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Growth and Decay *

F-LE.1, F-LE.2, F-LE.3, F-LE.5, F-LE.6, F-BF.1, F-BF.2,F-BF.3, F-BF.4, F-IF.4, F-IF.5, F-IF.9, NQ.1, A-SSE.1: Investigate situations that involve linear, quadratic, and exponential models, and use these models to solve problems. Recognize linear functions grow by equal differences over equal intervals; exponential functions grow by equal factors over equal intervals, and functions grow or decay by a percentage rate per unit interval. Interpret the inverse of functions, and model the inverse in graphs, tables, and equations. (CDE 2023, 42)

California Department of Education. 2023. Mathematics Framework Chapter 8. Sacramento, CA: California Department of Education.

Big Idea Success Criteria

The categories and their related standards below unpack the success criteria of this big idea.

Primary Standards

  • Construct and compare linear, quadratic, and exponential models and solve problems. (F-LE.1-6) (CDE 2013, 101)
    • F-LE.1. Distinguish between situations that can be modeled with linear functions and with exponential functions.
      1. Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals.
      2. Recognize situations in which one quantity changes at a constant rate per unit interval relative to another.
      3. Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another (CDE 2013, 92).
    • F-LE.2. Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table) (CDE 2013, 93).
    • F-LE.3. Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function (CDE 2013, 93).
    • F-LE.5. Interpret the parameters in a linear or exponential function in terms of a context. [Linear and exponential of form f(x) = bx + k] (CDE 2013, 93)
    • F-LE.6. Apply quadratic functions to physical problems, such as the motion of an object under the force of gravity. CA (CDE 2013, 101)
  • Build a function that models a relationship between two quantities. [For F-BF.1–2, linear, exponential, and quadratic] (F-BF.1–2) (CDE 2013, 92)
    • F-BF.1. Write a function that describes a relationship between two quantities.
      1. Determine an explicit expression, a recursive process, or steps for calculation from a context.
      2. 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 (CDE 2013, 92).
    • F-BF.2. Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms (CDE 2013, 92).
  • Build new functions from existing functions. [Linear, exponential, quadratic, and absolute value*; for F-BF.4a, linear only] (F-BF.3–4) (CDE 2013, 67)
    • 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); 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 (CDE 2013, 67)
    • F-BF.4. Find inverse functions.
      1. Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse (CDE 2013, 67).
  • Interpret functions that arise in applications in terms of the context. [Linear, exponential, and quadratic] (F-IF.4,5) (CDE 2013, 91)
    • F-IF.4. 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 (CDE 2013, 91).
    • F-IF.5. Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h 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 (CDE 2013, 92).
  • Analyze functions using different representations. [Linear, exponential, quadratic, absolute value*, step, piecewise-defined] (F-IF.9) (CDE 2013, 92)
    • F-IF.9. 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 (CDE 2013, 92).
  • Reason quantitatively and use units to solve problems. [Foundation for work with expressions, equations and functions] (N-Q.1) (CDE 2013, 121)
    • N-Q.1. 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 (CDE 2013, 121).
  • Interpret the structure of expressions. [Linear, exponential, and quadratic] (A-SSE.1) (CDE 2013, 98)
    • A-SSE.1. Interpret expressions that represent a quantity in terms of its context.
      1. Interpret parts of an expression, such as terms, factors, and coefficients.
      2. Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret P(1 + r)n as the product of P and a factor not depending on P (CDE 2013, 98).

*Note: The Big Idea of Growth and Decay in Algebra 1 focuses on linear, quadratic, and exponential functions.

Primary Standards

  • Build new functions from existing functions. [Linear, exponential, quadratic, and absolute value*; for F-BF.4a, linear only] (F-BF.4)(CDE 2013, 92)
    • F-BF.4. Find inverse functions.
      1. Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse (CDE 2013, 101).

*Note: The Big Idea of Growth and Decay in Algebra 1 focuses on linear, quadratic, and exponential functions.

California Department of Education. 2013. California Common Core State Standards. Sacramento, CA: California Department of Education.

Alternative Means of Expression

The following options give educators and IEP teams viable alternative means of expression a student could use when showing their understanding of this big idea. Much of the initiative team’s approach to identifying options centered on developing or adopting performance tasks to show what assessment might look like for this big idea.

Remember, LEAs adopt their own policies related to how a student meets the requirements for graduation. Educators and IEP teams should explore these resources with knowledge of these local policies.

General Guidance with Selecting Options for this Big Idea

For Category 1 of Growth and Decay, students can express models using visual/graphic representations (icons), physical manipulatives, or card sorts to match the units in the models (function, graph, data table, and rule). Once students have modeled the functions, they can describe the relationship’s growth or decline using verbal or written expression options and then graph the relationships on a digital graphing program such as Desmos, Geogebra, or a hand-held graphing calculator. Students can use these same expression options to express whether the relationship is linear, quadratic, or exponential.

For Category 2, students may use the same expression options from Category 1.

Sample Coursework

Project Created Performance Task

Alternate Means of Expression Option 1 is a performance task created by the project team that represents a viable alternate means of expression a school, district, teacher, or IEP team could utilize as an assessment option for this big idea.

Algebra 1 Growth and Decay (Teacher Guide)

This performance task evaluates students’ understanding of key concepts within the Algebra 1 Growth and Decay Big Idea. It is divided into parts, each targeting a specific component of the Big Idea(s). Each part offers accessible strategies and examples of how students can demonstrate proficiency with the concepts. Various tools, mediums, and connections are provided for teachers to customize the task to the unique needs, cultures, interests, and abilities of their students, promoting an inclusive and relevant educational experience.

When preparing this performance task, distinguish between the flexible and fixed elements to ensure students have multiple ways to demonstrate their knowledge without compromising the concepts’ depth and the rigor within the standards. Furthermore, educators should always consult the student’s Individualized Education Program (IEP) to ensure that all required accommodations and supplementary aids are provided during the assessment.

Growth and Decay: PT Teacher Guide

Mathematics

Algebra 1 *

This guide provides a sample performance task for this big idea created by the project team.

Comment

Algebra 1 Growth and Decay (Student Materials)

This document gives the companion student materials to the performance task fully described in the teacher guide. Please refer to the teacher guide linked as the option performance task for expanded details on appropriate and inappropriate supports for this task, as well as a list of potential alternate means of expression students could use when completing task items.

Growth and Decay: PT Student Materials

Mathematics

Algebra 1 *

These materials provide the student facing resources needed to deliver the option 1 performance task.

Comment

Performance Tasks

Alternate Means of Expression Option 2 represent either a single performance task or a set of performance tasks that have been curated from publicly available task repositories that can be used as a viable assessment option.

Performance Tasks Scoring Materials
Performance Task Materials
Performance Task Primary Source Documents

These performance tasks were gathered from publicly available performance task repositories, including the Mathematics Assessment Project (partnership with UC Berkeley, University of Nottingham, and the Shell Center for Mathematical Education), tied to the Common Core State Standards. According to the Mathematics Assessment Project, these “[performance] tasks are substantial, often involving several aspect of mathematics, and structured so as to ensure that all students have access to the problem. Students are guided through a “ramp” of increasing challenge to enable them to show the levels of performance they have achieved. While any of the mathematical practices may be required, these tasks especially feature MP2, MP6 and two others (MP3 – construct viable arguments and critique the reasoning of others; MP7 – look for and make use of structure).” These tasks exemplify different ways to assess a student’s understanding of the Big Ideas tied to Algebra 1. Educators should feel free to either use these tasks directly to assess students’ learning through alternate means of expression or to use these tasks as a model of different ways to assess student learning. These tasks are especially powerful when making real world connections to the Big Ideas and their related standards.

Overview Statement: This performance task is intended to help you assess how well students are able to interpret exponential and linear functions and in particular, to identify and help students who have the following difficulties:

  • Translating between descriptive, algebraic, tabular, and graphical representation of the functions.
  • Recognizing how and why a quantity changes per unit interval. To achieve these goals students work on simple and compound interest problems.

Growth and Decay: Public Performance Task

Mathematics

Algebra 1 *

This task assesses understanding of exponential and linear functions and rate of change.

Comment

Card Sort

Alternate Means of Expression Option 3 represent either a single performance tasks or a set of performance tasks that have been curated from publically avaible task repositories that can be used as a viable assessment option.

General Instructions for Completing the Card Sort Independently

  • Starting with a card from Set A: Begin by selecting a card from Set A.
  • Find its matching card from Set B: Carefully examine the cards in Set B to identify the one that corresponds or matches with the card from Set A.
  • Place cards side by side: As you make matches, place the paired cards side by side on a large sheet of paper. Avoid stacking them on top of each other so that you can easily see all your matches and make revisions if needed.
  • Explain your thinking: After making a match, take a moment to clearly and carefully explain your thought process. You can do this by writing down your explanation, recording it on a device, or explaining it aloud to yourself.
  • Repeat for Set C, D, etc. (if applicable): If the Card Sort includes an additional set of cards, repeat the above steps for this set as well.

Card Sort Scoring Materials
Card Sort Post-Assessment Task
Card Sort Materials
Card Sort Primary Source Materials

Card Sorts & Matching Activities are powerful tools in mathematics education. By using pre-existing representations, students can focus more on analyzing and making connections between mathematical concepts rather than exclusively spending time creating the representations themselves. This approach allows students to delve deeper into the mathematical content, fostering a deeper understanding of the connections between different concepts. Additionally, engaging in activities like Card Sorts & Matching aligns well with mathematical practices such as attending to precision (MP6) and looking for and making use of structure (MP7).

Generally when completing a card sort, students will need a cut-up copy of each “Card Set”, a large sheet of paper for making a poster (large enough to accommodate multiple sets of cards and space to write their justifications, a device to verbally record justifications, or a teacher to share justifications with), a glue stick, and (when noted) a graphing calculator to check answers. The Card Sorts typically have a blank section and/or blank cards for students to author the missing table, graph, algebraic rule, etc.

Overview Statement: This card sort is intended to help you assess how well students are able to interpret exponential and linear functions and in particular, to identify and help students who have the following difficulties:

  • Translating between descriptive, algebraic, tabular, and graphical representation of the functions.
  • Recognizing how and why a quantity changes per unit interval.

To achieve these goals students work on simple and compound interest problems.

Growth and Decay: Card Sort

Mathematics

Algebra 1 *

This card sort evaluates understanding of functions and their representations.

Comment

Bring Your Own Task (BYOT)

A Call to IEP Teams

We want students’ IEP team members to share their ideas regarding viable alternative means of expression pertaining to this big idea for students with disabilities, including those eligible for the CAA, these teams serve. IEP teams can define viable alternative means of expression for an individual student with an IEP, as long as these mediums meet the local requirements of the coursework.

A Call to Content-based Educators

In addition to IEP teams, we know secondary teachers and district curriculum leads have a wealth of experience and ideas related to innovative ways to assess students’ understanding of this content. We are interested in sample alternative means of expression this community sees as viable assessments of this big idea.

Please use the entry boxes below to share these ideas.

Important Note —These assessment tools will not be shared outside the review of the initiative team and will remain the intellectual property of the users who have made this submission. Furthermore, feedback or comments from the initiative team will not be given to uploaded content, nor does uploading materials imply that the alternative means of expression strategy is a viable option for this big idea.

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