digital Resources to Explore Mathematical Practices • Grade 6: Problem 4.1: reasoning With Even and odd numbers (Pearson Education, inc.

2014c, 64–66)

• Grade 7: Problem 1.3: From Sauna to Snowbank: Using a number Line (Pearson Education, inc. 2014a, 14–16) and Problem 1.4: in the chips: Using a chip Model (Pearson Education, inc. 2014a, 17–19)

• Grade 8: Problem 4.1: Vitruvian Man: relating body Measurements (Pearson Education, inc. 2014e, 82–84)

What mathematical practices will students use?

Mathematical Practice Used Where or how will students use it?




Using Teacher Place to Explore Mathematical Practices Used in an InvestigationLog on to teacher Place to locate the Standards for Mathematical Practice that are listed for your grade-level problem(s).

How does this compare to the What mathematical practices will students use? table that you completed in the previous activity?

Common Core Mathematical Practices Reflection in Student PlaceStudent Place’s ACTIVe-book functionalities include the following:

• Everything students write in Student Place’s ACTIVe-book is visible to the teacher.

• The teacher, student, and the student’s family can communicate through the ACTIVe-book.

• A teacher can grade an assignment with comments to the student or parent, and then the student or parent can write back with thoughts, questions, or another attempt at a solution to a problem.




(nYSED, n.d.)

Crosswalk of Common Core Instructional Shifts: Mathematics

6 Shifts: EngagenYwww.engageny.org

3 Shifts: Student achievement Partners www.achievethecore.org

1: Focus: Teachers use the power of the eraser and signifi cantly narrow and deepen the scope of how time and energy is spent in the math classroom. they do so in order to focus deeply on only the concepts that are prioritized in the standards so that students reach strong foundational knowledge and deep conceptual understanding and are able to transfer mathematical skills and understanding across concepts and grades.

2: Coherence: Principals and teachers carefully connect the learning within and across grades so that, for example, fractions or multiplication spiral across grade levels and students can build new understanding onto foundations built in previous years. teachers can begin to count on deep conceptual understanding of core content and build on it. Each standard is not a new event, but an extension of previous learning.

3: Fluency: Students are expected to have speed and accuracy with simple calculations; teachers structure class time and/or homework time for students to memorize, through repetition, core functions (found in the attached list of fl uencies) such as multiplication tables so that they are more able to understand and manipulate more complex concepts.

4: Deep Understanding: teachers teach more than “how to get the answer” and instead support students’ ability to access concepts from a number of perspectives so that students are able to see math as more than a set of mnemonics of discrete procedures. Students demonstrate deep conceptual understanding of core math concepts by applying them to new situations as well as writing and speaking about their understanding.

5: Application: Students are expected to use math and choose the appropriate concept for application even when they are not prompted to do so. teachers provide opportunities at all grade levels for students to apply math concepts in “real world” situations. teachers in content areas outside of math, particularly science, ensure that students are using math — at all grade levels — to make meaning of and access content.

6: Dual Intensity: Students are practicing and understanding. there is more than a balance between these two things in the classroom — both are occurring with intensty. teachers create opportunities for students to participate in “drills” and make use of those skills through extended application of math concepts. the amount of time and energy spent practicing and understanding learning environments is driven by the specifi c mathematical concept and therefore, varies throughout the given school year.

1: Focus strongly where the Standards focus

2: Coherence: Think across grades, and link to major topics within grades

3: Rigor: Require fl uency, application and deep understanding

=

=

=




Rigor in a CMP3 Lesson

Component of Rigor Example from Grade-Level CMP3 Lesson

Fluency

Deep Understanding

Application

Dual Intensity

(nYSED, n.d.)




Classroom Discourse“When a teacher succeeds in setting up a classroom in which students feel obligated to listen to one another, to make their own contributions clear and comprehensible, and to provide evidence for their claims, that teacher has set in place a powerful context for student learning.”

—Chapin, O’Connor, and Anderson (quoted in Van de Walle, Karp, and Bay-Williams 2013, 43)

Building a Mathematical CommunityThe following is an excerpt from Van de Walle, Karp, and Bay-Williams (2013, 44).




Best Practices for Promoting Mathematical Communities 1. Equity Sticks: Write the names of students on popsicle sticks, and place them in a cup or

bag. When calling on students to present or answer a question, randomly select a popsicle stick.

2. Excel® Random Generator: Enter the names of students into an Excel spreadsheet, and assign a student to use the random generator to select a student to present or answer a question. If possible, project the Excel random generator so that students see the fairness.

3. Student Fishbowl: Select students to act out group work, mathematical discourse, teamwork on a partner quiz, and so on while the other students observe and take notes on a graphic organizer. Conclude the Fishbowl activity with a whole-group discussion of how students in the fishbowl interacted and functioned.

4. Assigning Roles in Group Work: By formally assigning roles, students have a specific set of responsibilities. Some commonly used roles are timekeeper, Gatekeeper/taskmaster, Facilitator, Skeptic, Recorder, Summarizer, and Presenter.

5. Random Group Grading: randomly choose one member of a group to grade and assign that grade to the entire group. Assign zeros for copying.

6. Graded Individual Exit Slips: At the conclusion of the Summarize phase, have students independently complete graded exit slips to assess the day’s learning.

7. Three Before Me: Have a standing rule that a student must pose his or her question to three other students before he or she can ask the teacher.

8. Questioning to Promote Teamwork: Redirect all inquiries back to other group members in a form of a question. (For example, say, “What can you say about Tony’s question?”, or “Look at what Sara did. How can this help answer Tony’s question?”)

9. Display Student Work: Display students’ work, and refer to it to help students make connections between multiple representations and mathematical concepts.

10. Word Wall: Display a word wall with the definition, multiple representations, and examples of student work on the concept in context.

