1 Mathematics Framework

2 Adopted by the State Board of Education on July 12, 2023

3 Page 1 of 55

4 Mathematics Framework
5 Chapter 7: Mathematics: Investigating and
6 Connecting, Grades Six through Eight

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8 Mathematics Framework Chapter Seven: Mathematics: Investigating and Connecting,

9 Grades Six through Eight.................................................................................................
10 Introduction...................................................................................................................
11 The Importance of Middle School Math Experiences...................................................
12 Investigating and Connecting Mathematics..................................................................
13 Teaching the Big Ideas.................................................................................................
14 Designing Instruction to Investigate and Connect the Why, How, and What of
15 Mathematics..............................................................................................................
16 The Importance of the Drivers of Investigation and Content Connections..............10
17 The Importance of the Standards for Mathematical Practices................................12
18 Investigating and Connecting, Grades Six Through Eight..........................................16
19 Support for English Learners..................................................................................18
20 Content Connections Across the Big Ideas, Grades Six Through Eight.................21
21 The Big Ideas, Grades Six Through Eight..................................................................40
22 Conclusion..................................................................................................................49
23 Long Descriptions for Chapter 7.................................................................................50

24 Introduction

25 Mathematics in the middle grades, the focus of this chapter, builds on the foundational
26 understanding of math concepts students learned in the earlier grades, including place
27 value, arithmetic operations, fractions, geometric shapes and properties, and data and
28 measurement. Students’ solid grasp of these concepts supports their learning of the
29 major middle-grade topics: proportional reasoning, rational numbers, measurement in
30 geometrical and data science scenarios—concepts that, in turn, provide the base for
31 success in high school mathematics.

32 The critical element of success continues to be piquing students’ curiosity and interest
33 through engagement with meaningful and relevant math activities and experiences. As
34 this chapter discusses, students’ middle school experiences are pivotal in shaping
35 their attitudes toward math and self-perceptions as math learners. Combined with the
36 guidance they receive, those experiences determine whether or not students get on a

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37 pathway to high level math, crucially affecting their mathematics futures in high school
38 and beyond.

39 The Importance of Middle School Math Experiences

40 Middle grade mathematics experiences are pivotal not just because of the content
41 students are learning and the mathematical practices they are honing, but also
42 because this is a critical period in the development of their attitudes toward math and
43 perceptions of themselves as math learners. It is a time when students make choices
44 about mathematics coursework—or have those choices made for them—that have
45 long-term implications, including for their college and career achievements (Falco,
46 2019). Among these choices are whether some or all students begin the high school
47 math sequence with Algebra I or Mathematics I in middle school. As described in
48 chapters 8 and 9, this can enable students who are well-prepared to experience more
49 course-taking options in high school. At the same time, it is important that schools
50 organize for student success to ensure that lack of readiness or curriculum gaps do
51 not undermine foundational understanding for students who accelerate, or for others.
52 Whether middle schools decide to teach the CA CCSSM grade six through eight
53 curriculum to all or most students or to enable some or all students to accelerate
54 before high school, they should be prepared to close learning or curriculum gaps in
55 planful ways to enable maximum success for all students. Strategies for doing so are
56 detailed in chapter 9. Research conducted in middle school settings has found that
57 over the course of a school year many middle-grade students come to perceive
58 mathematics as less valuable and report reduced effort and persistence in in this
59 subject area (Pajares and Graham, 1999).

60 Middle school girls in particular tend to exhibit reduced self-efficacy around

61 mathematics (Falco, 2019), and that self-efficacy—belief in one’s own mathematics
62 ability—is a significant predictor of high school math success (Petersen and Hyde,
63 2017). Moreover, girls as a group, as well as African American and Latino students in
64 general, are underrepresented in Science, Technology, Engineering, and Mathematics
65 (STEM) fields. Students in these groups tend to experience significantly more

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66 academic barriers to mathematics exposure in the elementary and middle grades,

67 barriers that are negatively associated with high school math achievement (Williams et
68 al., 2016).

69 This framework is intended to help teachers ensure that the math experiences of all
70 their students are positive. It highlights the importance of keeping students actively
71 engaged in learning mathematics by piquing their curiosity and eliciting their interest
72 through math activities and experiences that students find meaningful and relevant.
73 The idea is for teachers to help students experience the “wonder, joy, and beauty of
74 math” (National Council of Teachers of Mathematics) and help students develop and
75 sustain a positive identity as capable mathematics learners.

76 As discussed in chapter one, teachers activate students’ curiosity and positive

77 disposition toward mathematics by providing a learning environment that inspires
78 wonder and affirms the connections among the math topics students encounter. As
79 they learn, students recognize that their learning is part of the magnificent and
80 coherent body of mathematical understanding. In this learning environment,
81 instruction shows students that their own thinking about mathematics matters and that
82 their differing backgrounds and capabilities contribute to a greater mathematical
83 understanding for everyone in the class. Students come to see that with every hard-
84 won realization, subtle and creative explanation, and deeper connection or complex
85 idea produced, their understanding is expanding and they are advancing as
86 developing mathematicians. In this environment. Teachers are champions of the
87 cause and facilitators of learning, rather than disseminators of information for students
88 to learn by rote.

89 Investigating and Connecting Mathematics

90 The goal of the California Common Core State Standards for Mathematics (CA
91 CCSSM) at every grade is for students to make sense of mathematics. To achieve
92 this goal, the framework recommends taking a “big ideas” approach to math teaching,
93 one in which mathematics is presented as a series of big ideas that enfold clusters of

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94 standards and connect concepts. As with transitional kindergarten through grade five
95 classrooms, for grades six through eight, the framework envisions mathematics
96 teaching and learning as a vibrant, multidimensional, interactive, and student-centered
97 endeavor of investigating and connecting big ideas. In that process, teachers focus on
98 ensuring that instruction meets the full range of student learning needs.

99 Starting in the earliest grades and throughout the middle and high school grade levels,
100 teachers design and carry out instruction that engages students in investigating the
101 big ideas and connecting content and mathematical practices within and across grade
102 levels and mathematical domains. This approach emphasizes students’ active
103 engagement in the learning process and provides students with frequent opportunities
104 for students to work with one another in connecting and communicating about the big
105 ideas.

106 Frequent opportunities for mathematical discourse, like implementing structured math
107 talks, create a climate for mathematical investigations, which promote understanding
108 (Sfard, 2007), language for communicating (Moschkovich, 1999) about mathematics,
109 and development of mathematical identities (Langer-Osuna and Esmonde, 2017).
110 Teachers create opportunities for students to construct mathematical arguments and
111 attend to, make sense of, and respond to the mathematical ideas of others. This
112 discourse, in turn, supports development of language proficiency in mathematics.

113 Ensuring frequent opportunities for mathematical discourse. As discussed in chapter

114 2, teachers facilitate student engagement and learning when they take an assets-
115 based approach to instruction—notably by cultivating a classroom environment that is
116 both culturally and linguistically responsive. Providing frequent opportunities for
117 mathematical discourse is one way of developing this type of climate for mathematical
118 investigations. Mathematical discourse can focus student thinking on tasks like
119 offering, explaining, and justifying mathematical ideas and strategies, as well as
120 attending to, making sense of, and responding to other people’s mathematical ideas.

121 Mathematical discourse entails communicating about mathematics with words,

122 gestures, drawings, manipulatives, representations, symbols, and other tools that are

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123 helpful for learning. In the early grades, for example, students might explore geometric
124 shapes, investigate ways to compose and decompose them, and reason with peers
125 about attributes of objects. Teachers’ orchestration of mathematical discussions (see
126 Stein and Smith, 2011), such as the reasoning segment of the activity in the example,
127 involves modeling mathematical thinking and communication, noticing and naming
128 students’ mathematical strategies, and orienting students to one another’s ideas.

129 Supporting development of language proficiency in mathematics. Mathematics is

130 considered by many to be a universal language, recognized throughout the world. All
131 California transitional kindergarten through grade twelve students are learning the
132 language of math, including its vocabulary. But students who are English learners
133 integrated in an English-only setting face the added challenge of learning mathematics
134 content and the language of instruction simultaneously. These students bring
135 experiences, perspectives, and ideas that enrich the classroom for all, and
136 instructional strategies that are designed to meet their needs, and that are aligned
137 with the California English Language Development Standards (CA ELD Standards),
138 support mathematical learning for all students.

139 Students who are English learners are most supported in learning the languages of
140 English and mathematics when they are given the opportunity to reason about
141 mathematics in small-group and whole-class discussions, listening to and connecting
142 with the ideas of other students (Zwiers, 2018). Students who engage in such
143 conversations develop these two important languages simultaneously. As Zwiers
144 points out, it is more productive to create engaging tasks that challenge students to
145 use reasoning, than to isolate particular words or use sentence starters: “We don’t
146 want to put the cart of language before the horse of understanding” (2018, 10).

147 Language development is supported when mathematical ideas are paired either
148 visually or physically with verbalizations. Tasks that show or require visual thinking
149 and that encourage discussion are ideal, and students can be encouraged to start
150 group work by asking each other, “How do you see the idea? How do you think about
151 this idea?” Support can also include the use of students’ first language.

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152 The English Learners Success Forum provides guidance on ways to develop
153 students’ language proficiency as they learn mathematics. That guidance
154 encompasses five focus areas (English Learners Success Forum, n.d.):

155 1. Interdependence of Mathematical Content, Practices, and Language;

156 2. Scaffolding and Supports for Simultaneous Development;
157 3. Mathematical Rigor Through Language;
158 4. Leveraging Students' Assets; and
159 5. Assessment of Mathematical Content, Practices, and Language.

160 (For more on support for students who are English learners, see the section
161 “Investigating and Connecting, Grades Six Through Eight,” below.)

162 Teaching the Big Ideas

163 As discussed in chapter two, teaching the big ideas of mathematics is one of the five
164 main components of teaching for equity and engagement. Big ideas are central to the
165 learning of mathematics, link numerous mathematics understandings into a coherent
166 whole, and provide focal points for student investigations (Charles, 2005). In this
167 framework, the big ideas are delineated by grade level and are the core content of
168 each grade. For example, in grade six there are 10 big ideas that form an organized
169 network of connections and relationships; the ideas are distance and direction, nets
170 and surface area, variability in data, relationships between variables, the shape of
171 distribution, graphing shapes, fraction relationships, model the world, patterns inside
172 numbers, and generalizing with multiple representations. The big ideas and their
173 connections for each middle-grade level are diagramed below in the section, “The Big
174 Ideas, Grades Six Through Eight.”

