 Secondary Mathematics I

Strand: FUNCTIONS - Building Linear or Exponential Functions (F.BF)

Build a linear or exponential function that models a relationship between two quantities (Standards F.BF.1-2). Build new functions from existing functions (Standard F.BF.3).
• 1,000 is half of 2,000
This real-life modeling task could serve as a summative exercise in which many aspects of students' knowledge of functions are put to work.
• Applications of Quadratic Functions video
This video introduces and explains the topic.
• Building a General Quadratic Function
This task is for instructional purposes only and builds on ''Building an explicit quadratic function.'' First, it is vital that students have worked through ''Building an explicit quadratic function'' before undertaking this task.
• Building a quadratic function from f(x)=x2
This is the first of a series of tasks aiming at understanding the quadratic formula in a geometric way in terms of the graph of a quadratic function. Here the student works with an explicit function and studies the impact of scaling and linear change of variables.
• Building an Explicit Quadratic Function by Composition
This task is intended for instruction and to motivate the task Building a General Quadratic Function. This task assumes that the students are familiar with the process of completing the square.
• Compounding with a 100% Interest Rate
This task provides an approximation, and definition, of e, in the context of more and more frequent compounding of interest in a bank account. This task is preliminary to F-LE Compounding Interest with a 5% Interest Rate which further develops the relationship between e and compound interest.
• Compounding with a 5% Interest Rate
This task develops reasoning behind the general formula for balances under continuously compounded interest. While this task itself specifically addresses the standard (F-BF), building functions from a context, a auxiliary purpose is to introduce and motivate the number e, which plays a significant role in the (F-LE) domain of tasks.
• Crude Oil and Gas Mileage
In this task students are asked to write expressions about the relation to the price of oil and gas mileage.
• Exploring Sinusoidal Functions
This task serves as an introduction to the family of sinusoidal functions. It uses a desmos applet to let students explore the effect of changing the parameters in y=Asin(B(x−h))+k on the graph of the function.
• Exponential Parameters
The task provides a reasonably straight-forward introduction to interpreting the parameters of an exponential function in terms of a modeling context. The task has students both generate an exponential expression from a contextual description, and in reverse, interpret parameters in a context from an algebraic expression.
• Flu on Campus
The purpose of this problem is to have students compose functions using tables of values only. Students are asked to consider the meaning of the composition of functions to solidify the concept that the domain of g contains the range of f.
• Function Flyer
The applet on this site allows the students to manipulate the graph of a function by changing the value of exponents, coefficients and constants.
• Function Matching
In this student interactive, from Illuminations, students demonstrate their understanding of function expressions by matching a function graph to a generated graph. Choose from several function types or select random and let the computer choose.
• FUNCTIONS - Building Linear or Exponential Functions (F.BF) - Sec Math I Core Guide
The Utah State Board of Education (USBE) and educators around the state of Utah developed these guides for the Secondary Mathematics I - Building Linear or Exponential Functions (F.BF)
• Generalizing Patterns: Table Tiles
This lesson unit is intended to help educators assess how well students are able to identify linear and quadratic relationships in a realistic context: the number of tiles of different types that are needed for a range of square tabletops.
• Graphit
With this interactive applet students are able to create graphs of functions and sets of ordered pairs on the same coordinate plane.
• Graphs of Compositions
This task addresses an important issue about inverse functions. In this case the function f is the inverse of the function g but g is not the inverse of f unless the domain of f is restricted.
• Identifying Even and Odd Functions
This task includes an experimental GeoGebra worksheet, with the intent that instructors might use it to more interactively demonstrate the relevant content material.
• Identifying Quadratic Functions (Vertex Form)
This task has students explore the relationship between the three parameters a, h, and k in the equation f(x)=a(xh)2+k and the resulting graph.
• Inductive Patterns
This collection of resources to teach graphing equations in slope intercept form includes warm-up exercises, a video presentation explaining the topic, practice exercises, worked examples, practice problems, and a review.