11. Sentence Starters Wall: Display a wall with sentence starters for seeking clarification, asking for help, acknowledging someone else’s idea, affirming someone else’s thoughts or ideas, reporting a partner’s idea, presenting a group’s idea, paraphrasing what someone else says, predicting, making a suggestion, giving an opinion, and respectfully disagreeing.

12. Think–Pair–Shares: Students have the opportunity to sort out their own ideas and discuss them with a peer before the whole class engages in a discussion on a mathematical question. This strategy reduces the anxiety around contributing to classroom discourse and promotes student-student interaction.

13. KWL Charts: Students activate prior knowledge and construct questions, and teachers formatively assess students’ thinking and prerequisite concepts.

14. Peer Assessment: Students manage the learning of their classroom, and the process promotes student-student mathematical discourse.




15. Student Behavior Self-Assessment: Use a rubric for students to grade how well their behavior was in accordance to the classroom norms.

16. Multiple Representations: as opposed to traditional direct instruction, encourage students to use their own math skills to solve problems and represent their solutions with any representation they can.

17. Talk Moves: When orchestrating classroom discussion, use revoicing, rephrasing, reasoning, elaboration, and wait time (Van de Walle, Karp, and Bay-Williams 2013, 43).

18. Star Student of the Week: Recognize students when they help other students and/or contribute to building a community of mathematicians in the classroom. Post students’ pictures with examples of why they have been awarded the honor.

19. Got It!–Almost–Not Yet: Have these three sections somewhere on a wall near the door where students exit, and have a small name tag (magnetic or pin-up) of each student’s name. ask students to place their name tags under the Got it!, almost, or not Yet to describe their level of understanding of the day’s learning or a specified question or concept. Use this as a formative assessment for planning.

20. Display a Daily Participation Grade: Develop a participation grading rubric that aligns with class norms. Make it simple and transparent, post participation grades daily for students to see, and make the weight enough for students to see that good participation grades will raise their averages.

21. What time is it?: Make a sign with Launch, Explore, Summarize, and any other phase(s) of a normal day (warm-up, exit slip, independent assessment, and so on). Label each phase with 0, 1, or 2, where 0 means no one talks, 1 means one person at a time talks, and 2 means everyone can use accountable talk. Use some way to indicate the phase of the lesson throughout each class.

22. Student Work Analysis: include presentation, discussion, and analysis of incorrect student work. Use this as both an opportunity for students to understand that incorrect solutions are just a way to rule out what does not work, as well as to make connections to the difference with correct solutions.

23. Equity Monitors: Have students designated as Equity Coaches monitor other students’ participation. these students can motivate, assist, or even assess participation depending on the makeup of the class.




Promoting a Mathematical Community

How to Build Mathematical Communities

How will you use a best practice to promote mathematical communities?

“Encourage student-student dialogue. . .”

“Encourage students to ask. . .”

“Ask follow-up questions. . .”

“call on students in. . .”

“Demonstrate to students. . .”

“Move students to more. . .”

“be sure all students. . .”

(Van de Walle, Karp, and Bay-Williams 2013, 44)




Discourse and Questioning: Delivering CMP3 with FidelityQuiz: To tell or not to tell? Answer True or False to each of the following questions:

_____ Symbolism and terminology should be introduced after concepts have been developed and as a means of expressing and labeling ideas.

_____ teachers can introduce important strategies if students do not use them, and must introduce the strategy as “just another strategy” rather than “the strategy”.

_____ Teachers clarify or formalize students’ methods by drawing attention to the connections between students’ methods.

_____ teachers can tell students any information that does not solve the problem and does not take away the need for students to reflect on the situation and develop solution methods they understand.

(Van de Walle, Karp, and Bay-Williams 2013, 44–45)

Orchestrating a Classroom Discussion

“classroom discourse refers to the interactions that occur throughout the lesson. Learning how to orchestrate an effective classroom discussion is quite complex. . .Note that the purpose is not for students to tell their answers and get validation from the teacher.”

—Van de Walle, Karp, and Bay-Williams (2013, 42)

Questions That Teachers Should Consider When Preparing for a Classroom Discussion • How can I help the students make sense of and appreciate the variety of methods that may

be used?

• How can I orchestrate the discussion so that students summarize their thinking about the problem?

• What questions can guide the discussion?

• What concepts or strategies need to be emphasized?

• What ideas do not need closure at this time?

• What definitions or strategies do we need to generalize?

• What connections and extensions can be made?

• What new questions might arise and how do I handle them?

• What can I do to follow up, practice, or apply the ideas after the summary?(Pearson Education, inc. 2014b)




How Talk Moves Promote the Standards for Mathematical Practice in Your Classroom

(Van de Walle, Karp, and Bay-Williams 2013, 43)

Standards for Mathematical Practice

Talk Moves That Help to Meet this Standard 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.

(nGa center and ccSSo 2010, 6–8)

Orchestrating Classroom Discourse 43

to see the varied approaches in how mathematics can be solved and see mathematics as something that they can do.

Discourse should occur throughout a lesson. When a problem is introduced, students can be asked what strategies they might use and why. By joining a group, you can model questions you want the students to ask each other and themselves. You can also model think-alouds, in which you discuss how you thought about the problem. These are critical for students with learning disabilities to support their thinking about a strategy because it makes explicit the reasoning process. In the upper grades, each group can have a designated monitor, whose job is to be the reflective ques-tioner (first modeled by you). In the discussion that occurs after students have solved the problem(s), students can reflect not just on their own strategy, but other’s strategies. Questions asking students if they would do it differently next time, which strategy made sense to them (and why), what caused problems for them, and how they overcame these stumbling blocks, are critical in developing mathe-matically proficient students. While many good questions are specific to the task being solved, some general questions can help students build understanding:

● What did you do that helped you understand the problem?

● Was there something in this problem that reminded you of another problem we’ve done?

● Did you find any numbers or information you didn’t need? How did you know that the information was not important?

● How did you decide what to do?

