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Designing Spaces

Grade 5, Mathematics

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Description of Unit

In this task, students study how mathematics applies to architecture and the creation of designed environments. Students explore how various designs relate to human needs and functions by observing their environments, drawing, graphing, and taking measurements. Students research a period of architecture, a particular style, or a famous architect of their choosing and analyze the way the architect uses geometric forms and mathematical patterns in his/her works. In small groups, students work together to design and build a variety of architectural spaces within the classroom using simple materials such as dowel rods and fabric. Finally, students present their works to an audience by leading guided tours through their structures during an event called the “Designer’s Showcase.”

This guide links the Designing Spaces unit to the Texas Essential Knowledge and Skills (TEKS) for fifth graders. Designing Spaces is a mathematics unit that allows students to study the interconnectedness of mathematics and visual art in the field of architecture. Designing Spaces has interdisciplinary connections to Art, Social Studies, and English Language Arts and Reading disciplines. For example, students will develop and organize ideas from the environment, as outlined in the Fine Arts TEKS, compose original texts, as outlined in the English Language Arts and Reading TEKS, and understand the positive and negative aspects of modifications to an environment, as described in the Social Studies TEKS. The following document includes the applicable TEKS and the details of the Designing Spaces unit. The final section of this document presents the applicable Texas College and Career Readiness Standards adopted by the Texas Higher Education Coordinating Board (THECB) on January 24, 2008.

Phase I. Learning Experiences

  1. Introduce students to the unit by discussing the career path of an architect. In what ways is an architect like an artist? In what ways is an architect like a mathematician? Discuss the importance of both mathematics and aesthetics when designing spaces. Share images of architectural drawings and blueprints, describing each feature and how it relates to the real space. Introduce students to the concepts of positive and negative space and how these spaces can be described geometrically.
  2. Break students into small groups and equip them with measuring tapes, yardsticks, or rulers and graph paper. Ask each group to construct a blueprint for the interior of the classroom by working backwards and measuring the interior of the current space as well as furniture and other features of the room. Students should indicate where windows and doors are on the blueprint. Provide students with hard-to-reach measurements, such as the height of the walls, and/or width and length of the ceiling. Student should include measurements and units on their drawings (e.g., 60 inches).
  3. Write a journal entry for each of three different spaces (e.g., the student’s bedroom, a favorite store or restaurant, and a library or gym). Describe the interior of each space:
    • What shapes exist in both the positive and negative spaces?
    • Describe the lighting in the space. How does natural light enter the room? How do artificial light sources provide illumination?
    • How might you describe your feelings when you are in the space? What characteristics of the space might contribute to evoking those feelings?
    • What physical materials do you see used in the construction of the space (e.g., wood, metal, carpet, fabric)?
    • What colors describe the space? What sensory information contributes to the space (e.g., sounds, smells, temperature)?
    • What are some of the activities that are most likely to occur in the space? How many individuals might the space reasonably accommodate at any one time? What features exist in the space?
    • If you were redesigning the space, what might you change and why?
  4. Introduce students to the concept of mathematical patterns such as the Fibonacci sequence, and in particular how this sequence leads to the Golden Ratio or Golden Mean. Also known as phi, this number appears throughout nature, and has been used for centuries by artists, musicians, poets, and architects in designing/composing their works. Guide students in constructing a Golden Rectangle on graph paper by shading in the number of squares in the Fibonacci sequence. Students may want to use color codes to indicate the sequential order of the squares (e.g., the first (red) = 1, the second (yellow) = 1, the third (orange) = 2).
  5. Using the Fibonacci Rectangle as a visual aid, ask students to take the sequence of Fibonacci numbers and turn them into ratios (i.e., take the length of each square in the sequence over the length of the square that occurs just before it). In other words, for the squares represented by Fibonacci numbers 5 and 8, the length of the second square (8) becomes the dividend (numerator) and the length of the first square (5) becomes the devisor (denominator). Now, find the quotient (result) of dividing these ratios . Repeat this exercise with each of the color-coded squares in the Golden Rectangle. Plot the quotients on graph paper and describe the results.
  6. Conduct a scavenger hunt looking for Fibonacci Rectangles (the Golden Mean) in the design of the classroom, furniture, or objects. How many designs, if any, come close to using these proportions? How would students redesign a piece of furniture in the classroom using the Golden Mean?
  7. Conduct Internet research on the Fibonacci Sequence, the Golden Mean and architectural structures.
  8. Take students on a field trip to an interesting architectural structure and draw the structure’s façade using a variety of drawing tools. Guide students in observing how light helps give the building or structure form. Help students identify mathematical terms in their drawings such as angels, arcs, and line segments.
  9. Invite a guest speaker from a local architectural firm into the classroom to discuss the interconnections between mathematics and art in the designing of real buildings and spaces.
  10. Discuss contemporary issues related to architecture, such as the issues of private versus public space, and the architect’s responsibility to the surrounding community.

