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Instructional Materials Evaluation Tool Integrated CME Project Mathematics ©2012
ALIGNMENT TO THE COMMON CORE STATE STANDARDS
Evaluators of materials should understand that at the heart of the Common Core State Standards is a substantial shift in mathematics instruction that demands the following:
1) Focus strongly where the Standards focus 2) Coherence: Think across grades/courses and link to major topics within a course 3) Rigor: In major topics, pursue conceptual understanding, procedural skill and fluency, and application with equal intensity.
Evaluators of materials must be well versed in the Standards related to the particular course, including understanding the Widely Applicable Prerequisites, how the content fits into the progressions in the Standards, and the expectations of the Standards with respect to conceptual understanding, fluency, and application. It is also recommended that evaluators refer to the Spring 2013 High School Publishers' Criteria for Mathematics while using this tool (achievethecore.org/publisherscriteria).
ORGANIZATION
SECTION I: NON-NEGOTIABLE ALIGNMENT CRITERIA
All submissions must meet all of the non-negotiable criteria at each grade level to be aligned to CCSS and before passing on to Section II.
SECTION II: ADDITIONAL ALIGNMENT CRITERIA AND INDICATORS OF QUALITY
The criteria in this section are additional alignment requirements that should be met by materials fully aligned with CCSS. A higher score in this section indicates that instructional materials are higher quality and more closely aligned to the Standards than instructional materials that have a lower score. Together, the non-negotiable criteria and the additional alignment criteria reflect the 8 criteria from the High School Publishers’ Criteria for Mathematics. The indicators of quality are taken from the High School Publishers’ Criteria as well. For more information on these elements, see achievethecore.org/publisherscriteria.
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SECTION I:
Non-Negotiable 1. FOCUS IN HIGH SCHOOL: In any single course, students and teachers using the materials as designed spend the majority of their time developing knowledge and skills that are widely applicable as prerequisites for postsecondary education.
Focus in High School True/False Evidence
1A. In an single course, students spend at least 50% of their time on Widely Applicable Prerequisites.
T F In Integrated CME Project, students spend at least 50% of their time on Widely Applicable Prerequisites. See Table 1.
1B. Student work in Geometry significantly involves applications/modeling as well as geometry applications that use algebra skills.
T F Geometry work is spread throughout all three courses of Integrated CME Project. Students gain extensive experience in picturing, drawing, constructing, and reasoning about shapes as shown in Course I, Chapter 7; Course II, Chapter 6, 7, 8, 9, and 10; and Course III, Chapter 9. Students bring together ideas from algebra 1 and geometry in Course I, Chapter 8 and Course III, Chapter 9.
1C. There are problems at a level of sophistication appropriate to high school (beyond mere review of middle school topics) that involve of the application
of knowledge and skills from grades 6–8 including:
• Applying ratios and proportional relationships. • Applying percentages and unit conversions, e.g.,
in the context of complicated measurement problems involving quantities with derived or compound units (such as mg/mL, kg/m3, acre-feet, etc).
• Applying basic function concepts, e.g., by interpreting the features of a graph in the content of an applied problem.
• Applying concepts and skills of geometric measurement, e.g., when analyzing a diagram of a schematic.
• Applying concepts and skills of basic statistics
and probability (see 6–8.SP).
• Performing rational number arithmetic fluently.
T F Examples of problems students solve at a level of sophistication appropriate to high school that involve application of knowledge and skills from
grades 6–8 can be found in the following chapters:
• Applying ratios and proportional relationships: Mathematics II, Chapter 7
• Applying percentages and unit conversions: Mathematics III, Chapter 9
• Applying basic functions concepts: Mathematics I, Chapter 5; Mathematics II, Chapter 7
• Applying concepts and skills of geometric measurement: Mathematics II, Chapter 7
• Applying concepts and skills of basic statistics and probability: Mathematics III, Chapter 3
• Performing rational number arithmetic fluently: Mathematics II, Chapter 1
To be aligned to the CCSSM, materials should devote the majority of class time developing knowledge and skills that are widely applicable as prerequisites for postsecondary education. All three of the T/F items above must be marked ‘true’ (T).
Meet? (Y/N) Yes
Justification/Notes
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SECTION I (continued):
Non-Negotiable 2. CONSISTENT, COHERENT CONTENT: Each course’s instructional materials are coherent and consistent with the content in the Standards.
Materials assess only at, or after,
the indicated grade level
Evidence
2A. Giving all students extensive work with course-level problems: Review of material from previous grades and courses is clearly identified as such to the teacher, and teachers and students can see what their specific responsibility is for the current year.
T F Integrated CME Project gives students extensive course-level work. Material from previous grades and courses is identified in the Teacher’s Edition. The Investigation and Chapter Roadmaps identify topic and chapter-level materials from previous grades and courses. The Lesson Overview identifies lesson-level material from previous grades and courses. At the beginning of each Investigation, both teachers and students are presented with specific learning goals and skills that will be developed through the ensuing lessons.
2B. Relating course-level concepts explicitly to prior knowledge from earlier grades and courses: The materials are designed so that prior knowledge becomes reorganized and extended to accommodate the new knowledge.
T F Integrated CME Project is designed to develop ideas thoroughly and then revisit them, as prior knowledge, only to extend and deepen. In particular, the first lesson of each Investigation, called Getting Started, is an exploratory lesson that previews the main ideas and activates prior knowledge that students can use and extend throughout the Investigation.
To be aligned to the CCSSM, materials for each course must be coherent and consistent with the content in the Standards. Both of the T/F items above must be marked ‘true’ (T).
Meet? (Y/N) Yes
Justification/Notes
The structure of Integrated CME Project is as follows: each book contains 8–10 chapters; each chapter contains a number of
Investigations that develop the chapter theme extensively; and each Investigation consists of several lessons.
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SECTION I (continued):
Non-Negotiable 3. RIGOR and BALANCE: Each grade’s instructional materials reflect the balances in the Standards and help students meet the Standards’ rigorous expectations, by helping students develop conceptual understanding, procedural skill and fluency, and application.
