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 TITLE : Unit 10: Making Connections SUGGESTED DURATION : 15 days

#### Unit Overview

Introduction
This unit bundles student expectations that address one-variable equations situations, total cost of repaying a loan, slope and y-intercept in proportional and non-proportional situations, direct variation, function representations, scatterplots, transformational geometry, and the Pythagorean Theorem. According to the Texas Education Agency, mathematical process standards including application, a problem-solving model, 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. The introduction to the grade level standards state, “While the use of all types of technology is important, the emphasis on algebra readiness skills necessitates the implementation of graphing technology.” Additionally, the availability of graphing technology is required during STAAR testing.

Prior to this Unit
In Unit 03, students represented, modeled, and solved one-variable equations with variables on both sides of the equal sign. In Unit 04, students used similar right triangles to develop an understanding of slope. Students explored that the ratio of the change in y-values and x-values is the same for any two points on the same line. Students used data from a table or graph to determine the rate of change or slope and the y-intercept in problem situations. In Unit 05, students distinguished between proportional and non-proportional linear situations and functions. They also solved problems involving direct variation. In Unit 06, students analyzed bivariate data within scatterplots and used trend lines to make predictions. In Unit 07, students explored the algebraic representations of reflections, translations, rotations, and dilations on the coordinate plane. In Unit 08, students were introduced to irrational numbers and solved problems involving the Pythagorean Theorem.

During this Unit
Students extend their understanding of solving equations to model and solve one-variable equations with variables on both sides of the equal sign. Students use data from a table or graph to determine the rate of change or slope and the y-intercept. Students specifically examine the relationship between the unit rate and slope of a line that represents a proportional linear situation. Students must identify functions using sets of ordered pairs, tables, mappings, and graphs. Students continue to examine characteristics of linear relationships through the lens of trend lines that approximate the relationship between bivariate sets of data. Observations include questions of association such as linear, non-linear, or no association. Students use trend lines that approximate the linear relationship between bivariate sets of data to make predictions. Students extend concepts of similarity to dilations on a coordinate plane as they compare and contrast a shape and its dilation(s). The concept of proportionality is revisited as students generalize the ratio of corresponding sides of a shape and its dilation as well as use an algebraic representation to explain the effect of a dilation on a coordinate plane. Properties of orientation and congruence are examined as students generalize the properties as they apply to rotations, reflections, translations, and dilations of two-dimensional figures on a coordinate plane. Students are expected to use an algebraic representation to explain the effect of translations, reflections over the x- or y-axis, dilations when a positive rational number scale factor is applied to a shape, and rotations limited to 90°, 180°, 270°, and 360°. The relationship between linear and area measurements of a shape and its dilation are also examined as students model the relationship and determine that the measurements are affected by both the scale factor and the dimension (one- or two-dimensional) of the measurement. Students use the Pythagorean Theorem and its converse to solve problems and apply these understandings to the coordinate plane as they determine the distance between two points on the coordinate plane. Financial literacy contexts, such as calculating and comparing simple and compound interest rates and how those rates affect earnings in a savings account or the total cost of repaying a loan or credit card, are embedded in this unit.

Other considerations: Reference the Mathematics COVID-19 Gap Implementation Tool Grade 8

After this Unit
In Algebra I, students will solve one-variable inequalities, including those for which the application of the distributive property is necessary and in which variables are included on both sides of the equation. Students will also calculate the rate of change of a linear function represented tabularly, graphically, or algebraically in context of mathematical and real-world problems. They will apply the concept of transformations as they examine parameter changes to linear and quadratic parent functions. In Geometry, students will generate and describe rigid and non-rigid transformations as they describe and perform transformations of figures in a plane using coordinate notations and determine the image or pre-image of a given two-dimensional figure under a composition of rigid and/or non-rigid transformations. Students will also identify and distinguish between reflection and rotational symmetry in a plane figure as well as apply the definition of congruence, in terms of rigid transformations, to identify congruent figures and their corresponding sides and angles.

Research
According to the Van de Walle, Karp, and Bay-Williams (2010), “Algebraic thinking or algebraic reasoning involves forming generalizations from experiences with number and computation, formalizing these ideas with the use of a meaningful symbol system, and exploring the concepts of pattern and functions. Far from a topic with little real-world use, algebraic thinking pervades all of mathematics and is essential for making mathematics useful in daily life” (p. 254). The National Council of Teachers of Mathematics (2010) states that “To develop a deep understanding of linear equations and linear functions, it is important for students to understand how different mathematical relationships between two quantities are reflected in the graph of the line that represents those relationships” (p. 18). They also conclude that, “In Grade 8, students are laying the foundation for many of the more sophisticated concepts they will learn in later grades. For example, students’ work with congruence and similarity will be applied when students learn about the various combinations of conditions that ensure congruent and similar triangles, such as the postulates and theorems” (p. 92).

National Council of Teachers of Mathematics. (2010). Focus in Grade 8: Teaching with Curriculum Focal Points. Reston, VA: National Council of Teacher of Mathematics, Inc.
Texas Education Agency & Texas Higher Education Coordinating Board. (2009). Texas college and career readiness standards. Retrieved from http://www.thecb.state.tx.us/index.cfm?objectid=E21AB9B0-2633-11E8-BC500050560100A9
Texas Education Agency. (2013). Texas response to curriculum focal points for kindergarten through grade 8 mathematics. Retrieved from https://www.texasgateway.org/resource/txrcfp-texas-response-curriculum-focal-points-k-8-mathematics-revised-2013
Van de Walle, J., Karp, K., & Bay-Williams, J. (2010).Elementary and middle school mathematics, teaching developmentally. Boston, MA: Allyn & Bacon.