Phase II. Independent Research

A. Research process

  1. Selecting a topic. Form small groups based on the selected architectural styles or architects students wish to explore. Topics to consider may include:
    • Frank Lloyd Wright
    • Le Corbusier
    • Walter Gropius
    • Frank Gehry
    • Philip Johnson
    • Renzo Piano
    Each group will design and construct a walk-through structure in the classroom using simple materials such as dowel rods and fabric. The teacher will provide the initial dimensions to each group for the structure’s footprint and height allowances. These structures should:
    • transform the space,
    • incorporate the Golden Mean in some way, and
    • reference the architectural style and/or architects studied.
    How might you describe these various styles of architecture from a mathematics standpoint? What shapes do you observe? How many of the structures seem to employ the Golden Mean or another mathematical pattern?
  2. Asking guiding questions. Once students have identified the period of architecture they wish to study, or the specific architect, they should think of three to five guiding questions to explore, such as:
    • What were the primary activities that occurred in these architectural spaces and how did the design lend itself to these functions?
    • How did the architect build the structure using geometric forms?
    • What types of materials were chosen to build the structure and why might the architect have selected these items?
    • How might you redesign this structure and what modifications might you make to it?
    • What elements of these designs will you incorporate into your original architectural design to be built within the classroom?
    • What sorts of modifications will you need to consider that allow for the simple materials available for your use?
    • How will you allocate resources for building the structure— human resources, material resources, and time resources?
    • What might a budget look like for your structure?
    • How will you design your structure so that it is safe to build and use within the classroom?
  3. Creating a research proposal. Working in groups, students brainstorm and create drawings for the structure on graph paper and/or using a visualization software program such as Google Sketch Up. Students incorporate the Golden Mean in some element of the design and explain their decisions.
  4. Conducting the research. Collaborate with the librarian to provide books and/or websites with visuals on architectural styles and famous architects. Observe the construction of local buildings, make sketches of building facades, and take tours of the insides of public/historic buildings such as capital buildings, courthouses, sports arenas or churches. Interview individuals who use these spaces to identify what they find most useful about the design and what they might change.
  5. Sharing findings. Each group presents its building proposal to the class. The proposal should include:
    • drawings and prototypes,
    • an estimated budget and timeline for construction,
    • considerations for safety,
    • explanation of the function and purpose of the structure,
    • references to other architects or styles,
    • considerations on how their structure will impact the surrounding space (classroom), and
    • the intended look and feel of the space.
    Once the students have made their presentations, the class may discuss the proposals, ask questions and/or make suggestions and finally issue “building permits” that allow the group to begin construction of the actual structural installation.

B. The product

Each group will build the structure they designed using simple materials. Groups should observe safety practices such as wearing hard hats and eye protection, and observing classroom rules with regards to sharp tools, climbing, and/or standing on structures. Once all groups have completed construction, the class may hold a “Designer’s Convention” that serves as an open house for other classes or the public to come in and learn about the projects.

C. Communication

Each group leads an audience through a tour of their structure, describing the function, features, the historical background or references, the mathematical patterns employed in the design, and the overall goal with regards to “look and feel” that they attempted to achieve.

D. A completed project consists of:

  1. A building proposal consisting of
    •  project goal and function of the structure,
    • drawings/schematics,
    • prototypes of the structure,
    • a budget,
    • a timeline,
    • impacts to the surrounding area,
    • references to other architects or styles, and
    • safety considerations.
  2. Research notes
  3. Fibonacci Rectangle Maps and Golden Mean graphs
  4. Photographs or video of the structures
  5. Videos, audio files, or written transcripts of the student-led tours of the structures during the “Designer’s Convention” including student responses to the Q&A sessions.

This unit may address the following TEKS.

110.7., English Language Arts and Reading, Grade 5, Adopted 2017

110.7(b)(5)
Developing and sustaining foundational language skills: listening, speaking, reading, writing, and thinking--self-sustained reading. The student reads grade-appropriate texts independently. The student is expected to self-select text and read independently for a sustained period of time.
110.7(b)(4)
Developing and sustaining foundational language skills: listening, speaking, reading, writing, and thinking--fluency. The student reads grade-level text with fluency and comprehension. The student is expected to use appropriate fluency (rate, accuracy, and prosody) when reading grade-level text.
110.7(b)(3)
Developing and sustaining foundational language skills: listening, speaking, reading, writing, and thinking--vocabulary. The student uses newly acquired vocabulary expressively.
110.7(b)(9)
Multiple genres: listening, speaking, reading, writing, and thinking using multiple texts--genres. The student recognizes and analyzes genre-specific characteristics, structures, and purposes within and across increasingly complex traditional, contemporary, classical, and diverse texts.
110.7(b)(1)(D)
work collaboratively with others to develop a plan of shared responsibilities.
110.7(b)(7)(F)
respond using newly acquired vocabulary as appropriate
110.7(b)(7)(G)
discuss specific ideas in the text that are important to the meaning.
110.7(b)(1)
Developing and sustaining foundational language skills: listening, speaking, discussion, and thinking--oral language. The student develops oral language through listening, speaking, and discussion.
110.7(b)(12)
Composition: listening, speaking, reading, writing, and thinking using multiple texts--genres. The student uses genre characteristics and craft to compose multiple texts that are meaningful.
110.7(b)(11)
Composition: listening, speaking, reading, writing, and thinking using multiple texts--writing process. The student uses the writing process recursively to compose multiple texts that are legible and uses appropriate conventions.
110.7(b)(13)
Inquiry and research: listening, speaking, reading, writing, and thinking using multiple texts. The student engages in both short-term and sustained recursive inquiry processes for a variety of purposes.

111.7, Mathematics, Grade 5

111.7(b)(1)
Mathematical process standards. The student uses mathematical processes to acquire and demonstrate mathematical understanding.
111.7(b)(4)
Algebraic reasoning. The student applies mathematical process standards to develop concepts of expressions and equations.
111.7(b)(5)
Geometry and measurement. The student applies mathematical process standards to classify two-dimensional figures by attributes and properties. The student is expected to classify two-dimensional figures in a hierarchy of sets and subsets using graphic organizers based on their attributes and properties.
111.7(b)(9)
Data analysis. The student applies mathematical process standards to solve problems by collecting, organizing, displaying, and interpreting data.

112.7., Science, Grade 5, Adopted 2021

112.7(b)(1)
Scientific and engineering practices. The student asks questions, identifies problems, and plans and safely conducts classroom, laboratory, and field investigations to answer questions, explain phenomena, or design solutions using appropriate tools and models. The student is expected to:
112.7(b)(2)
Scientific and engineering practices. The student analyzes and interprets data to derive meaning, identify features and patterns, and discover relationships or correlations to develop evidence-based arguments or evaluate designs. The student is expected to:
112.7(b)(1)(D)
use tools, including calculators, microscopes, hand lenses, metric rulers, Celsius thermometers, prisms, concave and convex lenses, laser pointers, mirrors, digital scales, balances, spring scales, graduated cylinders, beakers, hot plates, meter sticks, magnets, collecting nets, notebooks, timing devices, materials for building circuits, materials to support observations of habitats or organisms such as terrariums and aquariums, and materials to support digital data collection such as computers, tablets, and cameras to observe, measure, test, and analyze information;

113.16, Social Studies, Grade 5

113.16(b)(6)
Geography. The student uses geographic tools to collect, analyze, and interpret data.
113.16(b)(8)
Geography. The student understands the location and patterns of settlement and the geographic factors that influence where people live.
113.16(b)(9)
Geography. The student understands how people adapt to and modify their environment.
113.16(b)(21)
Culture. The student understands the relationship between the arts and the times during which they were created.
113.16(b)(24)
Social studies skills. The student applies critical-thinking skills to organize and use information acquired from a variety of valid sources, including electronic technology.
113.16(b)(25)
Social studies skills. The student communicates in written, oral, and visual forms.
113.16(b)(26)
Social studies skills. The student uses problem-solving and decision-making skills, working independently and with others, in a variety of settings.

117.117, Art, Grade 5

117.117(b)(1)
Foundations: observation and perception. The student develops and expands visual literacy skills using critical thinking, imagination, and the senses to observe and explore the world by learning about, understanding, and applying the elements of art, principles of design, and expressive qualities. The student uses what the student sees, knows, and has experienced as sources for examining, understanding, and creating artworks.
117.117(b)(2)
Creative expression. The student communicates ideas through original artworks using a variety of media with appropriate skills. The student expresses thoughts and ideas creatively while challenging the imagination, fostering reflective thinking, and developing disciplined effort and progressive problem-solving skills.
117.117(b)(3)
Historical and cultural relevance. The student demonstrates an understanding of art history and culture by analyzing artistic styles, historical periods, and a variety of cultures. The student develops global awareness and respect for the traditions and contributions of diverse cultures.
117.117(b)(4)
Critical evaluation and response. The student responds to and analyzes artworks of self and others, contributing to the development of lifelong skills of making informed judgments and reasoned evaluations.

