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.

8.2 
Number and operations. The student applies mathematical process standards to represent and use real numbers in a variety of forms. The student is expected to:


8.2A 
Extend previous knowledge of sets and subsets using a visual representation to describe relationships between sets of real numbers.
Supporting Standard

Extend
PREVIOUS KNOWLEDGE OF SETS AND SUBSETS USING A VISUAL REPRESENTATION
Including, but not limited to:
 Counting (natural) numbers – the set of positive numbers that begins at one and increases by increments of one each time {1, 2, 3, ..., n}
 Whole numbers – the set of counting (natural) numbers and zero {0, 1, 2, 3, ..., n}
 Integers – the set of counting (natural numbers), their opposites, and zero {–n, …, –3, –2, –1, 0, 1, 2, 3, ..., n}. The set of integers is denoted by the symbol Z.
 Rational numbers – the set of numbers that can be expressed as a fraction , where a and b are integers and b ≠ 0. The set of rational numbers is denoted by the symbol Q.
 Irrational numbers – the set of numbers that cannot be expressed as a fraction , where a and b are integers and b ≠ 0
 Real numbers – the set of rational and irrational numbers. The set of real numbers is denoted by the symbol ℜ.
 Visual representations of the relationships between sets and subsets of real numbers
To Describe
RELATIONSHIPS BETWEEN SETS OF REAL NUMBERS
Including, but not limited to:
 All counting (natural) numbers are a subset of whole numbers, integers, rational numbers, and real numbers.
 All whole numbers are a subset of integers, rational numbers, and real numbers.
 All integers are a subset of rational numbers and real numbers.
 All counting (natural) numbers, whole numbers, and integers are a subset of rational numbers and real numbers.
 Not all rational numbers are integers, whole numbers, or counting (natural) numbers.
 Terminating and repeating decimals are rational numbers but not integers, whole numbers, or counting (natural) numbers.
 All irrational numbers are a subset of real numbers.
 Real numbers include all rational numbers, integers, whole numbers, counting (natural) numbers, and irrational numbers.
 Not all real numbers are rational numbers.
Note(s):
 Grade Level(s):
 Grade 5 classified twodimensional figures in a hierarchy of sets and subsets using graphic organizers based on their attributes and properties.
 Grade 6 classified whole numbers, integers, and rational numbers using a visual representation such as a Venn diagram to describe relationships between sets of numbers.
 Grade 7 extended previous knowledge of sets and subsets using a visual representation to describe relationships between sets of rational numbers.
 Grade 8 introduces the set of irrational numbers as a subset of real numbers.
 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)

8.2B 
Approximate the value of an irrational number, including π and square roots of numbers less than 225, and locate that rational number approximation on a number line.
Supporting Standard

