__CURRICULUM TITLE__**:**Graphing Curves and their Applications in the Real World: Raw Data in an Inclusive Algebra II Classroom

__SUBJECT/TOPIC AREAS__**:**The Engineering Design Process (E.D.P.), Project-Based Learning (P.B.L.) and Universal Design for Learning (U.D.L.)

__KEY WORDS__**:**social constructivism | agency | meaning-making | strands of mathematical proficiency | conceptual understanding | procedural fluency | strategic competence | adaptive reasoning | productive disposition | creativity | formal critique | engagement in the classroom | engineering design process (E.D.P.) | project-based learning (P.B.L.) | universal design for learning (U.D.L.)

__CURRICULAR CONTEXT SUMMARY__**:**

Educators open to authentic, student-centered learning must provide robust scaffolds that are inclusive and enticing to students. In order for students to find their ‘edge’ and build their capacity to exercise ‘agency’ (MacLeod, 1987), teachers must develop cognitively challenging and socially relevant learning modules.

Drawing upon the ‘strands of mathematical proficiency’ (Kilpatrick et al., 2001), social constructivism (Vygotsky, 1978) and universal design for learning as the theoretical frameworks, this curriculum package represents an attempt at engaging all learners in hopes that what they learn will be cross-cutting, integral to S.T.E.A.M., and consistent with liberal arts education. Project-based learning (P.B.L.) and the engineering design process (E.D.P) are incorporated into this package as students learn about mathematical models, conic sections, exponential functions, and trigonometry while connecting the abstractions therein with the work of real scientists, music, digital software, scientific hardware, engineering projects, formal critique, and artistic works. The curriculum arc consists of eleven lessons and is designed to be situated either in the second or third quarter of the academic year. At the teacher's discretion, there may exist a one class period accoutrement woven into this curriculum package situated between the solar oven and string instrument threads (i.e. between lessons six and seven). This class period would treat students to a basic investigation into the mathematical underpinnings of exponential and logarithmic functions further explored in lesson seven.

__COMMON CORE STANDARDS ADDRESSED:__

*Thread: Solar Ovens*- The Mathematical Underpinning of Parabolas and Paraboloids
*or*The Physical Properties Behind Parabolic Solar Melting Devices - The Economics & Humanitarian Facets of Solar Ovens
- Developing Digital Renderings Using ‘Rhinoceros’
- Employing the Engineering Design Process to Parabolic Solar Ovens
- Engaging in Formal Critique

__Science__

- Reading: Literacy in Science and Technical subjects
*CCSS.ELA-Literacy.RST.9-10.4*(grade 9 - 10): Determine the meaning of symbols, key terms, and other domain-specific words and phrases as they are used in a specific scientific or technical context relevant to grades 9–10 texts and topics.*CCSS.ELA-Literacy.RST.11-12.4*(grade 11 - 12): Determine the meaning of symbols, key terms, and other domain-specific words and phrases as they are used in a specific scientific or technical context relevant to grades 11–12 texts and topics.*CCSS.ELA-Literacy.RST.9-10.5*(grade 9 - 10): Analyze the structure of the relationships among concepts in a text, including relationships among key terms (e.g., force, friction, reaction force, energy).

- Integration of Knowledge and Ideas
*CCSS.ELA-Literacy.RST.9-10.7*(grade 9 - 10): Translate quantitative or technical information expressed in words in a text into visual form (e.g., a table or chart) and translate information expressed visually or mathematically (e.g., in an equation) into words.*CCSS.ELA-Literacy.RST.11-12.7*(grade 11 - 12): Integrate and evaluate multiple sources of information presented in diverse formats and media (e.g., quantitative data, video, multimedia) in order to address a question or solve a problem.*CCSS.ELA-Literacy.RST.11-12.9*(grade 11 - 12): Synthesize information from a range of sources (e.g., texts, experiments, simulations) into a coherent understanding of a process, phenomenon, or concept, resolving conflicting information when possible.

__Mathematics__

- Create equations that describe numbers or relationships.
*CCSS.Math.Content.HSA-CED.A.2*(grade 9 - 12): Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

- Interpret functions that arise in applications in terms of the context.
*CCSS.Math.Content.HSF-IF.B.4*(grade 9 - 12): For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship.*Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.*(Modeling)

- Analyze functions using different representations.
*CCSS.Math.Content.HSF-IF.C.7*(grade 9 - 12): Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. (Modeling) a. Graph linear and quadratic functions and show intercepts, maxima, and minima. b. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. c. Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior. d. (+) Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior. e. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.*CCSS.Math.Content.HSF-IF.C.9*(grade 9 - 12): Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).*For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum.*

*Thread: String Instruments*- String Instrument Lab, Graphing Exponential & Logarithmic Functions and Developing Robust Mathematical Models
- Preparing to Construct String Instruments
- Constructing String Instruments
- Engaging in Formal Critique

__Speaking and Listening__

- Presentation of Knowledge and Ideas
*CCSS.ELA-Literacy.CCRA.SL.5*Make strategic use of digital media and visual displays of data to express information and enhance understanding of presentations.

__Science__

- Reading: Literacy in Science and Technical Subjects
*CCSS.ELA-Literacy.RST.9-10.7*(grade 9 - 10): Translate quantitative or technical information expressed in words in a text into visual form (e.g., a table or chart) and translate information expressed visually or mathematically (e.g., in an equation) into words.*CCSS.ELA-Literacy.RST.11-12.7*(grade 11 - 12): Integrate and evaluate multiple sources of information presented in diverse formats and media (e.g., quantitative data, video, multimedia) in order to address a question or solve a problem.*CCSS.ELA-Literacy.RST.9-10.8*(grade 9 - 10): Assess the extent to which the reasoning and evidence in a text support the author’s claim or a recommendation for solving a scientific or technical problem.*CCSS.ELA-Literacy.RST.11-12.9*grade 11 - 12): Synthesize information from a range of sources (e.g., texts, experiments, simulations) into a coherent understanding of a process, phenomenon, or concept, resolving conflicting information when possible.*(*

__Mathematics__

- Construct and compare linear, quadratic, and exponential models and solve problems.
*CCSS.Math.Content.HSF-LE.A.1*(grade 9 - 12): Distinguish between situations that can be modeled with linear functions and with exponential functions. - Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals. - Recognize situations in which one quantity changes at a constant rate per unit interval relative to another. - Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.*CCSS.Math.Content.HSF-LE.A.2*(grade 9 - 12): Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).*CCSS.Math.Content.HSF-LE.A.3*(grade 9 - 12): Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.*CCSS.Math.Content.HSF-LE.A.4*(grade 9 - 12): For exponential models, express as a logarithm the solution to*abct**= d*where*a, c*, and*d*are numbers and the base*b*is 2, 10, or e; evaluate the logarithm using technology.

- Interpret expressions for functions in terms of the situation they model.
*CCSS.Math.Content.HSF-LE.B.5*(grade 9 - 12): Interpret the parameters in a linear or exponential function in terms of a context.

*Thread: Collecting and Applying Real Data*- Model Eliciting Activity (M.E.A.)
- Trigonometry ‘Carousel’

__Reading__

- Integration of Knowledge and Ideas
*CCSS.ELA-Literacy.CCRA.R.7*Integrate and evaluate content presented in diverse media and formats, including visually and quantitatively, as well as in words.*CCSS.ELA-Literacy.CCRA.R.8*Delineate and evaluate the argument and specific claims in a text, including the validity of the reasoning as well as the relevance and sufficiency of the evidence.

__Writing__

- Production and Distribution of Writing
*CCSS.ELA-Literacy.CCRA.W.6*Use technology, including the Internet, to produce and publish writing and to interact and collaborate with others.

- Research to Build and Present Knowledge
*CCSS.ELA-Literacy.CCRA.W.9*Draw evidence from literary or informational texts to support analysis, reflection, and research.

__Speaking and Listening__

- Comprehension and Collaboration
*CCSS.ELA-Literacy.CCRA.SL.1*Prepare for and participate effectively in a range of conversations and collaborations with diverse partners, building on others’ ideas and expressing their own clearly and persuasively.*CCSS.ELA-Literacy.CCRA.SL.2*Integrate and evaluate information presented in diverse media and formats, including visually, quantitatively, and orally.*CCSS.ELA-Literacy.CCRA.SL.3*Evaluate a speaker’s point of view, reasoning, and use of evidence and rhetoric.

- Presentation of Knowledge and Ideas
*CCSS.ELA-Literacy.CCRA.SL.4*Present information, findings, and supporting evidence such that listeners can follow the line of reasoning and the organization, development, and style are appropriate to task, purpose, and audience.*CCSS.ELA-Literacy.CCRA.SL.5*Make strategic use of digital media and visual displays of data to express information and enhance understanding of presentations.

__Reading: Literacy in Science and Technical subjects__

- Craft and Structure
*CCSS.ELA-Literacy.RST.9-10.4*(grade 9 - 10):Determine the meaning of symbols, key terms, and other domain-specific words and phrases as they are used in a specific scientific or technical context relevant to grades 9–10 texts and topics.