 Quantitative relationships model problem situations efficiently and can be used to make generalizations, predictions, and critical judgements in everyday life. What patterns exist within different types of quantitative relationships and where are they found in everyday life? Why is the ability to model quantitative relationships in a variety of ways essential to solving problems in everyday life? Financial and economic knowledge leads to informed and rational decisions allowing for effective management of financial resources when planning for a lifetime of financial security.  Why is financial stability important in everyday life? What economic and financial knowledge is critical for planning for a lifetime of financial security? How can mapping one’s financial future lead to significant short and long-term benefits? How can current financial and economic factors in everyday life impact daily decisions and future opportunities?
Unit Understandings
and Questions
Overarching Concepts
and Unit Concepts
Performance Assessment(s)
• Understanding how two quantities vary together (covariation) in situations involving invariant (constant) relationships builds flexible functional reasoning in order to make predictions and critical judgements about the relationship.
• The unit rate can be determined from the graph of a proportional relationship and used to describe the constant rate of change, the slope of the line.
• How can a graph of a proportional relationship be used to interpret the slope of a line and the unit rate?
• The constant rate of change in y-values compared to the change in x-values, slope, and the y-intercept are the two characteristics that are used to define linear proportional and non-proportional situations.
• What relationship exists between the y-intercept and proportional and non-proportional problem situations?
• What is the process of determining the …
• y-intercept
• rate or change or slope
… of a set of data represented in a …
• table?
• graph?
• Proportional and non-proportional relationships can be presented using multiple representations, and those representations can be examined to distinguish between linear and non-linear proportional situations and identify attributes of linear relationships, all of which develops foundational concepts of functions.
• How can a trend line be used to …
• make predictions?
• describe a situation?
• What are the characteristics of a trend line that represents a …
• positive trend?
• negative trend?
• no trend?
• What are the characteristics of a function?
• How can sets of …
• ordered pairs
• tables
• mappings
• graphs
… be used to determine if a relationship is a function?
• What is the process for representing a linear relationship …
• verbally?
• with a table?
• with a graph?
• with an equation that simplifies to the form of y = mx + b?
• How are independent and dependent quantities related in a linear problem situation?
• What is the meaning of each of the variables in the equation y = mx + b?
• How are the table and graph of a linear problem situation related to an equation that simplifies to the form of y = mx + b?
• Understanding interest rates helps one make informed financial management decisions, which promotes a more secured financial future.
• What is the process for determining …
• simple interest?
• compound interest?
• How does the equation for simple interest differ from the equation of compound interest?
• How does understanding interest rates promote a more secured financial future?
• Proportionality
• Attributes of Linear Relations
• Slope
• y-intercept
• Statistics
• Predictions and inferences
• Data
• Statistical representations
• Ratios and Rates
• Slope
• Unit Rates
• Relationships and Generalizations
• Independent and dependent quantities
• Functions
• Linear proportional
• Linear non-proportional
• Representations
• Personal Financial Literacy
• Interest
• Simple
• Compound
• Associated Mathematical Processes
• Application
• Problem Solving Model
• Tools and Techniques
• Communication
• Representations
• Relationships
• Justification
 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.

 Quantitative relationships model problem situations efficiently and can be used to make generalizations, predictions, and critical judgements in everyday life. What patterns exist within different types of quantitative relationships and where are they found in everyday life? Why is the ability to model quantitative relationships in a variety of ways essential to solving problems in everyday life?
Unit Understandings
and Questions
Overarching Concepts
and Unit Concepts
Performance Assessment(s)
• Equations can be modeled, written, and solved using various methods to gain insight into the context of the situation and make critical judgments about algebraic relationships and flexible, efficient strategies.
• Why are expressions considered foundational to equations?
• How can a(n) …
• concrete model
• pictorial model
• algebraic representation
… be used to represent and solve an equation?
• What models effectively and efficiently represent how to solve equations?
• What is the process for solving an equation with variables on both sides, and how can the process be …
• described verbally?
• represented algebraically?
• When considering equations, …
• why is the variable isolated in order to solve?
• how are negative values represented in concrete and pictorial models?
• why must the solution be justified in terms of the problem situation?
• why does equivalence play an important role in the solving process?
• Why is it important to understand when and how to use standard algorithms?
• How does knowing more than one solution strategy build mathematical flexibility?
• Expressions, Equations, and Relationships
• Numeric and Algebraic Representations
• Expressions
• Equations
• Equivalence
• Operations
• Properties of operations
• Order of operations
• Representations
• Associated Mathematical Processes
• Application
• Problem Solving Model
• Tools and Techniques
• Communication
• Representations
• Relationships
• Justification
 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.

 Quantitative relationships model problem situations efficiently and can be used to make generalizations, predictions, and critical judgements in everyday life. What patterns exist within different types of quantitative relationships and where are they found in everyday life? Why is the ability to model quantitative relationships in a variety of ways essential to solving problems in everyday life? Geometric, spatial, and measurement reasoning are foundational to visualizing, analyzing, and applying relationships within and between scale, shapes, quantities, and spatial relations in everyday life. Why is developing geometric, spatial, and measurement reasoning essential? How does geometric, spatial, and measurement reasoning affect how one sees and works in the world?
Unit Understandings
and Questions
Overarching Concepts
and Unit Concepts
Performance Assessment(s)
• Understanding how two quantities vary together (covariation) and can be reasoned up and down in situations involving invariant (constant) relationships builds flexible proportional reasoning in order to make predictions and critical judgements about the relationship.
• Proportional relationships can be used to describe dilations by generalizing the ratios of corresponding sides of similar shapes and the relationship between the attributes of shape and its dilation in order to explain the effect of scale factor applied to two-dimensional figures algebraically.
• How is the algebraic representation used to describe the effect of a dilation affected when a scale factor …
• greater than 0 but less than 1
• equal to 1
• greater than 1
… is applied to a shape?
• Illustrating and analyzing geometric relationships in models and diagrams aid in representing attributes of geometric figures with quantifiable measures and equations in order to generalize geometric relationships and solve problems.
• What models and diagrams represent and explain the Pythagorean Theorem?
• What is the relationship between the legs and hypotenuse in a right triangle?
• How can the Pythagorean Theorem be used to solve problems?
• What is the process to determine the length of a leg when the measure of the hypotenuse and other leg is given?
• How can the converse of the Pythagorean Theorem be used to determine if a triangle is a right triangle?
• When reflecting a figure across the x-axis, why does the y-value always changes signs?
• When reflecting a figure across the y-axis, why does the x-value always changes signs?
• What algebraic representations generalize the effect of …
• translations over the x-axis?
• translations over the y-axis?
• reflections over the x-axis?
• reflections over the y-axis?
• rotations of 90°, 180°, 270°, and 360°?
• Proportionality
• Ratios and Rates
• Scale factors
• Relationships and Generalizations
• Proportional
• Geometric similarity
• Dilations
• Representations
• Expressions, Equations, and Relationships
• Geometric Representations
• Two-dimensional figures
• Geometric Relationships
• Formulas
• Pythagorean Theorem
• Converse of Pythagorean Theorem
• Measure relationships
• Geometric properties
• Operations
• Properties of operations
• Order of operations
• Representations
• Two-Dimensional Shapes
• Coordinate Plane
• Ordered pairs
• Location
• Transformations and Effects
• Translations
• Rotations
• Reflections
• Algebraic representations
• Associated Mathematical Processes
• Application
• Problem Solving Model
• Tools and Techniques
• Communication
• Representations
• Relationships
• Justification
 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.

#### MISCONCEPTIONS / UNDERDEVELOPED CONCEPTS

Misconceptions:

• Some students may think that an inequality can only have one solution value that makes the inequality true.
• Some students may think that whenever all terms with the variable cancel on both sides of the equal sign, then the equation has no solutions instead of possibly having an infinite number of solutions.

Underdeveloped Concepts:

• Some students may think that a constant term can be combined with a variable term.
• Some students may not relate the constant rate of change or unit rate to m in the equation y = mx + b.
• Some students may think that a constant rate of change always means the situation is proportional.
• Some students may think that a function can have multiple outputs (y) for the same input (x) or that a function cannot have multiple inputs (x) that corresponds to the same output (y).
• Students may think that the trend line has to begin at the origin or that if both numbers in the data set are decreasing, then it represents a negative trend.
• A positive trend and a negative trend may be confused with one another.
• A student may think a translation or a reflection does not create a congruent image.
• Some students may think the original figure is the image or vice versa, especially when dealing with dilations from a larger figure to a smaller figure.
• Some students may not relate the constant rate of change or unit rate to m in the equation y = mx + b.
• Some students may not associate the unit rate of a problem situation to the slope of the line that represents the problem situation.
• Students may think that all figures and images are drawn to scale rather than using the given measurements.

#### Unit Vocabulary

• Bivariate data – data relating two quantitative variables that can be represented by a scatterplot
• Center of dilaiton – a coordinate point that serves as the focal point for generating a dilation
• Coefficient – a number that is multiplied by a variable(s)
• Compound interest for an investment – interest that is calculated on the latest balance, including all compounded earned interest that has been added to the original principal investment
• Constant – a fixed value that does not appear with a variable(s)
• Dilation – a transformation in which an image is usually enlarged or reduced, depending on the scale factor, in such a way that the angles of the original figure and the dilated figure are congruent and the sides of the original figure and the dilated figure are proportional creating similar figures
• Equation – a mathematical statement composed of algebraic and/or numeric expressions set equal to each other
• Function – relation in which each element of the input (x) is paired with exactly one element of the output (y)
• Hypotenuse – the longest side of a right triangle, the side opposite the right angle
• Legs of a right triangle – the two shortest sides of a right triangle
• Linear relationship – a relationship with a constant rate of change represented by a graph that forms a straight line
• Mappings – the process of pairing input and output in a function. Mappings are usually demonstrated by a diagram consisting of two lists, usually in ovals, with arrows associating items from the first list to items in the second list.
• Order of operations – the rules of which calculations are performed first when simplifying an expression
• Positive rational numbers – the set of numbers that can be expressed as a fraction , where a and b are counting (natural) numbers
• Principal of an investment – the original amount invested
• Relation – a set of ordered pairs (x, y) where the x is associated with a specific y
• Right triangle – a triangle with one right angle (exactly 90°) and two acute angles
• Scale factor – the common multiplicative ratio between pairs of related data which may be represented as a unit rate
• Simple interest for an investment – interest paid on the original principal in an account, disregarding any previously earned interest
• Slope – the steepness of a line; rate of change in y (vertical) compared to change in x (horizontal), or or , denoted as m in y = mx + b
• Trend line – the line that best fits the data points of a scatterplot
• Unit rate – a ratio between two different units where one of the terms is 1
• Variable – a letter or symbol that represents a number
• y-intercepty coordinate of a point at which the relationship crosses the y-axis meaning the x coordinate is equal to zero, denoted as b in y = mx + b and the ordered pair (0, b)

Related Vocabulary:

 Annual Congruence Congruent Constant of proportionality Constant rate of change Converse Coordinate plane (coordinate grid) Correlation Corresponding angles Corresponding sides Dimension Enlargement Figure Formula Image Infinite solutions Input Interest rate Investment Linear Mapping Negative Negative trend No association No solution No trend Non-linear association Non-linear relationships Non-proportional One solution One-dimensional Ordered pair Orientation Origin Positive trend Preserve Prime notation Proportional Proportional relationship Pythagorean theorem Radical symbol Rate of change Right angle Rise Run Similarity Simplify Square Symmetry Transformation Tuition Two-dimensional Vertical x-axis x coordinate x-value y-axis y coordinate y-value
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 Center 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

Texas Education Agency – Texas Response to Curriculum Focal Points for K-8 Mathematics Revised 2013

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 Grade 8 Mathematics TEKS

Texas Instruments – Graphing Calculator Tutorials

TAUGHT DIRECTLY TEKS

TEKS intended to be explicitly taught in this unit.

TEKS/SE Legend:

• Knowledge and Skills Statements (TEKS) identified by TEA are in italicized, bolded, black text.
• Student Expectations (TEKS) identified by TEA are in bolded, black text.
• Student Expectations (TEKS) are labeled Readiness as identified by TEA of the assessed curriculum.
• Student Expectations (TEKS) are labeled Process standards as identified by TEA of the assessed curriculum.
• Portions of the Student Expectations (TEKS) that are not included in this unit but are taught in previous or future units are indicated by a strike-through.