Approximate
THE VALUE OF AN IRRATIONAL NUMBER, INCLUDING π AND SQUARE ROOTS OF NUMBERS LESS THAN 225
Including, but not limited to:
 Irrational numbers – the set of numbers that cannot be expressed as a fraction , where a and b are integers and b ≠ 0
 Rational number approximations of irrational numbers to the appropriate place value for context of mathematical and realworld problem situations
 Approximation symbol (≈)
 Square root – a factor of a number that, when squared, equals the original number
 Radical symbol ()
 represents the principal square root of x, the positive square root
 – represents the opposite of the principal square root of x, the negative square root
 Rational number approximations of square roots less than 225
 Integers
 Decimals
 Fractions
 Verification of rational number approximations of irrational numbers with a calculator
 Relationship between rational number approximations of perfect squares and irrational numbers
 Perfect squares of consecutive integers
Locate
RATIONAL NUMBER APPROXIMATIONS OF IRRATIONAL NUMBERS ON A NUMBER LINE
Including, but not limited to:
 Rational numbers – the set of numbers that can be expressed as a fraction , where a and b are integers and b ≠ 0. The set of rational numbers is denoted by the symbol Q.
 Irrational numbers – the set of numbers that cannot be expressed as a fraction , where a and b are integers and b ≠ 0
 All rational number approximations of irrational numbers can be located on a number line.
 Characteristics of a number line
 A number line begins as a line with predetermined intervals (or tick marks) with positions/numbers labeled.
 A minimum of two positions/numbers should be labeled.
 Numbers on a number line represent the distance from zero.
 The distance between the tick marks is counted rather than the tick marks themselves.
 The placement of the labeled positions/numbers on a number line determines the scale of the number line.
 Intervals between position/numbers are proportional.
 When reasoning on a number line, the position of zero may or may not be placed.
 When working with larger numbers, a number line without the constraint of distance from zero allows the ability to “zoomin” on the relevant section of the number line.
 Number lines extend infinitely in both directions (arrows indicate the number line continues infinitely).
 Numbers increase from left to right on a horizontal number line and from bottom to top on a vertical number line.
 Points to the left of a specified point on a horizontal number line are less than points to the right.
 Points to the right of a specified point on a horizontal number line are greater than points to the left.
 Points below a specified point on a vertical number line are less than points above.
 Points above a specified point on a vertical number line are greater than points below.
 Characteristics of an open number line
 An open number line begins as a line with no intervals (or tick marks) and no positions/numbers labeled.
 Numbers/positions are placed on the empty number line only as they are needed.
 When reasoning on an open number line, the position of zero is often not placed.
 When working with larger numbers, an open number line without the constraint of distance from zero allows the ability to “zoomin” on the relevant section of the number line.
 The placement of the first two numbers on an open number line determines the scale of the number line.
 Once the scale of the number line has been established by the placement of the first two numbers, intervals between additional numbers placed are approximately proportional.
 The differences between numbers are approximated by the distance between the positions on the number line.
 Open number lines extend infinitely in both directions (arrows indicate the number line continues infinitely).
 Numbers increase from left to right on a horizontal number line and from bottom to top on a vertical number line.
 Points to the left of a specified point on a horizontal number line are less than points to the right.
 Points to the right of a specified point on a horizontal number line are greater than points to the left.
 Landmark (or anchor) numbers may be placed on the open number line to help locate other numbers.
 Rational number approximations of irrational numbers
 Rational number approximations of square roots less than 225
 Integers
 Decimals
 Fractions
 Verification of rational number approximations of irrational numbers with a calculator
 Relationship between rational number approximations of perfect squares and irrational numbers
 Perfect squares of consecutive integers
Note(s):
 Grade Level(s):
 Grade 8 introduces approximating the value of an irrational number, including π and square roots of numbers less than 225, and locating that rational number approximation on a number line.
 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:
 I.B. Numeric Reasoning – Number sense and number concepts
 I.B.1. Use estimation to check for errors and reasonableness of solutions.

8.2D 
Order a set of real numbers arising from mathematical and realworld contexts.
Readiness Standard

Order
A SET OF REAL NUMBERS ARISING FROM MATHEMATICAL AND REALWORLD CONTEXTS
Including, but not limited to:
 Real numbers – the set of rational and irrational numbers. The set of real numbers is denoted by the symbol ℜ.
 Various forms of real numbers
 Rational numbers (positive or negative)
 Irrational numbers (positive or negative)
 Place value – the value of a digit as determined by its location in a number such as ones, tens, hundreds, one thousands, ten thousands, etc.
 Order numbers – to arrange a set of numbers based on their numerical value
 Number lines (horizontal/vertical)
 Numbers increase from left to right on a horizontal number line and from bottom to top on a vertical number line.
 Points to the left of a specified point on a horizontal number line are less than points to the right.
 Points to the right of a specified point on a horizontal number line are greater than points to the left.
 Points below a specified point on a vertical number line are less than points above.
 Points above a specified point on a vertical number line are greater than points below.
 Quantifying descriptor in mathematical and realworld problem situations (e.g., between two given numbers, greatest/least, ascending/descending, tallest/shortest, warmest/coldest, fastest/slowest, longest/shortest, heaviest/lightest, closest/farthest, oldest/youngest, etc.)
Note(s):
 Grade Level(s):
 Grade 6 ordered a set of rational numbers arising from mathematical and realworld contexts.
 Grade 8 introduces ordering a set of real numbers arising from mathematical and realworld contexts.
 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:
 I. Numeric Reasoning
 IX. Communication and Representation
 X. Connections