● How did you decide whether your answer was right?● Did you try something that didn’t work? How did you

figure out it was not going to work out?● Can something you did in this problem help you solve

other problems?

Notice these questions focus on the process as well as the answer, and what worked as well as what didn’t work. A bal-anced discussion helps students learn how to do mathematics.

In Classroom Discussions, a teacher resource describing how to implement effective discourse in the classroom, Chapin, O’Conner, and Anderson (2009) write, “When a teacher succeeds in setting up a classroom in which students feel obligated to listen to one another, to make their own contributions clear and comprehensible, and to provide evi-dence for their claims, that teacher has set in place a power-ful context for student learning” (p. 9). This is true for every student. There are no exceptions! Struggling learners often struggle because they have been denied the opportunity to explore and connect ideas. These authors share five “talk moves” that help a teacher to get students talking about mathematics (see Table 3.2).

The following exchange illustrates an example of dis-course with a small group of students discussing how to solve 27 – 19 = ____. The teacher is asking two students (Tyler and Aleah) to reconcile that they got different answers.

Tyler: Well, I added one to nineteen to get twenty. So then I did twenty-seven take away twenty and got seven. But I added one, so I needed to take one away from the seven, and I got six.

TAbLE 3.2PRoDuCTIvE TALk MovES FoR SuPPoRTIng CLASSRooM DISCuSSIonS

Talk Moves What It Means and Why Example Teacher Prompts

1.Revoicing Thismoveinvolvesrestatingthestatementasaquestioninordertoclarify,applyappropriatelanguage,andtoinvolvemorestudents.Itisanimportantstrategytoreinforcelanguageandenhancecompre-hensionforELLs.

“Youusedthehundredschartandcountedon?”“So,firstyourecordedyourmeasurementsinatable?”

2.Rephrasing Askingstudentstorestatesomeoneelse’sideasintheirownwordswillensurethatideasarestatedinavarietyofwaysandencouragestudentstolistentoeachother.

“WhocansharewhatRicardojustsaid,butusingyourownwords?”

3.Reasoning Ratherthanrestate,asintalkmove2,thismoveasksthestudentwhattheythinkoftheideaproposedbyanotherstudent.

“DoyouagreeordisagreewithJohanna?Why?”

4.Elaborating Thisisarequestforstudentstochallenge,addon,elaborate,orgiveanexample.Itisintendedtogetmoreparticipationfromstudents,deepenstudentunderstanding,andprovideextensions.

“Canyougiveanexample?”“DoyouseeaconnectionbetweenJulio’sideaandRhonda’sidea?”“Whatif...”

5.Waiting Ironically,one“talkmove”istonottalk.Quiettimeshouldnotfeeluncomfortable,butshouldfeellikethinkingtime.Ifitgetsawkward,askstudentstopair-shareandthentryagain.

“Thisquestionisimportant.Let’stakesometimetothinkaboutit.”

Source: BasedonChapin,S.,O’Conner,C.,&Anderson,N.(2009).Classroom Discussions: Using Math Talk to Help Students Learn(2nded.).Sausalito,CA:MathSolu-tions.Reprintedwithpermission.

102686 03 032-058 r1 mm.indd 43 11/30/11 8:48 AM




Analyzing Talk Moves

“the following exchange illustrates an example of discourse with a small group of students discussing how to solve 27 – 19 = ____. the teacher is asking two students (tyler and aleah) to reconcile that they got different answers.

tyler: Well, I added one to nineteen to get twenty. So then I did twenty-seven take away twenty and got seven. but i added one, so i needed to take one away from the seven, and i got six.

teacher: What do you think of that, Aleah?

aleah: that is not what i got.

teacher: Yes, I know that, but what do you think of Tyler’s explanation?

aleah: Well, it can’t be right, because i just counted up. i added one to nineteen to get twenty and then added seven more to get twenty-seven. So, I counted eight altogether. Six can’t be right.

teacher: Tyler, what do you think of Aleah’s explanation?

tyler: that makes sense, too. i should have counted.

teacher: So, do you think both answers are right?

tyler: no.

aleah: No. If it was twenty-seven minus twenty, the answer would be seven, because you count up seven. So, if it is nineteen, it has to be eight.

tyler: Oh, wait. I see something I did get the seven. . . . See, I got the twenty-seven take away twenty is seven. But then . . . I see . . . it’s twenty-seven take away nineteen. I took away twenty! i took away too many so i have to add one to the seven. i get eight, just like aleah! (kline, 2008, p. 148)

While this conversation is with two children, a similar style can be used in whole-class discussions, pushing students to help students make sense of what is correct and incorrect about their strategies.”


“What talk moves do you notice in this vignette?” (Van de Walle, Karp, and Bay-Williams 2013, 43)




Anticipating Questioning by Anticipating Student Thinking





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Questions to Ask Yourself When Planning a Lesson

Questions to Ask Your Students to Orchestrate Classroom Discussion




Use the following space to work out solutions to your grade-level problem:

• Grade 6: Problem 4.2: Using the Distributive Property (Pearson Education, inc. 2014c, 67–70)

• Grade 7: Problem 2.1: Extending addition to rational numbers (Pearson Education, inc. 2014a, 30–33)

• Grade 8: Problem 4.2: older and Faster: negative correlation (Pearson Education, inc. 2014e, 84–86)




Structuring the Explore and Summarize Phases of a CMP3 LessonMost problems in CMP3 require more than one set of Explore and Summarize phases. In other words, when planning a cMP3 lesson on a problem, teachers must plan how to structure Explore and Summarize phases around parts of a CMP3 problem.

Would you plan multiple Explore and Summarize phases for this problem? If so, after which parts of the problem?

top ten Uses for teachability.com

1. ___________________________________________________________________

2. ___________________________________________________________________

3. ___________________________________________________________________

4. ___________________________________________________________________

5. ___________________________________________________________________

6. ___________________________________________________________________

7. ___________________________________________________________________

8. ___________________________________________________________________

9. ___________________________________________________________________

10. ___________________________________________________________________




Revisit the Section 2 Big Questions • How can you use CMP3’s digital resources to plan and implement CMP3 lessons?