Specificity Legend:

• Supporting information / clarifications (specificity) written by TEKS Resource System are in blue text.
• Unit-specific clarifications are in italicized, blue text.
• Information from Texas Education Agency (TEA), Texas College and Career Readiness Standards (TxCCRS), Texas Response to Curriculum Focal Points (TxRCFP) is labeled.
• A Partial Specificity label indicates that a portion of the specificity not aligned to this unit has been removed.
TEKS# SE# TEKS SPECIFICITY
8.1 Mathematical process standards. The student uses mathematical processes to acquire and demonstrate mathematical understanding. The student is expected to:
8.1A Apply mathematics to problems arising in everyday life, society, and the workplace.
Process Standard

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.
• TxRCFP:
• Representing, applying, and analyzing proportional relationships
• Using expressions and equations to describe relationships, including the Pythagorean Theorem
• Making inferences from data
• TxCCRS:
• VII.D. Problem Solving and Reasoning – Real-world problem solving
• VII.D.1. Interpret results of the mathematical problem in terms of the original real-world situation.
• IX.A. Connections – Connections among the strands of mathematics
• IX.A.1. Connect and use multiple key concepts of mathematics in situations and problems.
• IX.A.2. Connect mathematics to the study of other disciplines.
• IX.B. Connections – Connections of mathematics to nature, real-world situations, and everyday life
• IX.B.1. Use multiple representations to demonstrate links between mathematical and real-world situations.
• IX.B.2. Understand and use appropriate mathematical models in the natural, physical, and social sciences.
• IX.B.3. Know and understand the use of mathematics in a variety of careers and professions.
8.1B 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.
Process Standard

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.
• TxRCFP:
• Representing, applying, and analyzing proportional relationships
• Using expressions and equations to describe relationships, including the Pythagorean Theorem
• Making inferences from data
• TxCCRS:
• I.B. Numeric Reasoning – Number sense and number concepts
• I.B.1. Use estimation to check for errors and reasonableness of solutions.
• V.A. Statistical Reasoning – Design a study
• V.A.1. Formulate a statistical question, plan an investigation, and collect data.
• VII.A. Problem Solving and Reasoning – Mathematical problem solving
• VII.A.1. Analyze given information.
• VII.A.2. Formulate a plan or strategy.
• VII.A.3. Determine a solution.
• VII.A.4. Justify the solution.
• VII.A.5. Evaluate the problem-solving process.
• VII.D. Problem Solving and Reasoning – Real-world problem solving
• VII.D.2. Evaluate the problem-solving process.
8.1C

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.

Process Standard

Select

TOOLS, INCLUDING PAPER AND PENCIL AND TECHNOLOGY AS APPROPRIATE, TO SOLVE PROBLEMS 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
• 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.
• TxRCFP:
• Representing, applying, and analyzing proportional relationships
• Using expressions and equations to describe relationships, including the Pythagorean Theorem
• Making inferences from data
• TxCCRS:
• I.B. Numeric Reasoning – Number sense and number concepts
• I.B.1. Use estimation to check for errors and reasonableness of solutions.
• V.C. Statistical Reasoning – Analyze, interpret, and draw conclusions from data
• V.C.2. Analyze relationships between paired data using spreadsheets, graphing calculators, or statistical software.
8.1D Communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams, graphs, and language as appropriate.
Process Standard

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.
• TxRCFP:
• Representing, applying, and analyzing proportional relationships
• Using expressions and equations to describe relationships, including the Pythagorean Theorem
• Making inferences from data
• TxCCRS:
• II.D. Algebraic Reasoning – Representing relationships
• II.D.1. Interpret multiple representations of equations, inequalities, and relationships.
• II.D.2. Convert among multiple representations of equations, inequalities, and relationships.
• VIII.A. Communication and Representation – Language, terms, and symbols of mathematics
• VIII.A.1. Use mathematical symbols, terminology, and notation to represent given and unknown information in a problem.
• VIII.A.2. Use mathematical language to represent and communicate the mathematical concepts in a problem.
• VIII.A.3. Use mathematical language for reasoning, problem solving, making connections, and generalizing.
• VIII.B. Communication and Representation – Interpretation of mathematical work
• VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
• VIII.B.2. Summarize and interpret mathematical information provided orally, visually, or in written form within the given context.
• VIII.C. Communication and Representation – Presentation and representation of mathematical work
• VIII.C.1. Communicate mathematical ideas, reasoning, and their implications using symbols, diagrams, models, graphs, and words.
• VIII.C.2. Create and use representations to organize, record, and communicate mathematical ideas.
• VIII.C.3. Explain, display, or justify mathematical ideas and arguments using precise mathematical language in written or oral communications.
• IX.B. Connections – Connections of mathematics to nature, real-world situations, and everyday life
• IX.B.1. Use multiple representations to demonstrate links between mathematical and real-world situations.
8.1E Create and use representations to organize, record, and communicate mathematical ideas.
Process Standard

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.
• TxRCFP:
• Representing, applying, and analyzing proportional relationships
• Using expressions and equations to describe relationships, including the Pythagorean Theorem
• Making inferences from data
• TxCCRS:
• VIII.B. Communication and Representation – Interpretation of mathematical work
• VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
• VIII.B.2. Summarize and interpret mathematical information provided orally, visually, or in written form within the given context.
• VIII.C. Communication and Representation – Presentation and representation of mathematical work
• VIII.C.1. Communicate mathematical ideas, reasoning, and their implications using symbols, diagrams, models, graphs, and words.
• VIII.C.2. Create and use representations to organize, record, and communicate mathematical ideas.
8.1F Analyze mathematical relationships to connect and communicate mathematical ideas.
Process Standard

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.
• TxRCFP:
• Representing, applying, and analyzing proportional relationships
• Using expressions and equations to describe relationships, including the Pythagorean Theorem
• Making inferences from data
• TxCCRS:
• VII.A. Problem Solving and Reasoning – Mathematical problem solving
• VII.A.1. Analyze given information.
• VIII.A. Communication and Representation – Language, terms, and symbols of mathematics
• VIII.A.1. Use mathematical symbols, terminology, and notation to represent given and unknown information in a problem.
• VIII.A.2. Use mathematical language to represent and communicate the mathematical concepts in a problem.
• VIII.A.3. Use mathematical language for reasoning, problem solving, making connections, and generalizing.
• VIII.B. Communication and Representation – Interpretation of mathematical work
• VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
• VIII.C. Communication and Representation – Presentation and representation of mathematical work
• VIII.C.1. Communicate mathematical ideas, reasoning, and their implications using symbols, diagrams, models, graphs, and words.
• VIII.C.2. Create and use representations to organize, record, and communicate mathematical ideas.
• VIII.C.3. Explain, display, or justify mathematical ideas and arguments using precise mathematical language in written or oral communications.
• IX.A. Connections – Connections among the strands of mathematics
• IX.A.1. Connect and use multiple key concepts of mathematics in situations and problems.
• IX.A.2. Connect mathematics to the study of other disciplines.
8.1G Display, explain, and justify mathematical ideas and arguments using precise mathematical language in written or oral communication.
Process Standard

