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 – Realworld problem solving
 VII.D.1. Interpret results of the mathematical problem in terms of the original realworld 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, realworld situations, and everyday life
 IX.B.1. Use multiple representations to demonstrate links between mathematical and realworld 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 problemsolving model that incorporates analyzing given information, formulating a plan or strategy, determining a solution, justifying the solution, and evaluating the problemsolving process and the reasonableness of the solution.
Process Standard

Use
A PROBLEMSOLVING MODEL THAT INCORPORATES ANALYZING GIVEN INFORMATION, FORMULATING A PLAN OR STRATEGY, DETERMINING A SOLUTION, JUSTIFYING THE SOLUTION, AND EVALUATING THE PROBLEMSOLVING PROCESS AND THE REASONABLENESS OF THE SOLUTION
Including, but not limited to:
 Problemsolving model
 Analyze given information
 Formulate a plan or strategy
 Determine a solution
 Justify the solution
 Evaluate the problemsolving 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 problemsolving process.
 VII.D. Problem Solving and Reasoning – Realworld problem solving
 VII.D.2. Evaluate the problemsolving 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 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.
 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, realworld situations, and everyday life
 IX.B.1. Use multiple representations to demonstrate links between mathematical and realworld 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 nonexamples, 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:


8.3A 
Generalize that the ratio of corresponding sides of similar shapes are proportional, including a shape and its dilation.
Supporting Standard

Generalize
THAT THE RATIO OF CORRESPONDING SIDES OF SIMILAR SHAPES ARE PROPORTIONAL, INCLUDING A SHAPE AND ITS DILATION
Including, but not limited to:
 Congruent – of equal measure, having exactly the same size and same shape
 Similar shapes – shapes whose angles are congruent and side lengths are proportional (equal scale factor)
 The order of the letters determines corresponding side lengths and angles.
 Notation for similar shapes
 Symbol for similarity (~) read as “similar to”
 Prime notation of image points
 Prime marks
 Ex: ABCD is the original figure or preimage and A'B'C'D' is the name of the image. A'B'C'D' is read as “A prime, B prime, C prime, D prime”.
 Generalizations of similarity
 The ratio of corresponding sides of similar shapes is proportional.
 Ratios comparing lengths within each shape or between shapes will determine if the shapes are similar.
 The reciprocal of the ratio of one side of a figure to the corresponding side of a proportional figure is the scale factor, which represents the change in the size of the figures.
 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)
Note(s):
 Grade Level(s):
 Grade 7 identified the critical attributes of similarity, including the generalization that the ratio of corresponding sides of similar figures are proportional.
 Grade 7 solved problems with similar shapes and scale drawings.
 Grade 8 introduces the term “dilation” with similar figures.
 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.D. Geometric and Spatial Reasoning – Measurements involving geometry and algebra
 III.D.3. Determine indirect measurements of geometric figures using a variety of methods.
 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.

8.3B 
Compare and contrast the attributes of a shape and its dilation(s) on a coordinate plane.
Supporting Standard

Compare, Contrast
THE ATTRIBUTES OF A SHAPE AND ITS DILATION(S) ON A COORDINATE PLANE
Including, but not limited to:
 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)
 Prime notation of image points
 Prime marks
 Ex: ABCD is the original figure or preimage and A'B'C'D' is the name of the image. A'B'C'D' is read as “A prime, B prime, C prime, D prime”.
 Coordinate plane (all four quadrants)
 Similar shapes – shapes whose angles are congruent and side lengths are proportional (equal scale factor)
 The order of the letters determines corresponding side lengths and angles.
 Notation for similar shapes
 Symbol for similarity (~) read as “similar to”
 Attributes of similar shapes
 Corresponding sides are proportional.
 Corresponding angles are congruent.
Note(s):
 Grade Level(s):
 Grade 7 identified the critical attributes of similarity, including the generalization that the ratio of corresponding sides of similar figures are proportional.
 Grade 8 introduces comparing and contrasting the attributes of a shape and its dilation(s) on a coordinate plane.
 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.

8.3C 
Use an algebraic representation to explain the effect of a given positive rational scale factor applied to twodimensional figures on a coordinate plane with the origin as the center of dilation.
Readiness Standard

Use
AN ALGEBRAIC REPRESENTATION TO EXPLAIN THE EFFECT OF A GIVEN POSITIVE RATIONAL SCALE FACTOR APPLIED TO TWODIMENSIONAL 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 preimage will result in the scale factor, k.
 Lines drawn through each point on the preimage 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 Level(s):
 Grade 8 introduces using an algebraic representation to explain the effect of a given positive rational scale factor applied to twodimensional 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.

