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 TITLE : Unit 04: Developing an Understanding of Slope and Y-Intercept SUGGESTED DURATION : 8 days

#### Unit Overview

Introduction
This unit bundles student expectations that address using tables and graphs to develop the understanding of slope and y-intercept. 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 Grade 7, students represented constant rates of change in mathematical and real-world problems given pictorial, tabular, verbal, numeric, graphical, and algebraic representations. Students determined the constant of proportionality within mathematical and real-world problems. Additionally, students represented linear relationships using verbal descriptions, tables, graphs, and equations that simplify to the form y = mx + b. Although students were formally introduced to the slope intercept form of y = mx + b, students did not explicitly study slope or -intercept.

During this Unit
Students use similar right triangles to develop an understanding of slope. This approach lends itself to the development of the formula for slope by determining the ratio of the change in y-values compared to the change in x-values is the same for any two points on the same line. Students use data from a table or graph to determine the rate of change or slope and the y-intercept.

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

After this Unit
In Unit 06, students will further examine slope and y-intercept through the lens of proportional and non-proportional situations. Students will interpret the unit rate as the slope of the line that models a proportional relationship. In Algebra I students will determine the slope of a line given a table of values, a graph, two points on the line, and an equation written in various forms, including y = mx + b, Ax + By = C, and yy1 = m(xx1). 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. Additionally, they 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.

In Grade 8, using similar right triangles to develop an understanding that slope, m, given as the rate comparing the change in y-values to the change in x-values, is the same for any two points (x1, y1) and (x2, y2) on the same line is STAAR Supporting Standard 8.4A. STAAR Readiness Standard 8.4C is identified as using data from a table or graph to determine the rate of change or slope and y-intercept in mathematical and real-world problems. Both of these standards are subsumed under the Grade 8 STAAR Reporting Category 2: Computations and Algebraic Relationships and the Grade 8 Texas Response to Curriculum Focal Points (TxRCFP): Representing, Applying and Analyzing Proportional Relationships. This unit is supporting the development of the Texas College and Career Readiness Standards (TxCCRS): I. Numeric Reasoning B1; II. Algebraic Reasoning D1, D2; III. Geometric and Spatial Reasoning B2, C1; V. Statistical Reasoning A1, C2; VI. Functions B1, C2; VII. Problem Solving and Reasoning A1, A2, A3, A4, A5, B1, C1, D1, D2; VIII. Communication and Representation A1, A2, A3, B1, B2, C1, C2, C3; IX. Connections A1, A2, B1, B2, B3.

Research
According to the National Mathematics Advisory Panel (2008), “Middle grade experience with similar triangles is most directly relevant for the study of Algebra. Sound treatments of the slope of a straight line and of linear functions depend logically on the properties of similar triangles” (p. 18). 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). Van de Walle and Lovin (2006) remark that “Representing functions in different ways can lead to analysis and understanding of that change. Students in the middle grades should develop an understanding of the multiple methods of expressing real-world functional relationships (words, graphs, equations, and tables). Working with these different representations of functions will allow students to develop a fuller understanding of this important concept” (p. 284).

National Council of Teachers of Mathematics. (2010). Focus in grade 8 teaching with curriculum focal points. Reston, VA: National Council of Teachers of Mathematics, Inc
National Mathematics Advisory Panel. (2008). Foundations for success: The final report of the national mathematics advisory panel. Washington, DC: U.S. Department of Education.
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., & Lovin, L. (2006). Teaching student-centered mathematics grades 5 – 8. Boston, MA: Pearson Education, Inc.

 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)
• Understanding how two quantities vary together (covariation) and can be reasoned up and down in situations involving invariant (constant) relationships builds flexible algebraic reasoning in order to make predictions and critical judgements about the relationship.
• Proportional and non-proportional linear relationships are represented by two attributes, the constant rate of change in y-values compared to the change in x-values, modeled using similar triangles to explain slope, and y-intercept.
• How can similar triangles be created from a line representing a linear situation?
• What relationship exists between similar triangles and the slope, m, of a line?
• How can similar triangles be used to generalize the formula for slope as the ratio of the change in the y-values to the change in the x-values?
• What relationship exists among the similar triangles?
• 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?
• Proportionality
• Attributes of Linear Relations
• Slope
• y-intercept
• Ratios and Rates
• Slope
• Relationships and Generalizations
• Equivalence
• Linear proportional
• Linear non-proportional
• Geometric similarity
• 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 the slope in a linear relationship is m = , since the x coordinate (horizontal) always comes before the y coordinate (vertical) in an ordered pair. Instead the correct representation of slope in a linear relationship is m = .
• Some students may think that the intercept coordinate is the zero term instead of the non-zero term, since intercepts are associated with zeros. In other words, students may think (0, 4) would be the x-intercept because the 0 is in the x coordinate.

Underdeveloped Concepts:

• Some students may confuse corresponding sides of similar triangles.

#### Unit Vocabulary

• Similar shapes – shapes whose angles are congruent and side lengths are proportional (equal scale factor)
• 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-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:

 Congruent Constant of proportionality Corresponding angles Corresponding sides Horizontal Input Linear Negative Non-proportional Origin Output Positive Proportional Rate of change Ratio Right angle Right triangle Rise Run Scale factor Similarity Undefined 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 Supporting 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, 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.
8.4 Proportionality. The student applies mathematical process standards to explain proportional and non-proportional relationships involving slope. The student is expected to:
8.4A Use similar right triangles to develop an understanding that slope, m, given as the rate comparing the change in y-values to the change in x-values, (y2 - y1)/ (x2 - x1), is the same for any two points (x1, y1) and (x2, y2) on the same line.
Supporting Standard

Use

SIMILAR RIGHT TRIANGLES TO DEVELOP AN UNDERSTANDING THAT SLOPE, m, GIVEN AS THE RATE COMPARING THE CHANGE IN y-VALUES TO THE CHANGE IN x-VALUES, , IS THE SAME FOR ANY TWO POINTS (x1, y1) AND (x2, y2) ON THE SAME LINE

Including, but not limited to:

• 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 right triangles when the longest sides of the triangles (sides opposite the right angles) lie on the same line
• Corresponding sides are proportional.
• Corresponding angles are congruent.
• Rate comparing the change in y-values to the change in x-values is the same for any two points on the same line.
• 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
• Slope is either positive, negative, zero, or undefined.
• Connections between similar right triangles and slope
• A right triangle can be formed from any two points on a line by drawing a vertical line from one point and a horizontal line from the other point until the lines intersect.
• Slope of the line containing the longest side (side opposite the right angle) of a right triangle is determined between the two vertices not forming the right angle.
• Reversal of the order of the points or  , yields the same slope.
• Not all longest sides (sides opposite the right angles) of similar right triangles have the same slope when placed on a coordinate plane.
• To have the same slope, similar right triangles must be placed on a coordinate plane having two points of their longest sides (sides opposite the right angles) that lie on the same line.

Note(s):

• Algebra I will determine the slope of a line given a table of values, a graph, two points on the line, and an equation written in various forms, including y = mx + b, Ax + By = C, and yy1 = m(xx1).
• 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.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.
• VI.B. Functions – Analysis of functions
• VI.B.1. Understand and analyze features of functions.
• 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.
8.4C 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.