This week we will examine the concept of a function, a fundamental concept underlying all of modern mathematics. You’re undoubtedly already familiar with functions in an intuitive sense: a function is something which, given

an input, produces an output. But you’ve probably never seen the formal definition of a function as it relates to set theory, which is what we’ll look at this week.

# 213 Results

This is a task from the Illustrative Mathematics website that is one part of a complete illustration of the standard to which it is aligned. Each task has at least one solution and some commentary that addresses important aspects of the task and its potential use.

- Subject:
- Functions
- Material Type:
- Activity/Lab
- Provider:
- Illustrative Mathematics
- Provider Set:
- Illustrative Mathematics
- Author:
- Illustrative Mathematics
- Date Added:
- 08/06/2015

The problem presents a context where a quadratic function arises. Careful analysis, including graphing, of the function is closely related to the context. The student will gain valuable experience applying the quadratic formula and the exercise also gives a possible implementation of completing the square.

- Subject:
- Mathematics
- Functions
- Material Type:
- Activity/Lab
- Provider:
- Illustrative Mathematics
- Provider Set:
- Illustrative Mathematics
- Author:
- Illustrative Mathematics
- Date Added:
- 05/01/2012

The problem statement describes a changing algae population as reported by the Maryland Department of Natural Resources. In part (a), students are expected to build an exponential function modeling algae concentration from the description given of the relationship between concentrations in cells/ml and days of rapid growth (F-LE.2).

- Subject:
- Mathematics
- Functions
- Material Type:
- Activity/Lab
- Provider:
- Illustrative Mathematics
- Provider Set:
- Illustrative Mathematics
- Author:
- Illustrative Mathematics
- Date Added:
- 05/01/2012

A work in progress, CK-12's Algebra I Second Edition is a clear presentation of algebra for the high school student. Topics include: Equations and Functions, Real Numbers, Equations of Lines, Solving Systems of Equations and Quadratic Equations.

- Subject:
- Algebra
- Functions
- Material Type:
- Textbook
- Provider:
- CK-12 Foundation
- Provider Set:
- CK-12 FlexBook
- Author:
- Andrew
- Annamaria
- Anne
- Eve
- Farbizio
- Gloag
- Rawley
- Date Added:
- 12/03/2010

CK-12 Foundation's Basic Algebra FlexBook is an introduction to the algebraic topics of functions, equations, and graphs for middle-school and high-school students.

- Subject:
- Algebra
- Functions
- Material Type:
- Textbook
- Provider:
- CK-12 Foundation
- Provider Set:
- CK-12 FlexBook
- Author:
- Farbizio, Annamaria
- Gloag, Andrew
- Gloag, Anne
- Kramer, Melissa
- Date Added:
- 09/21/2010

CK-12 Algebra Explorations is a hands-on series of activities that guides students from Pre-K to Grade 7 through algebraic concepts.

- Subject:
- Algebra
- Functions
- Material Type:
- Activity/Lab
- Lesson Plan
- Textbook
- Provider:
- CK-12 Foundation
- Provider Set:
- CK-12 FlexBook
- Author:
- Mary Cavanagh, Carol Findell, Carole Greenes
- Date Added:
- 10/04/2011

In this video segment from Cyberchase, Matt tries for a second time to arrange tables and chairs to accommodate 20 workers.

- Subject:
- Mathematics
- Algebra
- Functions
- Material Type:
- Lecture
- Provider:
- PBS LearningMedia
- Provider Set:
- PBS Learning Media: Multimedia Resources for the Classroom and Professional Development
- Author:
- U.S. Department of Education
- WNET
- Date Added:
- 09/23/2008

In this real world problem students solve questions based on the relationship between production costs and price.

- Subject:
- Mathematics
- Functions
- Material Type:
- Activity/Lab
- Provider:
- Illustrative Mathematics
- Provider Set:
- Illustrative Mathematics
- Author:
- Illustrative Mathematics
- Date Added:
- 05/01/2012

This task provides a real world context for interpreting and solving exponential equations. There are two solutions provided for part (a). The first solution demonstrates how to deduce the conclusion by thinking in terms of the functions and their rates of change. The second approach illustrates a rigorous algebraic demonstration that the two populations can never be equal.

