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 Instructional Focus DocumentMathematical Models with Applications
 TITLE : Unit 04: Applications of Algebra and Geometry in Art SUGGESTED DURATION : 12 days

#### Unit Overview

This unit bundles student expectations that address geometric transformations, symmetry, perspective drawings, similarity, proportional and non-proportional change, and trigonometric ratios as applied to fields of art and photography. Concepts are incorporated into both mathematical and real-world problem situations. According to the Texas Education Agency, mathematical process standards including application, tools and techniques, communication, representations, relationships, and justifications should be integrated (when applicable) with content knowledge and skills so that students are prepared to use mathematics in everyday life, society, and the workplace.

Prior to this unit, in Grade 6, students used scale factors involving ratios and rates to solve problems. In Grade 7, students studied similarity in relation to proportional change. In Grade 8, students studied similarity in angle relationships and applied scale factors proportionally to two-dimensional figures. In Geometry, students studied similarity, symmetry, transformations, applied scale factors to two-dimensional and three-dimensional figures to demonstrate proportional and non-proportional change, and applied trigonometric ratios.

During this unit, students use rigid transformations that maintain congruence (translation, reflection, rotation) and non-rigid, similarity transformations (dilations) to describe mathematical patterns and structure in art and photography. Students analyze reflectional and rotational symmetry as demonstrated in art and photography. Students use perspective drawings to represent three-dimensional relationships in two-dimensional representations using one- and two-point perspectives. Students use attributes and properties to analyze the surface area and volume of a three-dimensional figure. Students use scale factor(s) applied to one or two dimensions of a three-dimensional figure to demonstrate non-proportional change in surface area and volume. Students use a scale factor applied to all three dimensions of a three-dimensional figure to demonstrate proportional change in surface area and volume. Students generate scale models or drawings, using proportional and non-proportional change, for products in the field of art. Students determine the effects of cropping and sizing in photography. Students apply trigonometric ratios to determine lengths and angle measure in right triangles found in art.

After this unit, in Unit 05, students will use geometric transformations and proportions as applied to the field of music. Trigonometric ratios will be applied to solve problems in Precalculus. Throughout Math Models with Applications, students will be required to take given information or collected data and determine tools and methods needed to solve the problem situation. The concepts in this unit will be applied in subsequent mathematics courses.

This unit is supporting the development of the Texas College and Career Readiness Standards (TxCCRS): I. Numeric Reasoning B1; II. Algebraic Reasoning B1; III. Geometric Reasoning A3, B1, B2, B3; IV. Measurement Reasoning C1, C2, C3; VIII. Problem Solving and Reasoning; IX. Communication and Representation; X. Connections.

According to the Connections Standard for Grades 9-12 from the National Council of Teachers of Mathematics (NCTM), “Instructional programs from pre-kindergarten through grade 12 should enable students to:

• recognize and use connections among mathematical ideas;
• understand how mathematical ideas interconnect and build on one another to produce a coherent whole;
• recognize and apply mathematics in contexts outside of mathematics.

When students can see the connections across different mathematical content areas, they develop a view of mathematics as an integrated whole. As they build on their previous mathematical understandings while learning new concepts, students become increasingly aware of the connections among various mathematical topics. As students' knowledge of mathematics, their ability to use a wide range of mathematical representations, and their access to sophisticated technology and software increase, the connections they make with other academic disciplines, especially the sciences and social sciences, give them greater mathematical power” (NCTM, 2000, p. 354).

National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics: Connections standard for grades 9-12. Reston, VA: National Council of Teachers of Mathematics, Inc.
Education Policy Improvement Center (2009), Texas College and Career Standards, Austin, TX, University of Texas Printing.

#### OVERARCHING UNDERSTANDINGS and QUESTIONS

Application of attributes and measures of figures can be generalized to describe geometric relationships which can be used to solve problem situations.

• Why are attributes and measures of figures used to generalize geometric relationships?
• How can numeric patterns be used to formulate geometric relationships?
• Why is it important to distinguish measureable attributes?
• How do geometric relationships relate to other geometric relationships?
• Why is it essential to develop generalizations for geometric relationships?
• How are geometric relationships applied to solve problem situations?

Attributes and properties of two- and three-dimensional geometric shapes are foundational to developing geometric and measurement reasoning.