- Integration of Knowledge and Ideas
*CCSS.ELA-Literacy.RST.11-12.7*(grade 11 - 12):Integrate and evaluate multiple sources of information presented in diverse formats and media (e.g., quantitative data, video, multimedia) in order to address a question or solve a problem.*CCSS.ELA-Literacy.RST.11-12.9*(grade 11 - 12):Synthesize information from a range of sources (e.g., texts, experiments, simulations) into a coherent understanding of a process, phenomenon, or concept, resolving conflicting information when possible.

__Mathematics__

- Reason quantitatively and use units to solve problems
*CCSS.Math.Content.HSN-Q.A.1*(grade 9 - 12):Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.*CCSS.Math.Content.HSN-Q.A.2*(grade 9 - 12):Define appropriate quantities for the purpose of descriptive modeling.

- Interpret the structure of expressions
*CCSS.Math.Content.HSA-SSE.A.1*(grade 9 - 12): Interpret expressions that represent a quantity in terms of its context. (Modeling) - Interpret parts of an expression, such as terms, factors, and coefficients. - Interpret complicated expressions by viewing one or more of their parts as a single entity.*For example, interpret P(1+r)n as the product of P and a factor not depending on P.*

- Create equations that describe numbers or relationships
*CCSS.Math.Content.HSA-CED.A.2*(grade 9 - 12): Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

- Understand the concept of a function and use function notation
*CCSS.Math.Content.HSF-IF.A.2*(grade 9 - 12): Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.

- Interpret functions that arise in applications in terms of the context
*CCSS.Math.Content.HSF-IF.B.4*(grade 9 - 12): For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship.*Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.*(Modeling)

- Analyze functions using different representations
*CCSS.Math.Content.HSF-IF.C.7*(grade 9 - 12): Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. (Modeling) a. Graph linear and quadratic functions and show intercepts, maxima, and minima. b. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. c. Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior. d. (+) Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior. e. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.

- Extend the domain of trigonometric functions using the unit circle
*CCSS.Math.Content.HSF-TF.A.3*(grade 9 - 12): (+) Use special triangles to determine geometrically the values of sine, cosine, tangent for π/3, π/4 and π/6, and use the unit circle to express the values of sine, cosines, and tangent for*x*, π +*x*, and 2π –*x*in terms of their values for*x*, where*x*is any real number.*CCSS.Math.Content.HSF-TF.A.4*(grade 9 - 12): (+) Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.

- Model periodic phenomena with trigonometric functions
*CCSS.Math.Content.HSF-TF.B.5*(grade 9 - 12): Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline. (Modeling)*CCSS.Math.Content.HSF-TF.B.7*(grade 9 - 12): (+) Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions using technology, and interpret them in terms of the context. (Modeling)

__BIG IDEA/CONCEPT:__Connecting mathematical abstractions with student interests via a social constructivist learning unit developed with universal design for learning (U.D.L.), project-based learning (P.B.L.) and the engineering design process (E.D.P.) in mind.

__ENDURING UNDERSTANDING:__- Mathematics is relevant to humanitarian, economic, scientific and artistic work
- Mathematics is approachable and comprehensible by interpreting real data, connecting to student interests, enlisting multiple forms of expression, and critiquing peer work
- Mathematics is applicable to music, population studies, digital renderings, build projects (i.e. engineering) and scientific inquiry

__ESSENTIAL QUESTIONS THAT WILL BE CONSIDERED:__EQ1: What are the effects of the universal design for learning tenets (i.e. representation, action & expression, and engagement) on members of a high school Algebra II class?

EQ2: How does the introduction of an engineering design process lesson designed with diverse learners in mind affect their engagement in the classroom?

EQ3: What are the effects of collecting and interpreting real data on members of an Algebra II classroom?

EQ4: What are the effects on students’ perceptions of meaning when lesson plans incorporate formal critiques, multiple forms of expression, and connections to students’ personal interests?

__APPROPRIATE TECHNOLOGY AND TOOLS:__‘Desmos’ (an online grapher program) Large whiteboard

‘Rhinoceros’ (digital rendering software) Individual whiteboards

‘Google SketchUp’ (digital rendering software) 8.5 x 11 and 11 x 17 paper

‘Popplet’ (online brainstorming software) Expo markers

‘iMovie’ (digital video and media software) Colored markers

'VoiceThread' (digital audio recording software) Headphones (class set)

‘Google Docs’ (online collaborative word processing) Microphones (class set)

‘Google Hangouts’/‘Skype’ (for online calling) Aluminum foil

Microsoft Excel Aluminum tape

Exponential/logarithmic graph paper Aluminum flashing

Parabolic satellite dish Mirrored tiles

Non-parabolic solar oven Mirrors

Digital projector and screen Hand saws

Student and teacher laptops Electric drills

Colored pencils and pens Hammers

Scientific/graphing calculators Nails

Outside energy experts (e.g., electrical engineer) Screws

Outside biological expert (e.g., post-doc in cell biology) Chisels

School volunteer (e.g., mechanical engineer) Hot glue gun

School volunteer (e.g., architecture) Various dimensions of wood

__KNOWLEDGE = Content:__Principles of quadratic, exponential, logarithmic, and trigonometric curves

Connecting mathematical abstractions to student interests (e.g., music, humanitarian work, world issues, hands-on projects)