175 In the classroom, teachers engage students with the big ideas by designing instruction
176 around students’ investigations of intriguing experiences that are relevant to students’
177 grade level, background, and interests. Investigations motivate students to learn
178 focused, coherent, and rigorous mathematics. They also help teachers keep
179 instruction focused on the big ideas. Far from haphazard, investigations as envisioned

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180 in this framework are guided by a conception of the why, how, and what of
181 mathematics—a conception that makes connections across different aspects of
182 content and also connects content with mathematical practices.

183 Instructional materials should primarily focus on tasks that invite students to make
184 sense of important ideas, wonder in authentic contexts, and seek to investigate
185 mathematical questions. As students discuss mathematical ideas, their current
186 understandings may provide opportunities for rich discussion. Teachers who work
187 through an investigation themselves before their students embark on it can anticipate
188 and prepare students to take advantage of such moments. They can also note the ways
189 mathematical practices emerge in the investigations. It is important to remember that
190 teachers and students alike are doers of mathematics, and that their understanding of
191 the material evolves in the doing. For teachers, that doing includes lesson planning,
192 implementation, and reflection.

193 As students work through mathematical investigations, they and their teacher can
194 engage in discussions around the ideas that emerge in the investigation. The concepts
195 students identify and the connections they make are just as important as finding
196 answers.

197 Designing Instruction to Investigate and Connect the Why, How, and
198 What of Mathematics

199 To help teachers design instruction using the big-ideas approach, figure 7.1 maps out
200 the interplay at work when this conception is used to structure and guide student
201 investigations (see chapter 1). Three Drivers of Investigation (DIs)—sense-making,
202 predicting, and having an impact—provide the why of an activity. Eight Standards for
203 Mathematical Practice (SMPs) provide the how. And four Content Connections (CCs),
204 which ensure coherence throughout the grade levels, provide the what.

205 Figure 7.1 The Why, How, and What of Mathematics

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206

207 Long Description of figure 7.1

208 Figure 7.2 Drivers of Investigation, Standards for Mathematical Practices, and Content

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209 Connections

210

211 Long description of figure 7.2

212 The Importance of the Drivers of Investigation and Content

213 Connections

214 While chapter five focuses on the SMPs, this chapter and chapter six (grades TK–5)

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215 are organized around Drivers of Investigation and Content Connections. The three DIs
216 aim to ensure that there is always a reason to care about mathematical work and that
217 investigations allow students to make sense, predict, and/or affect the world. The four
218 CCs organize content and connect the big ideas—that is, provide mathematical
219 coherence—throughout the grades.

220 Drivers of Investigation

221 DI1: Make sense of the world (understand and explain)

222 DI2: Predict what could happen (predict)

223 DI3: Impact the future (affect)

224 Content Connections

225 CC1: Reasoning with data

226 CC2: Exploring changing quantities

227 CC3: Taking wholes apart, putting parts together

228 CC4: Discovering shape and space

229 To teach the grade level’s big ideas, a teacher will design instructional activities that
230 link one or more of the CCs with a DI—for example, link reasoning with data (CC1) to
231 predict what could happen (DI2), or link exploring changing quantities (CC2) to impact
232 the future (DI3). Because students actively engage in learning when they find purpose
233 and meaning in the learning, instruction should primarily involve tasks that invite
234 students to make sense of the big Ideas through investigation of questions in
235 authentic contexts.

236 An authentic activity or task is one in which students investigate or struggle with
237 situations or questions about which they actually wonder. Lesson design should be
238 built to elicit that wondering. For example, environmental issues on the school campus
239 or in the local community provide rich contexts for student investigations and
240 mathematical analysis, which, concurrently, help students develop their understanding
241 of California’s Environmental Principles and Concepts. A mathematics activity or task

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242 can be considered authentic if, as students attempt to understand the situation or
243 carry out the task, they see the need to learn or use the mathematical idea or strategy.

244 The four CCs are of equal importance; they are not meant to be addressed
245 sequentially. As captured in figure 7.2, Content Connections, Standards for
246 Mathematical Practices, and Drivers of Investigation, there is considerable crossover
247 between and among the practice standards and the content connections.

248 The content involved over the course of a single investigation cuts across several CA
249 CCSSM domains—for example, it may involve both Measurement and Data (MD),
250 Number and Operations in Base Ten (NBT), as well as Operations and Algebraic
251 Thinking (OA). Students simultaneously employ several of the SMPs as they conduct
252 their investigations.

253 The vignette Followed by a Whale exemplifies how the combination of DIs, CCs, and
254 SMPs can provide a powerful three-dimensional form of learning for students.
255 Integrating math with science and language arts, it features a task in which students
256 learn the real-life story of a swimmer who is followed by a baby whale. The swimmer
257 needs to decide if she should continue swimming to the shore, possibly beaching or
258 endangering the baby whale, or swim out to a nearby oil rig where the whale’s mother
259 may be located. To decide on a course of action, students must analyze proportional
260 relationships, add fractions, use ratio reasoning, compare two different functions, and
261 make use of data. They also need to account for ways their action will impact the
262 future.

263 The Importance of the Standards for Mathematical Practices

264 The CA CCSSM offer grade-level-specific guidelines for what mathematics topics are
265 considered essential to learn and for how students should engage in the discipline
266 using the SMPs. The SMPs reflect the habits of mind and of interaction that form the
267 basis of math learning—for example, reasoning, persevering in problem solving, and
268 explaining one’s thinking. The SMPs provide clear intent for the types of productive
269 actions and habits of thinking students engage in as they learn mathematics. As

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270 indicated by the joint statement released by the University of California, California
271 State University, and Community College systems (Intersegmental Committee of the
272 Academic Senates, 2013), the SMPs provide a sound foundation for the types of
273 mathematical work expected in higher education.

274 Standards for Mathematical Practice

275 SMP1. Make sense of problems and persevere in solving them

276 SMP2. Reason abstractly and quantitatively

277 SMP3. Construct viable arguments and critique the reasoning of others.

278 SMP4. Model with mathematics

279 SMP5. Use appropriate tools strategically

280 SMP6. Attend to precision

281 SMP7. Look for and make use of structure

282 SMP8. Look for and express regularity in repeated reasoning

283 To teach mathematics for understanding, it is essential to purposefully cultivate

284 students’ use of the practices. The introduction to the CA CCSSM is explicit on this
285 point. Identifying content standards and practice standards as two halves of a
286 powerful whole, it says effective mathematics instruction requires that the SMPs be
287 taught as carefully and intentionally as the content standards (CA CCSSM, 3). The
288 SMPs are designed to support students’ development across the school years.
289 Whether in the primary grade levels or high school, for example, students make sense
290 of problems and persevere to solve them (SMP.1).

291 The importance of the SMPs is discussed at length in chapter four, which provides
292 additional guidance on how teachers can cultivate students’ skillful use of the
293 practices. Using three interrelated SMPs for illustration, chapter four demonstrates
294 how teachers across the grade levels can incorporate key mathematical practices and
295 integrate them with each other to create powerful math experiences centered on
296 exploring, discovering, and reasoning. Such experiences enable students to develop

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297 and extend their skillful use of the practices as they move through the progression of
298 math content in the coming grade levels.

299 The SMPs are central to the mathematics classroom. From the earliest grades and on
300 through the middle and higher grade levels, mathematics requires that students make
301 sense of and work through problems, and students need the SMPs to successfully do
302 so.

303 What is a Model?

304 Modeling, as used in the CA CCSSM, is primarily about using mathematics to
305 describe the world. In elementary mathematics, a model might be a representation,
306 such as a math drawing or a situation equation (operations and algebraic thinking),
307 line plot, picture graph, or bar graph (measurement), or a building made of blocks
308 (geometry). In grades six and seven, a model could be a table or plotted line (ratio and
309 proportional reasoning) or box plot, scatter plot, or histogram (statistics and
310 probability). In grade eight, students begin to use functions to model relationships
311 between quantities. In high school, modeling becomes more complex, building on
312 what students have learned in kindergarten through grade eight.

313 Representations such as tables or scatter plots often serve as intermediate steps in
314 developing a model rather than serving as models themselves. The same
315 representations and concrete objects used as models of real-life situations are used to
316 understand mathematical or statistical concepts. The use of representations and
317 physical objects to understand mathematics is sometimes referred to as “modeling
318 mathematics,” and the associated representations and objects are sometimes called
319 “models.”

320 Readers are encouraged to review current information about modeling in the CCSS
321 progressions.

322 Because SMPs are linguistically demanding, as students learn and use them they
323 develop both skill in the practices and the language needed for fully engaging in the
324 discipline of mathematics. Regularly using the SMPs gives students opportunities to

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325 make sense of the specific linguistic features of the genres of mathematics, and to
326 produce, reflect on, and revise their own mathematical communications. That being
327 said, educators must remain aware of and provide support for students who may
328 grasp a concept, yet struggle to express their understanding. For students who are
329 English learners, as well as for students with other special learning needs, small-
330 group instruction can be useful for helping students develop the language needed for
331 engaging with the mathematical concepts and standards for an upcoming lesson.
332 (See chapter four for further discussion.)

333 As students use the SMPs, teachers have the opportunity to engage in formative
334 assessment and provide students with real-time feedback. Students can express an
335 idea in their own words, build a concrete model, illustrate their thinking pictorially,
336 and/or provide examples and possibly counter examples. A teacher might observe
337 them making connections between ideas or applying a strategy appropriately in
338 another related situation (Davis, 2006). Many useful indicators of deeper
339 understanding are actually embedded in the SMPs themselves. For example,
340 teachers can note when students analyze the relationships in a problem so that they,
341 the students, can understand the situation and identify possible ways to solve the
342 problem (SMP.1). Other examples of observable behaviors specified in the SMPs
343 include students’ abilities to use mathematical reasoning to justify their ideas (SMP.3);
344 draw diagrams of important features and relationships (SMP.4); select tools that are
345 appropriate for solving the particular problem at hand (SMP.5); and accurately identify
346 the symbols, units, and operations they use in solving problems (SMP.6).