• Why is it important to compare and contrast attributes and properties of two- and three-dimensional geometric shapes?
• How does analyzing the attributes and properties of two- and three-dimensional geometric shapes develop geometric and measurement reasoning?
Performance Assessment(s) Overarching Concepts
Unit Concepts
Unit Understandings
 Assessment information provided within the TEKS Resource System are examples that may, or may not, be used by your child’s teacher. In accordance with section 26.006 (2) of the Texas Education Code, "A parent is entitled to review each test administered to the parent’s child after the test is administered." For more information regarding assessments administered to your child, please visit with your child’s teacher.

Algebraic Reasoning

• Patterns/Rules

Geometric Reasoning

• Congruence
• Geometric Attributes/Properties
• Perspective Drawings
• Scale Factors
• Similarity
• Symmetry
• Transformations

Measurement Reasoning

• Angle
• Length

Associated Mathematical Processes

• Application
• Tools and Techniques
• Problem Solving Model
• Communication
• Representations
• Relationships
• Justification

Geometric relationships such as similarity, transformations, symmetry, and perspective can be used to model, describe, and solve problems in art and photography.

• Which geometric transformations maintain congruence, and how are congruent transformations applied in art and photography?
• Which geometric transformation demonstrates similarity, and how is similarity applied in art and photography?
• What is the purpose of symmetry, and how is it illustrated in art and photography?
• What are the different types of perspective, and how are they illustrated and applied in art and photography?
• How are similarity, transformations, symmetry, and perspective used to model problems in art and photography?
• How are similarity, transformations, symmetry, and perspective used to describe problems in art and photography?
• How are similarity, transformations, symmetry, and perspective used to solve problems in art and photography?
• How are the dimensions of photographs changed by cropping and sizing?
 Assessment information provided within the TEKS Resource System are examples that may, or may not, be used by your child’s teacher. In accordance with section 26.006 (2) of the Texas Education Code, "A parent is entitled to review each test administered to the parent’s child after the test is administered." For more information regarding assessments administered to your child, please visit with your child’s teacher.

Algebraic Reasoning

• Ratios
• Solve

Geometric Reasoning

• Geometric Attributes/Properties
• Scale Factors
• Two-Dimensional Figures
• Three-Dimensional Figures

Measurement Reasoning

• Dimensional Change
• Length
• Surface Area
• Units of Measure
• Volume

Associated Mathematical Processes

• Application
• Tools and Techniques
• Problem Solving Model
• Communication
• Representations
• Relationships
• Justification

Using scale factor(s) on one or two dimensions of a three-dimensional figure results in a non-proportional change in surface area and volume, whereas using a scale factor to all three dimensions of a three-dimensional figure results in a proportional change in surface area and volume.

• How are similar figures generated?
• When is dimensional change in three-dimensional figures proportional?
• When is dimensional change in three-dimensional figures non-proportional?
• What are some of the possible effects on the surface area and volume of an object when a scale factor is applied to just one of the dimensions?  Scale factor(s) on two dimensions?  Scale factor(s) three dimensions?
• How can the resulting effects on the measurements of the scaled object be predicted?
• How are scale factors used to create proportional and non-proportional three-dimensional designs in art and photography?
 Assessment information provided within the TEKS Resource System are examples that may, or may not, be used by your child’s teacher. In accordance with section 26.006 (2) of the Texas Education Code, "A parent is entitled to review each test administered to the parent’s child after the test is administered." For more information regarding assessments administered to your child, please visit with your child’s teacher.

Algebraic Reasoning

• Patterns/Rules
• Ratios
• Solve

Geometric Reasoning

• Geometric Attributes/Properties
• Trigonometric Ratios

Measurement Reasoning

• Angle Measures
• Length

Associated Mathematical Processes

• Application
• Tools and Techniques
• Problem Solving Model
• Communication
• Representations
• Relationships
• Justification

Trigonometric ratios are used to determine lengths and angles in figures composed of right triangles

• How are trigonometric ratios applied in art?
• How can trigonometric ratios be used to calculate lengths and angle measures in a right triangle?
• What determines which trigonometric ratio to use in a particular problem?

#### MISCONCEPTIONS / UNDERDEVELOPED CONCEPTS

Misconceptions:

• Some students may think that when a scale factor is applied to only one or two dimensions of a three-dimensional figure that the figure will change proportionally rather than changing non-proportionally.
• Some students may think that when a scale factor is applied to all dimensions in a figure, the surface and volume are changed by the same scale factor rather than the surface area and volume changing by the square and cube of the scale factor, respectively.
• Some students may think any repeated use of a shape arranged in a plane produces a tessellation rather than understanding that a tessellation must be the complete pattern that covers the entire plane leaving no spaces.
• Some students may think that taking the inverse of a trigonometric function is the same as taking the reciprocal of the trigonometric function.