• How do students use the Standards for Mathematical Practice in CMP3’s inquiry-based lessons?

• How can you orchestrate classroom discussions and build a mathematical community of learners in your CMP3 classroom?



Section 3: Assessment

Constant Assessment and Feedback“Extensive recent research has demonstrated convincingly that student learning improves significantly when teachers provide frequent feedback on their progress and when teachers use that assessment as a core input to their planning for instruction.”

—Wiliam quoted in Pearson Education, Inc. (2014b)

Dimensions of Learning Assessed in CMP3 • Content knowledge: Assessing content knowledge involves determining what students

know and what they are able to do.

• Mathematical disposition: A student’s mathematical disposition is healthy when he or she responds well to mathematical challenges and sees himself or herself as a learner and inventor of mathematics. Disposition also includes confidence, expectations, and metacognition (reflecting on and monitoring one’s own learning).

• Work habits: A student’s work habits are good when he or she is willing to persevere, contribute to group tasks, and follow tasks to completion. these valuable skills are used in nearly every career. To assess work habits, it is important to ask questions, such as “Are the students able to organize and summarize their work?” and “Are the students progressing in becoming independent learners?”


Assessment Features of CMP3 • Checkpoints – acE – notebooks and notebook checklist – Mathematical Reflections – Mathematical Practices – Looking back and Looking ahead – MathXL® for School (Web-based formative assessment tool)

• Surveys of Knowledge – Check-ups – Partner Quizzes – Unit tests – Self-Assessment – Projects

• Observations – Group work – Discussions – One-on-one interactions





How will you capture the assessment of your students?How will you collect and track the informal assessments of your CMP3 classroom?

What requirements will you have for students to demonstrate their thinking?




Organizing and Tracking Informal Assessments


Functionalities of Teacher Place That Support Collection and Tracking of Student Assessment • Teachers can take notes at any point in Teacher Place.

• Teachers can manage their classrooms by moving students into leveled groups, assigning them colors to indicate learner levels, and flagging them according to need.

• Teachers can enter notes on students.

• Teachers can use the clipboard function to take general notes on classes.

• Teachers can e-mail parents directly from Teacher Place.

88 Chapter 5 Building Assessment into Instruction

observed. Shorter periods of observation will focus on a particular cluster of concepts or skills or particular stu-dents. Over longer periods, you can note growth in math-ematical processes or practices, such as problem solving, representation, or reasoning. To use observation effec-tively, you should take seriously the following maxim: Do not attempt to record data on every student in a single class period.

Observation methods vary with the purposes for which they are used. Further, formats and methods of gathering observation data are going to be influenced by your indi-vidual teaching style and habits.

Anecdotal NotesOne system for recording observations is to write short notes either during or immediately after a lesson in a brief narrative style. One possibility is to have a card for each student. Some teachers keep the cards on a clipboard with each taped at the top edge (see Figure 5.6). Another option is to focus your observations on about five students a day. The students selected may be members of one or two

ObservationsAll teachers learn useful bits of information about their students every day. When the three-phase lesson format suggested in Chapter 3 is used, the flow of evidence about student performance increases dramatically, especially in the during and after portions of lessons. If you have a systematic plan for gathering this information while observing and listening to students, at least two very valu-able results occur. First, information that may have gone unnoticed is suddenly visible and important. Second, observation data gathered systematically can be added to other data and used in planning lessons, providing feed-back to students, conducting parent conferences, and determining grades.

Depending on what information you may be trying to gather, a single observation of a whole class may require several days to two weeks before all students have been

Michael

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Nov. 8 - Explained 2 ways

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FIgure 5.6 Preprinted cards for observation notes can be taped to a clipboard or folder for quick access.

Above and BeyondClear understanding. Communi-cates concept in multiple repre-sentations. Shows evidence of using idea without prompting.

Fraction whole made fromparts in rods and in sets.Explains easily.

On TargetUnderstands or is developingwell. Uses designated models.

Not There YetSome confusion or misunder-stands. Only models idea with help.

Needs help to doactivity. No con�dence.

John S. Mary

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Julie (rod) Lee (set)

George (set) J.B. (rod)

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Can make whole in eitherrod or set format (note).Hesitant. Needs prompt toidentify unit fraction.

Making Whole Given Fraction PartObservation Rubric

3/17

FIgure 5.7 Record names in a rubric during an activity or for a single topic over several days.

102686 05 078-093 r0 sr.indd 88 11/30/11 8:49 AM

Observations 89

Sharon V.

Statedproblem inown words

Showinggreater

reasonableness

Reluctant to useabstract models

Used pa�ernblocks to show

2/3 and 3/6

NO

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T

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RG

ET

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Understandsnumerator/denominator

FRACTIONS

Area models

Set models

Uses fractions inreal contexts

Estimates fractionquantities

Understands problem before beginning work

Is willing totake risks

Justifies results

PROBLEM SOLVING

FIgure 5.8 A focused computer-generated checklist and rubric can be printed for each student.

Topic:

Lalie

Pete

Names

Sid

Lakeshia

George

Pam

Maria

Mental ComputationAdding 2-digit numbers

On TargetNot There Yet Above and Beyond Comments

Can’t domentally

Has at leastone strategy

Uses di�erentmethods with

di�erent numbers

3-18-093-21-09

3-20-09 3-24-09

3-20-09+

Di�culty with problemsrequiring regroupingFlexible approachesusedCounts by tens, thenadds ones

Beginning to add thegroup of tens �rstUsing a postedhundreds chart3-24-09

FIgure 5.9 A full-class observation checklist can be used for longer-term objectives or for several days to cover a short-term objective.

cooperative groups. An alternative to cards is the use of large peel-off file labels, possibly preprinted with student names. The label notes are then moved to a more perma-nent notebook page for each student.

rubricsAnother possibility is to use your three- or four-point generic rubric on a reusable form as in Figure 5.7. Include space for content-specific indicators and another column to jot down names of students. A quick note or comment may be added to a name. This method is especially useful for planning purposes.