347 Students who regularly use the SMPs in their mathematical work develop mental
348 habits that enable them to approach novel problems, as well as routine procedural
349 exercises, and to solve them with confidence, understanding, and accuracy.
350 Specifically, recent research shows that an instructional approach focused on
351 mathematical practices may be important in supporting student achievement on
352 curricular standards and assessments and that it also contributes to students’ positive
353 affect and interest in mathematics (Sengupta-Irving and Enyedy, 2014).

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354 Investigating and Connecting, Grades Six Through Eight

355 In grades six through eight, students deepen their understanding of fractions
356 developed in the earlier grades, especially division of fractions, and develop an
357 understanding of ratios and proportions. These understandings bridge to a new type
358 of numbers—rational numbers—that are inclusive of all the number types students
359 have previously studied (whole numbers, integers, fractions, and decimals). Students
360 connect ratios, rates, and percentages and use these ideas to engage in proportional
361 reasoning as they solve authentic problems. By writing, interpreting, and using
362 expressions and equations, students can solve multistep problem situations. By
363 characterizing quantitative relationships using functions, they further develop
364 understanding of rates and changing quantities. Measurement and classification ideas
365 associated with two- and three-dimensional shapes and figures are connected to real-
366 world and algebraic representations. Measurement questions extend to the need for
367 measuring populations, using statistical inferences with sampling.

368 Rather than insisting on mastery of prior content or, especially, computational speed
369 and recall, it is important for teachers to focus on ensuring that students have access
370 to the content needed for the investigation at hand. (See chapter 3 for explanation of
371 the framework’s interpretation of fluency as flexibility in thinking, rather than only as
372 only speed in use of memorized facts.) Thus, tools that allow increased focus on
373 sense-making and building number sense should be readily available. These include
374 calculators and online tools, but also strategies and scaffolds centered on students
375 who are English learners. When used strategically, such tools allow students greater
376 access and can also support students’ completion of the mathematics assessments in
377 the California Assessment of Student Performance and Progress (CAASPP).

378 These tools do not replace the as-needed reinforcement or continued development of
379 students’ understanding of earlier grade-level concepts. However, instruction should
380 enable students to engage in grade-level investigations without a remedial precursor.
381 When grade-level activities entail a need for students to understand math content and
382 practices they have previously encountered, but perhaps not mastered, students have

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383 an incentive and are therefore more ready to revisit and deepen their understanding of
384 the earlier material.

385 A note on unfinished learning from previous grades: Students develop and learn at
386 different times and rates. For this or other reasons, some start a new grade level with
387 unfinished learning from earlier grade levels. In such cases, teachers should not
388 automatically assume these students to be low achievers, require interventions, or
389 need placement in a group that is learning standards from a lower grade level.
390 Instead, teachers need to identify students’ learning needs and provide appropriate
391 instructional support before considering interventions or any change in standards
392 taught. Figure 7.3 (adapted from Fossum, 2018) provides a helpful guide for
393 supporting for students with unfinished learning.

394 Figure 7.3 Supporting Students Who Have Unfinished Learning from Earlier Grades
Common Instructional Misstep Recommended Alternative
Blindly adhering to a pacing guide Use formative data to gauge student
calendar understanding and inform pacing

Halting whole-class instruction to provide Provide just-in-time support within each

a broad review of past material unit or during intervention
Trying to address every gap a student Prioritize and address the most essential
has prerequisite skills and understanding for
upcoming content
Trying to build missing understanding of Trace the learning progression, diagnose,
past material from the ground up or going and go back just enough to provide
too far back in the learning progression access to grade-level material
Re-teaching students using previously Provide a new experience to re-engage
failed methods and strategies students, as appropriate (San Francisco
Unified School District Mathematics
Department, 2015)
Disconnecting intervention from content Connect learning experiences in
students are learning in math class intervention and universal instruction
(CAST, 2018)
Choosing content for intervention based Focus on big ideas from current or
solely on students’ weakest areas previous grades as they relate to
upcoming content
Teaching all standards addressed in an Consider the Aspect of Rigor called for in
intervention in a step-by-step, procedural this framework (chapter 1) when
way designing and choosing tasks, activities,
or learning experiences

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Common Instructional Misstep Recommended Alternative

Over-reliance on computer programs in Facilitate rich learning experiences for
intervention students to complete unfinished learning
from previous or current grade

395 When students would benefit from extra support, it is advisable to offer them
396 opportunities to engage with math in ways that differ from their previous math
397 exposure—for example, by using more-visual approaches or using metaphorical
398 models, such as a pan balance as an equation. Teachers and administrators at the
399 middle-grade levels, as well as parents of students at these levels, are encouraged to
400 read Chapter 9: Structuring School Experiences for Equity and Engagement. That
401 chapter contains information for schools to consider as they structure activities,
402 classes, and schedules that can meet the many and varied needs of math learners at
403 these levels, including preparing all students for success in high school mathematics
404 courses and beyond.

405 Support for English Learners

406 While some students, indeed, lag in math mastery, for others, what appears to be lack
407 of understanding may be attributable, at least in part, to their inability to adequately
408 communicate their understanding. Here, too, providing appropriate instructional
409 support, for both content and language development is essential. In such cases,
410 teachers can use scaffolds and supports specifically oriented to students who are
411 English learners.

412 Instruction should always be designed to ensure that students at all levels of language
413 development can engage deeply with the important mathematical ideas of the
414 instruction (Walqui and van Lier, 2010). Principles and strategies for language
415 development—especially important for students who are English learners, but
416 valuable for all students—can be explored in Moschkovich (2013), Zwiers et al.
417 (2017), and Zwiers (2018), among many others.

418 Among instructional principles that enable engagement for students across the broad
419 spectrum of English language ability are

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420 ● focus on students’ mathematical reasoning, not accuracy in using language

421 (Moschkovich, 2013);
422 ● support sense-making (Zwiers et al., 2017);
423 ● optimize output and cultivate conversation (Zwiers et al., 2017);
424 ● use student conversations to foster reasoning and related language (Zwiers,
425 2018); and
426 ● maximize linguistic and cognitive meta-awareness (Zwiers et al., 2017).

427 Deliberate instructional routines can support the implementation of these principles.
428 The following recommendations reference the state’s ELD Standards they help
429 achieve:

430 1) Use iteration to help students develop stronger and clearer ideas and language
431 (e.g., through successive pair-shares; asking students to students convince
432 yourself, a friend, a skeptic). [CA ELD Standards Part I.A.1, Part I.B.5–6, Part
433 I.C.9–12, Part II.B.3–5].
434 2) Collect and display student thinking and sense-making language (e.g., gather
435 and show student discourse; use number and data talks) [CA ELD Standards
436 Part I.B.5–8].
437 3) Have students critique, correct, and clarify the work of others (e.g., ask them to
438 critique a partial or flawed response; have them use an always-sometimes-
439 never organizer to evaluate mathematical statements) [CA ELD Standards Part
440 I.A.1, Part I.B.5–6, Part I.C.9–12, Part II.B.3–5].
441 4) Create a need for students to communicate by distributing information within a
442 group (e.g., information-gap cards, games) [CA ELD Standards Part I.A.1–4,
443 Part I.B.5].
444 5) Have students to explore a context and to co-craft related questions and
445 problems. [CA ELD Standards Part 1.A.1–4, Part I.B.5].
446 6) Create opportunities for students to reflect on the way mathematical questions
447 are presented, and equip them with tools for negotiating meaning (e.g., three
448 reads; values/units chart) [CA ELD Standards Part I.5–8].
449 7) Foster students’ meta-awareness and ability to make connections between

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450 approaches, representations, examples, and language (e.g., have them use
451 compare-and-connect solution strategies; which one doesn’t belong?) [CA ELD
452 Standards Part I.A.1, Part I.B.5–6, Part I.C.9–12, Part II.B.3–5].
453 8) Support rich and inclusive discussions about mathematical ideas,
454 representations, contexts, and strategies (Use whole-class discussion
455 supports; have students do numbered heads together) (Zwiers et al., 2017) [CA
456 ELD Standards Part I.A.1–4].

457 The generic term “English learner” masks a great deal of variability in students’
458 experiences. For students at secondary level, some researchers (e.g., Freeman and
459 Freeman, 2002) group English learners by those who are newly arrived, which
460 generally means within the last four or five years, and have a pre-arrival history of
461 adequate schooling; those who are newly arrived and have a history of limited formal
462 schooling; and those who are designated as long-term English learners.

463 By understanding students’ lives outside of school and their previous schooling
464 experiences, schools can thoughtfully place students who are English learners in the
465 appropriate, and supported, setting for learning mathematics. Older students who had
466 sufficient opportunity for schooling before arriving are focused more on translating the
467 content and building on their existing mathematics literacy skills. Students who had
468 limited formal schooling benefit from programs with rich experiences in which they can
469 develop academic literacy, perhaps even reading. For both of these student types,
470 access to native language instruction serves as a bridge to instruction in English.
471 Dual-language, bilingual programs, and other blended approaches can offer
472 mathematics courses in a combination of students’ first language and of the target
473 language, in this case English.

474 Students who are long-term English learners tend to have somewhat different
475 characteristics from newly arrived English learners. Many are US-born and their entire
476 education experience has been in US schools. Although they are generally native
477 English speakers, English is not their home language and it’s for that reason they
478 were placed in language development programs in their early grades. A formal exit

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479 from most of these programs requires students to demonstrate English proficiency on
480 state-approved assessments and (usually) to also show on-grade-level academic
481 performance. Many students cannot meet this high threshold, causing them to remain
482 in language support programs for many years.

483 Among students who are long-term English learners, many can speak and write
484 conversational English, both with their peers and with teachers. In this sense, they
485 may be hard to distinguish from other English learners. These factors can mask their
486 literacy needs, obscuring the support they require to succeed academically. Students
487 who are long-term English learners tend to underperform and are often placed in
488 lower academic tracks. Understanding this can help schools develop mathematics
489 course pathways that are equitable, that provide the academic language scaffolding
490 needed by these students need, and that provide such support without putting them
491 into programs for students who are newly arrived English learners. Heterogeneous
492 classrooms can provide a broader array of access points and supports for both new
493 and long-term English learners.