Aspects of Rigor True/False Evidence
3A. Attention to Conceptual
Understanding: Materials develop conceptual understanding of key mathematical concepts, especially where called for in specific content standards or cluster headings.
T F Table 2 indicates the clusters in the CCSSM that call explicitly for the conceptual development of key concepts and the Integrated CME Project lessons where that development takes place.
3B. Attention to Procedural Skill and
Fluency: Materials give attention throughout the year to individual standards that set an expectation of procedural skill and fluency.
T F The Integrated CME Project approach to skill development and fluency is through orchestrated problem sets, in which students, by practicing skills such as factoring and simplifying, come to new mathematical insights. Students develop procedural fluency in the process of producing worthwhile mathematics.
3C. Attention to Applications: Materials are designed so that teachers and students spend sufficient time working with engaging applications/modeling.
T F Work with applications is woven into the problems presented in Integrated CME Project. The program revisits several scenarios in mathematics and several other fields throughout the three courses, each time taking on a new perspective. For example, the problem of determining a monthly payment on a loan appears throughout the program. Students use this content either as a launch into or an application of trial-and-error approximations, recursively defined functions, computer algebra systems, geometric series, and exponential functions. Additionally, the Mathematical Reflections that occur at the end of each Investigation provide students with engaging applications for the ideas they have learned.
3D. Balance: The three aspects of rigor are not always treated together, and are not always treated separately
T F In Integrated CME Project, conceptual understanding underpins fluency work; fluency is developed in the context of application through practice exercises. Within each Investigation and lesson, students think about and apply concepts in ways that deepen their conceptual understanding, which allows them to achieve fluency when appropriate.
To be aligned to the CCSSM, materials for each course must attend to each element of rigor and must represent the balance reflected in the Standards. All four of the T/F items above must be marked ‘true’ (T).’
Meet? (Y/N) Yes
Justification/Notes
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SECTION I (continued):
Non-Negotiable 4. PRACTICE-CONTENT CONNECTIONS: Materials meaningfully connect the Standards for Mathematical Content and the Standards for Mathematical Practice.
Practice-Content Connections True/False Evidence
4A. The materials connect the Standards for Mathematical Practice and the Standards for Mathematical Content.
T F Integrated CME Project has been built to incorporate the Standards for Mathematical Practice into its overall design. The fundamental principle of the program is developing mathematical Habits of Mind, which mirror and sometimes match the Mathematical Practices. For example, Developing Habits of Mind features provide multiple ways to think about lessons content, often emphasizing a habit of mind that has applications beyond the mathematics at hand. See Appendix A for a correlation between CME’s Habits of Mind and the Standards for Mathematical Practice.
4B. The developer provides a description or analysis, aimed at evaluators, which shows how materials meaningfully connect the Standards for Mathematical Practice to the Standards for Mathematical Content within each applicable course.
T F Appendix B offers an overview of the Standards for Mathematical Practice in this program.
To be aligned to the CCSSM, materials must connect the practice standards and content standards and the developer must provide a narrative that describes how the two sets of standards are meaningfully connected within the set of materials for each course. Both of the T/F items above must be marked ‘true’ (T).
Meet? (Y/N)
Yes
Justification/Notes
Materials must meet all four non-negotiable criteria listed above to be aligned to the CCSS and to continue to the evaluation in Section II.
# Met All of the Non-Negotiable criteria
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SECTION II ADDITIONAL ALIGNMENT CRITERIA AND INDICATORS OF QUALITY Materials must meet all four non-negotiable criteria listed above to be aligned to the CCSS and to continue to the evaluation in Section II. Section II includes additional criteria for alignment to the Standards as well as indicators of quality. Indicators of quality are scored differently from the other criteria: a higher score in this section indicates that materials are higher quality and more closely aligned to the Standards than instructional materials that have a lower score. Instructional materials evaluated against the criteria in Section II will be rated on the following scale: • 2 – (meets criteria): A score of 2 means that the materials meet the full intention of the criterion in all courses. • 1 – (partially meets criteria): A score of 1 means that the materials meet the full intention of the criterion for some courses or
meets the criterion in many aspects but not the full intent of the criterion. • 0 – (does not meet criteria): A score of 0 means that the materials do not meet many aspects of the criterion. For Section II parts A, B, and C, districts should determine the minimum number of points required for approval. Before evaluation, please review sections A – C, decide the minimum score according to the needs of your district, and write in the number for each section.
II(A) Alignment Criteria for Standards for Mathematical Content
Score Justification/Notes
1. Materials are consistent with the content in the Standards. Materials base courses on the content specified in the Standards.
2 1 0 Table 3 shows where standards are addressed in each of the three courses.
2. Materials foster coherence through connections in a single course, where appropriate and where required by the Standards.
2A. Materials included learning objectives that are visibly shaped by CCSSM cluster and domain headings.
2 1 0 The learning objectives throughout Integrated CME Project align closely to CCSSM cluster and domain headings. Table 4 shows an example of learning objectives and domain and cluster headings from Mathematics II, Chapter 2.
2B. Materials include learning problems and activities that serve to connect two or more clusters in a domain, or two or more domains in a category, or two or more categories, in cases where these connections are natural and important.
2 1 0 Table 4 shows a lesson, related categories, domains, clusters, standards, and learning objectives from Chapter 2 in Mathematics II. This sample serves as just one example of the comprehensive evidence of this criterion found in Integrated CME Project.
2C. Materials preserve the focus, coherence, and rigor of the Standards even when targeting specific objectives.
2 1 0 Each lesson maintains the same lesson structure, helping to preserve the focus, coherence and rigor of the standards. Furthermore, the organization and sequencing of topics are intentionally structured to promote focus, coherence, and rigor.
