introduction to modeling with functions assignment

• Contributors
• 1: Using IM Algebra
• 2: Frequently Asked Questions
• 1: Curriculum Components
• 2: Instructional Routines
• 2A: Contemplate then Calculate
• 2B: Connecting Representations
• 2C: Group Learning Routines
• 2D: Additional Instructional Routines
• 3: ELL and SpEd Student Support
• A1 U0: Introduction to Algebra I

A1 U1: Modeling with Functions

• A1 U2: Linear and Exponential Functions
• A1 U3: Linear Equations and Inequalities in One Variable
• A1 U4: Linear Equations and Inequalities in Two Variables
• A1 U5: Quadratic Functions
• A1 U6: Quadratic Equations
• A1 U7: Statistics
• Geo U0: Introduction to Geometry
• Geo U1: Tools of Geometry
• Geo U2: Proofs about Congruence
• Geo U3: Similarity and Proof
• Geo U4: Right Triangle Trigonometry
• Geo U5: Extending to Three Dimensions
• Geo U6: Coordinate Geometry
• Geo U6: Circles
• A2 U0: Introduction to Algebra II
• A2 U1: Families of Functions
• A2 U2: Exponential Functions
• A2 U3: Trigonometric Functions
• A2 U4: Rational and Polynomial Functions
• A2 U5: Probability
• A2 U6: Statistics (Inferences from Data)
• Resource: Quiz Banker
• Resource: Re-engagement
• Resource: Formative Assessment Lessons
• Resource: Cognitive Science
• Resource: Data Visualizations
• Resource: Math Stations
• Find Resources

Modeling with Functions

Understand the concept of a function and use function notation. Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If  f  is a function and  x  is an element of its domain, then  f ( x ) denotes the output of  f  corresponding to the input  x . The graph of  f  is the graph of the equation  y  =  f ( x ).

Understand the concept of a function and use function notation. Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.

Interpret functions that arise in applications in terms of the context.  For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.

Interpret functions that arise in applications in terms of the context. Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.  For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.

Interpret functions that arise in applications in terms of the context. Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.

Interpret linear models. Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.

Reason quantitatively and use units to solve problems.  Use units as a way to understand problems and to guide the solution of multi‐step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.

Reason quantitatively and use units to solve problems. Define appropriate quantities for the purpose of descriptive modeling.

Reason quantitatively and use units to solve problems.  Choose a level of accuracy appropriate to limitations on measurement when reporting quantities. The greatest precision for a result is only at the level of the least precise data point (example: if units are tenths and hundredths, then the appropriate level of precision is tenths). 

Analyze functions using different representations. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum.

Construct and compare linear, quadratic, and exponential models and solve problems. Distinguish between situations that can be modeled with linear functions and with exponential functions.

Construct and compare linear, quadratic, and exponential models and solve problems. Recognize situations in which one quantity changes at a constant rate per unit interval relative to another.

Construct and compare linear, quadratic, and exponential models and solve problems.   Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly,  quadratically, or (more generally) as a polynomial function.

Interpret expressions for functions in terms of the situation they model. Interpret the parameters in a linear or exponential function in terms of a context.

Build new functions from existing functions Identify the effect on the graph of replacing f ( x ) by f ( x ) + k , k f ( x ), f ( kx ), and f ( x + k ) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.

Purpose Statement for the Revision of Instructional Materials SY2020-21

Unit 1 Revision for Blended Instruction SY2020-21

Unit 1 Folder of Student Materials for SY2020-21

Spiraled Practice Problems for Unit 1

The resources linked below have NOT been modified for use in 2020-21, but they are available as a resource for teachers who are accustomed to the site. See the resources at the top of this unit page for guidance for 2020-21.

Note:  The previous promo links to the  Teacher Planning Notes for Unit 1 (PDF) ,  All Student Materials for Unit 1 (PDF) , and Printable from Unit 1 Activities (PDF)  are from SY2019-20. These files are included to make printing easier. However, the links are not live in this format. For the most updated version of materials and working links, scroll down to the Big Ideas and open the Google Doc versions, which are updated continuously.

Unit Resources:

Initial task see 2 items hide 2 items.

The Initial Task for a unit is intended to both preview the upcoming mathematics for a student and help teachers see how their students understand the mathematics prior to the unit.

Algebra I Archive

Initial Task - 200 Freestyle: 200 Freestyle - Teacher Materials

This page contains instructions on how to use the initial task, 200 Freestyle, to find out what your students already know about reading graphs.

Teacher Feedback

Please comment below with questions, feedback, suggestions, or descriptions of your experience using this resource with students.