494 Across these broad groups of students who are English learners, the basic
495 components of effective programs remain the same. Students should learn content
496 with rich thematic instruction that attends to big ideas and with challenging and
497 connected content, in a learning environment that incorporates collaboration,
498 feedback, language scaffolding, and respect for cultural diversity (Freeman and
499 Freeman, 2002).

500 Content Connections Across the Big Ideas, Grades Six Through
501 Eight

502 The big ideas for each grade level define the critical areas of instructional focus. By way
503 of the Content Connections, the big ideas unfold in a progression across the grade
504 levels, in accordance with the CA CCSSM principles of focus, coherence, and rigor.
505 Figure 7.4, Progression Chart of Big Ideas through Grades Six Through Eight, identifies
506 some of the big ideas for grades six through eight and indicates the CCs with which

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507 they are readily associated. The chart is followed by discussion of each CC, which
508 highlights specific associated SMPs, content standards, and activities.

509 Later in this section, find, for each grade level from six through eight, a figure with a
510 diagram of the big ideas for that grade level, as well as figure with a table of the CCs,
511 big ideas, and standards specific to the grade level.

512 Figure 7.4: Progression Chart of Big Ideas, Grade Levels Six Through Eight

Content Big Ideas: Grade Big Ideas: Grade Big Ideas: Grade
Connections Six Seven Eight
Reasoning with Variability in data Visualize Data explorations
Data Populations
Reasoning with The shape of Populations and Data graphs and
Data distributions samples tables
Reasoning with n/a Probability Models Interpret scatter
Data plots
Exploring Changing Fraction Proportional Multiple
Quantities relationships Relationships representations of
functions
Exploring Changing Patterns inside Unit rates in the Linear equations
Quantities numbers world
Exploring Changing Generalizing with Graphing Slopes and
Quantities multiple relationships intercepts
representations
Exploring Changing Relationships Scale Drawings Interpret scatter
Quantities between variables plots
Taking Wholes Model the world Shapes in the world Cylindrical
Apart, Putting Parts investigations
Together
Taking Wholes Nets and Surface 2-D and 3-D Pythagorean
Apart, Putting Parts Area connections explorations
Together
Taking Wholes n/a Angle relationships Big and small
Apart, Putting Parts numbers
Together
Discovering shape Nets and Surface Shapes in the world Shape, number,
and space Area and expressions

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Content Big Ideas: Grade Big Ideas: Grade Big Ideas: Grade
Connections Six Seven Eight
Discovering shape Distance and 2-D and 3-D Pythagorean
and space direction connections explorations
Discovering shape Graphing shapes Scale drawings Cylindrical
and space investigations
Discovering shape n/a Angle relationships Transformational
and space geometry

513 The following section explains the four Content Connections and provides examples
514 for each.

515 CC1: Reasoning with Data

516 Grades six through eight mathematics courses should give prominence to statistical
517 understanding and reasoning with and about data—reflecting the growing importance of
518 data in most mathematical situations that students will encounter in their lives. In
519 support of understanding and explaining their world, predicting it, and affecting it—all
520 Drivers of Investigation—students will carry out investigations using data they have
521 generated or have accessed from publicly available sources. These investigations help
522 students see data investigations as integral to their own lives, including other disciplines
523 they do or will study, such as science and social studies. Data-based investigations will
524 draw from Content Connections, such as exploring changing quantities.

525 An example of a data investigation that integrates learning in different subjects is

526 described in the vignette Crows, Seagulls, and School Lunches. The lesson described
527 in the vignette gives students opportunities to wonder about a situation that directly
528 affects them and generate questions based on what they wonder about: the gathering of
529 different kinds of birds at the student eating area during and after lunchtimes. Drawing
530 on environmental, scientific, and mathematical standards, the vignette also shows how
531 an authentic inquiry in which students collect and analyze data can cover all three
532 Drivers of Investigation, allowing students to reason with data in a way that helps them
533 make sense of the world, predict what could happen, and impact the future.

534 The CA CCSSM articulate a range of new expectations for data literacy, statistics, and

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535 data sense-making in the middle grades, some of which are new to teachers—who
536 are not likely to have been taught this content themselves. (For ways to ensure
537 teacher support to rethink mathematics teaching and acquire needed skills and
538 strategies, see chapter 10.) The content includes

539  data in the world: exploration, interpretation, decision making, ethics;

540  statistical variability: Describing, displaying, and comparing;
541  sampling to understand a population: randomness, bias, how many?;
542  multivariate thinking expressing dependence with functions and equations, to
543 answer the question “are they related?”; and
544  probability as the basis for data-based claims, the answer to the question,
545 “what are the chances?”

546 As in earlier grade levels, students experience quantitative modeling as a tool to help
547 them understand their worlds via a process that begins with wondering questions. The
548 middle grades also mark the beginning of the mathematical modeling cycle (Pelesko,
549 2015; see box What is a Model? above), and more formal instruction and
550 investigations with statistics, data science, and science (NGSS Lead States, 2013).
551 (See also chapter 5.)

552 One important aspect of data literacy is highlighting for students how many of them—
553 along with many adults—regularly surrender personal data, whether through apps,
554 online purchases, or interactive video games—and helping them investigate the
555 potential ramifications.

556 On the more positive side, students should develop an understanding of the new and
557 creative ways data can be displayed, beyond bar graphs or pie charts. Ideal data
558 visualizations to share with students to help develop their data literacy are interesting
559 and relevant to students and that also display data in new ways for students or that
560 have some quirks or features that make the visualization harder to read but may also
561 make it more engaging. Instruction oriented to data literacy can start with a data talk
562 (modeled after a number talk) that begins with a complex data visualization, in which
563 students are asked, what do you notice, what do you wonder, what is going on in this

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564 visualization. The New York Times section “What is Going on in this Graph?” (NY
565 Times, n.d.) provides current, topical, and novel representations of data that can serve
566 as strong examples for data talks.

567 Data talks provide a space for students to consider and interpret a variety of data and
568 data representations in a low-stakes, exploratory environment. After considering a
569 particular visualization, provided along with any necessary supports and scaffolds
570 (including those related to language) and enough time to process it, students discuss
571 what they notice. This helps them engage in conversations in which they describe
572 their observations and insights, including, for example, how the visual is structured,
573 one or more questions that the data are answering, or any questions about the data
574 that are not addressed by the visual or that, instead, are prompted the visual.
575 Teachers need not be experts in the content displayed in the visual; in fact, when
576 teachers field questions they cannot answer, they can use it as an opportunity to
577 model the curiosity that comes when an answer is not known. That modeling
578 demystifies a common notion among students that teachers have limitless knowledge;
579 by doing so, it reinforces the idea that understanding is an ongoing endeavor for
580 everyone and that curiosity is always an opportunity to understand more.

581 Data talks also offer a valuable way for teachers to be culturally responsive in their
582 instruction by bringing in student experiences and helping students develop critical
583 consciousness: the ability to identify, analyze, and solve real-world problems that
584 result in social inequalities. Further detail and ideas for the teaching of data literacy
585 and data science are given in chapter 5.

586 The vignette What’s a Fair Living Wage exemplifies lessons focused on a data
587 visualization. In this case, one that helps students see how mathematics can inform
588 their understanding of the world, including social justice issues (adapted from Berry et
589 al., 2020). As students explore cases of different wage earners and their ability to
590 make ends meet, they use systems of two linear equations to show how many hours
591 of work at minimum wage are needed to afford rent in different states in the US.

592 Teachers can poll their students to find out about their interests and build lessons

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593 about data visualization around those interests, both to motivate learning and to
594 bridge cultural divides in the classroom. While some students may not readily see
595 connections between mathematics and sports, for example, data visualization is a
596 powerful means of exploring performances of athletes. One example of a data
597 visualization that students may enjoy shows the basketball shots of Warrior’s point
598 guard Stephen Curry, as highlighted in chapter five; the data visualization is available
599 on the statistics and analytics website FiveThirtyEight (2015). Another sports-oriented
600 example, available from the National Collegiate Athletic Association, shares Division 1
601 women’s soccer games between 2017 and 2019 (approx. 6500 games). (YouCubed,
602 2020a.)

603 Online sources can provide rich datasets for students to explore and connect their
604 learning to the big mathematical ideas at their grade level. For example, figure 7.5
605 shows a data visualization and website link that a teacher might use with her students
606 to explore important environmental issues.

607 Figure 7.5 Environment-Oriented Data Visualization

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608

609 Long description of figure 7.5

610 Similarly, the Common Online Data Analysis Platform (CODAP, n.d.) enables students
611 to explore data sets. When having students do so, teachers can encourage them to
612 ask questions of the data—whether they are part of a data set teachers import into
613 CODAP or one of the datasets it provides. In any of the data investigations, students
614 can investigate patterns of association in bivariate data, visually exploring them by
615 dragging two variables to the different axes in the CODAP tool. Students can collect
616 survey data and compare the data with other previously collected survey data,

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617 drawing comparative inferences about two populations.

618 As the discussion and vignettes above illustrate, there are many ways in which middle
619 school students can be invited to be data explorers, learning about tools and
620 measures as they investigate questions they find interesting. In this way, students are
621 able to learn many of the common statistical and data science ideas formally
622 introduced in the middle grades—such as measures of center (mean, mode, median)
623 and spread (range), but also address the additional clusters of emphasis from the
624 standards included below (see chapter 5 for more information).

625 Content Connection 1 CA CCSSM Clusters of Emphasis

626 ● 6.SP: Develop understanding of statistical variability. Summarize and describe
627 distributions.
628 ● 7.SP: Use random sampling to draw inferences about a population. Draw
629 informal comparative inferences about two populations. Investigate chance
630 processes and develop, use, and evaluate probability models.
631 ● 8.SP: Investigate patterns of association in bivariate data.
632 ● 8.EE: Understand the connections between proportional relationships, lines and
633 linear equations.
634 ● 6.RP: Understand ratio concepts and use ratio reasoning to solve problems.
635 ● 7.RP Analyze proportional relationships and use them to solve real-world and
636 mathematical problems.