#### Unit Vocabulary

• Cropping – cutting off portions of a photograph
• Dilation – non-rigid, similarity transformation in which a figure is enlarged or reduced using a scale factor and a center of dilation
• One-point perspective (vanishing point) – a linear perspective that occurs when all parallel lines drawn away from the viewer appear to converge in a single vanishing point on the horizon line
• Perspective drawing – a type of drawing using lines and points that can represent a three-dimensional view on a two-dimensional surface
• Reflection - rigid transformation in which each point in a geometric figure is at an equal distance on the opposite side of a given line (line of symmetry)
• Rotation – rigid transformation where each point on the figure is rotated about a given point
• Sizing – a change in the dimensions of a photograph by a scale factor which may or may not be proportional
• Similar figures – shapes whose angles are congruent and side lengths are proportional (equal scale factor)
• Symmetry – refers to geometric figures or graphs consisting of two parts on either side of a point, line, or plane that are identical or congruent to each other
• Transformation – one to one mapping of points in a plane such that each point in the pre-image has a unique image and each point in the image has a pre-image
• Trigonometric ratios – a ratio of the measures of two sides of a right triangle based on their position in relation to an acute angle in the right triangle
• Translation – rigid transformation moving all points in a geometric figure the same distance and the same direction
• Two-point perspective – a linear perspective that occurs when all parallel lines drawn away from the viewer appear to converge in two different points on the horizon line that create a 90° angle from the point of projection or common vertex

Related Vocabulary:

 Cosine Inverse trigonometric ratios Non-proportional change Proportional change   Ratio Reflectional symmetry Rotational symmetry Scale factor Sine Surface area Tangent Tessellation Tessellation Three-dimensional figure Two-dimensional figure Vanishing point Volume
Unit Assessment Items System Resources Other Resources

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Unit Assessment Items that have been published by your district may be accessed through Search All Components in the District Resources tab. Assessment items may also be found using the Assessment Creator if your district has granted access to that tool.

System Resources may be accessed through Search All Components in the District Resources Tab.

Texas Higher Education Coordinating Board – Texas College and Career Readiness Standards (select CCRS from Standard Set dropdown menu)

Texas Education Agency – Mathematics Curriculum

Texas Education Agency – STAAR Mathematics Resources

Texas Education Agency Texas Gateway – Revised Mathematics TEKS: Vertical Alignment Charts

Texas Education Agency Texas Gateway – Mathematics TEKS: Supporting Information

Texas Education Agency Texas Gateway – Interactive Mathematics Glossary

Texas Education Agency Texas Gateway – Resources Aligned to Mathematical Models with Applications Mathematics TEKS

Texas Instruments – Graphing Calculator Tutorials

TEKS# SE# TEKS Unit Level Specificity

• Bold black text in italics: Knowledge and Skills Statement (TEKS)
• Bold black text: Student Expectation (TEKS)
• Strike-through: Indicates portions of the Student Expectation that are not included in this unit but are taught in previous or future unit(s)
• Blue text: Supporting information / Clarifications from TCMPC (Specificity)
• Blue text in italics: Unit-specific clarification
• Black text: Texas Education Agency (TEA); Texas College and Career Readiness Standards (TxCCRS)
M.1 Mathematical process standards. The student uses mathematical processes to acquire and demonstrate mathematical understanding. The student is expected to:
M.1A Apply mathematics to problems arising in everyday life, society, and the workplace.

Apply

MATHEMATICS TO PROBLEMS ARISING IN EVERYDAY LIFE, SOCIETY, AND THE WORKPLACE
Including, but not limited to:

• Mathematical problem situations within and between disciplines
• Everyday life
• Society
• Workplace

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• X. Connections
M.1B Use a problem-solving model that incorporates analyzing given information, formulating a plan or strategy, determining a solution, justifying the solution, and evaluating the problem-solving process and the reasonableness of the solution.

Use

A PROBLEM-SOLVING MODEL THAT INCORPORATES ANALYZING GIVEN INFORMATION, FORMULATING A PLAN OR STRATEGY, DETERMINING A SOLUTION, JUSTIFYING THE SOLUTION, AND EVALUATING THE PROBLEM-SOLVING PROCESS AND THE REASONABLENESS OF THE SOLUTION
Including, but not limited to:

• Problem-solving model
• Analyze given information
• Formulate a plan or strategy
• Determine a solution
• Justify the solution
• Evaluate the problem-solving process and the reasonableness of the solution

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• VIII. Problem Solving and Reasoning
M.1C Select tools, including real objects, manipulatives, paper and pencil, and technology as appropriate, and techniques, including mental math, estimation, and number sense as appropriate, to solve problems.

Select

TOOLS, INCLUDING REAL OBJECTS, MANIPULATIVES, PAPER AND PENCIL, AND TECHNOLOGY AS APPROPRIATE, AND TECHNIQUES, INCLUDING MENTAL MATH, ESTIMATION, AND NUMBER SENSE AS APPROPRIATE, TO SOLVE PROBLEMS
Including, but not limited to:

• Appropriate selection of tool(s) and techniques to apply in order to solve problems
• Tools
• Real objects
• Manipulatives
• Paper and pencil
• Technology
• Techniques
• Mental math
• Estimation
• Number sense

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• VIII. Problem Solving and Reasoning
M.1D Communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams, graphs, and language as appropriate.

Communicate

MATHEMATICAL IDEAS, REASONING, AND THEIR IMPLICATIONS USING MULTIPLE REPRESENTATIONS, INCLUDING SYMBOLS, DIAGRAMS, GRAPHS, AND LANGUAGE AS APPROPRIATE
Including, but not limited to:

• Mathematical ideas, reasoning, and their implications
• Multiple representations, as appropriate
• Symbols
• Diagrams
• Graphs
• Language

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• IX. Communication and Representation
M.1E Create and use representations to organize, record, and communicate mathematical ideas.

Create, Use

REPRESENTATIONS TO ORGANIZE, RECORD, AND COMMUNICATE MATHEMATICAL IDEAS
Including, but not limited to:

• Representations of mathematical ideas
• Organize
• Record
• Communicate
• Evaluation of the effectiveness of representations to ensure clarity of mathematical ideas being communicated
• Appropriate mathematical vocabulary and phrasing when communicating mathematical ideas

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• IX. Communication and Representation
M.1F Analyze mathematical relationships to connect and communicate mathematical ideas.

Analyze

MATHEMATICAL RELATIONSHIPS TO CONNECT AND COMMUNICATE MATHEMATICAL IDEAS
Including, but not limited to:

• Mathematical relationships
• Connect and communicate mathematical ideas
• Conjectures and generalizations from sets of examples and non-examples, patterns, etc.
• Current knowledge to new learning

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• X. Connections
M.1G Display, explain, and justify mathematical ideas and arguments using precise mathematical language in written or oral communication.

Display, Explain, Justify

MATHEMATICAL IDEAS AND ARGUMENTS USING PRECISE MATHEMATICAL LANGUAGE IN WRITTEN OR ORAL COMMUNICATION
Including, but not limited to:

• Mathematical ideas and arguments
• Validation of conclusions
• Displays to make work visible to others
• Diagrams, visual aids, written work, etc.
• Explanations and justifications
• Precise mathematical language in written or oral communication

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• IX. Communication and Representation
M.7 Mathematical modeling in fine arts. The student uses mathematical processes with algebra and geometry to study patterns and analyze data as it applies to fine arts. The student is expected to:
M.7A

Use trigonometric ratios and functions available through technology to model periodic behavior in art and music.

Use

TRIGONOMETRIC RATIOS AVAILABLE THROUGH TECHNOLOGY

Including, but not limited to:

• Trigonometric ratios – a ratio of the measures of two sides of a right triangle based on their position in relation to an acute angle in the right triangle
• Trigonometric ratios are used to find lengths in art, photography, etc.

Note(s):

• Geometry studied trigonometric ratios.
• Precalculus will study trigonometric ratios and functions.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS
• I. Numeric Reasoning
• B1 – Perform computations with real and complex numbers.
• II. Algebraic Reasoning
• B1 – Recognize and use algebraic (field) properties, concepts, procedures, and algorithms to combine, transform, and evaluate expressions.
• III. Geometric reasoning
• A3 – Recognize and apply right triangle relationships including basic trigonometry.
• B1 – Identify and apply transformations to figures.
• B2 – Identify the symmetries of a plane figure.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
M.7B Use similarity, geometric transformations, symmetry, and perspective drawings to describe mathematical patterns and structure in art and photography.

