Secondary Mathematics I

Strand: FUNCTIONS - Linear and Exponential (F.LE)

Construct and compare linear and exponential models and solve problems (Standards F.LE.1- 3). Interpret expressions for functions in terms of the situation they model. (Standard F.LE.5).

Standard F.LE.1

Distinguish between situations that can be modeled with linear functions and with exponential functions.

• Algae Blooms
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). The intent of part (b) is for students to gain an appreciation for the exponential growth exhibited despite an apparently modest growth rate of 1 cell division per day.
• ALGEBRA - Seeing Structure in Expressions (A.SSE) - 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 - Seeing Structure in Expressions (A.SSE)
• Basketball Bounces, Assessment Variation 1
This task asks students to analyze a set of data from a physical context, choose a model that reasonably fits the data, and use the model to answer questions about the physical context.
• Basketball Bounces, Assessment Variation 2
This task asks students to analyze a set of data from a physical context, choose a model that reasonably fits the data, and use the model to answer questions about the physical context.
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.
• Boiling Water
This task examines linear models for the boiling point of water as a function of elevation. Two sets of data are provided and each is modeled quite well by a linear function.
• Carbon 14 Dating, Variation 2
This exploratory task requires the student to use this property of exponential functions in order to estimate how much Carbon 14 remains in a preserved plant after different amounts of time.
• Choosing an appropriate growth model
The goal of this task is to examine some population data from a modeling perspective. Because large urban centers and their growth are governed by many complex factors, we cannot expect a simple model (linear, quadratic, or exponential) to give accurate values or predictions over large stretches of time. Deciding on an appropriate model is a delicate process requiring careful analysis.
• Comparing Exponentials
This task gives students an opportunity to work with exponential functions in a real world context involving continuously compounded interest. They will study how the base of the exponential function impacts its growth rate and use logarithms to solve exponential equations.
• Comparing Graphs of Functions
The goal of this task is to use appropriate tools to compare graphs of several functions. In addition, students are asked to study the structure of the different expressions to explain why these functions grow as they do.
• Dido and the Foundation of Carthage
The goal of this task is to interpret the mathematics behind a famous story from ancient mythology, giving rise to linear and quadratic expressions which model the story.
• Equal Differences over Equal Intervals 1
Students prove that linear functions grow by equal differences over equal intervals. They will prove this for equal intervals of length one unit, and note that in this case the equal differences have the same value as the slope.
• Equal Differences over Equal Intervals 2
Linear functions grow by equal differences over equal intervals. In this task students prove the property in general (for equal intervals of any length).
• Equal Factors over Equal Intervals
Examples in this task is designed to help students become familiar with this language "successive quotient". Depending on the students's prior exposure to exponential functions and their growth rates, instructors may wish to encourage students to repeat part (b) for a variety of exponential functions and step sizes before proceeding to the most general algebraic setting in part (c).
• Exploring Linear Functions: Representational Relationships
This lesson plan helps students better understand linear functions by allowing them to manipulate values and get a visual representation of the result.
• Exponential Functions
This task requires students to use the fact that the value of an exponential function f(x)=a⋅bx increases by a multiplicative factor of b when x increases by one. It intentionally omits specific values for c and d in order to encourage students to use this fact instead of computing the point of intersection, (p,q), and then computing function values to answer the question. This task is preparatory for standard F.LE.1a.
• Extending the Definitions of Exponents, Variation 2
The goal of this task is to develop an understanding of why rational exponents are defined as they are (N-RN.1), however it also raises important issues about distinguishing between linear and exponential behavior (F-LE.1c) and it requires students to create an equation to model a context (A-CED.2)
• Finding Linear and Exponential Models
The goal of this task is to present students with real world and mathematical situations which can be modeled with linear, exponential, or other familiar functions. In each case, the scenario is presented and students must decide which model is appropriate.
• FUNCTIONS - Linear and Exponential (F.LE) - 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 - Linear and Exponential (F.LE).
• Graphs and Functions
This lesson plan is designed to help the student understand how to plot functions on the Cartesian plane and how the graphing of functions leads to lines and parabolas.
• Identifying Functions
This task emphasizes the expectation that students know linear functions grow by constant differences over equal intervals and exponential functions grow by constant factors over equal intervals.
• Illegal Fish
This task asks students to interpret the relevant parameters in terms of the real-world context and describe exponential growth.
• In the Billions and Exponential Modeling
This problem provides an opportunity to experiment with modeling real data.
• In The Billions and Linear Modeling
This problem assumes students have completed several preliminary tasks about the fact that linear functions change by equal differences over equal intervals.
• Interesting Interest Rates
Given two bank interest rate scenarios, students will compare returns, write an expression for a balance, and create a table of values for the balances.
• 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.
• Lesson Starter: Populated Communities
Students will use statistics and probability knowledge, as well as critical thinking skills, to solve problems.
• Linear Functions
This task requires students to use the fact that on the graph of the linear function h(x)=ax+b, the y-coordinate increases by a when x increases by one. Specific values for a and b were left out intentionally to encourage students to use the above fact as opposed to computing the point of intersection, (p,q), and then computing respective function values to answer the question.
• Linear or exponential?
This task gives a variet of real-life contexts which could be modeled by a linear or exponential function. The key distinguishing feature between the two is whether the change by equal factors over equal intervals (exponential functions), or by a constant increase per unit interval (linear functions). The task could either be used as an assessment problem on this distinction, or used as an introduction to the differences between these very important classes of functions.
• Modeling: Having Kittens
This lesson unit is intended to help you assess how well students are able to interpret a situation and represent the constraints and variables mathematically, select appropriate mathematical methods to use, make sensible estimates and assumptions and investigate an exponentially increasing sequence.
• 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.
• More Complicated Functions: Introduction to Linear Functions
This lesson is designed to reinforce the concept of linear functions and ask students to write functions using English, tables and algebra.
• Perpendicular Lines video
This video introduces and explains perpendicular lines.
• Rising Gas Prices Compounding and Inflation
The purpose of this task is to give students an opportunity to explore various aspects of exponential models (e.g., distinguishing between constant absolute growth and constant relative growth, solving equations using logarithms, applying compound interest formulas) in the context of a real world problem with ties to developing financial literacy skills.
• Solving Problems with Linear and Exponential Models
The goal of this task is to provide examples of exponential and linear functions modeling different real world phenomena. Students will create the appropriate model and then use it to solve linear and exponential equations.
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.
• Triangular Numbers
The goal of this task is to work on producing a quadratic equation from an arithmetic context.
• US Population 1790-1860
This task focuses on the fact that exponential functions are characterized by equal successive quotients over equal intervals. This task can be used alongside F-LE Equal Factors over Equal Intervals.
• US Population 1982-1988
This task provides a preliminary investigation of mathematical modeling using linear functions. In particular, students are asked to make predictions using a linear model without ever writing down an equation for a line. As such, the task could be used to motivate or introduce the observation that linear functions are precisely those that change by constant differences over equal intervals.
• What functions do two graph points determine?
Given two points on a plane, students will demonstrate an understanding of unique linear function, unique exponential function, and quadratic function.

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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