Project Template

This project will be divided into three sections, and each section is approximately two weeks long and will focus on one of the three functions. At the beginning of each section, students will discover the properties of one of the three functions with different simulation applets on the internet. In class discussions, students can share ideas of what they’ve discovered and the teacher can point out any important properties that they missed. Then, the students will brainstorm on topics for the project as a class and ask questions about how aspects of different professions or events can be represented by a particular function. There will also be a guest speaker for each of the functions to talk their profession and how they use the particular function in their career. Next, the students are asked to divide into groups of four, pick their own unique topic relating to the given function and then using the library and internet they will gather information.

Driving Question

In what aspects of the real world are linear, quadratic and exponential functions used?

Students will be able to make the connection from theoretical to applied mathematics by discovering representations of linear, quadratic and exponential functions in the real world. They will become engage in mathematics and what to extend their knowledge by pursuing mathematics at a higher level. It will also encourage the community to get involved so that they can also make the connections in their lives.

In the past, mathematics has been regarded as a subject composed of just numbers, algorithms and theorems. The majority of students feel that they only have to memorize formulas to pass these mandatory courses in order to graduate. This disconnection between the classroom and the world outside of school has resulted in to our decline, as a nation, of mathematical understanding when compared to other countries around the world. Our traditional way of teaching has also caused women and minorities to become more disillusioned with mathematics and forfeit the idea of pursing mathematics to a higher degree (Boaler, 2002, p. 137-153). Students do not realize that mathematics and mathematical thinking are at the heart of real world situations because they were taught math as discrete subjects that have no relationship to each other or their actual lives.

Our project intends to combat this lack of practical significance so the students can see that application of mathematics in everyday activities. They will experience functions, not just as a separate unit in their journey through mathematics, but as an integral part of actual events and relationships in life. Students will have the opportunity to investigate what they feel is valuable and discover the mathematics behind it, thus engaging their interest and desire to learn. The NCTM also advocates this integrated approach.

“A coherent curriculum effectively organizes and integrates important mathematical ideas so that students can see how the ideas build on, or connect with, other ideas, thus enabling them to develop new understandings and skills…” (NCTM, 15).

As educators, we need to integrate the way in which mathematics represents real life situations into our teaching because students need to have an understanding of the mathematical concepts behind these situations. Our project allows students to do this, therefore, bridging the gap between algorithms and applications. In turn, this will engage students’ interest in mathematics and inspire them to achieve a higher education.

In this project, the students need to see applications of the three types of functions in various situations so the teacher needs to know some of these situations as examples or ideas. The main applications that we are going to use to introduce each of the functions is education and money, rocker launching and savings as examples for linear, quadratic and exponential, respectively. There is a linear relationship in the number of years that one goes to college and the amount of money they make at their job. The path of a projectile, specifically a rocket, traces the path of a quadratic function due to gravity. In finance, saving and investment, with a percent interest rate, have an exponential growth. The more invested early on, the greater the return is of your money in 20 or 30 years.

There are other uses of these functions that relate to things in the real world but the teacher might have to do some further research depending on his/her group of students. Some relationships that we came up with are:
Linear: prices of cars and the type of car, prices of rims and the size of the rim, constant acceleration.
Quadratics: the path of a baseball/basketball/football, wage/earnings
Exponential: growth of bacteria/rabbits, spread of diseases

The teacher must also have a thorough knowledge of Microsoft Excel and PowerPoint. They must also have some prior websites in mind that they can give to the students as a launching pad.

(3) Function concepts. Functions represent the systematic dependence of one quantity on another. Students use functions to represent and model problem situations and to analyze and interpret relationships.
(4) Relationship between equations and functions. Equations arise as a way of asking and answering questions involving functional relationships. Students work in many situations to set up equations and use a variety of methods to solve these equations.

(1) The student understands that a function represents a dependence of one quantity on another and can be described in a variety of ways. Following are performance descriptions.
(A) The student describes independent and dependent quantities in functional relationships.
(B) The student gathers and records data, or uses data sets, to determine functional (systematic) relationships between quantities.
(D) The student represents relationships among quantities using concrete models, tables, graphs, diagrams, verbal descriptions, equations, and inequalities.
(2) The student uses the properties and attributes of functions. Following are performance descriptions.
(A) The student identifies and sketches the general forms of linear (y = x) and quadratic (y = x2) parent functions.
(C) The student interprets situations in terms of given graphs or creates situations that fit given graphs.
(3) The student understands how algebra can be used to express generalizations and recognizes and uses the power of symbols to represent situations. Following are performance descriptions.
(A) The student uses symbols to represent unknowns and variables.
(4) The student understands the importance of the skills required to manipulate symbols in order to solve problems and uses the necessary algebraic skills required to simplify algebraic expressions and solve equations and inequalities in problem situations. Following are performance descriptions.
(A) The student finds specific function values, simplifies polynomial expressions, transforms and solves equations, and factors as necessary in problem situations.

(1) The student understands that linear functions can be represented in different ways and translates among their various representations. Following are performance descriptions.
(A) The student determines whether or not given situations can be represented by linear functions.
(C) The student translates among and uses algebraic, tabular, graphical, or verbal descriptions of linear functions.
(2) The student understands the meaning of the slope and intercepts of linear functions and interprets and describes the effects of changes in parameters of linear functions in real-world and mathematical situations. Following are performance descriptions.
(A) The student develops the concept of slope as rate of change and determines slopes from graphs, tables, and algebraic representations.
(B) The student interprets the meaning of slope and intercepts in situations using data, symbolic representations, or graphs.
(C) The student investigates, describes, and predicts the effects of changes in m and b on the graph of y = mx + b.
(D) The student graphs and writes equations of lines given characteristics such as two points, a point and a slope, or a slope and y-intercept.

(1) The student understands that the graphs of quadratic functions are affected by the parameters of the function and can interpret and describe the effects of changes in the parameters of quadratic functions. Following are performance descriptions.
(A) The student determines the domain and range values for which quadratic functions make sense for given situations.
(B) The student investigates, describes, and predicts the effects of changes in a on the graph of y = ax2.
(C) The student investigates, describes, and predicts the effects of changes in c on the graph of y = x2 + c.
(D) For problem situations, the student analyzes graphs of quadratic functions and draws conclusions.
(2) The student understands there is more than one way to solve a quadratic equation and solves them using appropriate methods. Following are performance descriptions.
(A) The student solves quadratic equations using concrete models, tables, graphs, and algebraic methods.
(B) The student relates the solutions of quadratic equations to the roots of their functions.
(3) The student understands there are situations modeled by functions that are neither linear nor quadratic and models the situations. Following are performance descriptions.
(A) The student uses patterns to generate the laws of exponents and applies them in problem-solving situations.
(B) The student analyzes data and represents situations involving inverse variation using concrete models, tables, graphs, or algebraic methods.
(C) The student analyzes data and represents situations involving exponential growth and decay using concrete models, tables, graphs, or algebraic methods.

Understand patterns, relations, and functions

• Generalize patterns using explicitly defined and recursively defined functions;
• Understand relations and functions and select, convert flexibly among, and use various
representations for them;
• Analyze functions of one variable by investigating rates of change, intercepts, zeros,
asymptotes, and local and global behavior;
• Understand and perform transformations such as arithmetically combining, composing,
and inverting commonly used functions, using technology to perform such
operations on more-complicated symbolic expressions;
• Understand and compare the properties of classes of functions, including exponential,
polynomial, rational, logarithmic, and periodic functions;
• Interpret representations of functions of two variables

Analyze change in various contexts

• Approximate and interpret rates of change from graphical and numerical data.

Basic operations and concepts

Students demonstrate a sound understanding of the nature and operation of technology systems.
Students are proficient in the use of technology.

Social, ethical, and human issues

Students understand the ethical, cultural, and societal issues related to technology.
Students practice responsible use of technology systems, information, and software.
Students develop positive attitudes toward technology uses that support lifelong learning, collaboration, personal pursuits, and productivity.

Technology productivity tools

Students use technology tools to enhance learning, increase productivity, and promote creativity.
Students use productivity tools to collaborate in constructing technology-enhanced models, prepare publications, and produce other creative works.

Technology communications tools

Students use telecommunications to collaborate, publish, and interact with peers, experts, and other audiences.
Students use a variety of media and formats to communicate information and ideas effectively to multiple audiences.

Technology research tools

Students use technology to locate, evaluate, and collect information from a variety of sources.
Students use technology tools to process data and report results.
Students evaluate and select new information resources and technological innovations based on the appropriateness for specific tasks.

Technology problem-solving and decision-making tools

Students use technology resources for solving problems and making informed decisions.
Students employ technology in the development of strategies for solving problems in the real world.

Muddiest Point – To see what the students are unclear about
Concept Map - To see if the students are making connections between the function and their topic
Project Prospectus – So the students can map out how they are going to proceed with the project
Productive Time Logs - So the teacher and the student can see how much work they are getting done while in the classroom
Punctuated Lectures – To see if the students are listening and understanding others presentations
Application Cards – To brainstorm the applications that a function can be used for

Functions in the Real World

Understand patterns, relations, and functions

Analyze change in various contexts