8.6 
Expressions, equations, and relationships. The student applies mathematical process standards to develop mathematical relationships and make connections to geometric formulas. The student is expected to:


8.6C 
Use models and diagrams to explain the Pythagorean theorem.
Supporting Standard

Use
MODELS AND DIAGRAMS TO EXPLAIN THE PYTHAGOREAN THEOREM
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: sum of the squares of the legs equals the square of the hypotenuse
 Formula: a^{2} + b^{2} = c^{2}, where a and b represent the legs and c represents the hypotenuse
 Models and diagrams
 Square tiles
 Grid paper
 Tangrams
Note(s):
 Grade Level(s):
 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.D. Algebraic Reasoning – Representing relationships
 II.D.1. Interpret multiple representations of equations, inequalities, and relationships.
 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.

8.7 
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.
Readiness Standard

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
 a^{2} + b^{2} = c^{2}, 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
 a^{2} + b^{2} = c^{2}, where a and b represent the legs of a right triangle and c represents the hypotenuse
Note(s):
 Grade Level(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.

8.7D 
Determine the distance between two points on a coordinate plane using the Pythagorean Theorem.
Supporting Standard

Determine
THE DISTANCE BETWEEN TWO POINTS ON A COORDINATE PLANE USING THE PYTHAGOREAN THEOREM
Including, but not limited to:
 Coordinate plane (coordinate grid) – a twodimensional plane on which to plot points, lines, and curves
 Axes – the vertical and horizontal lines that act as a reference when plotting points on a coordinate plane
 Intersecting lines – lines that meet or cross at a point
 Origin – the starting point in locating points on a coordinate plane
 Quadrants – any of the four areas created by dividing a plane with an xaxis and yaxis
 Attributes of the coordinate plane
 Two number lines intersect perpendicularly to form the axes, which are used to locate points on the plane.
 The horizontal number line is called the xaxis.
 The vertical number line is called the yaxis.
 The xaxis and the yaxis cross at 0 on both number lines and that intersection is called the origin.
 The ordered pair of numbers corresponding to the origin is (0, 0)
 Four quadrants are formed by the intersection of the xand yaxes and are labeled counterclockwise with Roman numerals.
 Iterated units are labeled and shown on both axes to show scale.
 Intervals may or may not be increments of one.
 Intervals may or may not include decimal or fractional amounts.
 Relationship between ordered pairs and attributes of the coordinate plane
 A pair of ordered numbers names the location of a point on a coordinate plane.
 Ordered pairs of numbers are indicated within parentheses and separated by a comma. (x, y).
 The first number in the ordered pair represents the parallel movement on the xaxis, left or right starting at the origin.
 The second number in the ordered pair represents the parallel movement on the yaxis, up or down starting at the origin.
 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
 a^{2} + b^{2} = c^{2}, 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.
 Generalizations from points on a coordinate plane
 A right triangle can be formed from any two points on a nonhorizontal, nonvertical line by drawing a vertical line from one point and a horizontal line from the other point until the lines intersect.
 The Pythagorean Theorem can be used to determine the distance between two points on a coordinate plane.
Note(s):
 Grade Level(s):
 Grade 8 introduces determining the distance between two points on a coordinate plane using the Pythagorean Theorem.
 Geometry will derive and use the distance formula to verify geometric relationships and 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 onevariable equations or inequalities in problem situations. The student is expected to:


8.8D 
Use informal arguments to establish facts about the angle sum and exterior angle of triangles, the angles created when parallel lines are cut by a transversal, and the angleangle criterion for similarity of triangles.
Supporting Standard