Initial Task - 200 Freestyle: 200 Freestyle - Student Materials

Big Idea 1: Rate of change describes how one quantity changes with respect to another. See 2 items Hide 2 items

Big Idea 1: Rate of change describes how one quantity changes with respect to another.: Core Resource: Sorting Sequences

• Supplementary Resources for Big Idea 1 1 week

Big Idea 2: A graph is a visual representation of the relationship between variables of a function. See 2 items Hide 2 items

Big Idea 2: A graph is a visual representation of the relationship between variables of a function.: Core Resource: Studying Situations With Graphs

• Supplementary Resources for Big Idea 2 1 week

Big Idea 3: Functions can be represented in multiple, equivalent ways. See 2 items Hide 2 items

Big Idea 3: Functions can be represented in multiple, equivalent ways.: Core Resource: Function Features

• Supplementary Resources for Big Idea 3 1 week

Big Idea 4: Functions belonging to the same family share similar graphs, behaviors, and characteristics. See 2 items Hide 2 items

Big Idea 4: Functions belonging to the same family share similar graphs, behaviors, and characteristics.: Core Resource: Function Family Features

• Supplementary Resources for Big Idea 4 1 week

Formative Assessment Lesson See 2 items Hide 2 items

A Formative Assessment Lesson (also known as a Classroom Challenge) is a carefully designed lesson that both supports teachers in understanding how students make sense of the unit's mathematics and offers students opportunities to revisit and deepen their understanding of that mathematics.

Formative Assessment Lesson: Functions and Situations

A Classroom Challenge ( aka formative assessment lesson) is a classroom-ready lesson that supports formative assessment. The lesson’s approach first allows students to demonstrate their prior understandings and abilities in employing the mathematical practices, and then involves students in resolving their own difficulties and misconceptions through structured discussion.

Formative Assessment Lesson: Supporting Materials for Functions and Situations

Re-engagement See 1 item Hide 1 item

Re-engagement means going back to a familiar problem or task and looking at it again in different ways, with a new lens, or going deeper into the mathematics. This is often done by showing examples of student work and providing prompts to help students think about the mathematical ideas differently.  This guide  provides more information on how to design re-engagement lessons for your students which you can use at any time during a unit where you think it will be helpful for students to revisit a specific mathematical idea before moving on.

Re-engagement: Re-engagement for Unit 1

Re-engagement means going back to a familiar problem or task and looking at it again in different ways, with a new lens, or going deeper into the mathematics. This is often done by showing examples of student work and providing prompts to help students think about the mathematical ideas differently.  This guide  provides more information on how to design re-engagement lessons for your students, which you can use at any time during a unit, where you think it will be helpful for students to revisit a specific mathematical idea before moving on.

End of Unit Assessment See 4 items Hide 4 items

The End of Unit Assessment is intended to surface how students understand the mathematics in relation to the end of year goal of a Regents examination. To support retention, the end of unit assessments are intentionally designed with spiralled questions from previous units.

End of Unit Assessment: End of Unit Assessment for Unit 1

After this unit, how prepared are your students for the end-of-course Regents examination?  The end of unit assessment is designed to surface how students understand the mathematics in the unit.  It includes spiralled multiple choice and constructed response questions, comparable to those on the end-of-course Regents examination.  A rich task, that allows for multiple entry points and authentic assessment of student learning, may be available for some units and can be included as part of the end of unit assessment.  All elements of the end of unit assessment are aligned to the NYS Mathematics Learning Standards and PARCC Model Frameworks prioritization. 

This Performance Task can be used as the Part IV item of the Algebra Unit 1 End of Unit Assessment. Bike Ride can provide additional information about how students apply their understanding of reading graphs and interpreting its meaning.

Quiz Banker creates student-ready editable quiz and answer documents based on an item bank of over 2500 NY state exam questions.

This link takes you to the New Visions Cloudlab, where you can get the Quiz Banker add-on and watch videos about how to use this Google Sheets tool (note: you need to add Quiz Banker to your Google tools once; if you have previously installed the add-on, it appears in the "Add-ons" menu in Google Sheets).

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• Modeling with Functions

Key Questions

If a value changes based on its relationship to some other value, then a function can (often) be used to model the change in that primary value.

Some examples might help.

Example 1 Suppose you purchase a $1000 Guaranteed Investment Certificate with an annual (compounded) return rate of 3% and you want to know the value of that GIC (after having held it for some number of years).

We can model the value of your GIC using a function: #v(t) = 1000 xx (1 + 0.03)^t#

(where #t# is the number of years you have owned the GIC). Even if you put in some value for #t# , the function is not really a monetary value (you can't buy anything with the number that comes out of the function, but it reflects or " models " that value.

Example 2 If you drop an object the speed at which that object is falling (ignoring air resistance and assuming it doesn't hit something else) changes with the distance that the object has already fallen.

Specifically, if the distance is measured in feet and the speed in feet per second , then we could model the speed of our falling object dependent upon the distance it has fallen by a function: #s(d) = 2 xx d#

#2 xx d# is just a number, but it is a number we can think of as being related to a specific speed. In that sense, #s(d)# is a function that models speed depending upon distance ( #d# ).