MUST HAVE ----- POINTS IN SECTION II(A) FOR APPROVAL Score: 14
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SECTION II ADDITIONAL ALIGNMENT CRITERIA AND INDICATORS OF QUALITY
II(B) Alignment Criteria for Standards for Mathematical
Practice Score Justification/Notes
3. Focus and Coherence via Practice Standards:
Materials promote focus and coherence by connecting practice standards with content that is emphasized in the Standards.
2 1 0 The lesson structure of Integrated CME Project provides a connection between Mathematical Practices and the content emphasized in the Standards. The Developing Habits of Mind feature appears in lessons, connecting mathematical Habits of Mind, which reflect the Mathematical Practices, to the lesson content that meets the content that is emphasized in the Standards. In-Class Experiments further deepen this connection, providing students with questions, procedures, and calculations that helps them understand the basis of key concepts, while promoting Habits of Mind.
4. Careful Attention to Each Practice Standard: Materials attend to the full meaning of each practice standard.
2 1 0 Mathematical Habits of Mind are Integrated CME Project’s fundamental organizing principle, which allows the program to fully attend to the Standard for Mathematical Practice. Appendix A provides a correlation between the CME Habits of Mind and the Mathematical Practices. Appendix B provides a summary of how the Mathematical Practices are attended to throughout Integrated CME Project.
5. Emphasis on Mathematical Reasoning: Materials support the Standards' emphasis on mathematical reasoning by:
5A. Materials prompt students to construct viable arguments and critique the arguments of other concerning key course-level mathematics that is detailed in the content standards (cf. MP.3).
2 1 0 Integrated CME Project provides students frequent opportunities to construct viable arguments and critique the reasoning of others. For example, What’s Wrong Here exercises ask students to find, critique, and
correct errors in reasoning—mistakes that typically highlight common
errors. For Discussion features also provide opportunity for students to talk through questions or problems surrounding the concepts explored in each question.
5B. Materials engage students in problem solving as a form of argument.
2 1 0 The role of proof and justification as means for establishing results, communicating ideas, and discovering new conjectures is explored in all three courses. Students not only read and understand logical arguments, they construct such arguments themselves. Both What’s Wrong Here exercises and For Discussions features, as noted above, provide students opportunities to use argumentation as a form of problem solving.
5C. Materials explicitly attend to the specialized language of mathematics.
2 1 0 Throughout Integrated CME Project, the habit of using precise language is not only a mechanism for effective communication; it is a tool for understanding. One device for achieving this are the Minds in Action dialogues. The students in these features often struggle with how to say what they know in precise language, serving as a model for students on how to use the specialized language of mathematics.
MUST HAVE ----- POINTS IN SECTION II(B) FOR APPROVAL Score: 10
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SECTION II - ADDITIONAL ALIGNMENT CRITERIA AND INDICATORS OF QUALITY
II(C) Indicators of Quality Score Justification/Notes
6. Materials support the uses of technology as called for in the Standards.
2 1 0 Technology is integrated throughout Integrated CME Project, with an appendix at the end of each course keyed to the TI-Nspire handhelds.
7. The underlying design of the materials distinguishes between problems and exercises. In essence the difference is that in solving problems, students learn new mathematics, whereas in working exercises, students apply what they have already learned to build mastery. Each problem or exercise has a purpose.
2 1 0 The lessons structure of the program clearly distinguishes problems and exercises. In-Class Experiments allow students to learn new mathematics through problems. Three groups of exercises, which appear in each lesson, For You to Explore and Check Your Understanding, On Your Own, and Maintain Your Skills, provide students opportunities to apply what they’ve learned.
8. Design of assignments is not haphazard: exercises are given in intentional sequences.
2 1 0 The sequencing of lessons and topics is based on the careful planning of the CME authors. The exercises for these lessons are also structured intentionally to provide individual students with ability-appropriate practice.
9. There is variety in the pacing and grain size of content coverage.
2 1 0 The number of days devoted to each cluster varies depending on the grain size and content coverage of the cluster. In addition to the Pacing Guide, teachers are given suggestions, depending on student response, for varying the pacing of each lesson in the Lesson Overview.
10. There is variety in what students produce. For example, students are assigned to produce answers and solutions, but also, in a course-appropriate way, arguments and explanations, diagrams, mathematical models, etc.
2 1 0 “Experience before formality” is a guiding principle of Integrated CME Project. As such, students spend plenty of time playing with concepts, producing not only answers and solutions, but developing their own questions, constructing new conjectures, and communicating ideas.
11. Lessons are thoughtfully structured and support the teacher in leading the class through the learning paths at hand, with active participation by all students in their own learning and in the learning of their classmates.
2 1 0 The lesson structure of the Integrated CME Project fully supports teachers. It also encourages active participation from all students through features such as Minds in Action and For Discussion.
12. There are separate teacher materials that support and reward teacher study including, but not limited to: discussion of the mathematics of the units and the mathematical point of each lesson as it relates to the organizing concepts of the unit, discussion on student ways of thinking and anticipating a variety of students responses, guidance on lesson flow, guidance on questions that prompt students thinking, and discussion of desired mathematical behaviors being elicited among students.
2 1 0 The Daily Planner Section, found at the beginning of each chapter, always includes a Math Background feature that explains the mathematics in each chapter. Both Chapter Roadmaps, appearing at the beginning of each chapter, and Investigation Roadmaps, appearing at the beginning of each Investigation, provide the organizing concepts of chapters and topics as they relate to each lesson. The Lesson Overview, appearing at the beginning of each lesson, provides the mathematical point of each lesson.
13. Manipulatives are faithful representations of the mathematical objects they represent.
2 1 0 Integrated CME Project does not contain any manipulatives.
14. Manipulatives are connected to written methods. 2 1 0 Integrated CME Project does not contain any manipulatives.
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II(C) Indicators of Quality Score Justification/Notes
15. Materials are carefully reviewed by qualified individuals, whose names are listed, in an effort to ensure freedom from mathematical errors, age-level appropriateness, freedom from bias, and freedom from unnecessary language complexity.