• Inductive Reasoning
This collection of resources to teach graphing equations in slope intercept form includes warm-up exercises, a video presentation explaining the topic, practice exercises, worked examples, practice problems, and a review.
• Inductive Reasoning video
This video introduces and explains the topic.
• Introduction to the Materials (Math 1)
Introduction to the Materials in the Mathematics One of the The MVP classroom experience begins by confronting students with an engaging task and then invites them to grapple with solving it. As students ideas emerge, take form, and are shared, the teacher orchestrates the student discussions and explorations towards a focused mathematical goal. As conjectures are made and explored, they evolve into mathematical concepts that the community of learners begins to embrace as effective strategies for analyzing and solving problems.
• Kimi and Jordan
In the middle grades, students have lots of experience analyzing and comparing linear functions using graphs, tables, symbolic expressions, and verbal descriptions. In this task, students may choose a representation that suits them and then reason from within that representation. When used in instruction, this task provides opportunities to compare representations and to make connections among them.
• Lake Algae
The purpose of this task is to introduce students to exponential growth. While the context presents a classic example of exponential growth, it approaches it from a non-standard point of view. Instead of giving a starting value and asking for subsequent values, it gives an end value and asks about what happened in the past.
• Medieval Archer
This task addresses the first part of standard F-BF.3: “Identify the effect on the graph of replacing f(x) by f(x)+k, kf(x), f(kx), and f(x+k) for specific values of k (both positive and negative).” Here, students are required to understand the effect of replacing x with x+k, but this task can also be modified to test or teach function-building skills involving f(x)+k, kf(x), and f(kx) in a similar manner.
• Module 1: Sequences - Student Edition (Math 1)
The Mathematics Vision Project, Secondary Math One Module 1, Sequences is written as two intertwined learning cycles that begin by alternating from arithmetic sequence to geometric sequences, so students can compare and contrast features as they represent both types of sequences with tables, graphs, story contexts, diagrams, and equations.
• Module 1: Sequences - Teacher Notes (Math 1)
The Mathematics Vision Project, Secondary Math One Module 1 Teacher Notes, Sequences is written as two intertwined learning cycles that begin by alternating from arithmetic sequence to geometric sequences, so students can compare and contrast features as they represent both types of sequences with tables, graphs, story contexts, diagrams, and equations.
• Module 2: Linear & Exponential Functions - Student Edition (Math 1)
The Mathematics Vision Project, Secondary Math One Module 2, Linear and Exponential Functions, begins with a learning cycle that introduces contexts with continuous domains and defining linear functions as having a constant rate of change and exponential functions as having a constant ratio over equal intervals.
• Module 2: Linear & Exponential Functions - Teacher Notes (Math 1)
Mathematics Vision Project, Secondary Math One Module 2 Teacher Notes, Linear and Exponential Functions, begins with a learning cycle that introduces contexts with continuous domains and defining linear functions as having a constant rate of change and exponential functions as having a constant ratio over equal intervals.
• Module 3: Features of Functions - Student Edition (Math 1)
The Mathematics Vision Project, Secondary Math One Module 3, Features of Functions, is the culminating functions module in Secondary Math I. In this module, students broaden their thinking about functions to relationships that are not either linear or exponential.
• Module 8: Connecting Algebra & Geometry - Student Edition (Math 1)
The Mathematics Vision Project, Secondary Math One Module 8, Connecting Algebra and Geometry, students use the Pythagorean Theorem to find the distance between two points and to derive the distance formula.
• Module 8: Connecting Algebra & Geometry - Teacher Notes (Math 1)
The Mathematics Vision Project, Secondary Math One Module 8 Teacher Notes, Connecting Algebra and Geometry, students use the Pythagorean Theorem to find the distance between two points and to derive the distance formula.
• Multi-Function Data Flyer
The applet in this lesson allows students to plot ordered pairs and then change the values in order to observe the effects of those changes.
• Polynomials
This collection of resources to teach graphing equations in slope intercept form includes warm-up exercises, a video presentation explaining the topic, practice exercises, worked examples, practice problems, and a review.
• Polynomials video
This video introduces and explains the topic.
• Representing Patterns
This collection of resources to teach graphing equations in slope intercept form includes warm-up exercises, a video presentation explaining the topic, practice exercises, worked examples, practice problems, and a review.
• Representing Patterns video
This video introduces tables and graphs as representations of patterns.
• Representing Polynomials
This lesson unit is intended to help educators assess how well students are able to translate between graphs and algebraic representations of polynomials.
This is a two-part assessment task. Part A asks students to interpret the symbolic representation of a transformation on a function. Part B is about connecting the representations of a function. Use the navigation at the upper right of this page to access the task.
• Secondary I Textbook
Secondary I Textbook is composed of modules that are aligned with the Utah Core State Standards for Mathematics. Each lesson begins with a worthwhile task that has been designed to develop mathematical understanding, solidify that understanding, or allow for practice of the new concepts, while focusing on the mathematical goals of the chosen learning cycle.
• Skeleton Tower
This problem is a quadratic function example.
Susie is organizing the printing of tickets for a show. She has collected prices from several printers. The student's task is to use graphs and algebra to advise Susie on how to choose the best printer.
Kate makes patchwork cushions using right triangles made from squares of material. In this task, students must investigate number patterns and to find a rule, or a formula, that will help Kate figure out the number of squares she needs for cushions of different sizes.
Susie is organizing the printing of tickets for a show. She has collected prices from several printers. The student's task is to use graphs and algebra to advise Susie on how to choose the best printer.
In this task, students will look for rules which let you work out how many blocks of different colors are needed to make different sized patterns.
In this task, students will look for rules which let them work out how many blocks of different colors are needed to make different sized patterns.
In this task, students must work out a rule for calculating the total number of cubes needed to build towers of different heights.
In this task, students must work out how many whole, half and quarter tiles tiles are needed to cover the tops of tables of different sizes.
• Sum of Functions
The intent of this problem is to have students think about how function addition works on a fundamental level, so formulas have been omitted on purpose. Although this problem does not ask students to "write a function that describes a relationship between two quantities", it can provide students with understandings preparatory for F.BF.1b.
• Summer Intern
Students are given the following task and asked to write an expression. "You have been hired for a summer internship at a marine life aquarium. Part of your job is diluting brine for the saltwater fish tanks. The brine is composed of water and sea salt, and the salt concentration is 15.8% by mass, meaning that in any amount of brine the mass of salt is 15.8% of the total mass."
• Susita's Account
This task asks students to determine a recursive process from a context. Students who study computer programming will make regular use of recursive processes.
• Temperature Conversions
Unit conversion problems provide a rich source of examples both for composition of functions (when several successive conversions are required) and inverses (units can always be converted in either of two directions).
• The Canoe Trip, Variation 1
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.
• The Canoe Trip, Variation 2
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.
• Tidal Waves (pdf)
Students analyze a problem faced by the captain of a shipping vessel. Students may use a range of functions to model the situation and reflect on their usefulness. Because trigonometric functions can be useful, this task would be particularly appropriate for students who have had an introduction to graphing sine and cosine functions.
• Transforming the graph of a function
Like "Building functions: concrete case'' this task examines, in a graphical setting, the impact of adding a scalar, multiplying by a scalar, and making a linear substitution of variables on the graph of a function f. The setting here is abstract as there is no formula for the function f. The focus is therefore on understanding the geometric impact of these three operations.
• Writing Expressions and Equations video
How to write an equation using what we know to solve a problem we don't know. http://www.uen.org - in partnership with Utah State Board of Education (USBE) and Utah System of Higher Education (USHE).  Send questions or comments to USBE Specialist - Lindsey  Henderson and see the Mathematics - Secondary website. For general questions about Utah's Core Standards contact the Director - Jennifer  Throndsen.

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