- Subject:
- Mathematics
- Functions
- Material Type:
- Activity/Lab
- Provider:
- Illustrative Mathematics
- Provider Set:
- Illustrative Mathematics
- Author:
- Illustrative Mathematics
- Date Added:
- 05/01/2012

This task could be put to good use in an instructional sequence designed to develop knowledge related to students' understanding of linear functions in contexts. Though students could work independently on the task, collaboration with peers is more likely to result in the exploration of a range of interpretations.

- Subject:
- Mathematics
- Functions
- Material Type:
- Activity/Lab
- Provider:
- Illustrative Mathematics
- Provider Set:
- Illustrative Mathematics
- Author:
- Illustrative Mathematics
- Date Added:
- 05/01/2012

This task involves a fairly straightforward decaying exponential. Filling out the table and developing the general formula is complicated only by the need to work with a fraction that requires decisions about rounding and precision.

- Subject:
- Mathematics
- Functions
- Material Type:
- Activity/Lab
- Provider:
- Illustrative Mathematics
- Provider Set:
- Illustrative Mathematics
- Author:
- Illustrative Mathematics
- Date Added:
- 05/01/2012

This task describes two linear functions using two different representations. To draw conclusions about the quantities, students have to find a common way of describing them. We have presented three solutions (1) Finding equations for both functions. (2) Using tables of values. (3) Using graphs.

- Subject:
- Mathematics
- Functions
- Material Type:
- Activity/Lab
- Provider:
- Illustrative Mathematics
- Provider Set:
- Illustrative Mathematics
- Author:
- Illustrative Mathematics
- Date Added:
- 05/01/2012

The purpose of this task is for students to interpret two distance-time graphs in terms of the context of a bicycle race. There are two major mathematical aspects to this: interpreting what a particular point on the graph means in terms of the context, and understanding that the "steepness" of the graph tells us something about how fast the bicyclists are moving.

- Subject:
- Mathematics
- Functions
- Material Type:
- Activity/Lab
- Provider:
- Illustrative Mathematics
- Provider Set:
- Illustrative Mathematics
- Author:
- Illustrative Mathematics
- Date Added:
- 05/01/2012

This book will initiate you into an esoteric world. You will learn and apply the methods of thought that mathematicians use to verify theorems, explore mathematical truth and create new mathematical theories. This will prepare you for advanced mathematics courses, for you will be better able to understand proofs, write your own proofs and think critically and inquisitively about mathematics.

- Subject:
- Functions
- Material Type:
- Textbook
- Provider:
- Virginia Commonwealth University
- Author:
- Richard Hammack
- Date Added:
- 01/01/2009

This task provides an exploration of a quadratic equation by descriptive, numerical, graphical, and algebraic techniques. Based on its real-world applicability, teachers could use the task as a way to introduce and motivate algebraic techniques like completing the square, en route to a derivation of the quadratic formula.

- Subject:
- Mathematics
- Algebra
- Functions
- Material Type:
- Activity/Lab
- Provider:
- Illustrative Mathematics
- Provider Set:
- Illustrative Mathematics
- Author:
- Illustrative Mathematics
- Date Added:
- 05/01/2012

This task is for instructional purposes only and builds on ``Building an explicit quadratic function.''

- Subject:
- Mathematics
- Algebra
- Functions
- Material Type:
- Activity/Lab
- Provider:
- Illustrative Mathematics
- Provider Set:
- Illustrative Mathematics
- Author:
- Illustrative Mathematics
- Date Added:
- 08/20/2012

This is the first of a series of task aiming at understanding the quadratic formula in a geometric way in terms of the graph of a quadratic function.

- Subject:
- Mathematics
- Functions
- Material Type:
- Activity/Lab
- Provider:
- Illustrative Mathematics
- Provider Set:
- Illustrative Mathematics
- Author:
- Illustrative Mathematics
- Date Added:
- 08/15/2012

This task is intended for instruction and to motivate "Building a general quadratic function.''

- Subject:
- Mathematics
- Functions
- Material Type:
- Activity/Lab
- Provider:
- Illustrative Mathematics
- Provider Set:
- Illustrative Mathematics
- Author:
- Illustrative Mathematics
- Date Added:
- 08/26/2013

Full course of Algebra 1 is presented online by Georgia Virtual Learning. Audio, video, text, games and activities are included to engage ninth grade students in learning.