Use

SIMILARITY, GEOMETRIC TRANSFORMATIONS, SYMMETRY, AND PERSPECTIVE DRAWINGS

Including, but not limited to:

• Similar figures – shapes whose angles are congruent and side lengths are proportional (equal scale factor)
• Geometric transformations
• Transformation – one to one mapping of points in a plane such that each point in the pre-image has a unique image and each point in the image has a pre-image
• Dilation – non-rigid, similarity transformation in which a figure is enlarged or reduced using a scale factor and a center of dilation
• Reflection – rigid transformation in which each point in a geometric figure is at an equal distance on the opposite side of a given line (line of symmetry)
• Translation – rigid transformation moving all points in a geometric figure the same distance and the same direction
• Rotation – rigid transformation where each point on the figure is rotated about a given point
• Perspective drawing – a type of drawing using lines and points that can represent a three-dimensional view on a two-dimensional surface
• One-point perspective (vanishing point) – a linear perspective that occurs when all parallel lines drawn away from the viewer appear to converge in a single vanishing point on the horizon line
• Two-point perspective – a linear perspective that occurs when all parallel lines drawn away from the viewer appear to converge in two different points on the horizon line that create a 90° angle from the point of projection or common vertex
• Symmetry – refers to geometric figures or graphs consisting of two parts on either side of a point, line, or plane that are identical or congruent to each other
• Property of congruence
• Congruence is preserved when a two-dimensional figure is transformed and the image is identical in shape and size.
• Congruence is not preserved when a two-dimensional figure is transformed and the image is not identical in shape and/or identical in size.
• Property of orientation of the vertices
• Orientation of the vertices of an image is determined by naming the vertices in the same order as the corresponding vertices of its pre-image and not determined by a figure’s direction or a figure’s size.
• Orientation of the vertices is preserved in translation, rotations, and dilations.
• Orientation of the vertices is not preserved in reflections.

To Describe

MATHEMATICAL PATTERNS AND STRUCTURE IN ART AND PHOTOGRAPHY

Including, but not limited to:

• Symmetry and transformations
• Cropping and sizing photographs
• Cropping – cutting off portions of a photograph
• Sizing –  a change in the dimensions of a photograph by a scale factor which may or may not be proportional
• Perspective in paintings and diagrams

Note(s):

• Geometry studied trigonometric ratios.
• Precalculus will study trigonometric ratios and functions.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS
• I. Numeric Reasoning
• B1 – Perform computations with real and complex numbers.
• II. Algebraic Reasoning
• B1 – Recognize and use algebraic (field) properties, concepts, procedures, and algorithms to combine, transform, and evaluate expressions.
• III. Geometric reasoning
• B1 – Identify and apply transformations to figures.
• B2 – Identify the symmetries of a plane figure.
• B3 – Use congruence transformations and dilations to investigate congruence, similarity, and symmetries of plane figures.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
M.7D Use scale factors with two-dimensional and three-dimensional objects to demonstrate proportional and non-proportional changes in surface area and volume as applied to fields.

Use

SCALE FACTORS WITH TWO-DIMENSIONAL AND THREE-DIMENSIONAL OBJECTS

Including, but not limited to:

• Scale factors applied to all dimensions of a figure or object.
• Creates a figure or object that is similar to the original
• Scale factors applied to only one dimension of a two-dimensional figure or object or applied to one or two dimensions of a three-dimensional figure or object.
• Creates a figure or object that is not similar to the original.
• Different scale factors applied to each of the dimensions of a two-dimensional or three-dimensional figure or object.
• Creates a figure or object that is not similar to the original.

To Demonstrate

PROPORTIONAL AND NON-PROPORTIONAL CHANGES IN SURFACE AREA AND VOLUME AS APPLIED TO FIELDS

Including, but not limited to:

• Drawing, painting
• Photography
• Sculpture

Notes:

• Grades 6 and 7 studied proportions.
• Grades 2 through 5 studied perimeter.
• Geometry studied perimeter, area, surface area, and volume.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS
• I. Numeric Reasoning
• B1 – Perform computations with real and complex numbers.
• II. Algebraic Reasoning
• B1 – Recognize and use algebraic (field) properties, concepts, procedures, and algorithms to combine, transform, and evaluate expressions.
• III. Geometric reasoning
• B1 – Identify and apply transformations to figures.
• B2 – Identify the symmetries of a plane figure.
• B3 – Use congruence transformations and dilations to investigate congruence, similarity, and symmetries of plane
• IV. Measurement reasoning
• C1 – Find the perimeter and area of two-dimensional figures
• C2 – Determine the surface area and volume of three-dimensional figures.
• C3 – Determine indirect measurements of figures using scale drawings, similar figures, Pythagorean Theorem, and basic geometry.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections