Standard 2 : Relations and Functions (Archived)



This document was generated on CPALMS - www.cpalms.org

Draw and interpret graphs of relations. Understand the notation and concept of a function, find domains and ranges, and link equations to functions.

General Information

Number: MA.912.A.2
Title: Relations and Functions
Type: Standard
Subject: X-Mathematics (former standards - 2008) - Archived
Grade: 912
Body of Knowledge: Algebra

Related Benchmarks

This cluster includes the following benchmarks
Code Description
MA.912.A.2.1: Create a graph to represent a real-world situation.
MA.912.A.2.2: Interpret a graph representing a real-world situation.
MA.912.A.2.3: Describe the concept of a function, use function notation, determine whether a given relation is a function, and link equations to functions.
MA.912.A.2.4: Determine the domain and range of a relation.
MA.912.A.2.5: Graph absolute value equations and inequalities in two variables.
MA.912.A.2.6: Identify and graph common functions (including but not limited to linear, rational, quadratic, cubic, radical, absolute value).
MA.912.A.2.7: Perform operations (addition, subtraction, division, and multiplication) of functions algebraically, numerically, and graphically.
MA.912.A.2.8: Determine the composition of functions.
MA.912.A.2.9: Recognize, interpret, and graph functions defined piece-wise with and without technology.
MA.912.A.2.10: Describe and graph transformations of functions
MA.912.A.2.11: Solve problems involving functions and their inverses.
MA.912.A.2.12: Solve problems using direct, inverse, and joint variations.
MA.912.A.2.13: Solve real-world problems involving relations and functions.


Related Access Points

This cluster includes the following access points.

Independent

Access Point Number Access Point Title
MA.912.A.2.In.a: Organize data from real-world situations into categories, identify the labels, and display in simple bar, line, and circle graphs.
MA.912.A.2.In.b: Interpret simple bar, line, and circle graphs representing data from real-world situations.
MA.912.A.2.In.c: Identify the mathematical relationship (function) and the type of information represented in a function table or simple graph.
MA.912.A.2.In.d: Use function tables and simple graphs to determine the mathematical relationship between two numbers representing real-world situations.
MA.912.A.2.In.e: Use function tables and simple graphs to determine the mathematical relationship between two numbers representing real-world situations.

Supported

Access Point Number Access Point Title
MA.912.A.2.Su.a: Organize data from real-world situations into categories, identify the labels, and display in pictographs and bar graphs.
MA.912.A.2.Su.b: Identify which categories have the largest, smallest, or the same amount in pictographs and bar graphs representing real-world situations.
MA.912.A.2.Su.c: Identify number patterns and relationships using physical and visual models representing real-world situations.

Participatory

Access Point Number Access Point Title
MA.912.A.2.Pa.a: Count objects, pictures, or symbols used in a pictograph or chart and identify total to 10.
MA.912.A.2.Pa.b: Compare sets to 10 of objects, pictures, or symbols using one-to-one correspondence and identify which has more or less.


Related Resources

Vetted resources educators can use to teach the concepts and skills in this topic.

Lesson Plans

Name Description
Linear Motion:

In this activity students will learn the relationship between:

  • Distance and displacement
  • Velocity and speed
  • Vectors and scalars
  • Acceleration

and demonstrate their knowledge through group presentations.

Visualization of Social Networks with Node Graphs:

This lesson introduces the concept of node graphs for the purpose of visualizing social networks.

The lesson is presented with an introductory physical activity where students create a living graph. Students, building on their existing knowledge regarding common graph types, learn how node graphs can be used to visualize data from social networks.

Students will then participate in a simulated contagious infection event and will accurately record data about the transmission of the disease. These data will be used to construct a single computer file to be used to create a single node graph for describing the network. Students will then be responsible for understanding how to interpret the resulting network graph in the context of the activity.

Professional Developments

Name Description
Domain and Range of a Function: Khan Academy video Tutorial.

Definition of domain and range. The tutorial uses six different examples to demonstrate linear, quadratic, and rational functions with consideration to the domain and range.
Mathematical Modeling: Insights into Algebra, Teaching for Learning: This professional development resource provides a rich collection of information to help teachers engage students more effectively in mathematical modeling. It features videos of two complete lessons with commentary, background information on effective teaching, modeling, and lesson study, full lesson plans to teach both example lessons, examples of student work from the lessons, tips for effective teaching strategies, and list of helpful resources.
  • In Lesson 1 students use mathematical models (tables and equations) to represent the relationship between the number of revolutions made by a "driver" and a "follower" (two connected gears in a system), and they will explain the significance of the radii of the gears in regard to this relationship.
  • In Lesson 2 students mathematically model the growth of populations and use exponential functions to represent that growth.

Unit/Lesson Sequences

Name Description
Direct and Inverse Variation: "Lesson 1 of two lessons teaches students about direct variation by allowing them to explore a simulated oil spill using toilet paper tissues (to represent land) and drops of vegetable oil (to simulate a volume of oil). Lesson 2 teaches students about inverse variation by exploring the relationship between the heights of a fixed amount of water poured into cylindrical containers of different sizes as compared to the area of the containers' bases." from Insights into Algebra 1 - Annenberg Foundation.
Variables and Patterns of Change: Translating Words Into Symbols; Linear Equations: Lesson Plan 1: Miles of Tiles - The Pool Border Problem, students will recognize patterns and represent situations using algebraic notation and variables. Lesson Plan 2: Cups and Chips - Solving Linear Equations Using Manipulatives, students use manipulatives to represent visually the steps they take to obtain a solution to an algebraic equation. They develop an understanding of the connections between the solution involving manipulatives and the symbolic solution. Students work in teams of four. Site includes a Topic Overview, Lesson Plans, Student Work, Teaching Strategies, Resources, and a video of Workshop 1; Part 1.
Quadratic Functions: Workshop 4: Lesson 1 of two lessons requires students to explore quadratic functions by examining the family of functions described by y = a (x - h)squared+ k. In Lesson 2 students explore quadratic functions by using a motion detector known as a Calculator Based Ranger (CBR) to examine the heights of the different bounces of a ball. Students will represent each bounce with a quadratic function of the form y = a (x - h)squared + k. Background information, resources, references and videos of the lessons are included. Students work in teams of four.

Video/Audio/Animations

Name Description
Basic Linear Function:

This video demonstrates writing a function that represents a real-life scenario.

MIT BLOSSOMS - Fabulous Fractals and Difference Equations :

This learning video introduces students to the world of Fractal Geometry through the use of difference equations. As a prerequisite to this lesson, students would need two years of high school algebra (comfort with single variable equations) and motivation to learn basic complex arithmetic. Ms. Zager has included a complete introductory tutorial on complex arithmetic with homework assignments downloadable here. Also downloadable are some supplemental challenge problems. Time required to complete the core lesson is approximately one hour, and materials needed include a blackboard/whiteboard as well as space for students to work in small groups. During the in-class portions of this interactive lesson, students will brainstorm on the outcome of the chaos game and practice calculating trajectories of difference equations.

MIT BLOSSOMS - Is Bigger Better? A Look at a Selection Bias that Is All Around Us: This learning video addresses a particular problem of selection bias, a statistical bias in which there is an error in choosing the individuals or groups to make broader inferences. Rather than delve into this broad topic via formal statistics, we investigate how it may appear in our everyday lives, sometimes distorting our perceptions of people, places and events, unless we are careful. When people are picked at random from two groups of different sizes, most of those selected usually come from the bigger group. That means we will hear more about the experience of the bigger group than that of the smaller one. This isn't always a bad thing, but it isn't always a good thing either. Because big groups "speak louder," we have to be careful when we write mathematical formulas about what happened in the two groups. We think about this issue in this video, with examples that involve theaters, buses, and lemons. The prerequisite for this video lesson is a familiarity with algebra. It will take about one hour to complete, and the only materials needed are a blackboard and chalk. The downloadable Teacher's Guide found on the same page as the video, provides suggestions for classroom activities during each of the breaks between video segments.
Fitting a Line to Data:

Khan Academy tutorial video that demonstrates with real-world data the use of Excel spreadsheet to fit a line to data and make predictions using that line.

Virtual Manipulative

Name Description
Equation Grapher:

This interactive simulation investigates graphing linear and quadratic equations. Users are given the ability to define and change the coefficients and constants in order to observe resulting changes in the graph(s).

Worksheets

Name Description
Practice with Dimensional Analysis:

This is a worksheet that can be used for students individually or as a cooperative learning resource for practice with dimensional analysis. Answers are in red as a separate copy of the worksheet.

Quadratic Functions:

This worksheet gives students one place to show all transformations (reflections, vertical stretches/compressions, and translations) for the quadratic function. The worksheet also has a place for domain and range for each transformation.



Student Resources

Vetted resources students can use to learn the concepts and skills in this topic.

Video/Audio/Animations

Title Description
Basic Linear Function:

This video demonstrates writing a function that represents a real-life scenario.

MIT BLOSSOMS - Fabulous Fractals and Difference Equations :

This learning video introduces students to the world of Fractal Geometry through the use of difference equations. As a prerequisite to this lesson, students would need two years of high school algebra (comfort with single variable equations) and motivation to learn basic complex arithmetic. Ms. Zager has included a complete introductory tutorial on complex arithmetic with homework assignments downloadable here. Also downloadable are some supplemental challenge problems. Time required to complete the core lesson is approximately one hour, and materials needed include a blackboard/whiteboard as well as space for students to work in small groups. During the in-class portions of this interactive lesson, students will brainstorm on the outcome of the chaos game and practice calculating trajectories of difference equations.

Fitting a Line to Data:

Khan Academy tutorial video that demonstrates with real-world data the use of Excel spreadsheet to fit a line to data and make predictions using that line.

Virtual Manipulative

Title Description
Equation Grapher:

This interactive simulation investigates graphing linear and quadratic equations. Users are given the ability to define and change the coefficients and constants in order to observe resulting changes in the graph(s).



Parent Resources

Vetted resources caregivers can use to help students learn the concepts and skills in this topic.

Video/Audio/Animation

Title Description
Fitting a Line to Data:

Khan Academy tutorial video that demonstrates with real-world data the use of Excel spreadsheet to fit a line to data and make predictions using that line.