ChecklistsTo cut down on writing and to help focus your attention, a checklist with several specific processes or content objec-tives can be devised and duplicated for each student (see Figure 5.8). Regardless of the checklist format, a place for comments should be included.

Another format involves listing all students in a class on a single page or not more than three pages (see Figure 5.9). Across the top of the page are specific abilities or deficiencies to look for. (These can be based on learning progressions or trajectories.) Pluses and minuses, checks, or codes corresponding to your general rubric can be entered in the grid. A full-class checklist is more likely to be used for long-term objectives. Topics that might be appropriate for this format include problem-solving pro-cesses, communication skills, and such subject areas as basic fact fluency or computational estimation skills. Dat-ing entries or noting specific activities observed is also helpful.

102686 05 078-093 r0 sr.indd 89 11/30/11 8:49 AM

Observations 89

Sharon V.

Statedproblem inown words

Showinggreater

reasonableness

Reluctant to useabstract models

Used pa�ernblocks to show

2/3 and 3/6

NO

T T

HE

RE

YE

T

ON

TA

RG

ET

AB

OV

E A

ND

BE

YO

ND

CO

MM

EN

TS

NAME:

Understandsnumerator/denominator

FRACTIONS

Area models

Set models

Uses fractions inreal contexts

Estimates fractionquantities

Understands problem before beginning work

Is willing totake risks

Justifies results

PROBLEM SOLVING

FIgure 5.8 A focused computer-generated checklist and rubric can be printed for each student.

Topic:

Lalie

Pete

Names

Sid

Lakeshia

George

Pam

Maria

Mental ComputationAdding 2-digit numbers

On TargetNot There Yet Above and Beyond Comments

Can’t domentally

Has at leastone strategy

Uses di�erentmethods with

di�erent numbers

3-18-093-21-09

3-20-09 3-24-09

3-20-09+

Di�culty with problemsrequiring regroupingFlexible approachesusedCounts by tens, thenadds ones

Beginning to add thegroup of tens �rstUsing a postedhundreds chart3-24-09

FIgure 5.9 A full-class observation checklist can be used for longer-term objectives or for several days to cover a short-term objective.

cooperative groups. An alternative to cards is the use of large peel-off file labels, possibly preprinted with student names. The label notes are then moved to a more perma-nent notebook page for each student.

rubricsAnother possibility is to use your three- or four-point generic rubric on a reusable form as in Figure 5.7. Include space for content-specific indicators and another column to jot down names of students. A quick note or comment may be added to a name. This method is especially useful for planning purposes.

ChecklistsTo cut down on writing and to help focus your attention, a checklist with several specific processes or content objec-tives can be devised and duplicated for each student (see Figure 5.8). Regardless of the checklist format, a place for comments should be included.

Another format involves listing all students in a class on a single page or not more than three pages (see Figure 5.9). Across the top of the page are specific abilities or deficiencies to look for. (These can be based on learning progressions or trajectories.) Pluses and minuses, checks, or codes corresponding to your general rubric can be entered in the grid. A full-class checklist is more likely to be used for long-term objectives. Topics that might be appropriate for this format include problem-solving pro-cesses, communication skills, and such subject areas as basic fact fluency or computational estimation skills. Dat-ing entries or noting specific activities observed is also helpful.

102686 05 078-093 r0 sr.indd 89 11/30/11 8:49 AM




Organizing and Tracking Formal Assessment and Student ThinkingSorting Assessment Features by Dimensions of Student Learning

Dimension of Learning Assessment Features by Dimension of Learning

Content Knowledge:assessing content knowledge involves determining what students know and what they are able to do.

Mathematical Disposition: a student’s mathematical disposition is healthy when he or she responds well to mathematical challenges and sees himself or herself as a learner and inventor of mathematics. Disposition also includes confidence, expectations, and metacognition (reflecting on and monitoring one’s own learning).

Work Habits: a student’s work habits are good when he or she is willing to persevere, contribute to group tasks, and follow tasks to completion. these valuable skills are used in nearly every career. to assess work habits, it is important to ask questions, such as, “Are the students able to organize and summarize their work?” and “Are the students progressing in becoming independent learners?”


CMP3 Assessment FeaturesCheckpoints: these are assessment tools that give teachers and students an opportunity to check student understanding at key points in the unit.

1. Applications-Connections-Extensions (ACE) by assigning acE exercises as homework, teachers can assess each student’s developing knowledge of concepts and skills.

2. Notebooks and Notebook Checklist Many teachers require their students to keep organized notebooks, which include homework, notes from class, vocabulary, and assessments. Each unit includes a checklist to help students organize their notebooks before they turn them in for teacher feedback.




3. Mathematical Reflections A set of summarizing questions, called Mathematical Reflections, occurs at the end of each Investigation. These questions can help teachers assess students’ developing conceptual knowledge and skills in the investigation.

4. Looking Back this Unit review feature includes two to four problems that ask students to explain their reasoning. Collectively, the pieces have students summarize and connect what they have learned within and across units.

5. MathXL for School This is a web-based formative assessment tool.

Surveys of Knowledge: these assessments provide teachers with a broad view of student knowledge both during a unit and at the end of a unit.

6. Check-ups These are short, individual assessment instruments. Check-up questions tend to be less complex and more skill-oriented than questions on quizzes and unit tests.

7. Partner Quizzes Each unit has at least one partner quiz. Quiz questions are richer and more challenging than check-up questions.

8. Unit Tests Each unit includes a test that is intended to be an individual assessment. the test informs teachers about a student’s ability to apply, refine, modify, and possibly extend the mathematical knowledge and skills acquired in the unit.