8.10 
Twodimensional shapes. The student applies mathematical process standards to develop transformational geometry concepts. The student is expected to:


8.10A 
Generalize the properties of orientation and congruence of rotations, reflections, translations, and dilations of twodimensional shapes on a coordinate plane.
Supporting Standard

Generalize
THE PROPERTIES OF ORIENTATION AND CONGRUENCE OF ROTATIONS, REFLECTIONS, TRANSLATIONS, AND DILATIONS OF TWODIMENSIONAL SHAPES ON A COORDINATE PLANE
Including, but not limited to:
 Properties of orientation
 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 preimage and not determined by a figure’s direction or a figure’s size.
 Orientation of the vertices is preserved when a twodimensional figure is transformed and the preimage and image either both have clockwise orientation or both have counterclockwise orientation.
 Translations preserve orientation of the vertices.
 Rotations preserve orientation of the vertices.
 Dilations preserve orientation of the vertices.
 Orientation of the vertices is not preserved when a twodimensional figure is transformed and the preimage and image are such that one has clockwise orientation and the other has counterclockwise orientation.
 Reflections do not preserve orientation of the vertices.
 A change in orientation of the vertices implies a change in the orientation of the figure.
 Orientation of the figure
 Orientation of a figure is determined by the position of the figure on the plane. It is determined by how the figure appears on the plane including the position of the vertices of the figure or any distinguishing mark.
 Orientation of the figure is preserved when a twodimensional figure is transformed and the preimage and image both face the same direction on the plane.
 Translations preserve orientation of the figure.
 Dilations preserve orientation of the figure.
 Orientation of the figure is not preserved when a twodimensional figure is transformed and the preimage and image are such that they do not face the same direction on the plane.
 Rotations do not preserve orientation of the figure.
 Exception, rotations of 360° do preserve orientation of the figure.
 Reflections do not preserve orientation of the figure.
 A change in the orientation of the figure may not mean a change in the orientation of the vertices.
 Property of congruence
 Congruence is preserved when a twodimensional figure is transformed and the image is identical in shape and identical in size.
 Congruence is not preserved when a twodimensional figure is transformed and the image is not identical in shape and/or identical in size.
 Prime notation of image points
 Prime marks
 Ex: ABCD is the original figure or preimage and A'B'C'D' is the name of the image. A'B'C'D' is read as “A prime, B prime, C prime, D prime”.
 Coordinate plane (all four quadrants)
 Transformation and properties of orientation and congruence
 Rotation – a transformation frequently described as a turn of a figure around a designated point
 Origin as center of rotation
 Reflection – a transformation frequently described as a flip or a mirror image of the original figure
 Translation – a transformation frequently described as a slide of a figure
 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)
 Generalizations of the properties of orientation considering only one transformation
 Orientation of the vertices is preserved for rotations, translations, and dilations.
 Orientation of the vertices is not preserved for reflections.
 Orientation of the figure is preserved for translations and dilations.
 Orientation of the figure is not preserved for reflections and rotations, except for rotations of 360°.
 Generalization of the property of congruence considering only one transformation
 Congruence is preserved for rotations, reflections, translations, and dilations with a scale factor of 1.
 Congruence is not preserved for dilations
 Enlargements for positive scale factors greater than 1
 Reductions for positive scale factors greater than 0 but less than 1
Note(s):
 Grade Level(s):
 Grade 8 introduces generalizing the properties of orientation and congruence of rotations, reflections, translations, and dilations of twodimensional shapes on a coordinate plane.
 Geometry introduces rotations and dilations that may or may not be about the origin.
 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.

