- MAFS.912.F-BF.1.1: Write a function that describes a relationship between two quantities. ★
- Determine an explicit expression, a recursive process, or steps for calculation from a context.
- Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model.
- Compose functions. For example, if T(y) is the temperature in the atmosphere as a function of height, and h(t) is the height of a weather balloon as a function of time, then T(h(t)) is the temperature at the location of the weather balloon as a function of time.
- MAFS.912.F-BF.1.2: Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms. ★
|
Generated on 6/11/2026 at 9:48 PM |
The webpage this document was printed/exported from can be found at the following URL:
https://staging.cpalms.org//PreviewIdea/Preview/1530
https://staging.cpalms.org//PreviewIdea/Preview/1530
Standard 1: Build a function that models a relationship between two quantities. (Algebra 1 - Supporting Cluster) (Algebra 2 - Major Cluster) Archived
Clusters should not be sorted from Major to Supporting and then taught in that order. To do so would strip the coherence of the mathematical ideas and miss the opportunity to enhance the major work of the grade with the supporting clusters.
Cluster Standards
This cluster includes the following benchmarks.
Visit the specific benchmark webpage to find related instructional resources.
Cluster Information
Number: MAFS.912.F-BF.1
Title: Build a function that models a relationship between two quantities. (Algebra 1 - Supporting Cluster) (Algebra 2 - Major Cluster)
Type: Cluster
Subject: Mathematics - Archived
Grade: 912
Cluster: Functions: Building Functions
Cluster Access Points
This cluster includes the following Access Points.
- MAFS.912.F-BF.1.AP.1a: Select a function that describes a relationship between two quantities (e.g., relationship between inches and centimeters, Celsius Fahrenheit, distance = rate x time, recipe for peanut butter and jelly- relationship of peanut butter to jelly f(x)=2x, where x is the quantity of jelly, and f(x) is peanut butter.
- MAFS.912.F-BF.1.AP.2a: Write arithmetic sequences with an explicit formula.
- MAFS.912.F-BF.1.AP.2b: Select the function that models the arithmetic sequence written recursively.
- MAFS.912.F-BF.1.AP.2c: Write geometric sequences with an explicit formula
. - MAFS.912.F-BF.1.AP.2d: Select the function that models the geometric sequence written recursively.
Cluster Resources
Vetted resources educators can use to teach the concepts and skills in this topic.
Formative Assessments
- Furniture Purchase: Students are asked to write two explicit functions given verbal descriptions in a real-world context, compose the two functions, and explain the meaning in context.
- How Much Bacteria?: Students are asked to write and combine an exponential and a constant function from a verbal description to use when answering a related context question.
- Giveaway: Students are asked to write an explicit function rule given a verbal description of a functional relationship in a real-world context and are then asked to use the function rule to answer a question.
- Saving for a Car: Students are asked to write an explicit function rule given a verbal description of a functional relationship in a real-world context and are then asked to use the function rule to answer specific questions.
Lesson Plans
- Modeling: Rolling Cups: This lesson unit is intended to help you assess how well students are able to choose appropriate mathematics to solve a non-routine problem, generate useful data by systematically controlling variables and develop experimental and analytical models of a physical situation.
- Generalizing Patterns: Table Tiles: This enrichment lesson assesses how well students can 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. In particular, this lesson aims to identify and help students who have difficulties with choosing an appropriate, systematic way to collect and organize data, examining the data and looking for patterns; finding invariance and covariance in the numbers of different types of tile, generalizing using numerical, geometrical, or algebraic structure, and describing and explaining findings clearly and effectively.
- Movie Theater MEA: In this Model Eliciting Activity, MEA, students create a plan for a movie theater to stay in business. Data is provided for students to determine the best film to show, and then based on that decision, create a model of ideal sales. Students will create equations and graph them to visually represent the relationships.
Model Eliciting Activities, MEAs, are open-ended, interdisciplinary problem-solving activities that are meant to reveal students’ thinking about the concepts embedded in realistic situations. Click here to learn more about MEAs and how they can transform your classroom.
- Plants versus Pollutants Model Eliciting Activity: The Plants versus Pollutants MEA provides students with an open-ended problem in which they must work as a team to design a procedure to select the best plants to clean up certain toxins. This MEA requires students to formulate a phytoremediation-based solution to a problem involving cleaning of a contaminated land site. Students are provided the context of the problem, a request letter from a client asking them to provide a recommendation, and data relevant to the situation. Students utilize the data to create a defensible model solution to present to the client.
Model Eliciting Activities, MEAs, are open-ended, interdisciplinary problem-solving activities that are meant to reveal students’ thinking about the concepts embedded in realistic situations. Click here to learn more about MEAs and how they can transform your classroom.
Problem-Solving Tasks
- Temperatures in Degrees Fahrenheit and Celsius: Temperature conversions provide a rich source of linear functions which are encountered not only in science but also in our every day lives when we travel abroad. The first part of this task provides an opportunity to construct a linear function given two input-output pairs. The second part investigates the inverse of a linear function while the third part requires reasoning about quantities and/or solving a linear equation.
- Compounding with a 100% Interest Rate: This task leads to an approximation and definition of the irrational number e, using the context of examining more and more frequent compounding of interest in a bank account. The approach is computational.
- A Sum of Functions: In this example, students are given the graph of two functions and are asked to sketch the graph of the function that is their sum. The intent is that students develop a conceptual understanding of function addition. Multiple solution methods are provided.
- 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.
- Crude Oil and Gas Mileage: This task introduces students to composite functions. First, students write simple expressions for problems involving distance per unit of volume, and then students compose the related functions to answer a question based on a real-life event.
- Flu on Campus: The context of this example is the spread of a flu virus on campus and the related sale of tissue boxes sold. Students interpret the composite function and determine values simply by using the tables of values.
- Compounding with a 5% Interest Rate: This task develops the reasoning behind the general formula for balances under continuously compounded interest. While this task itself specifically addresses building functions from a context, an auxiliary purpose is to introduce and motivate the number e.
- Temperature Conversions: This task explores function composition in the context of temperature conversions. Students interpret what x, f(g(x)), and g(f(x)) represent, determine whether these compositions have meaningful real-world interpretations, compute f(g(x)) using given formulas, and finally derive a new function h that converts Fahrenheit to Kelvin.
- Susita's Account: This task models a real-world financial scenario involving exponential growth with a constant subtraction. Students write a recursive equation for Susita’s account balance and use it to determine her initial amount based on a given condition.
- Summer Intern: This task engages students in modeling a real-world situation involving salt concentration and dilution. It integrates algebraic reasoning, function notation, and graph interpretation in a practical context. Students will use proportions of mass and volume to create ideal brine for saltwater fish tanks. It also asks students to compare graphs.
- Skeleton Tower: Students explore a quadratic relationship derived from an arithmetic sequence by analyzing a patterned tower built with cubes. They calculate the number of cubes in one tower, extend the pattern to 12 iterations, and then generalize the relationship for any number of iterations, n, using an algebraic expression.
- 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.
- The Canoe Trip, Variation 2: This task leads 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 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 the context.
Professional Development
- 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.
Tutorials
- Geometric sequence or progression: We will learn how to write a geometric sequence.
- Geometric series: Geometric series
- Finding the nth term in a recursively defined sequence: Finding the 4th term in recursively defined sequence
Video/Audio/Animation
- Basic Linear Function: This video demonstrates writing a function that represents a real-life scenario.