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.
• TxRCFP:
• Representing, applying, and analyzing proportional relationships
• Using expressions and equations to describe relationships, including the Pythagorean Theorem
• Making inferences from data
• TxCCRS:
• VII.A. Problem Solving and Reasoning – Mathematical problem solving
• VII.A.4. Justify the solution.
• VII.B. Problem Solving and Reasoning – Proportional reasoning
• VII.B.1. Use proportional reasoning to solve problems that require fractions, ratios, percentages, decimals, and proportions in a variety of contexts using multiple representations.
• VII.C. Problem Solving and Reasoning – Logical reasoning
• VII.C.1. Develop and evaluate convincing arguments.
• VIII.A. Communication and Representation – Language, terms, and symbols of mathematics
• VIII.A.3. Use mathematical language for reasoning, problem solving, making connections, and generalizing.
• VIII.B. Communication and Representation – Interpretation of mathematical work
• VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
• VIII.B.2. Summarize and interpret mathematical information provided orally, visually, or in written form within the given context.
• VIII.C. Communication and Representation – Presentation and representation of mathematical work
• VIII.C.3. Explain, display, or justify mathematical ideas and arguments using precise mathematical language in written or oral communications.
8.3 Proportionality. The student applies mathematical process standards to use proportional relationships to describe dilations. The student is expected to:
8.3C Use an algebraic representation to explain the effect of a given positive rational scale factor applied to two-dimensional figures on a coordinate plane with the origin as the center of dilation.

Use

AN ALGEBRAIC REPRESENTATION TO EXPLAIN THE EFFECT OF A GIVEN POSITIVE RATIONAL SCALE FACTOR APPLIED TO TWO-DIMENSIONAL FIGURES ON A COORDINATE PLANE WITH THE ORIGIN AS THE CENTER OF DILATION

Including, but not limited to:

• Positive rational numbers – the set of numbers that can be expressed as a fraction , where a and b are counting (natural) numbers
• Various forms of positive rational numbers
• Counting (natural) numbers
• Decimals
• Fractions
• Percents
• Scale factor – the common multiplicative ratio between pairs of related data which may be represented as a unit rate
• Dilation – a transformation in which an image is usually enlarged or reduced, depending on the scale factor, in such a way that the angles of the original figure and the dilated figure are congruent and the sides of the original figure and the dilated figure are proportional creating similar figures
• Enlargement (scale factor >1)
• Reduction (0 < scale factor < 1)
• Congruent (scale factor = 1)
• Coordinate plane (all four quadrants)
• Center of dilation – a coordinate point that serves as the focal point for generating a dilation
• The ratio of the distance from the center of dilation to any point on the image compared to the distance from the center of dilation to the corresponding point on the pre-image will result in the scale factor, k.
• Lines drawn through each point on the pre-image and its corresponding image point will intersect at the center of dilation.
• Origin as center of dilation
• Algebraic representation to describe effects of dilations
• (x, y) → (kx, ky), where k is the scale factor used to dilate a figure about the origin
• Various representations of dilations
• Verbal
• Graphical
• Tabular
• Algebraic

Note(s):

• Grade 8 introduces using an algebraic representation to explain the effect of a given positive rational scale factor applied to two-dimensional figures on a coordinate plane with the origin as the center of dilation.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxRCFP:
• Representing, applying, and analyzing proportional relationships
• TxCCRS:
• III.B. Geometric and Spatial Reasoning – Transformations and symmetry
• III.B.1. Identify transformations and symmetries of figures.
• III.B.2. Use transformations to investigate congruence, similarity, and symmetries of figures.
• III.C. Geometric and Spatial Reasoning – Connections between geometry and other mathematical content strands
• III.C.1. Make connections between geometry and algebraic equations.
• VII.B. Problem Solving and Reasoning – Proportional reasoning
• VII.B.1. Use proportional reasoning to solve problems that require fractions, ratios, percentages, decimals, and proportions in a variety of contexts using multiple representations.
8.4 Proportionality. The student applies mathematical process standards to explain proportional and non-proportional relationships involving slope. The student is expected to:
8.4B Graph proportional relationships, interpreting the unit rate as the slope of the line that models the relationship.

Graph

PROPORTIONAL RELATIONSHIPS, INTERPRETING THE UNIT RATE AS THE SLOPE OF THE LINE THAT MODELS THE RELATIONSHIP

Including, but not limited to:

• Unit rate – a ratio between two different units where one of the terms is 1
• Slope – the steepness of a line; rate of change in y (vertical) compared to change in x (horizontal), or or , denoted as m in y = mx + b
• Linear proportional relationship
• Linear
• Passes through the origin (0, 0)
• Represented by y = kx
• Constant of proportionality represented as • When b = 0 in y = mx + b, then k = the slope, m
• Graphing unit rate from various representations
• Verbal
• Numeric
• Tabular(horizontal/vertical)
• Symbolic/algebraic
• Connections between unit rate in proportional relationships to the slope of a line

Note(s):

• Algebra I will calculate the rate of change of a linear function represented tabularly, graphically, or algebraically in context of mathematical and real-world problems.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxRCFP:
• Representing, applying, and analyzing proportional relationships
• TxCCRS:
• VI.C. Functions – Model real-world situations with functions
• VI.C.2. Develop a function to model a situation.
• VII.B. Problem Solving and Reasoning – Proportional reasoning
• VII.B.1. Use proportional reasoning to solve problems that require fractions, ratios, percentages, decimals, and proportions in a variety of contexts using multiple representations.
• VIII.C. Communication and Representation – Presentation and representation of mathematical work
• VIII.C.1. Communicate mathematical ideas, reasoning, and their implications using symbols, diagrams, models, graphs, and words.
• IX.B. Connections – Connections of mathematics to nature, real-world situations, and everyday life
• IX.B.1. Use multiple representations to demonstrate links between mathematical and real-world situations.
8.4C Use data from a table or graph to determine the rate of change or slope and y-intercept in mathematical and real-world problems.