Disseminating knowledge and vetting ideas through engineering, multi-sensory experiences and creative processes & products

__SKILLS = Power verbs:__Gaining comfort using digital and woodworking tools

Familiarity with NASA’s Beginning Engineering, Science and Technology (B.E.S.T.) protocol for the engineering design process (‘ask’, ‘imagine’, ‘plan’, ‘create’, ‘experiment’, ‘improve’)

Amplifying ‘conceptual understanding’, ‘procedural fluency’, ‘strategic competence’, ‘adaptive reasoning’, and a ‘productive disposition’ (i.e. the ‘strands of mathematical proficiency’; Kilpatrick et al., 2001)

Providing useful feedback to peers

Utilizing constructive peer and teacher criticism to drive robust, iterative work

__DISPOSITIONS = Attitude:__Growth-oriented

Focusing on proximal goals

An “inclination to see mathematics as sensible, useful, and worthwhile coupled with a belief in diligence and one’s own efficacy” (Kilpatrick et al., 2001, p. 5)

Intellectually curious

Intrinsically motivated

Gravitating towards resourcefulness

Willing to compassionately collaborate

Perseverant in the face of nascent and continued challenges

Self-advocacy

__ASSESSMENTS:__

**Formative:**- Quick writes and lists of as many ideas from last class
- Imagining and drawing two or more ways to construct various prototypes
- Sharing ideas and drawings with another pair of students and the whole class
- Recording observations
- Describing mathematical relationships
- Calculating solutions to relevant warm-up questions
- Experimenting with the reflectivity, malleability, pliability, and elasticity of wood, aluminum foil, aluminum tape, aluminum flashing, glass, reflective mirrors, necks of guitars, etc..
- Deciding how to communicate findings to the rest of the class
- Communicating team progress
- Diagramming or explaining a tactic used individually or within a team while working towards an answer
- Responding to instructional videos
- Explaining the origin in differences between experimental and theoretical values
- Participation in online and group games
- Offering constructive feedback during formal critiques of student work

*Summative:*1. “Know-Want to Know-Learned” (i.e. “K.W.L.”): Students write or record responses about what they knew about said topics before class, what they still want to know about, and what they learned through the class period.

2. Exit Slip: Computing, diagramming, or beginning to draw with the hopes of reaching important developmental nodes

3. Portfolios

4. Engineering prototypes modified through peer feedback

__OBJECTIVES FROM 6 FACETS + MISCONCEPTIONS:__As a result of working within this curriculum, students will be able to:

*Explain:*Students will be encouraged to link mathematical abstractions regarding quadratic, exponential, logarithmic, and trigonometric functions to digital renderings, parabolic solar ovens, pitch of notes, fret patterns on string instruments, population studies, marine science, economics, and humanitarian aid.

*Apply:*Members of the class will be supported through engineering design projects emphasizing iterative work and constellating around parabolic solar ovens and string instruments. Students will apply their understanding of mathematics to provide well-reasoned commentary to classmates during formal critique, to their development of ad campaigns promoting their designs, and to decisions they make amidst interactive simulations.

*Interpret:*Students will identify what each of the mathematical functions looks like in two-dimensions as well as three-dimensions, translate schematic drawings into built prototypes, and develop conjectures about their carbon footprint and potential downstream effects.

*Empathize:*Students will research the labors and health hazards associated with collecting fuel, sterilizing drinking water, and cooking food in developing nations. Class participants will learn about the challenges associated with developing solar ovens capable of achieving sterilizing temperatures.

*Gain Perspective:*Students will learn that engineering is always a work in progress and that feedback is critical to pushing practical, health-related and artistic efforts forward.

**Gain Self Knowledge:**

Students will develop ‘agency’ (MacLeod, 1987) by working with woodworking tools, using digital software and engaging in metacognition regularly to assess what they know, want to know, and have learned.

*Overcome The Naïve View of: (Misconceptions):*Math is only intended for a few types of learners

Math is abstract and does not have relevance in disciplines like the arts and humanities

Math is not creative

Math classes emphasize memorization and regurgitation

References

Kilpatrick, J. , Swafford, J. , & Findell, B. (2001).

*Adding It Up: Helping children learn*

*mathematics*. Washington, DC: National Academy Press.

MacLeod, J. (1987).

*Ain’t no making it. Leveled Aspirations in a Low Income*

*Neighborhood*. Westview Press: Colorado.