2 1 0 Integrated CME Project authorship team includes Al Cuoco, whose paper on mathematical habits of mind is listed as one of the works consulted by the writers of the Common Core State Standards, and Bowen Kerins. These materials have been field tested in classrooms and reviewed to ensure age-level appropriateness, freedom from errors, bias, and unnecessary language complexity.
16. The visual design isn't distracting or chaotic, but supports students in engaging thoughtfully with the subject.
2 1 0 The design of Integrated CME Project is interesting, supporting students as they interact with concepts.
17. Support for English Language Learners and other special populations is thoughtful and helps those students meet the same standards as all other students. The language in which problems are posed is carefully considered.
2 1 0 One of the most important pedagogical structures in Integrated CME Project is the low-threshold, high-ceiling approach. Each chapter starts with activities that are accessible to all students. Each chapter ends with problems that will challenge even the most advanced students. This approach results in flexibility that allows for program use in a variety of settings.
MUST HAVE ----- POINTS IN SECTION II(C) FOR APPROVAL Score: 20
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FINAL EVALUATION
SECTION PASS/FAIL Final Justification/Notes
Section 1 - Non-Negotiable Criteria 1–4 P
Section II(A) - Alignment Criteria for Standards for Mathematical Content
P
Section II(B) - Alignment Criteria for Standards for Mathematical Practice
P
Section II(C) - Indicators of Quality P
FINAL DECISION FOR THIS MATERIAL PURCHASE? Y/N
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Table 1 Time Spent on Widely Applicable Prerequisites
Course Widely Applicable Prerequisites Days Spent on Clusters
% of Total Time Spent on Clusters
I Number and Quantity (includes clusters: N-RN: A, B; N-Q: A) 12 10%
Algebra (includes clusters: A-SSE: A, B; A-APR: A, B, C, D; A-CED: A; A-REI: A, B, C, D)
36 31%
Functions (includes clusters: F-IF: A, B, C; F -BF: A; F-LE: A) 21 18%
Geometry (includes clusters: G-CO: A, C; G-SRT: B, C) 13 11%
Statistics and Probability (includes clusters: S-IC: A; S-ID: A, C) 2 1%
Total: 84 72%
Number of days in Course I: 115
II Number and Quantity (includes clusters: N-RN: A, B; N-Q: A) 15 10%
Algebra (includes clusters: A-SSE: A, B; A-APR: A, B, C, D; A-CE: A; A-REI: A, B, C, D)
42 28%
Functions (includes clusters: F-IF: A, B, C; F-BF: A; F-LE: A) 23 15%
Geometry (includes clusters: G-CO: A, C; G-SRT: B, C) 15 10%
Statistics and Probability (includes clusters: S-IC: A; S-ID: A, C) 1 0.6%
Total: 96 64%
Number of days in Course II: 149
III Number and Quantity (includes clusters: N-RN: A, B; N-Q: A) 11 9%
Algebra (includes clusters: A-SSE: A, B; A-APR: A, B, C, D; A-CED: A; A-REI: A, B, C, D)
31 25%
Functions (includes clusters: F-IF: A, B, C; F-BF: A; F-LE: A) 41 34%
Geometry (includes clusters: G-CO: A, C; G-SRT: B, C) 1 0.8%
Statistics and Probability (includes clusters: S-IC: A; S-ID: A, C) 8 6%
Total: 92 76%
Number of days in Course III: 120
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Table 2 Concept Clusters
Standard Integrated CME Project Lessons
Number and Quantity
N-VM.C.10 (+) Understand that the zero and identity matrices play a role in matrix addition and multiplication similar to the role of 0 and 1 in the real numbers. The determinant of a square matrix is nonzero if and only if the matrix has a multiplicative inverse.
Mathematics I: Lessons H.10, H.11
Algebra
A-APR.A.1 Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.
Mathematics II: Lesson 2.06
A-APR.D.7 (+) Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression; add, subtract, multiply, and divide rational expressions.
Mathematics III: Lesson 1.1, 7.07
A-REI.D.10 Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line).
Mathematics I: Lessons 3.01, 3.03, 3.06;
Mathematics II: Lessons 4.07, 4.12, 4.15–
4.17, 10.02, 10.04, H.19, H.20, H.23, H.24, H.25; Mathematics III: Lessons 1.01, 1.02, 1.03, 8.02, 8.09
Functions
F-IF.A.1 Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x).
Mathematics I: Lessons 5.01, 5.03, 5.04; Mathematics II: Lessons 4.07
F-BF.B.5 (+) Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.
Mathematics III: Lessons 8.07–8.09, 8.13,
8.14
F-TF.A.1 Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.
Mathematics II: Lesson 8.07; Mathematics III: Lessons 5.01, 5.02
F-TF.B.6 (+) Understand that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows its inverse to be constructed.
Mathematics III: Lesson H.04
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Standard Integrated CME Project Lessons
Geometry
G-SRT.C.6 Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.
Mathematics II: Lessons 9.08, H.12; Mathematics III: Lesson 4.00
G-SRT.D.11 (+) Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g., surveying problems, resultant forces).
Mathematics III: Lessons 4.12, 4.13
Statistics and Probability
S-IC.A.1 Understand statistics as a process for making inferences about population parameters based on a random sample from that population.
Mathematics III: Lessons 3.06, 3.10
S-CP.A.2 Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities, and use this characterization to determine if they are independent.
Mathematics II: Lesson 5.02; Mathematics III: Lesson 3.02
S-CP.A.3 Understand the conditional probability of A given B as P(A and B)/P(B), and interpret independence of A and B as saying that the conditional probability of A given B is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B.