Use
INFORMAL ARGUMENTS TO ESTABLISH FACTS ABOUT THE ANGLE SUM AND EXTERIOR ANGLE OF TRIANGLES, THE ANGLES CREATED WHEN PARALLEL LINES ARE CUT BY A TRANSVERSAL, AND THE ANGLEANGLE CRITERION FOR SIMILARITY OF TRIANGLES
Including, but not limited to:
 Angle – two rays with a common end point (the vertex)
 Degree – the measure of an angle where each degree represents of a circle
 Unit measure labels as “degrees” or with symbol for degrees (°)
 Adjacent angles – two nonoverlapping angles that share a common vertex and exactly one ray
 Complementary angles – two angles whose degree measures have a sum of 90°
 Supplementary angles – two angles whose degree measures have a sum of 180°
 Triangle – a polygon with three sides and three vertices
 Interior angles of a triangle – angles that are inside of a triangle, formed by two sides of the triangle
 Exterior angles of a triangle – angles that are outside of a triangle between one side of a triangle and the extension of the adjacent side
 Informal arguments to establish facts about triangles
 The sum of the measures of the interior angles of a triangle equals 180º.
 Adjacent interior and exterior angles create a supplementary pair of angles (the sum of the measures equals 180º).
 An exterior angle is equal to the sum of the two nonadjacent interior angles or the remote interior angles.
 The sum of the measures of the exterior angles, one at each vertex, of a triangle equals 360°.
 Equations to represent the relationships between interior and/or exterior angles and to determine a missing angle measure
 Congruent angles – angles whose angle measurements are equal
 Arc(s) on angles are usually used to indicate congruency (one set of congruent angles would have 1 arc, another set of congruent angles would have 2 arcs; etc.).
 Arcs and tick marks on angles can be used to indicate congruency (one set of congruent angles would have 1 arc with 1 tick mark, another set of congruent angles would have 1 arc with 2 tick marks; etc.).
 Vertical angles – a pair of nonadjacent, nonoverlapping angles formed by two intersecting lines creating angles that are opposite and congruent to each other
 Parallel lines – lines that lie in the same plane, never intersect, and are always the same distance apart
 Various orientations including vertical, horizontal, diagonal, and parallel lines of even, uneven, or offset lengths
 Lines that are parallel may or may not contain parallel markings.
 Transversal – a line that intersects two or more lines
 Alternate interior angles
 When two parallel lines are cut by a transversal, congruent alternate interior angles are formed on opposite sides of the transversal and inside the parallel lines.
 Same side interior angles
 When two parallel lines are cut by a transversal, supplementary same side interior angles are formed between the parallel lines on the same side of the transversal.
 Alternate exterior angles
 When two parallel lines are cut by a transversal, congruent alternate exterior angles are formed on opposite sides of the transversal and outside the parallel lines.
 Same side exterior angles
 When two parallel lines are cut by a transversal, supplementary same side exterior angles are formed outside the parallel lines on the same side of the transversal.
 Corresponding angles
 When two parallel lines are cut by a transversal, corresponding angles (one interior angle and one exterior angle) are formed on the same side of the transversal and on the same side of the parallel lines.
 Informal arguments to establish facts about the angles created when parallel lines are cut by a transversal
 Angleangle criterion for triangles – if two angles in one triangle are congruent to two angles in another triangle, then the measures of the third angle in both triangles are congruent
 Informal arguments to establish facts about the angleangle criterion for similarity of triangles
 The sum of the measures of the interior angles of a triangle is 180º. If two angles of one triangle are congruent to two angles of another triangle, then the measures of the third angles of the triangles must also be congruent, meaning the two triangles are similar. Therefore, if two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar.
Note(s):
 Grade Level(s):
 Grade 7 wrote and solved equations using geometry concepts, including the sum of the angles in a triangle, and angle relationships.
 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:
 III.A. Geometric and Spatial Reasoning – Figures and their properties
 III.A.2. Form and validate conjectures about one, two, and threedimensional figures and their properties.
 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.3. Determine indirect measurements of geometric figures using a variety of methods.
 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.