9. Self-Assessment After every unit, students complete a self-assessment, summarizing the mathematics they learned in the unit and the ideas with which they are still struggling. the self-assessment also asks students to provide examples of what they did in class to add to the learning of the mathematics.

10. Project at least three units in each grade include projects that can be used to replace or supplement the unit test. Projects give teachers an opportunity to assign tasks that are more product/performance- based than those on traditional tests. Project tasks are typically open-ended and allow students to engage in independent work and to demonstrate broad understanding of ideas in the unit.

Observations: the curriculum provides teachers with numerous opportunities to assess student understanding by observing students during group work and class discussions.

11. Group Work Many problems provide the opportunity to observe students as they “do mathematics,” applying their knowledge, exhibiting their mathematical disposition, and displaying their work habits as they contribute to group tasks.

12. Class Discussions the summary portion of each problem and the Mathematical Reflections at the end of each Investigation provide ongoing opportunities to assess students’ understanding through class discussions.

13. One-on-one interactions In the Explore phase, teachers can assess students during one-on-one interactions.





Student NotebooksExample 1: Some teachers ask students to organize their binders in sections:

i. textbook

II. Journal (a notebook with a record of the in-class problems and summary notes)

III. Mathematical Reflections

IV. Vocabulary (a student-created Mathematics Dictionary)

V. Homework (ACE questions and/or additional practice problems)

VI. Assessments (individual check-up quizzes, partner quizzes, unit tests and/or projects)

Example 2: Other teachers have found that the following simpler organization is effective:

i. a front pocket for the current work in an investigation

II. The textbook and a Mathematics Dictionary in the three-ring binder

III. A back pocket for assessed student work to include in-class problems, ACE, Mathematical Reflections, and quizzes

How will you require your students to organize their work?




Revisit Section 3 Big Questions • How will you use CMP3 assessments to evaluate the three dimensions of learning?

• How will you organize and track your students’ informal assessments?

• How can students organize their notebooks so that they can reflect on their learning and you can easily assess it?



Section 4: Planning a CMP3 Unit

CMP3 Units

Connected Mathematics Project, 3rd Edition (CMP3)

Grade 6 Before Grade 6 After

•PrimeTime

•BitsandPiecesI

•ShapesandDesigns

•BitsandPiecesII

•CoveringandSurrounding

•BitsandPiecesIII

•HowLikelyIsIt?

•DataAboutUs

•PrimeTime

•ComparingBitsandPieces

•Let’sBeRational

•CoveringandSurrounding

•DecimalOperations

•VariablesandPatterns

•DataAboutUs

Changes from Connected Mathematics Project 2 to Connected Mathematics Project 3

What Changed for Grade 6?

•Prime Time:Newcontentonproperties,includingDistributiveProperty.

•TheCCSSemphasisoftheconceptofratioatGrade6ledtoreworkingBits and Pieces IintoComparing Bits and Pieces.

•Let’s Be RationalisbasedonBits and Pieces II,withremovalofsomecontentandadditionaldevelopmentofdivisionoffractions,includingemphasisonestimation.AnewInvestigationprovidesalgebraicsupportonpropertiesofoperations.

•Covering and SurroundingcutsbackonareaandmovesinvestigationofcirclestoGrade7.Additionalcontentincludesnets,surfacearea,andvolumeofprismswithfractionalsidelengths.

•Decimal Opsisanewunitthataddressesthedecimalfluencystandards.

•How Likely Is It?hasbeenremovedtoreflecttheeliminationofprobabilityatGrade6intheCCSS.

•TheCCSScallsfortheintroductiontoalgebraandthegraphingoflinesatGrade 6,andsoVariables and PatternswasmovedtotheendofGrade6.

•Data About UshasbeenrevisedtocutbackonthedifferenttypesofgraphsandtoaddcontentthatwasformerlyinSamples and Populations.



Section 4: Planning a CMP3 Unit to Achieve the Focus, Coherence, and Rigor of the Common Core

Connected Mathematics Project 3


•VariablesandPatterns

•StretchingandShrinking

•ComparingandScaling

•AccentuatetheNegative

•MovingStraightAhead

•FillingandWrapping

•WhatDoYouExpect?

•DataDistributions

•ShapesandDesigns

•AccentuatetheNegative

•StretchingandShrinking

•ComparingandScaling

•MovingStraightAhead

•WhatDoYouExpect?

•FillingandWrapping

•SamplesandPopulations


•Becauseofthede-emphasisofGeometryinGrade6,Shapes and DesignshasbeenmovedtothebeginningofGrade7.

•Accentuate the Negativeincludesnewcontentonrationalnumbers.Throughouttheunit,studentsperformoperationswithfractions,decimals,andintegers.

•Comparing and ScalinghasbeenrevisedtomorecompletelydeveloptheideasandstrategiesinitiatedinStretching and Shrinking.Studentsuseproportionalreasoningincontextsotherthangeometriccontextsanddevelopadditionalstrategiesforsolvingproportions,includingefficientscalingandcommondenominators.Moretimeisspentondevelopingproportionalreasoningandconnectionsamongrates,ratios,ratetables,andproportions.Theconstantofproportionalityisintroducedandprovidesconnectionstothenextunit,Moving Straight Ahead.

•SomeofthecontentfromCMP2’sHow Likely Is It?hasbeenaddedtotheGrade7probabilityunitWhat Do You Expect?

•SomeofthedataunitSamples and PopulationshasbeenmovedfromGrade8toGrade7,withnewmaterialadded.

•Filling and WrappingpicksupcirclesfromCovering and Surrounding,andaddsvolumeandsurfaceareaofcylinders,cones,andspheres.