128.7., Spanish Language Arts and Reading, Grade 5, Adopted 2017

128.7(b)(4)
Developing and sustaining foundational language skills: listening, speaking, reading, writing, and thinking--fluency. The student reads grade-level text with fluency and comprehension. The student is expected to use appropriate fluency (rate, accuracy, and prosody) when reading grade-level text.
128.7(b)(1)(D)
work collaboratively with others to develop a plan of shared responsibilities.
128.7(b)(7)(F)
respond using newly acquired vocabulary as appropriate
128.7(b)(7)(G)
discuss specific ideas in the text that are important to the meaning.
128.7(b)(3)(D)
identify, use, and explain the meaning of idioms, adages, and puns
128.7(b)(3)(B)
use context within and beyond a sentence to determine the relevant meaning of unfamiliar words or multiple-meaning words;
128.7(b)(3)(A)
use print or digital resources to determine meaning, syllabication, pronunciation, and word origin;
128.7(b)(3)(C)
identify the meaning of and use base words with affixes, including trans-, super-, anti-, semi-, -logía, -ificar, -ismo, and -ista and roots, including audi, crono, foto, geo, and terr;
128.7(b)(9)
Multiple genres: listening, speaking, reading, writing, and thinking using multiple texts--genres. The student recognizes and analyzes genre-specific characteristics, structures, and purposes within and across increasingly complex traditional, contemporary, classical, and diverse texts.
128.7(b)(5)
Developing and sustaining foundational language skills: listening, speaking, reading, writing, and thinking--self-sustained reading. The student reads grade-appropriate texts independently. The student is expected to self-select text and read independently for a sustained period of time.
128.7(b)(11)(D)
edit drafts using standard Spanish conventions, including:
128.7(b)(11)(D)(viii)
subordinating conjunctions to form complex sentences;
128.7(b)(11)(D)(ix)
capitalization of initials, acronyms, and organizations;
128.7(b)(11)(D)(x)
punctuation marks, including commas in compound and complex sentences, em dash for dialogue, italics and underlining for titles and emphasis, and quotation marks for titles
128.7(b)(11)(D)(xi)
correct spelling of words with grade-appropriate orthographic patterns and rules
128.7(b)(11)(D)(vii)
pronouns, including personal, possessive, objective, reflexive, prepositional, and indefinite;
128.7(b)(11)(D)(v)
conjunctive adverbs;
128.7(b)(11)(D)(i)
complete simple and compound sentences with subject-verb agreement and avoidance of splices, run-ons, and fragments;
128.7(b)(11)(D)(ii)
irregular verbs;
128.7(b)(11)(D)(iii)
collective nouns;
128.7(b)(11)(D)(iv)
adjectives, including those indicating origin, and their comparative and superlative forms;
128.7(b)(11)(D)(vi)
prepositions and prepositional phrases and their influence on subject-verb agreement;
128.7(b)(11)(B)(i)
organizing with purposeful structure, including an introduction, transitions, and a conclusion
128.7(b)(12)(D)
compose correspondence that requests information.
128.7(b)(12)(B)
compose informational texts, including brief compositions that convey information about a topic, using a clear central idea and genre characteristics and craft;
128.7(b)(13)(G)
develop a bibliography
128.7(b)(13)(H)
use an appropriate mode of delivery, whether written, oral, or multimodal, to present results.
128.7(b)(13)(E)
demonstrate understanding of information gathered;
128.7(b)(13)(D)
understand credibility of primary and secondary sources;
128.7(b)(13)(A)
generate and clarify questions on a topic for formal and informal inquiry;
128.7(b)(13)(B)
develop and follow a research plan with adult assistance;
128.7(b)(13)(C)
identify and gather relevant information from a variety of sources;
128.7(b)(1)(B)
follow, restate, and give oral instructions that include multiple action steps;
128.7(b)(1)(A)
listen actively to interpret verbal and non-verbal messages, ask relevant questions, and make pertinent comments;
128.7(b)(1)
Developing and sustaining foundational language skills: listening, speaking, discussion, and thinking--oral language. The student develops oral language through listening, speaking, and discussion.
128.7(b)(1)(C)
give an organized presentation employing eye contact, speaking rate, volume, enunciation, natural gestures, and conventions of language to communicate ideas effectively

This unit may address the following Texas College and Career Readiness Standards.

Science:

S.I.C.1
Collaborate on joint projects.
S.I.D.1
Demonstrate literacy in computer use.
S.I.D.2
Use computer models, applications, and simulations.
S.I.E.1
Use several modes of expression to describe or characterize natural patterns and phenomena. These modes of expression include narrative, numerical, graphical, pictorial, symbolic, and kinesthetic.
S.I.E.2
Use essential vocabulary of the discipline being studied.
S.II.A.3
Understand ratios, proportions, percentages, and decimal fractions, and translate from any form to any other.
S.II.A.4
Use proportional reasoning to solve problems.
S.II.C.1
Understand simple vectors, vector notations, and vector diagrams, and carry out simple calculations involving vectors.
S.II.C.4
Understand basic geometric principles.
S.II.D.1
Use dimensional analysis in problem solving.
S.III.A.1
Use correct applications of writing practices in scientific communication.
S.III.B.3
Recognize scientific and technical vocabulary in the field of study and use this vocabulary to enhance clarity of communication.
S.III.B.4
List, use , and give examples of specific strategies before, during, and after reading to improve comprehension.
S.III.C.1
Prepare and present scientific/technical information in appropriate formats for various audiences.
S.III.D.1
Use search engines, databases, and other digital electronic tools effectively to locate information.
S.III.D.2
Evaluate quality, accuracy, completeness, reliability, and currency of information from any source.
S.V.E.2
Use scale to relate models and structures.
S.X.E.1
Describe the different uses for land (land management).