637 CC2: Exploring Changing Quantities

638 Counting, organizing, adding, subtracting, multiplying, and dividing quantities have
639 been of primary importance for much of students’ mathematics experiences in
640 transitional kindergarten through grade five. In grade six, students are introduced to
641 the idea that, in many cases, quantities act in concert rather than alone. Developing
642 an understanding of how quantities can vary together begins with the transition from
643 part-to-whole ratios (fractions) to other ratios that can be written in fraction form. The
644 understanding of part: whole fractions established in grade levels three through five
645 provides students with the foundation they need to explore other ratios, rates, and

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646 percents in grades six through eight. In grade six, students’ prior understanding of
647 multiplication and division of whole numbers and fraction concepts, such as
648 equivalence and fraction operations, contribute to their study of ratios, unit rates, and
649 proportional relationships. In grade seven, students deepen their proportional
650 reasoning as they investigate proportional relationships, determine unit rates, and
651 work with two-variable equations. In grade eight, they build on their work with unit
652 rates from grade six and proportional relationships from grade seven to compare
653 graphs, tables, and equations of proportional relationships and form a pivotal
654 understanding for the slope of a line as a type of unit rate. This learning progression
655 culminates in grade eight with students’ introduction to functions as one of the most
656 important types of co-varying relationships between two quantities. In a sense, in
657 grades six through eight, students transition from an understanding of quantities as
658 independent of one another to quantities that vary together.

659 Through investigations in this connected content area, students build many concrete
660 examples of functions. CC2 connects easily with CC1: reasoning with data, through
661 many rich modeling and statistics investigations. Specific contextualized examples of
662 functions are crucial precursors to students’ work with such categories of functions as
663 linear, exponential, quadratic, polynomial, and rational and to the abstract notion of
664 function. Notice that the name of the CC calls out changing quantities, not changing
665 numbers. In considering how quantities change, as opposed to strictly numbers, a
666 greater variety of contexts and representations is possible, as are connections among
667 quantities (e.g., relating paint and area). Functions referring to authentic contexts give
668 students concrete representations that can support reasoning, providing multiple entry
669 paths and reasoning strategies and require engaging in SMP.2 (Reason abstractly
670 and quantitatively). Authentic contexts also help maintain and build connections
671 between mathematical ideas and students’ lives.

672 Ratios and Proportions

673 Education research over the past several decades has focused on students’
674 understanding of ratios and proportional situations, largely because of the crucial
675 bridge that ratios and proportions form between fractions (in elementary grades) and

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676 linear relationships (in high school grades). The type of thinking that students exhibit
677 as they work on proportional situations is known as proportional reasoning, which
678 Lamon (2012) defines as “reasoning up and down in situations in which there exists
679 an invariant (constant) relationship between two quantities that are linked and varying
680 together” (3). The researcher also points out that this type of reasoning goes well
681 beyond simply setting up or solving equations of the form a/b = c/d (see also chapter 3
682 for issues that arise when cross-multiplying).

683 Lamon (1993) has characterized two dimensions of proportional reasoning, in general,
684 as relative thinking and unitizing. Lamon describes relative thinking as the ability to
685 compare quantities in problem situations, while unitizing is the ability to shift the
686 perception of the unit (or whole/unit whole) to incorporate composite units. Activities
687 and problems that foster the development of these dimensions of proportional
688 reasoning should be utilized as possible. In general, emphasis should be placed on
689 students’ ability to recognize the connections between representations of the
690 quantities in problems and the connections between solution strategies, rather than on
691 solely finding answers.

692 Carpenter et al. (1999) proposes four stages of students’ development of proportional
693 reasoning:

694 ● Level 1: Students focus on random calculations or additive differences in ratio

695 work.
696 ● Level 2: Students perceive a ratio as a single unit and can scale up or down the
697 ratio, in a multiplicative or additive fashion, by scale factors that are whole
698 numbers.
699 ● Level 3: Students still conceive of a ratio as a single unit, but they can scale the
700 ratio by non-integer amounts.
701 ● Level 4: Students recognize and make use of the relationship within a ratio and
702 between two equivalent ratios.

703 Steinthorsdottir and Sriraman’s 2009 investigation of the learning of proportions by

704 middle-grade girls yielded evidence in support of providing sequenced tasks at all four

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705 levels to support students’ development of proportional reasoning. Specifically, the

706 researchers found that tasks helping students to think about multiplicative
707 relationships both between and within ratios were beneficial for students’ learning of
708 proportions. The norms of productive discourse and provision of appropriate
709 scaffolding further supported the learning.

710 Relative Thinking

711 The approaches described in the grade six vignette Mixing Paint illustrate the relative
712 thinking described by Lamon (2012) and demonstrate a progression from
713 understanding of ratios to understanding proportional reasoning by focusing on
714 connections between differing viewpoints of the problem. In the vignette, students are
715 given a recipe for Orange Sunglow Paint that calls for three parts of yellow paint to
716 four parts of red paint. They are asked: How many parts of yellow are needed to make
717 a batch that uses 20 parts of red paint?

718 Algebra
719 Algebra is often taught through symbols and symbol manipulation, but students
720 benefit from approaching content in different ways. Mathematics with which students
721 engage visually and through words is especially important to combine with number
722 and symbol work. Approaching algebra visually enables students to see mathematics
723 as a creative and connected subject. One of the most well-known and effective
724 lessons for introducing students to thinking about algebra visually, and for helping
725 them understand algebraic equivalence, is the “border problem.”

726 In this activity, students are asked to look briefly at a border around a square, like one
727 shown in figure 7.6, below. Both the square and the border consist of some number of
728 smaller squares, and students are asked to work out how many there are in the
729 border, without counting them (see also Boaler and Humphreys, 2005). To prevent
730 students from having time to count the squares, it is important that they can see the
731 figure only briefly.

732 Students determine many different answers for the number of squares on the border
733 —40, 38, and the correct answer of 36 are typical. A variety of responses—along with

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734 the teacher’s specific open-ended questions and academic conversation sentence
735 frames—give teachers the opportunity to ask students to justify different answers, to
736 construct viable arguments, and to critique the reasoning of others. As the lesson
737 progresses, students think numerically and then verbally and eventually algebraically
738 about ways to describe the number of squares in any border and the different ways in
739 which students see the number. These different ways of seeing the border offer an
740 opportunity for students to develop multiple ways of seeing and flexibly understanding
741 algebraic equivalence.

742 Figure 7.6 Squares with Borders for Use in the Border Problem

743

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744 Long description of figure 7.6

745 Source (with full lesson plan): YouCubed, 2018.

746 Illustrating another aspect of teaching CC2, the vignette Equivalent Expressions—
747 Integrated ELD and Mathematics portrays a teacher using a particular lesson to
748 employ formative assessment strategies that allow him to gauge how well his students
749 currently understand whether two expressions are equivalent.

750 CC3: Taking Wholes Apart and Putting Parts Together

751 Students enter the middle-grade levels with many experiences of taking wholes apart
752 and putting parts together:

753 ● Decomposing numbers by place value

754 ● Assembling sub-products in an area representation of two-digit by two-digit
755 multiplication
756 ● Finding area of a plane figure by decomposing into rectangular or triangular
757 pieces
758 ● Exploring polygons and polyhedra in terms of faces, edges, vertices, and angles

759 Decomposing challenges and ideas into manageable pieces and assembling
760 understanding of smaller parts into understanding of a larger whole are fundamental
761 aspects of doing mathematics. Often these processes are closely tied with SMP.7 (Look
762 for and make use of structure). This CC spans and connects many typically separate
763 content clusters in number, algebra, and shape and space. Decomposing an area
764 computation into parts can lead to an algebraic formulation as a quadratic expression, in
765 which the terms in the expression have actual geometric meaning for students.

766 It is common to hear teacher stories of students who “know how to do all the parts, but
767 can’t put them together.” Mathematics textbooks often handle this challenge by doing
768 the intellectual work of assembly for the students (perhaps assuming that by reading
769 repeated examples, students will eventually be able to replicate). Word problems that
770 provide or identify the exact or relevant mathematical information for students, sub-

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771 problems that lay out intermediate calculations and all the reasoning, and references to
772 worked examples that are almost identical to the problem a student must work are all
773 ways of allowing students to avoid the need to assemble understanding rather than
774 developing that ability.

775 Problems that are presented with insufficient or mathematically extraneous information;
776 investigations that require students to decide how to decompose a problem, splitting
777 their work into discrete segments, and, then assemble understanding at the conclusion;
778 and problems that require piecing together factors affecting mathematical behavior
779 (such as the function assembly problems in the high school section of chapter 4) are all
780 ways to engage in this CC.

781 This CC can serve as a vehicle for student exploration of larger scale problems and
782 projects, many of which will intersect with other CCs as well. Investigations in this CC
783 require students to decompose challenges into manageable pieces and assemble
784 understanding of smaller parts into understanding of a larger whole. When students
785 conduct an investigation related to this CC, it is crucial that decomposing and assembly
786 be a student task, not one that is taken on by teacher or text. It’s helpful to have
787 students start with a simpler problem. In solving that simpler problem, students can gain
788 insight into the essential aspects of larger scale problems. Mathematicians also
789 regularly draw visual representations of relationships even when the ideas being
790 explored are not geometric (Su, 2020).

791 In grades six through eight, this CC will be especially important, and helpful, as
792 students develop understanding of the number system, Pythagorean theorem,
793 scientific notation, and angles.

794 Unitizing
795 In the problem that is the basis for the snapshot Building Apartments, students unpack
796 the notion of what constitutes the whole (also called the unit or unit whole). While
797 identifying the whole is fundamental to understanding fractions in grades three
798 through five (as described in chapters 3 and 6), it also is essential to students as they
799 make sense of proportional situations (Lamon, 2012).

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800 Snapshot: Building Apartments

801 Grade Level/Course: Grade six

802 Drivers of Investigation: 2, Predict What Could Happen

803 Content Connections: 3, Taking Wholes Apart and Putting Parts Together

804 Standards for Mathematical Practice: 1, Make sense of problems and persevere in
805 solving them; 2, Reason abstractly and quantitatively; 3, Construct viable arguments
806 and critique the reasoning of others; 4, Model with mathematics; 7, Look for and make
807 use of structure; 8, Look for and express regularity in repeated reasoning

808 Content Connection 3 CA CCSSM Clusters of Emphasis

809 ● 6.NS: Apply and extend previous understandings of multiplication and division to
810 divide fractions by fractions. Compute fluently with multi-digit numbers and find
811 common factors and multiples. Apply and extend previous understandings of
812 numbers to the system of rational numbers.
813 ● 6.EE: Apply and extend previous understandings of arithmetic to algebraic
814 expressions. Reason about and solve one-variable equations and inequalities.
815 ● 7.EE: Use properties of operations to generate equivalent expressions. Solve
816 real-life and mathematical problems using numerical and algebraic expressions
817 and equations.
818 ● 6.RP: Understand ratio concepts and use ratio reasoning to solve problems.
819 ● 7.RP Analyze proportional relationships and use them to solve real-world and
820 mathematical problems.
821 ● 7.NS: Apply and extend previous understandings of operations with fractions to
822 add, subtract, multiply and divide rational numbers.
823 ● 8.NS: Know that there are numbers that are not rational, and approximate them
824 by rational numbers.
825 ● 8.EE: Work with radicals and integer exponents. Understand the connections
826 between proportional relationships, lines and linear equations. Analyze and solve
827 linear equations and pairs of simultaneous linear equations.

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828 Relevant CA CCSSM Content Clusters/Standards:

829 ● 6.EE: Apply and extend previous understandings of arithmetic to algebraic
830 expressions. Reason about and solve one-variable equations and inequalities.
831 ● 6.RP: Understand ratio concepts and use ratio reasoning to solve problems.
832 ● 7.RP Analyze proportional relationships and use them to solve real-world and
833 mathematical problems.

834 Background: Ms. K often begins the day with her homeroom students by exploring
835 school and community events and happenings. Today, she notices an article in the
836 local newspaper about how bird nesting houses are being built in the park by the river.
837 The birds of this species are highly social and prefer to have a variety of enclosures
838 for mating and rearing their young. Since her class has been working on ratio and
839 proportion problems, she sees an opportunity to connect an understanding of
840 ornithology, specifically how environmental factors can influence organisms’ growth
841 (NGSS, MS-LS-1-5), with an understanding of the relevant mathematics for that week.
842 She asks students to work with a partner, and she poses the following situation for her
843 class:

844 After analyzing local bird populations of a particular species, scientists

845 determined that, in order to meet the bird community’s needs, multichamber
846 houses are needed. Every time they build three single-chamber houses, they
847 should build four two-chamber houses and one three-chamber house.
848 (adapted from Lamon, 1993)

849 Ms. K then asks each student pair to draft three questions about the given situation,
850 after which she collects the questions on the board. She notices many of them touch
851 on how many total chambers there could be or how many total birds can be
852 accommodated. Because she has encouraged students to ask questions, they are
853 able to develop their natural curiosity about ways that numbers, and groups of
854 numbers, fit together. Many of the initial questions are about the reasons why some
855 bird species like to live communally, which allows for a fascinating comparison
856 between the preferred living arrangements of these particular birds and those of
857 people. Four questions in particular seem fruitful to explore mathematically. After the

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858 class helps her further clarify them, she writes the final questions on the board and
859 has each student pair choose one that the partners will investigate. She gives them 20
860 minutes, after which they will report their findings to the class by making a small
861 poster. The questions are:

862 1. Why do the birds like to build nests in the houses like this? Why not all
863 single chamber houses, for example?
864 2. How many houses of each kind are needed to accommodate a certain
865 number of birds (like 50, 100 or 150)? Is there a pattern between the
866 number of houses and number of birds?
867 3. How many birds could be accommodated if a certain number of the houses
868 (like 50 or 100) are built? Is there a relationship between the number of
869 birds and number of houses?
870 4. If the park only allows for a certain number of houses (like 50,100, 150) to
871 be built, how many of each kind should there be? Is there a relationship
872 between number of houses and how many of each kind?

873 As students work in pairs, Ms. K notices that many are drawing tables and diagrams to
874 organize their work. In thinking about this problem, students need to be mindful of the
875 many types of units (groups) involved in it: groups of each size house, groups of eight
876 houses, total group of birds, total group of chambers, total group of houses. In
877 attending to these different types of units (or wholes), students develop the
878 understanding that there is flexibility in allocating how many parts are in a whole, and
879 that this flexibility offers a new perspective when engaging in proportional reasoning.

880 (end snapshot)

881 CC4: Discovering Shape and Space

882 Students need mathematical tools to explore and understand the shape and space of
883 the physical world; as such, teachers should continue to offer instruction that
884 motivates such explorations. As in other aspects of math teaching and learning,
885 maintaining connection to concrete situations and authentic questions is crucial and
886 this content area could be investigated with any of the three Drivers of Investigation, to
887 help students understand, predict or affect.

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888 Geometric problem situations encourage different modes of thought than do

889 numerical, algebraic, and computational situations. It is important to realize that “visual
890 thinking” or “geometric reasoning” is as legitimate as algebraic or computational
891 thinking; and, for some students, geometric thinking can provide access more readily
892 than other modes to rich mathematical work (Driscoll et al., 2007). The CA CCSSM
893 support this visual thinking by defining congruence and similarity in terms of dilations
894 and rigid motions of the plane, and through its emphasis on physical models,
895 transparencies, and geometry software.

896 As emphasized throughout this framework, flexibility in moving between different

897 representations and points of view brings great mathematical power. Students should
898 not experience geometry primarily as a way to formalize visual thinking into algebraic
899 or numerical representations. Instead, they should have occasion to gain insight into
900 situations presented numerically or algebraically by transforming them into geometric
901 representations, as well as the more common algebraic or numerical representations
902 of geometric situations. For example, students can use similar triangles to explore
903 questions about integer-coordinate points on a line presented algebraically (Driscoll et
904 al., 2017).

905 In grades three through five, students develop many foundational notions of two- and
906 three-dimensional geometry, such as area (including surface area of three-
907 dimensional figures), perimeter, angle measure, and volume. Shape and space work
908 in grades six through eight is largely about connecting these notions to each other, to
909 students’ lives, and to other areas of mathematics.

910 In grade six, for example, two-dimensional and three-dimensional figures are related
911 to each other via nets and surface area (6.G.4), two-dimensional figures are related to
912 algebraic representation via coordinate geometry (6.G.3), and volume is connected to
913 fraction operations by exploring the size of a cube that could completely pack a
914 shoebox with fractional edge lengths (6.G.2). (The vignette Learning About Shapes
915 Through Sponge Art describes a sixth grade teacher supporting students in learning
916 about shapes, using molding clay, since she has seen students struggle with 2-D

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917 representations of 3-D shapes as they were learning about surface area and volume.)
918 In grade seven, relationships between angle or side measurements of two-
919 dimensional figures and their overall shape (7.G.2), between three-dimensional
920 figures and their two-dimensional slices (7.G.3), between linear and area
921 measurements of two-dimensional figures (7.G.4), and between geometric concepts
922 and real-world contexts (7.G.6) are all important foci.

923 In grade eight, two important relationships between different plane figures—
924 congruence and similarity—are defined and explored in depth and used as contexts
925 for reasoning in the manner discussed in chapter 4. The Pythagorean Theorem is
926 developed as a relationship between an angle measure in a triangle and the area
927 measures of three squares (8.G.6). Also, in grade eight, several clusters in the
928 Expressions and Equations standards domain should sometimes be approached from
929 a geometric point of view, with algebraic representations coming later: In an
930 investigation, proportional relationships between quantities can be first encountered
931 as a graph, leading to natural questions about points of intersection (8.EE.7, 8.EE.8)
932 or the meaning of slope (8.EE.6).

933 The Big Ideas, Grades Six Through Eight

934 The foundational mathematics content—that is, the big ideas—progresses through
935 transitional kindergarten through grade twelve in accordance with the CA CCSSM
936 principles of focus, coherence, and rigor. As students explore and investigate the big
937 ideas, they will engage with many different content standards and come to understand
938 the connections between them.

939 Each grade-level-specific big-idea diagram below (figures 7.7, 7.9, and 7.11) shows
940 the ideas as colored circles of varying sizes. A circle’s size indicates the relative
941 importance of the idea it represents, as determined by the number of connections that
942 particular idea has with other ideas. Big ideas are considered connected to one
943 another when they enfold two or more of the same standards; the greater the number
944 of standards one big idea shares with other big ideas, collectively, the more connected

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945 and important the idea is considered to be.

946 Each big idea diagram is followed by a figure (figures 7.8, 7.10, and 7.12,
947 respectively) that reiterates the grade-specific big ideas and, for each idea, shows
948 associated content connections and content standards, as well as providing some
949 detail on how content standards can be addressed in the context of the CCs described
950 in this framework. The figures provide a deeper look into how each big idea at each
951 grade level is situated within a broader content connection, and how each big idea
952 includes several CA CCSSM content standards. Given this nesting, teaching to these
953 big ideas can be seen as an efficient form of standards-aligned instruction.

954 It should be said that there are many interpretations of big ideas in mathematics, and
955 those presented in these figures are one variation. Providing mathematics teachers
956 with adequate release time to collaborate with colleagues and engage in discussions
957 around their vision of big ideas at their grade level or in a particular course can enable
958 them to create rich, deep tasks that invite students to explore and grapple with those
959 big ideas (Arbaugh and Brown, 2005).

960 Figure 7.7 Grade Six Big Ideas

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961

962 Long description of figure 7.7

963 Figure 7.8 Grade Six Content Connections, Big Ideas, and Content Standards

Content Big Idea Grade Six Content Standards

Connection
Reasoning with Variability in Data SP.1, SP.5, SP.4: Investigate real world data
Data sources, ask questions of data, start to understand
variability - within data sets and across different
forms of data, consider different types of data, and
represent data with different representations.
Reasoning with The Shape of SP.2, SP.3, SP.5: Consider the distribution of data
Data Distributions sets - look at their shape and consider measures of
center and variability to describe the data and the
situation which is being investigated.

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Content Big Idea Grade Six Content Standards

Connection
Exploring Fraction NS.1, RP.1, RP.3: Understand fractions divided by
Changing Relationships fractions, thinking about them in different ways
Quantities (e.g., how many 1/3 are inside 2/3?), considering
the relationship between the numerator and
denominator, using different strategies and visuals.
Relate fractions to ratios and percentages.
Exploring Patterns inside NS.4, RP.3: Consider how numbers are made up,
Changing Numbers exploring factors and multiples, visually and
Quantities numerically.
Exploring Generalizing with EE.6, EE.2, EE.7, EE.3, EE.4, RP.1, RP.2, RP.3:
Changing Multiple Generalize from growth or decay patterns, leading
Quantities Representations to an understanding of variables. Understand that a
variable can represent a changing quantity or an
unknown number. Analyze a mathematical situation
that can be seen and solved in different ways and
that leads to multiple representations and
equivalent expressions. Where appropriate in
solving problems, use unit rates.
Exploring Relationships EE.9, EE.5, RP.1, RP.2, RP.3, NS.8, SP.1, SP.2:
Changing Between Use independent and dependent variables to
Quantities Variables represent how a situation changes over time,
recognizing unit rates when it is a linear
relationship. Illustrate the relationship using tables,
4 quadrant graphs and equations, and understand
the relationships between the different
representations and what each one communicates.
Taking Wholes Model the World NS.3, NS.2, NS.8, RP.1, RP.2, RP.3: Solve and
Apart, Putting model real world problems. Add, subtract, multiply,
Parts Together and divide multi-digit numbers and decimals, in real-
world and mathematical problems - with sense
making and understanding, using visual models and
algorithms.
Taking Wholes Nets and Surface EE.1, EE.2, G.4, G.1, G.2, G.3: Build and
Apart, Putting Area decompose 3-D figures using nets to find surface
Parts Together area. Represent volume and area as expressions
involving whole number exponents.
and
Discovering
Shape and Space

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Content Big Idea Grade Six Content Standards

Connection
Discovering Distance and NS.5, NS.6, NS.7, G.1, G.2, G.3, G.4: Students
Shape and Space Direction experience absolute value on numbers lines and
relate it to distance, describing relationships, such
as order between numbers using inequality
statements.
Discovering Graphing Shapes G.3, G.1, G.4, NS.8, EE.2: Use coordinates to
Shape and Space represent the vertices of polygons, graph the
shapes on the coordinate plane, and determine side
lengths, perimeter, and area.

964 Figure 7.9 Grade Seven Big Ideas

965

966 Long description of figure 7.9

967 Figure 7.10 Grade Seven Content Connections, Big Ideas, and Content Standards

43
43

Content Big Idea Grade Seven Content Standards

Connection
Reasoning with Populations SP.1, SP.2, RP.1, RP.2, RP.3, NS.1, NS.2, NS.3, EE.3:
Data and Samples Study a population by taking random samples and
determine if the samples accurately represent the
population.
● Analyze and critique reports by examining the
sample and the claims made to the general
population
● Use classroom simulations and computer
software to model repeated sampling, analyzing
the variation in results.
Reasoning with Visualize SP.3, SP.4, NS.1, NS.2, NS.3, EE.3: Draw comparative
Data Populations inferences about populations - consider what visual plots
show, and use measures of center and variability
● Students toggle between the mathematical
results and their meaningful interpretation with
their given context, considering audiences,
implications, etc.
Reasoning with Probability SP.5, SP.6, SP.7, SP.8, RP.1, RP.2, RP.3, NS.1, NS.2,
Data Models NS.3, EE.3: Develop a probability model and use it to
find probabilities of events and compound events,
representing sample spaces and using lists, tables, and
tree diagrams.
● Compare observed probability and expected
probability.
● Explore potential bias and over-representation in
real world data sets, and connect to dominating
narratives and counter narratives used in public
discourse.
Exploring Proportional EE.2, EE.3, RP.1, RP.2, RP.3: Explore, understand, and
Changing Relationships use proportional relationships: - using fractions, graphs,
Quantities and tables.
Exploring Unit Rates in RP.1, RP.2, RP.3, EE1, EE.2, EE.3, EE.4: Solve real
Changing the World world problems using equations and inequalities, and
Quantities recognize the unit rate within representations.
Exploring Graphing EE.4, RP.1, RP.2, RP.3: Solve problems involving
Changing Relationships proportional relationships that can lead to graphing using
Quantities geometry software and making sense of solutions.

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Content Big Idea Grade Seven Content Standards

Connection
Taking Wholes 2-D and 3-D G.1, G.2, G.3, NS.1, NS.2, NS.3: Draw and construct
Apart, Putting Connections shapes, slice 3-D figures to see the 2-D shapes.
Parts Together Compare and classify the figures and shapes using area,
surface area, volume, and geometric classifications for
and
triangles, polygons, and angles. Make sure to measure
Discovering with fractions and decimals, using technology for
Shape and calculations
Space
Taking Wholes Angle G.5, G.6, NS.1, NS.2, NS.3: Explore relationships
Apart, Putting Relationships between different angles, including complementary,
Parts Together supplementary, vertical, and adjacent, recognizing the
and relationships as the measures change. For example,
angles A and B are complementary. As the measure of
Discovering angle, A increases, the measure of angle B decreases.
Shape and
Space
Discovering Scale G.1, EE.2, EE.3, EE.4, NS.2, NS.3, RP.1, RP.2, RP.3:
Shape and Drawings Solve problems involving scale drawings and construct
Space geometric figures using unit rates to accurately represent
real world figures. (Use technology for drawing)
and
Exploring
Changing
Quantities
Discovering Shapes in the G.1, G.2, G.3, G.4, G.5, G.6, NS.1, NS.2, NS.3: Solve
Shape and World real life problems involving triangles, quadrilaterals,
Space polygons, cubes, right prisms, and circles using angle
measures, area, surface area, and volume.
and
Exploring
Changing
Quantities

968 Figure 7.11 Grade Eight Big Ideas

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969

970 Long description of figure 7.11

971 Figure 7.12 Grade Eight Content Connections, Big Ideas, and Content Standards

Content Big Idea Grade Eight Content Standards

Connection
Reasoning with Interpret Scatter SP.1, SP.2, SP.3, EE.2, EE.5, F.1, F.2, F.3:
Data plots Construct and interpret data visualizations,
including scatter plots for bivariate measurement
and
data using two-way tables. Describe patterns
Exploring noting whether the data appear in clusters, are
Changing linear or nonlinear, whether there are outliers,
Quantities and if the association is negative or positive.
Interpret the trend(s) in change of the data points
over time.

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Content Big Idea Grade Eight Content Standards

Connection
Reasoning with Data, Graphs, and SP.3, SP.4, EE.2, EE.5, F.3, F.4, F.5: Construct
Data Tables graphs of relationships between two variables
(bivariate data), displaying frequencies and
relative frequencies in a two-way table.
● Use graphs with categorical data to help
students describe events in their lives,
looking at patterns in the graphs.
Reasoning with Data Explorations SP.1, SP.2, SP.3, SP.4, EE.4, EE.5, F.1, F.2,
Data F.3, F.4, F.5: Conduct data explorations, such as
the consideration of seafloor spreading, involving
large data sets and numbers expressed in
scientific notation, including integer exponents for
large and small numbers using technology.
● Identify a large dataset and discuss the
information it contains
● Identify what rows and columns
represent in a spreadsheet
Exploring Linear Equations EE.5, EE.7, EE.8, F.2, F.4, F.5: Analyze slope
Changing and intercepts and solve linear equations
Quantities including pairs of simultaneous linear equations
through graphing and tables and using
technology.
Exploring Multiple EE.5, EE.6, EE.7: Move between different
Changing Representations representations of linear functions (i.e., equation,
Quantities of Functions graph, table, and context), sketch and analyze
graphs, use similar triangles to visualize slope
and rate of change with equations containing
rational number coefficients.
Exploring Slopes and EE.5, SP.1, SP.2, SP.3: Construct graphs using
Changing Intercepts bivariate data, comparing the meaning of parallel
Quantities and non-parallel slopes with the same or different
y-intercepts using technology.
Taking Wholes Cylindrical G.9, G.6, G.7, G.8, NS.1, NS.2: Solve real world
Apart, Putting Investigations problems with cylinders, cones, and spheres.
Parts Together Connect volume and surface area solutions to
the structure of the figures themselves (e.g., why
and
and how is the area of a circle formula used to
Discovering find the volume of a cylinder?). Show visual
Shape and proofs of these relationships, through modeling,
Space building, and using computer software.

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Content Big Idea Grade Eight Content Standards

Connection
Taking Wholes Pythagorean G.7, G.8, NS.1, NS.2, EE.1, EE.2: Conduct
Apart, Putting Explorations investigations in the coordinate plane with right
Parts Together triangles to show that the areas of the squares of
each leg combine to create the square of the
and
hypotenuse and name this as the Pythagorean
Discovering Theorem. Using technology, use the
Shape and Pythagorean Theorem to solve real world
Space problems that include irrational numbers.
Taking Wholes Big and Small EE.1, EE.2, EE.3, EE.4, NS.1, NS.2: Use
Apart, Putting Numbers scientific notation to investigate problems that
Parts Together include measurements of very large and very
small numbers. Develop number sense with
integer exponents (e.g., 1/27 =1/33 = 3-3).
Discovering Shape, Number, G.9, G.6, G.7, G.8, EE.1, EE.2, NS.1, NS.2:
Shape and and Expressions Compare shapes containing circular measures to
Space prisms. Note that cubes and squares represent
unit measures for volume and surface area. See
and use the connections between integer
exponents and area and volume.
Discovering Transformational G.1, G.2, G.3, G.4, G.5, G.6, G.7, G.8: Plot two
Shape and Geometry dimensional figures on a coordinate plane, using
Space geometry software, noting similarity when
dilations are performed and the corresponding
angle measures maintain congruence. Perform
translations, rotations, and reflections and notice
when shapes maintain congruence.

972 Conclusion

973 The middle grades are critical years when students often decide whether they want to
974 continue or dis-identify with mathematics. This chapter outlines a vision for middle
975 school mathematics that engages students in problems that elicit curiosity about the
976 world and prompt wondering about mathematical relationships. Mathematical
977 explorations that students encounter in middle school can give them opportunities to
978 appreciate mathematics, leading them to include math in their future plans. Classroom
979 discussions can allow development of self-awareness as well as collaboration and
980 social-emotional skills, as they learn to value the perspectives of others. Discussions

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981 of mathematical ideas also support all students, including English learners, in learning
982 the language of mathematics.