Mathematics II: Lessons 5.03, 5.04
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Table 3 Standards Coverage in Integrated CME Project
CCSSM Standards Course I Lessons Course II Lessons Course III Lessons
N-RN.A.1
1.06, 1.09–1.11
N-RN.A.2 1.04, 1.10, 1.11 8.11 N-RN.B.3 1.05
N-Q.A.1
2.17, 3.05, 3.08, 3.12, 4.04, 4.12 6.03, 6.06, 6.07, 6.09, 6.10
3.07, 4.07, 4.13, 4.19, 4.20, 7.01, 7.02, 9.10
8.05, 8.08, 8.09, 8.10, 7.10, 2.01, 2.04, 4.11, 3.07, 3.15-3.17
N-Q.A.2 3.12, 6.09 7.10, 9.12 N-Q.A.3 6.03, 6.08 6.09 3.12, 7.10 N-CN.A.1 3.02, 3.05 6.01, 6.02, 6.09 N-CN.A.2 3.05, 6.01, 6.02 6.01, 6.02, 6.09 N-CN.A.3 (+) 3.05, 6.01 6.01, 6.02
N-CN.B.4 (+) H.01, H.02, H.03 6.02, 6.03, 6.05, 6.06, 6.07, 6.08
N-CN.B.5 (+) H.03, H.06, H.07 6.03, 6.04, 6.06, 6.08 N-CN.B.6 (+) H.04 N-CN.C.7 3.11, 3.12, H.07 6.05, 6.06 N-CN.C.8 (+) 3.12 6.06, 6.07 N-CN.C.9 (+) 3.12, H.07 6.07, 6.08 N-VM.A.1 (+) H.02, H.03 H.02, H.03, H.05, H.06 N-VM.A.2 (+) H.02 N-VM.A.3 (+) N-VM.B.4 (+) H.02, H.03 H.03 N-VM.B.4a (+) H.02, H.03 H.03 N-VM.B.4b (+) H.02, H.03 N-VM.B.4c (+) H.02, H.03 H.03 N-VM.B.5 (+) H.02, H.03 H.03, H.06 N-VM.B.5a (+) H.03 N-VM.B.5b (+) H.06 N-VM.C.6 (+) H.06, H.07, H.09, H.12 N-VM.C.7 (+) H.07 N-VM.C.8 (+) H.07, H.08, H.09 N-VM.C.9 (+) H.09, H.10, H.11 N-VM.C.10 (+) H.10, H.11 N-VM.C.11 (+) H.12, H.13 N-VM.C.12 (+) H.13
A-SSE.A.1
2.02, 2.03, 2.05, 4.02 2.01, 2.02, 2.05, 2.07, 2.08, 2.10, 3.04, 4.13
1.02, 1.04, 1.05, 1.06, 1.07, 1.08, 1.09, 1.10, 2.01, 2.03, 2.06, 2.09, 5.04, 5.07, 5.08, 6.03, 7.10, 7.11, H.07, H.08
A-SSE.A.1a 2.01, 2.02, 2.05, 2.07, 2.08, 2.10
1.04, 5.07, 6.03
A-SSE.A.1b 2.05, 4.02 2.11, 3.04 1.02, 1.06, 1.07, 1.08, 1.09,
1.10, 2.06, 5.04, H.08
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CCSSM Standards Course I Lessons Course II Lessons Course III Lessons
A-SSE.A.2
2.17, 4.02 1.07, 1.08, 1.09, 1.11, 2.02, 2.03, 2.09, 2.10, 2.11, 3.11, 3.12, 3.13
1.08, 1.09, 1.10, H.08, 6.07
A-SSE.B.3
5.16 1.07, 1.08, 1.09, 1.11, 2.02, 2.03, 2.04, 2.10, 2.11, 2.12, 3.06, 3.07, 3.11, 3.12, 3.13, 10.02, 10.04, H.25
1.05, 1.06, 1.08, 1.09, 1.10, 2.06, 5.04, 7.01, 7.02, 7.03, 7.04, 8.07, 8.08, 8.09
A-SSE.B.3a 2.02, 2.03, 2.04, 2.10, 2.11, 3.06
1.06, 1.08, 5.04, 10.02
A-SSE.B.3b 2.12, 3.06, 3.07 10.04, H.25 A-SSE.B.3c 5.16 1.07, 1.08, 1.09, 1.11 A-SSE.B.4 2.01, 2.04, 2.08 2.02, 2.11, 2.12, 2.13, 7.10 A-APR.A.1 2.06 A-APR.B.2 1.05, 1.06, 7.03, 7.04, 7.05
A-APR.B.3 2.03, 2.10, 2.11, 2.12, 3.02, 3.04, 3.06, 3.07, 3.08
1.06, 1.08, 1.09, 7.01, 7.02
A-APR.C.4 2.01, 3.09 1.06, 6.01, 6.02 A-APR.C.5 (+) 2.01, 2.06, 5.01, 5.06 2.15, 2.16, 3.12, 3.16 A-APR.D.6 1.05, 7.03, 7.07, 7.08 A-APR.D.7 (+) 1.11, 7.07
A-CED.A.1
2.09, 2.10, 2.11, 2.14, 2.15, 2.16, 4.09, 4.11, 5.02, 5.07, 5.15, 5.17, 6.07
3.05, 3.06, 3.07, 4.12, 4.13
A-CED.A.2 2.17, H.06, H.07, H.09 3.05 7.03, 7.05, 8.12
A-CED.A.3 2.17, 3.04, 3.13, 4.04, 4.09, 4.10, 4.11, 4.12
2.02, 2.03, 2.04, 3.05 , 3.06
A-CED.A.4 2.17, 4.02, 4.03 2.01, 2.02, 2.03, 2.04 H.08 A-REI.A.1 2.10, 2.11, 2.12, 2.13, 4.02 3.07 5.04 A-REI.A.2 1.06 1.08, 1.10, 7.06, 9.08, 9.09
A-REI.B.3 2.08, 2.11, 2.12, 2.13, 2.16, 2.17, 4.09, 4.11
A-REI.B.4 2.03, 2.10, 2.11, 2.12, 3.01, 3.02, 3.03, 3.04, 3.09, 3.12, 4.06
1.08, 5.04, 9.0, 9.11
A-REI.B.4a 2.12 3.01, 3.02, 2.12 6.01, 1.08
A-REI.B.4b 2.03, 2.10, 2.11, 2.12, 3.02, 3.03, 3.04, 3.10, 3.11, 3.12
6.01
A-REI.C.5 4.08 A-REI.C.6 3.06, 4.05, 4.06, 4.07, 4.08 H.08, H.09, H.10 A-REI.C.7 3.06, 4.10, 4.12 A-REI.C.8 (+) H.10 H.10, H.11 A-REI.C.9 (+) H.10 H.11
A-REI.D.10
3.01, 3.03, 3.06 4.07, 4.12, 4.15, 4.16, 4.17, 10.02, 10.04, H.19, H.20, H.23, H.24, H.25
1.01, 1.02, 1.03, 8.02, 8.09
A-REI.D.11 3.07, 4.09, 4.11, 4.10 10.01, 4.12 1.02, 1.03, 1.08, 7.03, 7.05,
8.02, 8.098.12 A-REI.D.12 4.12 F-IF.A.1 5.01, 5.03, 5.04 4.07
16
CCSSM Standards Course I Lessons Course II Lessons Course III Lessons
F-IF.A.2 5.04, 5.05, 5.06, 5.12, 5.13, 5.14
4.07, 4.08, 8.13
F-IF.A.3 5.12, 5.13, 5.15 4.01, 4.02 2.10, 7.10
F-IF.B.4
5.05, 5.06, 5.12, 5.14, 5.17 3.06, 3.07, 3.08, 4.14, 4.15, 4.16, 4.17, H.18, H.19, H.20
5.06, 5.07, 5.08, 7.10, 7.02, 7.03, 7.04, 7.05, 7.07, 8.02, 8.09, 8.14
F-IF.B.5 3.06, 4.07 5.05 F-IF.B.6 3.12, 6.08 4.03, 4.04 5.05, 7.03, 7.05, 8.10
F-IF.C.7
3.08, 3.09, 4.02, 4.03, 4.10, 5.06, 5.16, 5.17
3.06, 3.07, 3.08, 4.14, 4.15, 4.16, 4.17, H.19, H.20
1.01, 1.03 5.03, 5.06, 5.07, 5.08, 7.01, 7.02, 7.03, 7.05, 7.06, 7.07, 8.02, 8.09, 8.10, 8.13, H.03, H.04, H.05
F-IF.C.7a 3.07, 3.08, 4.02, 4.03, 5.06 3.07, 3.08, 4.14, 4.15, 4.16,
4.17
F-IF.C.7b 3.09 4.07, 4.09
F-IF.C.7c 3.06, 3.07, 3.08, 4.15, 4.16, 4.17
1.01, 1.03, 7.01, 7.02, 7.03, 7.04, 7.05
F-IF.C.7d 4.14 7.01, 7.02, 7.03, 7.04, 7.05,
7.07
F-IF.C.7e
5.16, 5.17 4.12 4.07, 4.08, 5.06, 5.07, 5.08, 8.09, 8.10, 8.11, 8.12, 8.13, 8.14
F-IF.C.8 4.03, 5.01, 5.06, 5.17 2.01, 2.02 7.01, 7.02, 7.04 F-IF.C.8a 2.12, 3.06, 3.07, 3.08 F-IF.C.8b 5.15, 5.16, 5.17 4.13 8.03, 8.04, 8.05 F-IF.C.9 5.05, 5.06, 5.14 3.06, 3.07, 3.08
F-BF.A.1 5.13, 5.14 4.07, 4.12, 4.13 2.09, 7.10, 7.11, 8.02, 8.03,
8.04
F-BF.A.1a 5.02, 5.03, 5.05, 5.08, 5.09 2.09, 3.01, 3.03, 3.04, 5.08,
7.10, 8.02, 8.03, 8.04 F-BF.A.1b 5.04 4.12, 4.13 1.02, 1.03 F-BF.A.1c 5.04 4.06, 4.07 8.14, H.04 F-BF.A.2 5.13, 5.14, 5.15 4.02, 4.03, 4.04, 4.11, 8.01 2.0, 2.09, 2.10, 2.11
F-BF.B.3 3.07, 3.08, 3.09, 8.07 4.14, 4.16, 4.17 5.03, 5.06, 5.07, 5.08, 7.01,
7.03 F-BF.B.4 5.04 4.09 8.13, 8.14, H.04 F-BF.B.4a 4.09 8.14, H.04 F-BF.B.4b (+) 4.09 8.14, H.04 F-BF.B.4c (+) 4.09 8.13, 8.14, H.04 F-BF.B.4d (+) 4.09 H.04 F-BF.B.5 (+) 8.07, 8.08, 8.09, 8.13, 8.14 F-LE.A.1 5.14, 5.17 4.11 F-LE.A.1a 5.13, 5.15 4.03, 4.04, 4.11, 4.13 F-LE.A.1b 5.15 4.03, 4.04 F-LE.A.1c 5.15 4.13 8.03, 8.04, 8.05, 8.11
F-LE.A.2 3.13, 4.02, 4.03, 5.16 4.04, 4.12, 4.13 2.09, 2.10, 2.11, 8.02, 8.03,
8.04, 8.05
17
CCSSM Standards Course I Lessons Course II Lessons Course III Lessons
F-LE.A.3 5.17 8.05 F-LE.A.4 8.07, 8.08, 8.13 F-LE.A.5 4.04, 4.06, 5.16 8.13 F-TF.A.1 8.07 5.01, 5.02 F-TF.A.2 H.14 4.02, 5.01, 5.02, 5.03
F-TF.A.3 (+) H.13, H.14, H.15 4.01, 4.02, 4.03, 5.01, 5.02,
5.03, 5.05, H.02, H.03 F-TF.A.4 (+) H.16 4.04, 5.03, 5.04, H.02, H.03 F-TF.B.5 5.06, 5.08 F-TF.B.6 (+) H.04
F-TF.B.7 (+) H.19, H.20, H.17, H.16 4.04, 4.05, 4.07, 4.08, 5.04, 5.06, 5.08, H.04