Social Studies:

SS.I.A.1
 Use the tools and concepts of geography appropriately and accurately.
SS.I.A.2
 Analyze the interaction between human communities and the environment.
SS.I.A.3
 Analyze how physical and cultural processes have shaped human communities over time.
SS.I.A.6
 Analyze the relationship between geography and the development of human communities.
SS.IV.A.1
 Identify and analyze the main idea(s) and point(s)-of-view in sources.
SS.IV.A.2
 Situate an informational source in its appropriate contexts (contemporary, historical, cultural).
SS.IV.A.3
 Evaluate sources from multiple perspectives.
SS.IV.A.4
 Understand the differences between a primary and secondary source and use each appropriately to conduct research and construct arguments.
SS.IV.A.5
 Read narrative texts critically.
SS.IV.A.6
 Read research data critically.
SS.IV.B.1
 Use established research methodologies.
SS.IV.B.3
 Gather, organize, and display the results of data and research.
SS.IV.B.4
 Identify and collect sources.
SS.IV.C.1
 Understand and interpret presentations (e.g., speeches, lectures, informal presentations) critically.
SS.V.A.1
 Use appropriate oral communication techniques depending on the context or nature of the interaction.
SS.V.A.2
 Use conventions of standard written English.
SS.V.B.1
 Attribute ideas and information to source materials and authors.

Cross-Disciplinary Standards:

CDS.I.A.1
 Engage in scholarly inquiry and dialogue.
CDS.I.A.2
 Accept constructive criticism and revise personal views when valid evidence warrants.
CDS.I.B.1
 Consider arguments and conclusions of self and others.
CDS.I.B.4
 Support or modify claims based on the results of an inquiry.
CDS.I.C.1
 Analyze a situation to identify a problem to be solved.
CDS.I.C.2
 Develop and apply multiple strategies to solve a problem.
CDS.I.C.3
 Collect evidence and data systematically and directly relate to solving a problem.
CDS.I.D.1
 Self-monitor learning needs and seek assistance when needed.
CDS.I.D.2
 Use study habits necessary to manage academic pursuits and requirements.
CDS.I.D.3
 Strive for accuracy and precision.
CDS.I.D.4
 Persevere to complete and master tasks.
CDS.I.E.1
 Work independently.
CDS.I.E.2
 Work collaboratively.
CDS.I.F.1
 Attribute ideas and information to source materials and people.
CDS.I.F.2
 Evaluate sources for quality of content, validity, credibility, and relevance.
CDS.II.A.2
 Use a variety of strategies to understand the meanings of new words.
CDS.II.A.3
 Identify the intended purpose and audience of the text.
CDS.II.A.4
 Identify the key information and supporting details.
CDS.II.A.5
 Analyze textual information critically.
CDS.II.A.6
 Annotate, summarize, paraphrase, and outline texts when appropriate.
CDS.II.A.7
 Adapt reading strategies according to structure of texts.
CDS.II.A.8
 Connect reading to historical and current events and personal interest.
CDS.II.B.1
 Write clearly and coherently using standard writing conventions.
CDS.II.B.2
 Write in a variety of forms for various audiences and purposes.
CDS.II.C.2
 Explore a research topic.
CDS.II.C.3
 Refine research topic based on preliminary research and devise a timeline for completing work.
CDS.II.C.4
 Evaluate the validity and reliability of sources.
CDS.II.C.5
 Synthesize and organize information effectively.
CDS.II.C.6
 Design and present an effective product.
CDS.II.C.7
 Integrate source material.
CDS.II.C.8
 Present final product.
CDS.II.D.1
 Identify patterns or departures from patterns among data.
CDS.II.D.3
 Present analyzed data and communicate findings in a variety of formats.
CDS.II.E.1
 Use technology to gather information.
CDS.II.E.2
 Use technology to organize, manage, and analyze information.
CDS.II.E.3
 Use technology to communicate and display findings in a clear and coherent manner.
CDS.II.E.4
 Use technology appropriately.