- Subject:
- Algebra
- Functions
- Material Type:
- Activity/Lab
- Full Course
- Reading
- Provider:
- Georgia Department of Education
- Provider Set:
- Georgia Virtual Learning
- Date Added:
- 08/02/2013

This task was developed by high school and postsecondary mathematics and health sciences educators, and validated by content experts in the Common Core State Standards in mathematics and the National Career Clusters Knowledge & Skills Statements. It was developed with the purpose of demonstrating how the Common Core and CTE Knowledge & Skills Statements can be integrated into classroom learning - and to provide classroom teachers with a truly authentic task for either mathematics or CTE courses.

- Subject:
- Health, Medicine and Nursing
- Algebra
- Functions
- Material Type:
- Activity/Lab
- Lesson Plan
- Provider:
- National Association of State Directors of Career Technical Education Consortium
- Provider Set:
- Career Technical Education
- Date Added:
- 07/27/2012

This task was developed by high school and postsecondary mathematics and health sciences educators, and validated by content experts in the Common Core State Standards in mathematics and the National Career Clusters Knowledge & Skills Statements. It was developed with the purpose of demonstrating how the Common Core and CTE Knowledge & Skills Statements can be integrated into classroom learning - and to provide classroom teachers with a truly authentic task for either mathematics or CTE courses.

- Subject:
- Health, Medicine and Nursing
- Algebra
- Functions
- Material Type:
- Activity/Lab
- Lesson Plan
- Provider:
- National Association of State Directors of Career Technical Education Consortium
- Provider Set:
- Career Technical Education
- Date Added:
- 07/26/2012

This course begins with a review of algebra specifically designed to help and prepare the student for the study of calculus, and continues with discussion of functions, graphs, limits, continuity, and derivatives. The appendix provides a large collection of reference facts, geometry, and trigonometry that will assist in solving calculus problems long after the course is over. Upon successful completion of this course, the student will be able to: calculate or estimate limits of functions given by formulas, graphs, or tables by using properties of limits and LĺÎĺ_ĺĚĺ_hopitalĺÎĺ_ĺĚĺ_s Rule; state whether a function given by a graph or formula is continuous or differentiable at a given point or on a given interval and justify the answer; calculate average and instantaneous rates of change in context, and state the meaning and units of the derivative for functions given graphically; calculate derivatives of polynomial, rational, common transcendental functions, and implicitly defined functions; apply the ideas and techniques of derivatives to solve maximum and minimum problems and related rate problems, and calculate slopes and rates for function given as parametric equations; find extreme values of modeling functions given by formulas or graphs; predict, construct, and interpret the shapes of graphs; solve equations using NewtonĺÎĺ_ĺĚĺ_s Method; find linear approximations to functions using differentials; festate in words the meanings of the solutions to applied problems, attaching the appropriate units to an answer; state which parts of a mathematical statement are assumptions, such as hypotheses, and which parts are conclusions. This free course may be completed online at any time. It has been developed through a partnership with the Washington State Board for Community and Technical Colleges; the Saylor Foundation has modified some WSBCTC materials. (Mathematics 005)

- Subject:
- Calculus
- Functions
- Material Type:
- Assessment
- Full Course
- Reading
- Syllabus
- Textbook
- Provider:
- The Saylor Foundation
- Date Added:
- 04/16/2012

This course is an introduction to contemporary calculus and is the first of a three-part sequence. In this course students explore the concepts, applications, and techniques of Calculus - the mathematics of change. Calculus has wide-spread application in science, economics and engineering, and is a foundation college course for further work in these areas. This is a required class for most science and mathematics majors.Login: guest_oclPassword: ocl

- Subject:
- Calculus
- Functions
- Material Type:
- Full Course
- Homework/Assignment
- Lecture Notes
- Lesson Plan
- Reading
- Syllabus
- Provider:
- Washington State Board for Community & Technical Colleges
- Provider Set:
- Open Course Library
- Date Added:
- 10/31/2011

CK-12 Foundation's Single Variable Calculus FlexBook introduces high school students to the topics covered in the Calculus AB course. Topics include: Limits, Derivatives, and Integration.

- Subject:
- Calculus
- Functions
- Material Type:
- Textbook
- Provider:
- CK-12 Foundation
- Provider Set:
- CK-12 FlexBook
- Date Added:
- 10/06/2009

CK-12's Texas Instrument Calculus Student Edition is a useful companion to a Calculus course, offering extra assignments and opportunities for students to understand course material through their graphing calculator.

- Subject:
- Calculus
- Functions
- Material Type:
- Textbook
- Provider:
- CK-12 Foundation
- Provider Set:
- CK-12 FlexBook
- Author:
- Jordan, Lori
- Date Added:
- 12/15/2010

CK-12's Texas Instruments Calculus Teacher's Edition is a useful companion to a Calculus course, offering extra assignments and opportunities for students to understand course material through their graphing calculator.

- Subject:
- Calculus
- Functions
- Material Type:
- Textbook
- Provider:
- CK-12 Foundation
- Provider Set:
- CK-12 FlexBook
- Author:
- Jordan, Lori
- Date Added:
- 12/17/2010

We told you about the derivatives of many functions, but you might want proof that what we told you is actually true. That's what this tutorial tries to do!

- Subject:
- Calculus
- Functions
- Material Type:
- Lecture
- Provider:
- Khan Academy
- Provider Set:
- Khan Academy
- Date Added:
- 05/30/2013

CK-12 Calculus Teacher's Edition covers tips, common errors, enrichment, differentiated instruction and problem solving for teaching CK-12 Calculus Student Edition. The solution guide is available upon request.

- Subject:
- Calculus
- Functions
- Material Type:
- Textbook
- Provider:
- CK-12 Foundation
- Provider Set:
- CK-12 FlexBook
- Author:
- Dreyfuss, Andrew
- Narasimhan, Ramesh
- Prolo, Jared
- Date Added:
- 06/24/2011

The purpose of this task is to give students practice constructing functions that represent a quantity of interest in a context, and then interpreting features of the function in the light of that context. It can be used as either an assessment or a teaching task.

- Subject:
- Mathematics
- Functions
- Material Type:
- Activity/Lab
- Provider:
- Illustrative Mathematics
- Provider Set:
- Illustrative Mathematics
- Author:
- Illustrative Mathematics
- Date Added:
- 05/01/2012

The primary purpose of this task is to lead students to a numerical and graphical understanding of the behavior of a rational function near a vertical asymptote, in terms of the expression defining the function. The canoe context focuses attention on the variables as numbers, rather than as abstract symbols.

- Subject:
- Mathematics
- Functions
- Material Type:
- Activity/Lab
- Provider:
- Illustrative Mathematics
- Provider Set:
- Illustrative Mathematics
- Author:
- Illustrative Mathematics
- Date Added:
- 05/01/2012

The task requires the student to use logarithms to solve an exponential equation in the realistic context of carbon dating, important in archaeology and geology, among other places. Students should be guided to recognize the use of the natural logarithm when the exponential function has the given base of e, as in this problem. Note that the purpose of this task is algebraic in nature -- closely related tasks exist which approach similar problems from numerical or graphical stances.

- Subject:
- Mathematics
- Functions
- Archaeology
- Material Type:
- Activity/Lab
- Provider:
- Illustrative Mathematics
- Provider Set:
- Illustrative Mathematics
- Author:
- Illustrative Mathematics
- Date Added:
- 05/01/2012

In the task "Carbon 14 Dating'' the amount of Carbon 14 in a preserved plant is studied as time passes after the plant has died. In practice, however, scientists wish to determine when the plant died and, as this task shows, this is not possible with a simple measurement of the amount of Carbon 14 remaining in the preserved plant. The equation for the amount of Carbon 14 remaining in the preserved plant is in many ways simpler here, using 12 as a base.