F-TF.C.8 F-TF.C.9 (+) H.21 4.09, H.07 G-CO.A.1 7.08, 8.02, 8.12, 8.13 8.06, 10.0 G-CO.A.2 8.05, 8.06, 8.07, 8.08, 8.10 7.02, 7.16 G-CO.A.3 8.08 G-CO.B.4 8.06, 8.07, 8.08 G-CO.B.5 8.06, 8.07, 8.08, 8.10 G-CO.C.6 8.01, 8.02, 8.09 G-CO.C.7 8.04, 8.09 6.0 G-CO.C.8 8.04, 8.09 8.0 G-CO.D.9 6.05, 6.06, 6.13 G-CO.D.10 7.05, 7.06 6.02, 6.07, 6.13, 6.18, 9.04 G-CO.D.11 6.17, 6.18 G-CO.E.12 7.06 G-CO.E.13 7.09 8.03 G-SRT.A.1 7.07, 7.08 G-SRT.A.1a 7.08 G-SRT.A.1b 7.02, 7.08
G-SRT.A.2 7.03, 7.04, 7.07, 7.08, 7.14, 7.15
G-SRT.A.3 7.15 G-SRT.B.4 7.11, 7.12, 9.03 G-SRT.B.5 7.15, 9.01, 9.02, 9.03 G-SRT.C.6 9.08, H.12 4.0 G-SRT.C.7 9.08, H.12 4.0, H.04, H.05, H.07, H.08 G-SRT.C.8 9.07, 9.08, 9.09, H.12 4.0 G-SRT.D.9 (+) 9.09 4.10, 4.11 G-SRT.D.10 (+) 4.12, 4.13
G-SRT.D.11 (+) 4.12, 4.13
G-C.A.1 8.07
G-C.A.2 6.02, 8.08, 8.10, 8.12
G-C.A.3 8.11
G-C.A.4 (+) 8.12
G-C.B.5 8.07 8.05
G-GPE.A.1 10.0, 10.04
18
CCSSM Standards Course I Lessons Course II Lessons Course III Lessons
G-GPE.A.2 H.24 9.0, 9.10, 9.12
G-GPE.A.3 (+) H.24, H.25 9.10, 9.11, 9.12
G-GPE.B.4 8.11, 8.13 9.05, 10.02, 10.03 9.0
G-GPE.B.5 4.07, 8.12, 8.13
G-GPE.B.6 8.11, H.05 9.05
G-GPE.B.7 8.11, 8.13 9.05
G-GMD.A.1 8.02, 8.03, 9.10, 9.12, 9.13
G-GMD.A.2 (+) 9.12, 9.13, 9.14
G-GMD.A.3 9.10 9.12, 9.13, 9.14
G-GMD.B.4 9.12, H.22, H.23 9.08, 9.09
G-MG.A.1 9.1
G-MG.A.2 7.16 9.07
G-MG.A.3 9.05, 9.06
S-ID.A.1 6.03, 6.04 3.15, 3.16
S-ID.A.2 6.02, 6.04 3.07
S-ID.A.3 6.04
S-ID.A.4 3.15, 3.16, 3.17
S-ID.B.5 6.05
S-ID.B.6 6.06, 6.08 H.01, H.02, H.03, H.04
S-ID.B.6a 6.06, 6.08 H.01, H.02, H.03, H.04
S-ID.B.6b H.01, H.02, H.03, H.04
S-ID.B.6c H.01, H.02, H.03, H.04
S-ID.C.7 4.04, 3.12, 6.08
S-ID.C.8 6.06 H.03, H.04
S-ID.C.9 6.06
S-IC.A.1 3.06, 3.10
S-IC.A.2 5.01 3.01, 3.09, 3.10
S-IC.B.3 3.11
S-IC.B.4 3.06, 3.10, 3.13, 3.14
S-IC.B.5 3.11
S-IC.B.6 3.17
S-CP.A.1 5.02 3.02
S-CP.A.2 5.02 3.02
S-CP.A.3 5.03, 5.04
S-CP.A.4 5.03, 5.04
S-CP.A.5 5.03, 5.04
S-CP.B.6 5.03, 5.04
S-CP.B.7 5.02 3.02
S-CP.B.8 (+) 5.03, 5.04
S-CP.B.9 (+) 5.02, 5.05, 5.06 3.02, 3.03, 3.04, 3.09, 3.12
S-MD.A.1 (+) 5.02, 5.06 3.02, 3.04, 3.16
S-MD.A.2 (+) 5.06, 5.07 3.04, 3.05
S-MD.A.3 (+) 5.06 3.04, 3.06, 3.09, 3.12
S-MD.A.4 (+) 3.06, 3.07
S-MD.B.5 (+) 5.06, 5.07 3.04, 3.05
19
CCSSM Standards Course I Lessons Course II Lessons Course III Lessons
S-MD.B.5a (+) 5.06, 5.07 3.04, 3.05
S-MD.B.5b (+) 3.06, 3.08, 3.12
S-MD.B.6 (+) 5.01, 5.02 3.01, 3.02
S-MD.B.7 (+) 5.07 3.05
Table 4 Coherent Connections Across Categories, Domains, and Clusters
Lesson Lesson Objectives Domain and Cluster
Standards Connections
Course II, Lesson 2.02: Form and Function—Showing Expressions are Equivalent
• Use basic rules and moves to transform expressions and determine whether different expressions define the same function.
• Factor expressions by identifying a common factor.
• Apply identities when solving problems.
A.SSE – Interpret the structure of expressions. A.CED – Create equations that describe numbers or relationships. F.IF – Analyze functions using different representations.
A.SSE.A.1, ASSE.A.2, A.CED.A.4, F.IF.C.8
In this lesson, students learn about identities. Students use the basic rules of algebra to prove that when two expressions are equal under basic rules, they define functions that always produce the same output for any input.
20
Appendix A Connecting Habits of Mind to the Standards for Mathematical Practice
Mathematical Practice Mathematical Habits of Mind
#1 Make sense of problems and persevere in solving them
• Performing thought experiments • Expecting math to make sense
#2 Reason abstractly and quantitatively • Finding and explaining patterns • Creating and using representations • Generalizing from examples
• “Delayed evaluations” — seeking form in calculations
• Purposefully transforming and interpreting expressions • Seeking and specifying structural similarities
#3 Construct viable arguments and critique the reasoning of others
• Expecting math to make sense • Extending operations to preserve rules for calculating
#4 Model with mathematics • Creating and using representations
• “Delayed evaluations” — seeking form in calculations
#5 Use appropriate tools strategically • Seeking and specifying structural similarities • Purposefully transforming and interpreting expressions
#6 Attend to precision • Expecting mathematics to make sense • Seeking and expressing regularity in repeated calculations
#7 Look for and make use of structure • “Delayed evaluations” — seeking form in calculations
• “Chunking” (changing variables in order to hide complexity) • Reasoning about and picturing calculations and operations • Extending operations to preserve rules for calculating • Purposefully transforming and interpreting expressions • Seeking and specifying structural similarities
#8 Look for and express regularity in repeated reasoning
• Seeking and expressing regularity in repeated calculations • Generalizing from examples • Finding and explaining patterns • Purposefully transforming and interpreting
21
Appendix B Overview of the Standards for Mathematical Practice in Integrated CME Project
MP1: Make sense of problems and persevere in solving them:
Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, “Does this make sense?” They can understand the approaches of others to solving complex problems and identify correspondences between different approaches. The “experience before formality” principle in Integrated CME Project has sense-making as one its main goals. Definitions and theorems are capstones, not foundations, and students spend plenty of time playing with ideas in simple and transparent cases before the ideas are formalized and made precise.
MP2: Reason abstractly and quantitatively:
Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize—to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents—and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects. The whole “contextualize-decontextualize” dialectic is another major theme of Integrated CME Project. Students first interact with variables as placeholders for numbers, and problems always refer back to the quantities that are represented. As students progress through the courses (and the program), they notice that symbols become contexts in their own right, so that, for example, they eventually study the system of polynomials as a structure with its own internal logic.
MP3: Construct viable arguments and critique the reasoning of others:
Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose. Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and—if there is a flaw in an argument—explain what it is. Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades. Later, students learn to determine domains to which an argument applies. Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments.
The role of proof and justification—as means for establishing results, communicating ideas, and discovering new conjectures
—is emphasized in all three courses. A recurring type of problem across the program, What’s Wrong Here, asks students to
critique and repair errors in reasoning.
22
MP4: Model with mathematics:
Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts, and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose. Students in Integrated CME Project build mathematical models using functions, equations, graphs, tables, and technology. They abstract from data to build equations that model a host of phenomena. One of the distinguishing features of Integrated CME Project is that students learn how to calculate a line of best fit using quadratic functions.
MP5: Use appropriate tools strategically
Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. For example, mathematically proficient high school students analyze graphs of functions and solutions generated using a graphing calculator. They detect possible errors by strategically using estimation and other mathematical knowledge. When making mathematical models, they know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of concepts. Students in Integrated CME Project use an array of tools ranging from physical devices to experiment with geometric optimization problems to technological tools. Technology is integrated throughout the program, with an appendix at the end of each book keyed to the TI-Nspire handhelds.
MP6: Attend to precision
Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. They are careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context. In the elementary grades, students give carefully formulated explanations to each other. By the time they reach high school they have learned to examine claims and make explicit use of definitions. Throughout CME, the habit of using precise language is not only a mechanism for effective communication; it’s a tool for understanding. One device for helping them do this is the recurring “Minds in Action” dialogues that feature students wrestling with how to say what they know in precise mathematical language. Another design feature is the introduction of vocabulary only when students need it. For example, students work with polynomials informally early in the program, but terminology such as monomial, degree, and coefficient are not introduced until they start studying these things formally.
MP7: Look for and make use of structure
Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 x 8 equals the well remembered 7 x 5 + 7 x 3, in preparation for learning about the distributive property. In the expression x
2 + 9x + 14, older students can see the 14 as 2 x 7 and the 9 as 2 + 7. They recognize the
significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as
23
single objects or as being composed of several objects. For example, they can see 5 - 3(x - y)2 as 5 minus a positive number times a
square and use that to realize that its value cannot be more than 5 for any real numbers x and y. Students look for regularity and structure in almost every lesson of Integrated CME Project. Our whole approach to completing the square, “chunking” to remove terms in the factoring lessons, and transforming area formulas is devoted to seeing and making use of structure in algebraic expressions. Students also build mathematical structures in the more traditional sense: they study the algebra of functions, and they are introduced to complex numbers as polynomials with an extra simplification rule. Students also create algebraic groups (without calling them that) of roots of unity via “remainder arithmetic” with polynomials.
MP8: Look for and express regularity in repeated reasoning
Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts. Upper elementary students might notice when dividing 25 by 11 that they are repeating the same calculations over and over again, and conclude they have a repeating decimal. By paying attention to the calculation of slope as they repeatedly check whether points are on the line through (1, 2) with slope 3, middle school students might abstract the equation (y - 2)/ (x - 1) = 3. Noticing the regularity in the way terms cancel when expanding (x - 1)(x + 1), (x - 1)(x
2 + x + 1), and (x - 1)(x
3 + x
2 + x + 1) might lead them to the
general formula for the sum of a geometric series. As they work to solve a problem, mathematically proficient students maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results. This is an explicit habit of mind that is given serious attention in Integrated CME Project, especially in the courses that contain algebra. Our approach to finding equations and functions that model situations, to finding equations of lines, to fitting functions to data, and to finding and establishing algebraic identities all revolve around this practice. Often, students look at a problem and have no idea how to start. A common mantra in Integrated CME Project is “try it with number.” Mathematics is open to experiment, and general results, or at least conjectures, often spring from trying examples, looking for regularity, and seeking what seem to be general trends.