PearsonSchool.com 800-848-9500 Copyright Pearson Education, Inc., or its affiliates. All rights reserved. M

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•ThinkingwithMathematicalModels

•LookingforPythagoras

•Growing,Growing,Growing

•Frogs,Fleas,andPaintedCubes

•Kaleidoscopes,Hubcaps,andMirrors

•SayItwithSymbols

•TheShapesofAlgebra

•SamplesandPopulations

•ThinkingwithMathematicalModels

•LookingforPythagoras

•Growing,Growing,Growing

•Frogs,Fleas,andPaintedCubes

•Butterflies,Pinwheels,andWallpaper

•SayItwithSymbols

• It’sIntheSystem

•FormalizingFunctions


•AtGrade8,thecorrelationand2-waytableanalysisofSamples and PopulationsisnowaddressedattheendofThinking With Mathematical Models.

- Looking for Pythagoras hasbeenrevisedtoincludecuberoots,repeatingdecimals,terminatingdecimals,andirrationalnumbers.

- InGrowing, Growing, Growing,scientificnotationisintroduced.Contenthasbeenaddedtofocusontherulesofexponentsandequivalentexpressionsusingexponents.Theunitendswithanexplorationoftheeffectsofgrowthratesandy-interceptsonthegraphsofexponentialfunctions.

•Butterflies, Pinwheels, and WallpaperisarevisionofKaleidoscopes, Hubcaps, and MirrorsthatcorrespondstotheGrade8standardsoncongruenceandsimilarity.

•Anewunit,It’s in the System, addressesthestandardsonsystemsofequations.

•TwounitsareforteacherswhowanttoteachAlgebra1atGrade8:Frogs, Fleas, and Painted Cubesand Formalizing Functions(workingtitle).

(Pearson Education, inc. 2014, n.d.a., 1–3)




Content of CMP3 Units by Grade LevelCMP3 Unit Content Change to Align with CCSSM Standard(s)

Grade 6

Prime TimeProperties, including the Distributive Property

Comparing Bits and PiecesEmphasis on ratios

Covering and Surroundingnets, surface area, and volume of prisms with fractional sides

Decimal OpsDecimal fluency

Variables and Patternsalgebra and graphing of lines

Data About UsVariability, characteristics of data distributions, and measures of variation and central tendency

Grade 7

Accentuate the Negativerational numbers and operations with fractions, decimals and integers

Comparing and ScalingUse of proportional reasoning outside of geometry; connections among rates, ratios, rate tables, and proportions; and the constant of proportionality

What Do You Expect?Probability as likelihood of an event, approximating probability, and the probability of compound events

Samples and PopulationsUsing random sampling to understand and draw inferences about a population

Filling and Wrappingcontent on circles and volume and surface are of cylinders, cones, and spheres

Grade 8

Thinking with Mathematical ModelsCorrelation and two-way table analysis

Looking for Pythagorascube roots, repeating decimals, terminating decimals, and irrational numbers

Growing, Growing, GrowingScientific notation and rules of exponents

Butterflies, Pinwheels, and Wallpapercongruence and similarity

It’s in the SystemSystems of equations

(Pearson Education, inc. 2014, n.d.a., 1–3)




Unit 1 Breakdown • Grade 6: Prime Time

• Grade 7: Accentuate the Negative

• Grade 8: Thinking with Mathematical Models

For this activity, break down Unit 1 of the grade that you teach. Use the following example of part of a breakdown of the Say it with Symbols unit.

Say It with SymbolsInvestigation 1:

1.1 Essential Idea(s): This is a new feature for CMP3; Use the graphic organizer from the Toolkit for new York city to identify Essential idea(s).Focus Question: This is a new feature for CMP3; Use the graphic organizer from the Toolkit for new York city to identify the Focus Question.Conduct Summarize phases after students explore each of the following: a1, a2, a3, b1, b2, cUse the following as exit slips, homework, and warm-up problems: ACE (1, 2, 18-20): 1, 2, either 18 or 19; use 20 as a next day warm-up

1.2 Essential Idea(s): This is a new feature for CMP3; Use the graphic organizer from the Toolkit for new York city to identify Essential idea(s).Focus Question: This is a new feature for CMP3; Use the graphic organizer from the Toolkit for new York city to identify the Focus Question.Conduct Summarize phases after students explore each of the following: a1, a2, a3, a4, b, cUse the following as exit slips, homework, and warm-up problems: acE (3, 4, 25, 26): 3, 25, 26; use 4 as a next day warm-up

1.3 Essential Idea(s): This is a new feature for CMP3; Use the graphic organizer from the Toolkit for new York city to identify Essential idea(s).Focus Question: This is a new feature for CMP3; Use the graphic organizer from the Toolkit for new York city to identify the Focus Question.Conduct Summarize phases after students explore each of the following: b, c, DUse the following as exit slips, homework, and warm-up problems: ACE (5, 6): 6; use 5 as an exit ticket

1.4 Essential Idea(s): This is a new feature for CMP3; Use the graphic organizer from the Toolkit for new York city to identify Essential idea(s).Focus Question: This is a new feature for CMP3; Use the graphic organizer from the Toolkit for new York city to identify the Focus Question.Conduct Summarize phases after students explore each of the following: a4, b4, D4, E2Use the following as exit slips, homework, and warm-up problems: acE (8, 9, 12–14, 51): 8 or 9, 12; use 13 as an exit slip, 14 as a warm-up, and use 51 for extension or class review prior to a test, or a partner quiz




Investigation 2:

2.1 Essential Idea(s): This is a new feature for CMP3; Use the graphic organizer from the Toolkit for new York city to identify Essential idea(s).

Focus Question: This is a new feature for CMP3; Use the graphic organizer from the Toolkit for new York city to identify the Focus Question.

Conduct Summarize phases after students explore each of the following: a2, b3, c (linear or nonlinear)

Use the following as exit slips, homework, and warm-up problems: acE (2, 3–5, 13–15): 3, 4, 13, 2 (for next day warm-up)

2.2 continue this process for the entire Unit 1 for the grade that you teach. . .