8.10B 
Differentiate between transformations that preserve congruence and those that do not.
Supporting Standard

Differentiate
BETWEEN TRANSFORMATIONS THAT PRESERVE CONGRUENCE AND THOSE THAT DO NOT
Including, but not limited to:
 Property of congruence
 Congruence is preserved when a twodimensional figure is transformed and the image is identical in shape and identical in size.
 Congruence is not preserved when a twodimensional figure is transformed and the image is not identical in shape and/or identical in size.
 Generalization of the property of congruence considering only one transformation
 Congruence is preserved for rotations, reflections, translations, and dilations with a scale factor of 1.
 Congruence is not preserved for dilations
 Enlargements for positive scale factors greater than 1
 Reductions for positive scale factors greater than 0 but less than 1
 Prime notation of image points
 Various representations of transformations to determine congruence (verbal, graphical, tabular, algebraic)
 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)
 Reflection – a transformation frequently described as a flip or a mirror image of the original figure
 Algebraic notation
 Reflection across the vertical axis
 Reflection across the horizontal axis
 Translation – a transformation frequently described as a slide of a figure
 Algebraic notation
 Translation h units horizontally
 Translation k units vertically
 Translation h units horizontally and k units vertically
 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)
 Positive, rational number scale factors
 Algebraic notation
 Dilation of scale factor k
Note(s):
 Grade Level(s):
 Grade 8 introduces differentiating between transformations that preserve congruence and those that do not.
 Geometry introduces rotations and dilations that may or may not be about the origin.
 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.

8.10C 
Explain the effect of translations, reflections over the x or yaxis, and rotations limited to 90°, 180°, 270°, and 360° as applied to twodimensional shapes on a coordinate plane using an algebraic representation.
Readiness Standard

Explain
THE EFFECT OF TRANSLATIONS, REFLECTIONS OVER THE x OR yAXIS, AND ROTATIONS LIMITED TO 90°, 180°, 270°, AND 360° AS APPLIED TO TWODIMENSIONAL SHAPES ON A COORDINATE PLANE USING AN ALGEBRAIC REPRESENTATION
Including, but not limited to:
 Prime notation of image points
 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
 Translation k units vertically
 Translation h units horizontally and k units vertically
 Reflection – a transformation frequently described as a flip or a mirror image of the original figure
 Algebraic notation
 Reflection across a vertical axis (yaxis)
 Reflection across a horizontal axis (xaxis)
 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 Level(s):
 Grade 8 introduces explaining the effect of translations, reflections over the x or yaxis, and rotations limited to 90°, 180°, 270°, and 360° as applied to twodimensional 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.

8.10D 
Model the effect on linear and area measurements of dilated twodimensional shapes.
Supporting Standard

Model
THE EFFECT ON LINEAR AND AREA MEASUREMENTS OF DILATED TWODIMENSIONAL SHAPES
Including, but not limited to:
 Linear measurement
 Perimeter – a linear measurement of the distance around the outer edge of a figure
 Circumference – a linear measurement of the distance around a circle
 Perimeter and circumference are onedimensional linear measures.
 Positive rational number side lengths
 Area – the measurement attribute that describes the number of square units a figure or region covers
 Area is a twodimensional square unit measure.
 Positive rational number side lengths
 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)
 Model of the effect on linear and area measurements of dilated twodimensional figures
 Dilating a twodimensional figure by a scale factor, recording the linear and area measurements of the figure and image, and determining the relationship between the scale factor and measurements
 Multiplying linear dimensions of a twodimensional figure by a constant scale factor results in a proportional onedimensional measure (perimeter/circumference).
 Multiplying linear dimensions of a twodimensional figure by a constant scale factor results in a twodimensional measure (area) that is equivalent to the original area multiplied by the scale factor squared.
 Generalizations of the effects on linear and area measurements of dilated twodimensional figures
 Linear measurements of a figure dilated by a scale factor of a, result in linear measurements of its image multiplied by a.
 Linear measurements of a figure dilated by a scale factor of a, result in area measurements of its image multiplied by a^{2}.
Note(s):
 Grade Level(s):
 Grade 8 introduces modeling the effect on linear and area measurements of dilated twodimensional shapes.
 Geometry describes how changes in the linear dimensions of a shape affect the two and threedimensional measures.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxRCFP:
 Representing, applying, and analyzing proportional relationships.
 TxCCRS:
 I.C. Numeric Reasoning – Systems of measurement
 I.C.1. Select or use the appropriate type of method, unit, and tool for the attribute being measured.
 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.
 III.D. Geometric and Spatial Reasoning – Measurements involving geometry and algebra
 III.D.1. Find the perimeter and area of twodimensional figures.
 III.D.2. Determine the surface area and volume of threedimensional figures.
 III.D.3. Determine indirect measurements of geometric figures using a variety of methods.
 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.