Use

DATA FROM A TABLE OR GRAPH TO DETERMINE THE RATE OF CHANGE OR SLOPE AND y-INTERCEPT IN MATHEMATICAL AND REAL-WORLD PROBLEMS

Including, but not limited to:

• Slope – the steepness of a line; rate of change in y (vertical) compared to change in x (horizontal), or or , denoted as m in y = mx + b
• Determining rate of change or slope from various representations
• Table (horizontal/vertical)
• Graph
• Connections between unit rate, rate of change, and slope in mathematical and real-world problems
• y-intercept – y coordinate of a point at which the relationship crosses the y-axis meaning the x coordinate is equal to zero, denoted as b in y = mx + b and the ordered pair (0, b)
• Determining y-intercept from various representations
• Table (horizontal/vertical)
• Graph
• Connections between the “starting point” (the output value when the input value is 0) and y-intercept in mathematical and real-world problem situations
• Linear proportional relationship
• Linear
• Passes through the origin (0, 0)
• Represented by y = kx
• Constant of proportionality represented as • When b = 0 in y = mx + b, then k = the slope, m
• Linear non-proportional relationship
• Linear
• Does not pass through the origin (0, 0)
• Represented by y = mx + b, where b ≠ 0
• Constant slope represented as m = or m = or m = Note(s):

• Algebra I will calculate the rate of change of a linear function represented tabularly, graphically, or algebraically in context of mathematical and real-world problems.
• Algebra I will graph linear functions on the coordinate plane and identify key features, including x-intercept, y-intercept, zeros, and slope, in mathematical and real-world problems.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxRCFP:
• Representing, applying, and analyzing proportional relationships
• TxCCRS:
• VI.B. Functions – Analysis of functions
• VI.B.1. Understand and analyze features of functions.
• VI.C. Functions – Model real-world situations with functions
• VI.C.2. Develop a function to model a situation.
• IX.B. Connections – Connections of mathematics to nature, real-world situations, and everyday life
• IX.B.1. Use multiple representations to demonstrate links between mathematical and real-world situations.
8.5 Proportionality. The student applies mathematical process standards to use proportional and non-proportional relationships to develop foundational concepts of functions. The student is expected to:
8.5D Use a trend line that approximates the linear relationship between bivariate sets of data to make predictions.

Use

A TREND LINE THAT APPROXIMATES THE LINEAR RELATIONSHIP BETWEEN BIVARIATE SETS OF DATA TO MAKE PREDICTIONS

Including, but not limited to:

• Bivariate data – data relating two quantitative variables that can be represented by a scatterplot
• Characteristics of bivariate data that suggests a linear relationship
• Linear proportional relationship
• Linear
• Passes through the origin (0, 0)
• Represented by y = kx
• Constant of proportionality represented as • When b = 0 in y = mx + b, then k = the slope, m.
• Linear non-proportional relationship
• Linear
• Does not pass through the origin (0, 0)
• Represented by y = mx + b, where b ≠ 0
• Constant slope represented as m = or m = or m = • Graph of data suggests a constant rate of change between the independent and dependent values
• Trend line – the line that best fits the data points of a scatterplot
• A tool for making predictions by approximating the linear relationship between bivariate sets of data
• A trend line contains most of the data points and/or is situated so that the data points are evenly distributed above and below the line.
• Given or collected data
• Analysis of parts of data representation
• Title
• Labels
• Scales
• Graphed data
• Predictions of independent value when given a dependent value using a trend line that approximates the linear relationship
• Predictions of dependent value when given an independent value using a trend line that approximates the linear relationship

Note(s):

• Grade 8 introduces using a trend line that approximates the linear relationship between bivariate sets of data to make predictions.
• Algebra I will calculate, using technology, the correlation coefficient between two quantitative variables and interpret this quantity as a measure of the strength of the linear association.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxRCFP:
• Representing, applying, and analyzing proportional relationships
• TxCCRS:
• V.B. Statistical Reasoning – Describe data
• V.B.4. Describe patterns and departure from patterns in the study of data.
• VI.C. Functions – Model real-world situations with functions
• VI.C.1. Apply known functions to model real-world situations.
• VII.A. Problem Solving and Reasoning – Mathematical problem solving
• VII.A.1. Analyze given information.
• VIII.C. Communication and Representation – Presentation and representation of mathematical work
• VIII.C.1. Communicate mathematical ideas, reasoning, and their implications using symbols, diagrams, models, graphs, and words.
• IX.B. Connections – Connections of mathematics to nature, real-world situations, and everyday life
• IX.B.1. Use multiple representations to demonstrate links between mathematical and real-world situations.
8.5G Identify functions using sets of ordered pairs, tables, mappings, and graphs.

Identify

FUNCTIONS USING SETS OF ORDERED PAIRS, TABLES, MAPPINGS, AND GRAPHS

Including, but not limited to:

• Relation – a set of ordered pairs (x, y) where the x is associated with a specific y
• Function – relation in which each element of the input (x) is paired with exactly one element of the output (y)
• Distinguish between relations and functions
• All functions are relations but not all relations are functions.
• Various representations
• Sets of ordered pairs
• Tables (horizontal/vertical)
• Mappings – the process of pairing input and output in a function. Mappings are usually demonstrated by a diagram consisting of two lists, usually in ovals, with arrows associating items from the first list to items in the second list.
• Visual representation of a relation or a pairing of inputs with outputs
• Arrows connect inputs to corresponding outputs
• Can be used to quickly determine if a relation is a function
• Graphs
• Vertical line test can be used to determine if a relation is a function when graphing

Note(s):

• Grade 8 introduces identifying functions using sets of ordered pairs, tables, mappings, and graphs.
• Algebra I will introduce function notation.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxRCFP:
• Representing, applying, and analyzing proportional relationships
• TxCCRS:
• VI.A. Functions – Recognition and representation of functions
• VI.A.1. Recognize if a relation is a function.
8.5I Write an equation in the form y = mx + b to model a linear relationship between two quantities using verbal, numerical, tabular, and graphical representations.