Revisit Section 4 Big Questions • What content are you expected to teach this year?

• What are the big ideas of Unit 1 at the grade level that you teach?

• How will you structure the Explore and Summarize phases of the lessons in Unit 1, and what homework will you assign?



Reflection and Closing

Homework to Complete Prior to Teacher Session 2 • Work with a colleague to complete the Investigation problems for Unit 1. Try to solve

each problem in as many ways as possible. Write down any questions that you would ask students for each solution strategy. Post your solution strategies and questions to one problem to teachability.com. observe, and comment on at least one other posted problem.

Suggested Ideas • Select a lesson from CMP3. Write a one-page narrative of how you envision the lesson

will play out in the classroom. collaborate with participants from the intensive Sessions to compare and contrast the narratives.

• Before Teacher Session 2, read “Progressions for the Common Core State Standards in Mathematics: numbers and operations in base ten” and “Progressions for the common core State Standards in Mathematics: number and operations—Fractions.” complete the guiding template as you read. these documents contain the vital information on the development, tools, and strategies of the common core.



Appendix

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Appendix



Referencescawleti, Gordon, ed. 2004. Handbook of Research on Improving Student Achievement, 3rd ed. arlington, Va:

Educational research Service.connected Mathematics Project, and Michigan State University. 2013a. “cMP3 6th Grade overview.” Presentation given

at the 16th annual cMP Users’ conference, Michigan State University, February 2013. accessed May 14, 2013. http://connectedmath.msu.edu/conferences/users/cMP3_6thGrade_overview_Users.pdf.

———. 2013b. “cMP3 7th Grade overview.” Presentation given at the 16th annual cMP Users’ conference, Michigan State University, February 2013. accessed May 14, 2013. http://connectedmath.msu.edu/conferences/users/cMP3_7thGrade_overview_Users.pdf.

———. 2013c. “cMP3 8th Grade overview.” Presentation given at the 16th annual cMP Users’ conference, Michigan State University, February 2013. accessed May 14, 2013. http://connectedmath.msu.edu/conferences/users/cMP3_8thGrade_overview_Users.pdf.

common core State Standards initiative. 2012. “Mathematics.” accessed May 30, 2013. http://www.corestandards.org/Math.

common core Standards Writing team, the. 2011a. “Progressions for the common core State Standards in Mathematics (draft): k–5, number and operations in base ten.” tools for the common core (blog). accessed June 25, 2013. http://commoncoretools.files.wordpress.com/2011/04/ccss_progression_nbt_2011_04_073.pdf.

———. 2011b. “Progressions for the common core State Standards in Mathematics (draft): 3–5 number and operations—Fractions”. Tools for the Common Core (blog). accessed June 25, 2013. http://commoncoretools.files.wordpress.com/2011/08/ccss_progression_nf_35_2011_08_12.pdf.

Grouws, Douglas, and kristin cebulla. 2000. Improving Student Achievement in Mathematics. accessed october 22, 2011. http://www.iaoed.org/files/prac04e.pdf.

Lappan, Glenda, Dennis rankin, and Jeanne rast. 2013. “cMP3 assessment Features: constant assessment and Feedback.” Pearson Education, Inc.; 7 min., 15 sec. MP4.

National Governors Association Center for Best Practices (NGA Center), Council of Chief State School Officers (ccSSo). 2010. “common core State Standards for Mathematics.” Washington, Dc: national Governors Association Center for Best Practices, Council of Chief State School Officers. Accessed May 1, 2013. http://www.corestandards.org/assets/ccSSi_Math%20Standards.pdf.

new York State Education Department (nYSED). n.d. “crosswalk of common core instructional Shifts: Mathematics.” accessed May 28, 2013. http://schools.nyc.gov/NR/rdonlyres/9375E046-3913-4AF5-9FE3-D21BAE8FEE8D/0/commoncoreinstructionalShifts_Mathematics.pdf.

New York State Mathematics Common Core Workgroup. n.d. “P-12 Common Core Learning Standards for Mathematics.” new York, nY: new York State Mathematics common core Workgroup. accessed May 28, 2013. http://www.p12.nysed.gov/ciai/common_core_standards/pdfdocs/nysp12cclsmath.pdf.

Pearson Education, inc. n.d.a. CMP3: Changes from connected Mathematics Project 2 to connected Mathematics Project 3. Upper Saddle river, nJ: Pearson Education, inc.

———. n.d.b. CMP3: Core Middle Grades Mathematics Program. Upper Saddle river, nJ: Pearson Education, inc.———. 2014a. CMP3: Accentuate the Negative: Integers and Rational Numbers. Upper Saddle river, nJ: Pearson

Education, inc. ———. 2014b. CMP3 Program and Implementation Guide. Upper Saddle river, nJ: Pearson Education, inc. ———. 2014c. CMP3: Prime Time: Factors and Multiples. Upper Saddle river, nJ: Pearson Education, inc. ———. 2014d. CMP3 Sampler. Upper Saddle river, nJ: Pearson Education, inc.———. 2014e. CMP3: Thinking with Mathematical Models: Linear and Inverse Variation. Upper Saddle river, nJ:

Pearson Education, inc.Phillips, Elizabeth. 2013. “What’s New?” Presentation given at the 16th Annual CMP Users’ Conference, Michigan State

University, February 2013. accessed May 14, 2013. http://connectedmath.msu.edu/conferences/users/.

Van de Walle, John A., Karen S. Karp, and Jennifer M. Bay-Williams. 2013. Elementary and Middle School Mathematics: Teaching Developmentally. Upper Saddle river, nJ: Pearson Education, inc.

Wiliam, Dylan. 2007. “keeping Learning on track: classroom assessment and the regulation of Learning.” in F. k. Lester Jr. (Ed.), Second Handbook of Mathematics Teaching and Learning. Greenwich, ct: information age Publishing.

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