English Language Arts:

ELA.I.A.5
Edit writing for audience, purpose, context, and style, assuring that it conforms to Standard American English, when appropriate.
ELA.I.A.2
Generate ideas, gather information, and manage evidence relevant to the topic and purpose.
ELA.I.A.3
Evaluate relevance, quality, sufficiency, and depth of preliminary ideas and information; organize material generated; and formulate a thesis or purpose statement.
ELA.II.B.4
Make inferences about the denotative and connotative meanings of unfamiliar words using context clues.
ELA.II.B.3
Use reference guides to confirm the meanings of new words or concepts.
ELA.II.B.1
Identify new words and concepts acquired through study of their relationships to other words and concepts.
ELA.II.A.2
Use text features to form an overview of content and to locate information.
ELA.II.A.1
Use effective reading strategies to determine a written work’s purpose and intended audience.
ELA.III.A.2
Engage in reasoned dialogue, including with people who have different perspectives.
ELA.III.A.1
Participate actively, effectively, and respectfully in one-on-one oral communication as well as in group discussions.
ELA.IV.A.2
Listen critically and respond appropriately.
ELA.IV.A.1
Use a variety of active listening strategies to enhance comprehension.
ELA.V.C.1
Integrate and organize material effectively.
ELA.V.C.3
Follow relevant rules governing attribution.
ELA.V.C.2
Use and attribute source material ethically.
ELA.V.B.3
Assess the relevance and credibility of sources.
ELA.V.B.1
Explore and collect a range of potential sources.
ELA.V.A.1
Articulate and investigate research questions.
ELA.V.A.2
Explore and refine a research topic.
ELA.V.A.3
Devise a plan for completing work on time.
ELA.V.B.2
Distinguish between and among primary and secondary sources.

Mathematics:

M.III.A.2
Form and validate conjectures about one-, two-, and three-dimensional figures and their properties.
M.III.B.1
Identify transformations and symmetries of figures.
M.VIII.C.2
ate and use representations to organize, record, and communicate mathematical ideas.
M.VIII.C.3
Explain, display, or justify mathematical ideas and arguments using precise mathematical language in written or oral communications.
M.VIII.C.1
Communicate mathematical ideas, reasoning, and their implications using symbols, diagrams, models, graphs, and words.
M.VIII.B.1
Model and interpret mathematical ideas and concepts using multiple representations.
M.VIII.A.1
Use mathematical symbols, terminology, and notation to represent given and unknown information in a problem.
M.VIII.A.2
Use mathematical language to represent and communicate the mathematical concepts in a problem.
M.VIII.A.3
Use mathematical language for reasoning, problem solving, making connections, and generalizing.
M.VIII.B.2
Summarize and interpret mathematical information provided orally, visually, or in written form within the given context.
M.V.B.1
Classify types of data.
M.V.B.2
Construct appropriate visual representations of data.
M.VI.C.1
Apply known functions to model real-world situations.
M.VI.B.1
Understand and analyze features of a functions.
M.VI.B.2
Algebraically construct and analyze new functions.
M.VII.D
Real-world problem solving
M.VII.C.2
Understand attributes and relationships with inductive and deductive reasoning.
M.VII.A.1
Analyze given information.
M.VII.D.1
Interpret results of the mathematical problem in terms of the original real-world situation.
M.VII.D.2
Evaluate the problem-solving process.
M.VII.A.2
Formulate a plan or strategy.
M.IX.B.3
Know and understand the use of mathematics in a variety of careers and professions.
M.IX.B.2
Understand and use appropriate mathematical models in the natural, physical, and social sciences.
M.IX.B.1
Use multiple representations to demonstrate links between mathematical and real-world situations.
M.IX.A.1
Connect and use multiple key concepts of mathematics in situations and problems.
M.IX.A.2
Connect mathematics to the study of other disciplines.
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