Write

AN EQUATION IN THE FORM y = mx + b TO MODEL A LINEAR RELATIONSHIP BETWEEN TWO QUANTITIES USING VERBAL, NUMERICAL, TABULAR, AND GRAPHICAL REPRESENTATIONS

Including, but not limited to:

• Slope – the steepness of a line; rate of change in y (vertical) compared to change in x (horizontal), or or , denoted as m in y = mx + b
• y-intercept – y coordinate of a point at which the relationship crosses the y-axis meaning the x coordinate is equal to zero, denoted as b in y = mx + b and the ordered pair (0, b)
• Linear relationship – a relationship with a constant rate of change represented by a graph that forms a straight line
• One quantity is dependent on the other
• Two quantities may be directly proportional to each other
• Can be classified as a positive or negative relationship
• Can be expressed as a pair of values that can be graphed as ordered pairs
• Graph of the ordered pairs matching the relationship will form a line
• Linear non-proportional relationship
• Linear
• Does not pass through the origin (0, 0)
• Represented by y = mx + b, where b ≠ 0
• Constant slope represented as m = or m = or m = • Equations in the form y = mx + b to represent relationships between two quantities
• Rational number coefficients and constants
• Various representations
• Verbal
• Numerical
• Tabular (horizontal/vertical)
• Graphical

Note(s):

• Grade 7 represented linear relationships using verbal descriptions, tables, graphs, and equations that simplify to the form y = mx + b.
• Algebra I will write linear equations in two variables given a table of values, a graph, and a verbal description.
• Algebra I will use, translate, and make connections among algebraic, tabular, graphical, or verbal descriptions of linear functions.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxRCFP:
• Representing, applying, and analyzing proportional relationships
• TxCCRS:
• II.D. Algebraic Reasoning – Representing relationships
• II.D.2. Convert among multiple representations of equations, inequalities, and relationships.
• VIII.B. Communication and Representation – Interpretation of mathematical work
• VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
• VIII.C. Communication and Representation – Presentation and representation of mathematical work
• VIII.C.1. Communicate mathematical ideas, reasoning, and their implications using symbols, diagrams, models, graphs, and words.
8.7 Expressions, equations, and relationships. The student applies mathematical process standards to use geometry to solve problems. The student is expected to:
8.7C Use the Pythagorean Theorem and its converse to solve problems.

Use

THE PYTHAGOREAN THEOREM AND ITS CONVERSE TO SOLVE PROBLEMS

Including, but not limited to:

• Right triangle – a triangle with one right angle (exactly 90°) and two acute angles
• Legs of a right triangle – the two shortest sides of a right triangle
• Hypotenuse – the longest side of a right triangle, the side opposite the right angle
• Pythagorean Theorem
• Verbal
• The sum of the squares of the legs of a right triangle equals the square of the hypotenuse.
• Formula
• a2 + b2 = c2, where a and b represent the legs of a right triangle and c represents the hypotenuse
• When solving for a, b, or c both the positive and negative numerical values should be considered, but since the applications are measurements the negative values do not apply.
• Converse of Pythagorean Theorem
• Verbal
• If the sum of the squares of the two shortest sides of a triangle equals the square of the third side, then the triangle is a right triangle.
• Formula
• a2 + b2 = c2, where a and b represent the legs of a right triangle and c represents the hypotenuse

Note(s):

• Grade 8 introduces using the Pythagorean Theorem and its converse to solve problems.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxRCFP:
• Using expressions and equations to describe relationships, including the Pythagorean Theorem
• TxCCRS:
• II.A. Algebraic Reasoning – Identifying expressions and equations
• II.A.1. Explain the difference between expressions and equations.
• III.A. Geometric and Spatial Reasoning – Figures and their properties
• III.A.3. Recognize and apply right triangle relationships including basic trigonometry.
• III.D. Geometric and Spatial Reasoning – Measurements involving geometry and algebra
• III.D.3. Determine indirect measurements of geometric figures using a variety of methods.
8.8 Expressions, equations, and relationships. The student applies mathematical process standards to use one-variable equations or inequalities in problem situations. The student is expected to:
8.8C Model and solve one-variable equations with variables on both sides of the equal sign that represent mathematical and real-world problems using rational number coefficients and constants.

Model, Solve

ONE-VARIABLE EQUATIONS WITH VARIABLES ON BOTH SIDES OF THE EQUAL SIGN THAT REPRESENT MATHEMATICAL AND REAL-WORLD PROBLEMS USING RATIONAL NUMBER COEFFICIENTS AND CONSTANTS

Including, but not limited to:

• Equation – a mathematical statement composed of algebraic and/or numeric expressions set equal to each other
• Variable – a letter or symbol that represents a number
• One variable on both sides of the equation
• Coefficient – a number that is multiplied by a variable(s)
• Integers
• Decimals (positive or negative)
• Fractions (positive or negative)
• Constant – a fixed value that does not appear with a variable(s)
• Integers
• Decimals (positive or negative)
• Fractions (positive or negative)
• Characteristics of equations
• Equates two expressions
• Equality of the variable
• One solution
• Equality words and symbol
• Equal to, =
• Relationship of order of operations within an equation
•  Order of operations – the rules of which calculations are performed first when simplifying an expression
• Parentheses/brackets: simplify expressions inside parentheses or brackets in order from left to right
• Exponents: rewrite in standard numerical form and simplify from left to right
• Multiplication/division: simplify expressions involving multiplication and/or division in order from left to right
• Addition/subtraction: simplify expressions involving addition and/or subtraction in order from left to right
• Models to solve one-variable equations with variables on both sides of the equal sign (concrete, pictorial, algebraic)
• Solutions to one-variable equations with variables on both sides of the equal sign from mathematical and real-world problem situations
• Possible solutions
• One real solution
• No solution
• Infinite solutions (all real solutions)

Note(s):

• Grade 7 modeled and solved one-variable, two-step equations and inequalities.
• Algebra I will solve linear equations in one variable, including those for which the application of the distributive property is necessary and for which variables are included on both sides.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxRCFP:
• Using expressions and equations to describe relationships, including the Pythagorean Theorem
• TxCCRS:
• II.A. Algebraic Reasoning – Identifying expressions and equations
• II.A.1. Explain the difference between expressions and equations.
• II.C. Algebraic Reasoning – Solving equations, inequalities, and systems of equations and inequalities
• II.C.3. Recognize and use algebraic properties, concepts, and algorithms to solve equations, inequalities, and systems of linear equations and inequalities.
• II.D. Algebraic Reasoning – Representing relationships
• II.D.2. Convert among multiple representations of equations, inequalities, and relationships.
• VII.A. Problem Solving and Reasoning – Mathematical problem solving
• VII.A.3. Determine a solution.
• VII.D. Problem Solving and Reasoning – Real-world problem solving
• VII.D.1. Interpret results of the mathematical problem in terms of the original real-world situation.
• VIII.A. Communication and Representation – Language, terms, and symbols of mathematics
• VIII.A.1. Use mathematical symbols, terminology, and notation to represent given and unknown information in a problem.
• VIII.A.3. Use mathematical language for reasoning, problem solving, making connections, and generalizing.
• VIII.C. Communication and Representation – Presentation and representation of mathematical work
• VIII.C.1. Communicate mathematical ideas, reasoning, and their implications using symbols, diagrams, models, graphs, and words.
• IX.B. Connections – Connections of mathematics to nature, real-world situations, and everyday life
• IX.B.1. Use multiple representations to demonstrate links between mathematical and real-world situations.
• IX.B.2. Understand and use appropriate mathematical models in the natural, physical, and social sciences.
8.10 Two-dimensional shapes. The student applies mathematical process standards to develop transformational geometry concepts. The student is expected to:
8.10C Explain the effect of translations, reflections over the x- or y-axis, and rotations limited to 90°, 180°, 270°, and 360° as applied to two-dimensional shapes on a coordinate plane using an algebraic representation.

Explain

THE EFFECT OF TRANSLATIONS, REFLECTIONS OVER THE x- OR y-AXIS, AND ROTATIONS LIMITED TO 90°, 180°, 270°, AND 360° AS APPLIED TO TWO-DIMENSIONAL SHAPES ON A COORDINATE PLANE USING AN ALGEBRAIC REPRESENTATION

Including, but not limited to:

• Prime notation of image points
• Prime marks
• Coordinate plane (all four quadrants)
• Single transformations
• Effects of transformations as algebraic representations
• Translation – a transformation frequently described as a slide of a figure
• Algebraic notation
• Translation h units horizontally
• (x, y) → (x + h, y)
• Translation k units vertically
• (x, y) → (x, y + k)
• Translation h units horizontally and k units vertically
• (x, y) → (x + h, y + k)
• Reflection – a transformation frequently described as a flip or a mirror image of the original figure
• Algebraic notation
• Reflection across a vertical axis (y-axis)
• (x, y) → (–x, y)
• Reflection across a horizontal axis (x-axis)
• (x, y) → (x, –y)
• Rotation –  a transformation frequently described as a turn of a figure around a designated point
• Origin as center of rotation
• Algebraic notation
• Rotation of 90° counterclockwise around the origin: (x, y) → (–y, x)
• Same as a rotation of 270° clockwise around the origin: (x, y) → (–y, x)
• Rotation of 180º counterclockwise around the origin: (x, y) → (–x, –y)
• Same as a rotation of 180º clockwise around the origin: (x, y) → (–x, –y)
• Rotation of 270º counterclockwise around the origin: (x, y) → (y, –x)
• Same as a rotation of 90º clockwise around the origin: (x, y) → (y, –x)
• Rotation of 360° counterclockwise or clockwise around the origin: (x, y) → (x, y)
• Determine the transformation performed from a graphed set of figures.
• Graph a transformation based on a given rule.

Note(s):

• Grade 8 introduces explaining the effect of translations, reflections over the x- or y-axis, and rotations limited to 90°, 180°, 270°, and 360° as applied to two-dimensional shapes on a coordinate plane using an algebraic representation.
• Geometry introduces rotations and dilations that may or may not be about the origin.
• Geometry introduces composite transformations.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxRCFP:
• Grade Level Connections (reinforces previous learning and/or provides development for future learning)
• TxCCRS:
• III.B. Geometric and Spatial Reasoning – Transformations and symmetry
• III.B.1. Identify transformations and symmetries of figures.
• III.B.2. Use transformations to investigate congruence, similarity, and symmetries of figures.
• III.C. Geometric and Spatial Reasoning – Connections between geometry and other mathematical content strands
• III.C.1. Make connections between geometry and algebraic equations.
8.12 Personal financial literacy. The student applies mathematical process standards to develop an economic way of thinking and problem solving useful in one's life as a knowledgeable consumer and investor. The student is expected to:
8.12D Calculate and compare simple interest and compound interest earnings.

Calculate, Compare

SIMPLE INTEREST AND COMPOUND INTEREST EARNINGS

Including, but not limited to:

• Principal of an investment – the original amount invested
• Simple interest for an investment – interest paid on the original principal in an account, disregarding any previously earned interest
• Compound interest for an investment – interest that is calculated on the latest balance, including all compounded interest that has been added to the original principal investment
• Formulas for interest from STAAR Grade 8 Mathematics Reference Materials
• Simple interest
• I = Prt, where I represents the interest, P represents the principal amount deposited, r represents the interest rate in decimal form, and t represents the number of years the principal amount is deposited
• Compound interest
• A = P(1+ r)t, where A represents the total accumulated amount including the principal and earned compounded interest, P represents the principal amount deposited, r represents the interest rate in decimal form, and t represents the number of years the principal amount is deposited
• Comparing simple and compound interest earnings

Note(s):

• Grade 7 calculated and compared simple interest and compound interest earnings.
• Grade 8 solves real-world problems comparing how interest rate and loan length affect the cost of credit.
• Algebra I will refer to 1 + r in the compound interest formula, A = P(1 + r)t, as the factor and will be given the variable b.
• Mathematical Models with Applications will introduce analyzing compound interest for multiple compounding periods within a year.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxRCFP:
• Financial Literacy
• TxCCRS:
• I.A. Numeric Reasoning – Number representations and operations
• I.A.1. Compare relative magnitudes of rational and irrational numbers, and understand that numbers can be represented in different ways.
• II.A. Algebraic Reasoning – Identifying expressions and equations
• II.A.1. Explain the difference between expressions and equations.
• IX.A. Connections – Connections among the strands of mathematics
• IX.A.1. Connect and use multiple key concepts of mathematics in situations and problems.
• IX.A.2. Connect mathematics to the study of other disciplines.
• IX.B. Connections – Connections of mathematics to nature, real-world situations, and everyday life
• IX.B.2. Understand and use appropriate mathematical models in the natural, physical, and social sciences.
• IX.B.3. Know and understand the use of mathematics in a variety of careers and professions. 