- Subject:
- Mathematics
- Functions
- Material Type:
- Activity/Lab
- Provider:
- Illustrative Mathematics
- Provider Set:
- Illustrative Mathematics
- Author:
- Illustrative Mathematics
- Date Added:
- 08/21/2012

This problem introduces the method used by scientists to date certain organic material. It is based not on the amount of the Carbon 14 isotope remaining in the sample but rather on the ratio of Carbon 14 to Carbon 12. This ratio decreases, hypothetically, at a constant exponential rate as soon as the organic material has ceased to absorb Carbon 14, that is, as soon as it dies. This problem is intended for instructional purposes only. It provides an interesting and important example of mathematical modeling with an exponential function.

- Subject:
- Mathematics
- Functions
- Material Type:
- Activity/Lab
- Provider:
- Illustrative Mathematics
- Provider Set:
- Illustrative Mathematics
- Author:
- Illustrative Mathematics
- Date Added:
- 08/21/2012

This exploratory task requires the student to use a property of exponential functions in order to estimate how much Carbon 14 remains in a preserved plant after different amounts of time.

- Subject:
- Mathematics
- Functions
- Material Type:
- Activity/Lab
- Provider:
- Illustrative Mathematics
- Provider Set:
- Illustrative Mathematics
- Author:
- Illustrative Mathematics
- Date Added:
- 05/01/2012

This simple task assesses whether students can interpret function notation. The four parts of the task provide a logical progression of exercises for advancing understanding of function notation and how to interpret it in terms of a given context.

- Subject:
- Mathematics
- Functions
- Material Type:
- Activity/Lab
- Provider:
- Illustrative Mathematics
- Provider Set:
- Illustrative Mathematics
- Author:
- Illustrative Mathematics
- Date Added:
- 05/01/2012

This task presents a real world situation that can be modeled with a linear function best suited for an instructional context.

- Subject:
- Mathematics
- Functions
- Material Type:
- Activity/Lab
- Provider:
- Illustrative Mathematics
- Provider Set:
- Illustrative Mathematics
- Author:
- Illustrative Mathematics
- Date Added:
- 05/01/2012

This task is intended strictly for instructional purposes with the goal of building understandings of linear relationships within a meaningful and, hopefully, somewhat familiar context.

- Subject:
- Mathematics
- Functions
- Material Type:
- Activity/Lab
- Provider:
- Illustrative Mathematics
- Provider Set:
- Illustrative Mathematics
- Author:
- Illustrative Mathematics
- Date Added:
- 05/01/2012

This lab is an investigation into exponential and logarithmic functions. The lab helps students understand the visual graph of exponential functions and logarithmic functions. It also shows how transformations are shown on a graph as well as the connection between logarithmic and exponential functions. The lab uses desmos.com and takes 30-45 minutes.

- Subject:
- Algebra
- Functions
- Material Type:
- Activity/Lab
- Author:
- Dr. Mary Alice Smeal
- Date Added:
- 06/03/2019

his book is an introduction to combinatorial mathematics, also known as combinatorics. The book focuses especially but not exclusively on the part of combinatorics that mathematicians refer to as “counting.” The book consists almost entirely of problems. Some of the problems are designed to lead you to think about a concept, others are designed to help you figure out a concept and state a theorem about it, while still others ask you to prove the theorem. Other problems give you a chance to use a theorem you have proved. From time to time there is a discussion that pulls together some of the things you have learned or introduces a new idea for you to work with. Many of the problems are designed to build up your intuition for how combinatorial mathematics works. There are problems that some people will solve quickly, and there are problems that will take days of thought for everyone. Probably the best way to use this book is to work on a problem until you feel you are not making progress and then go on to the next one. Think about the problem you couldn’t get as you do other things. The next chance you get, discuss the problem you are stymied on with other members of the class. Often you will all feel you’ve hit dead ends, but when you begin comparing notes and listening carefully to each other, you will see more than one approach to the problem and be able to make some progress. In fact, after comparing notes you may realize that there is more than one way to interpret the problem. In this case your first step should be to think together about what the problem is actually asking you to do. You may have learned in school that for every problem you are given, there is a method that has already been taught to you, and you are supposed to figure out which method applies and apply it. That is not the case here. Based on some simplified examples, you will discover the method for yourself. Later on, you may recognize a pattern that suggests you should try to use this method again.

- Subject:
- Mathematics
- Functions
- Material Type:
- Textbook
- Provider:
- Dartmouth College
- Author:
- Kenneth Bogart
- Date Added:
- 01/01/2004