983 Long Descriptions for Chapter 7

984 Figure 7.1 The Why, How, and What of Mathematics (accessible
985 version)

Why How What

Drivers of Investigation Standards for Content Connections
Mathematical Practice
In order to… Students will… While…
DI1. Make Sense of SMP1. Make Sense of CC1. Communicating
the World Problems and Stories with Data
(Understand and Persevere in Solving CC2. Exploring
Explain) them Changing Quantities
DI2. Predict What SMP2. Reason CC3. Taking Wholes
Could Happen Abstractly and Apart, Putting Parts
(Predict) Quantitatively Together
DI3. Impact the Future SMP3. Construct Viable CC4. Discovering
(Affect) Arguments and Shape and Space
Critique the
Reasoning of Others
SMP4. Model with
Mathematics
SMP5. Use Appropriate
Tools Strategically
SMP6. Attend to
Precision
SMP7. Look for and
Make Use of
Structure
SMP8. Look for and
Express Regularity in
Repeated Reasoning

986 Return to figure 7.1 graphic

987 Figure 7.2 Drivers of Investigation, Mathematical Practices, and

988 Content Connections

989 A spiral graphic shows how the Drivers of Investigation (DIs), Standards for

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990 Mathematical Practice (SMPs) and Content Connections (CCs) interact. The DIs are
991 listed under “In order to...”: Make Sense of the World (Understand and Explain);
992 Predict What Could Happen (Predict); Impact the Future (Affect). The SMPs are listed
993 under “Students will...”: Make sense of problems and persevere in solving them;
994 Reason abstractly and quantitatively; Construct viable arguments and critique the
995 reasoning of others; Model with mathematics; Use appropriate tools strategically;
996 Attend to precision; Look for and make use of structure; Look for and express
997 regularity in repeated reasoning. Finally, the CCs are listed under, “While...”:
998 Communicating Stories with Data; Exploring Changing Quantities; Taking Wholes
999 Apart, Putting Parts Together; Discovering Shape and Space. Return to figure 7.2
1000 graphic

1001 Figure 7.5 Environment-Oriented Data Visualization

1002 Bar graphs display with left axis representing number of endangered species on the
1003 IUCN Red List, by class, labeled from 0 to 15,000. Bottom axis is in years, 2007 to
1004 2019, with one bar every three years, so five bars total. The bars increase in height to
1005 indicate that the number of endangered species has risen from 2007 to 2019. The
1006 height of the first bar, for year 2007, is 7,851, the second bar is 9,618 for year 2010,
1007 the third bar is 11,212 for year 2013, the fourth bar is 12,630 for year 2016, and the
1008 height of the last bar, for year 2019, is 14,234. The bars are also color coded to
1009 indicate more specificity for eight types of species: mammals, reptiles, amphibians,
1010 birds, insects, mollusks, fish and other (including other invertebrate [spineless]
1011 animals, such as crustaceans, corals, and arachnids [spiders, scorpions]). Source:
1012 IUCN Red List. Link is provided at bottom of graphic:
1013 https://www.statista.com/chart/17122/number-of-threatened-species-red-list/ Return to
1014 figure 7.5 graphic

1015 Figure 7.6 Squares with Borders for Use in the Border Problem

1016 Six rectangles include two squares each. Squares include borders comprised of
1017 various shadings. Rectangle one includes two squares shaded to indicate 10 + 10 + 8
1018 + 8 and n + (n – 2) + n + n (n – 2). Rectangle two includes two squares shaded 10 + 9

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1019 + 9 + 8 and n + 2(n – 1) + (n – 2). Rectangle three includes two squares shaded 4 × 8
1020 + 4 and 4(n – 2) + 4. Rectangle five includes two squares shaded 9 + 9 + 9 + 9 = 9 × 4
1021 and (n – 1) × 4. Rectangle five includes two squares shaded 4 × 10 – 4 and 4n – 4.
1022 Rectangle six includes two squares shaded (10 × 10) – (8 × 8) and n squared – (n – 2)
1023 squared. Return to figure 7.6 graphic

1024 Figure 7.7 Grade Six Big Ideas

1025 The graphic illustrates the connections and relationships of some sixth-grade
1026 mathematics concepts. Direct connections include:

1027  Variability in Data directly connects to: The Shape of Distributions, Relationships
1028 Between Variables

1029  The Shape of Distributions directly connects to: Relationships Between

1030 Variables, Variability in Data

1031  Fraction Relationships directly connects to: Patterns Inside Numbers,

1032 Generalizing with Multiple Representations, Model the World, Relationships
1033 Between Variables

1034  Patterns Inside Numbers directly connects to: Fraction Relationships,

1035 Generalizing with Multiple Representations, Model the World, Relationships
1036 Between Variables

1037  Generalizing with Multiple Representations directly connects to: Patterns Inside
1038 Numbers, Fraction Relationships, Model the World, Relationships Between
1039 Variables, Nets & Surface Area, Graphing Shapes

1040  Model the World directly connects to: Fraction Relationships, Relationships
1041 Between Variables, Patterns Inside Numbers, Generalizing with Multiple
1042 Representations, Graphing Shapes

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1043  Graphing Shapes directly connects to: Model the World, Generalizing with
1044 Multiple Representations, Relationships Between Variables, Distance &
1045 Direction, Nets & Surface

1046  Nets & Surface directly connects to: Graphing Shapes, Generalizing with Multiple
1047 Representations, Distance & Direction

1048  Distance & Direction directly connects to: Graphing Shapes, Nets & Surface Area

1049  Relationships Between Variables directly connects to: Variability in Data, The
1050 Shape of Distributions, Fraction Relationships, Patterns Inside Numbers,
1051 Generalizing with Multiple Representations, Model the World, Graphing Shapes
1052 Return to figure 7.7 graphic

1053 Figure 7.9 Grade Seven Big Ideas

1054 The graphic illustrates the connections and relationships of some seventh-grade
1055 mathematics concepts. Direct connections include:

1056  Angle Relationships directly connects to: Scale Drawings, 2D & 3D Connections,
1057 Populations & Samples, Proportional Relationships, Shapes in the World,
1058 Visualize Populations, Probability Models

1059  Scale Drawings directly connects to: 2D & 3D Connections, Graphing

1060 Relationships, Populations & Samples, Unit Rates in the World, Proportional
1061 Relationships, Visualize Populations, Probability Models, Angle Relationships

1062  Graphing Relationships directly connects to: Populations & Samples, Unit Rates
1063 in the World, Proportional Relationships, Probability Models, Scale Drawings

1064  2D & 3D Connections directly connects to: Scale Drawings, Angle Relationships,
1065 Probability Models, Proportional Relationships, Visualize Populations, Shapes in
1066 the World, Populations & Samples

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1067  Populations & Samples directly connects to: 2D & 3D Connections, Scale
1068 Drawings, Angle Relationships, Probability Models, Proportional Relationships,
1069 Visualize Populations, Shapes in the World, Unit Rates in the World, Graphing
1070 Relationships

1071  Unit Rates in the World directly connects to: Populations & Samples, Graphing
1072 Relationships, Scale Drawings, Proportional Relationships, Probability Models,
1073 Visualize Populations

1074  Shapes in the World directly connects to: Populations & Samples, 2D & 3D
1075 Connections, Proportional Relationships, Scale Drawings, Angle Relationships,
1076 Probability Models, Visualize Populations

1077  Visualize Populations directly connects to: 2D & 3D Connections, Scale

1078 Drawings, Angle Relationships, Probability Models, Proportional Relationships,
1079 Populations & Samples, Shapes in the World, Unit Rates in the World

1080  Probability Models directly connects to: 2D & 3D Connections, Scale Drawings,
1081 Angle Relationships, Proportional Relationships, Visualize Populations, Shapes
1082 in the World, Unit Rates in the World, Graphing Relationships, Populations &
1083 Samples

1084  Proportional Relationships directly connects to: 2D & 3D Connections, Scale

1085 Drawings, Angle Relationships, Probability Models, Populations & Samples,
1086 Visualize Populations, Shapes in the World, Unit Rates in the World, Graphing
1087 Relationships
1088 Return to figure 7.9 graphic

1089 Figure 7.11 Grade Eight Big Ideas

1090 The graphic illustrates the connections and relationships of some eighth-grade
1091 mathematics concepts. Direct connections include:

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 53

1092  Data Explorations directly connects to: Slopes & Intercepts, Linear Equations,
1093 Multiple Representations of Functions, Data Graphs & Tables, Interpret Scatter
1094 plots, Big & Small Numbers

1095  Slopes & Intercepts directly connects to: Linear Equations, Multiple
1096 Representations of Functions, Data Graphs & Tables, Interpret Scatter plots,
1097 Data Explorations

1098  Linear Equations directly connects to: Slopes & Intercepts, Data Explorations,
1099 Multiple Representations of Functions, Data Graphs & Tables, Interpret Scatter
1100 plots

1101  Multiple Representations of Functions directly connects to: Data Graphs &
1102 Tables, Interpret Scatter plots, Data Explorations, Slopes & Intercepts, Linear
1103 Equations

1104  Data Graphs & Tables directly connects to: Multiple Representations of
1105 Functions, Linear Equations, Slopes & Intercepts, Data Explorations, Interpret
1106 Scatter plots, Shape Number & Expressions, Big & Small Numbers, Pythagorean
1107 Explorations

1108  Pythagorean Explorations directly connects to: Data Graphs & Tables, Interpret
1109 Scatter plots, Cylindrical Investigations, Transformational Geometry, Shape
1110 Number & Expressions, Big & Small Numbers

1111  Big & Small Numbers directly connects to: Pythagorean Explorations, Data
1112 Graphs & Tables, Interpret Scatter plots, Data Explorations, Cylindrical
1113 Investigations, Transformational Geometry, Shape Number & Expressions

1114  Shape Number & Expressions directly connects to: Big & Small Numbers,
1115 Pythagorean Explorations, Data Graphs & Tables, Interpret Scatter plots,
1116 Cylindrical Investigations

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1117  Transformational Geometry directly connects to: Big & Small Numbers,
1118 Pythagorean Explorations, Cylindrical Investigations

1119  Cylindrical Investigations directly connects to: Big & Small Numbers,
1120 Pythagorean Explorations, Shape Number & Expressions, Transformational
1121 Geometry

1122  Interpret Scatter plots directly connects to: Data Explorations, Slopes &
1123 Intercepts, Linear Equations, Multiple Representations of Functions, Data Graphs
1124 & Tables, Pythagorean Explorations, Big & Small Numbers, Shape Number &
1125 Expressions
1126 Return to figure 7.11 graphic

California Department of Education, October 2023

55