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 Instructional Focus DocumentMathematical Models with Applications
 TITLE : Unit 02: Geometric Relationships in Architecture and Engineering SUGGESTED DURATION : 6 days

#### Unit Overview

This unit bundles student expectations that address the Pythagorean Theorem, special right triangle relationships, and trigonometric ratios to calculate distances and find angle measures as applied to fields of architecture and engineering. 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 8, students modeled the Pythagorean Theorem and used the Pythagorean Theorem and it's converse to solve problems and to determine distances on a coordinate plane. In Geometry, students applied special right triangle relationships and the Pythagorean Theorem, including Pythagorean triples, to solve problems. In Geometry, students also used the trigonometric ratios sine, cosine, and tangent to find lengths of sides and measures of angles in right triangles in real-world problem situations.

During this unit, students apply the Pythagorean Theorem, special right triangles relationships (30°– 60°– 90° or 45°– 45°– 90°), and trigonometric ratios to solve real-world problems. Students write representative equations using the appropriate theorem or trigonometric ratio, including the inverses of the trigonometric ratios, to find missing values and solve real-world problems in architecture and engineering.

After this unit, students will solve measurement problems, including problems involving proportional and non-proportional change as applied to real-world problems. Students will apply the Pythagorean Theorem, special right triangle relationships, and trigonometric ratios, as appropriate, to determine lengths or distances needed to solve measurement problems. Throughout the fields of Mathematical 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 also 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; VII. Functions A2; 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

Geometric and spatial reasoning are necessary to describe and analyze geometric relationships in mathematics and the real-world.

• Why are geometric and spatial reasoning necessary in the development of an understanding of geometric relationships?
• Why is it important to visualize and use diagrams to effectively communicate/illustrate geometric relationships?

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?
• How do different systems of measure relate to one another?
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
• Ratios/Rates
• Solve

Geometric Reasoning

• Geometric Attributes/Properties
• Relationships
• Pythagorean Theorem
• Two-Dimensional Figures

Measurement Reasoning

• Angle Measures
• Distance/Length
• Trigonometric Ratios
• Units of Measure

Associated Mathematical Processes

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

Diagrams can be used to visualize and illustrate geometric relationships and aid in solving problems in architecture and engineering.

• Why are diagrams necessary for visualizing the geometric relationships found in the problem situation?
• How are diagrams used to organize information from the problem situation?
• How do diagrams aid in calculations when solving problems?

Geometric tools such as Pythagorean Theorem, special right triangle relationships, and trigonometric ratios are used to determine lengths and angles in right triangles.

• How can the Pythagorean Theorem be used to calculate distances?
• How can the Pythagorean Theorem be used to determine if the triangle is a right triangle?
• For which particular geometric figures can Pythagorean Theorem be used?
• What information is needed in order to use the Pythagorean Theorem?
• How can the special right triangle relationships be used to calculate distances?
• For which particular geometric figures can special right triangle relationships be used?
• What information is needed to use special right triangle relationships to solve problems?
• How can trigonometric ratios be used to calculate distances and angle measures in a right triangle?
• For what particular figures can trigonometric ratios be used to solve problems?
• What information is needed to use trigonometric ratios to solve problem situations?

#### MISCONCEPTIONS / UNDERDEVELOPED CONCEPTS

Misconceptions:

• Some students may think that the Pythagorean Theorem can be used for all triangles rather than just for right triangles.
• Some students think that if an angle in a triangle is given, then the hypotenuse is the side opposite that angle rather than always the side opposite the right angle.
• Some students may be confused when determining the side adjacent to the side opposite an angle.

Underdeveloped Concepts:

• Students may use the calculator in radians since that is the default, instead of changing the calculator setting to degrees when calculating trigonometric functions.

#### Unit Vocabulary

• Pythagorean Theorem – in Euclidean geometry, relationship between the lengths of the sides of a right triangle. The square of the length of the hypotenuse is equal to the sum of the quares of the lengths of the other two sides.
• Special Right-Triangles – right triangles which have angles that measure 30°-60°-90° or 45°-45°-90°
• 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

Related Vocabulary:

 Adjacent angle Adjacent side Angle of depression Angle of elevation Cosine of an angle Hypotenuse Isosceles right  triangle Leg of a triangle Opposite side Opposite angle Right triangle Inverse trigonometric functions Sine of an angle Tangent of an angle
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.6 Mathematical modeling in science and engineering. The student applies mathematical processes with algebra and geometry to study patterns and analyze data as it applies to architecture and engineering. The student is expected to:
M.6C Use the Pythagorean Theorem and special right-triangle relationships to calculate distances.

Use

THE PYTHAGOREAN THEOREM AND SPECIAL RIGHT-TRIANGLE RELATIONSHIPS

AS APPLIED TO SCIENCE AND ENGINEERING

Including, but not limited to:

• Pythagorean Theorem – in Euclidean geometry, relationship between the lengths of the sides of a right triangle. The square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.
• Symbolic notation: c2a2b2, where c is the length of the hypotenuse and a and b are the lengths of the other two sides
• Special right-triangles – right triangles which have angles that measure 30°– 60°– 90° or 45°– 45°– 90° degrees
• 30°– 60°– 90° Triangle
• Length of the hypotenuse is twice the length of the short leg.
• Length of the long leg is  times the length of the short leg.
• 45°– 45°– 90° Triangle
• Isosceles right triangle
• Legs are equal in length.
• Length of the hypotenuse is  times the length of the legs.

To Calculate

DISTANCES

Including, but not limited to:

• Lengths of sides of a triangle
• Pythagorean Theorem: a2b2c2
• Triangular models in real-world problem situations

Note(s):

• Grade 8 used Pythagorean Theorem and its converse to solve problems and find distances on the coordinate plane
• Geometry proved the Pythagorean Theorem and used it in problem solving and other proofs.
• Geometry studied special right triangles, 30°– 60°– 90° and 45°– 45°– 90° and used the relationships to solve problems
• 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.
• VII. Functions
• A2 – Recognize and distinguish between different types of functions.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
M.6D Use trigonometric ratios to calculate distances and angle measures as applied to fields.
Use

TRIGONOMETRIC RATIOS

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
• Inverse trigonometric functions can be used to find the measure of an acute angle given the length of two sides of a right triangle.
• If sin∠ , then ∠ = sin–1.
• The inverse of sin∠ may also be written as arc sin.
• If cos∠ = , then ∠ = cos–1.
• The inverse of cos∠ may also be written as arccos.
• If tan∠ = , then ∠ = tan–1.
• The inverse of tan∠ may also be written as arctan.

To Calculate

DISTANCES AND ANGLE MEASURES AS APPLIED TO FIELDS

Including, but not limited to:

• Engineering and Surveying
• Construction (civil engineering)

Note(s):

• Algebra I studied parent functions f(x) = x, f(x) = x2, and f(x) = .
• Geometry studied trigonometric ratios and used them to find distances and angle measures in problems.
• Precalculus will use trigonometric ratios to solve real-world problems.
• 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.
• VII. Functions
• A2 – Recognize and distinguish between different types of functions.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections