# Modeling with Functions *

**N-Q.1, N-Q.2, N-Q.3, F-BF.1, F-IF.1, F-IF.2, F-IF.4, F-LE.5, S-ID.7, A-CED.1, A-CED.2, A-CED.3, A-SSE.1:** Build functions that model relationships between two quantities, including examples with inequalities; using units and different representations. Describe and interpret the relationships modeled using visuals, tables, and graphs. (CDE 2023, 56)

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

**Reason quantitatively and use units to solve problems. [Foundation for work with expressions, equations, and functions] (N-Q.1–3) (CDE 2013, 90)**- 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, 90).
- N-Q.2. Define appropriate quantities for the purpose of descriptive modeling (CDE 2013, 90).
- N-Q.3. Choose a level of accuracy appropriate to limitations on measurement when reporting quantities (CDE 2013, 90).

**Create equations that describe numbers or relationships. [Linear and exponential (integer inputs only); for A.CED.3, linear only] (A-CED.1–3) (CDE 2013, 90)**- A-CED.1. Create equations and inequalities in one variable including ones with absolute value and use them to solve problems.
*Include equations arising from linear and quadratic functions, and simple rational and exponential functions.*CA (CDE 2013, 90) - A-CED.2. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales (CDE 2013, 90).
- A-CED.3. Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context.
*For example, represent inequalities describing nutritional and cost constraints on combinations of different foods*(CDE 2013, 90).

- A-CED.1. Create equations and inequalities in one variable including ones with absolute value and use them to solve problems.
**Build a function that models a relationship between two quantities. [For F.BF.1, linear and exponential (integer inputs)] (F-BF.1) (CDE 2013, 92)**- F-BF.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*(CDE 2013, 92).

- F-BF.1. Write a function that describes a relationship between two quantities.

### Primary Standards

**Interpret the structure of expressions. [Linear expressions and exponential expressions with integer exponents] (A-SSE.1) (CDE 2013, 90)**- A-SSE.1. Interpret expressions that represent a quantity in terms of its context.
- Interpret parts of an expression, such as terms, factors, and coefficients.
- 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, 90).

- A-SSE.1. Interpret expressions that represent a quantity in terms of its context.
- Understand the concept of a function and use function notation. [Learn as general principle. Focus on linear and exponential (integer domains) and on arithmetic and geometric sequences.] (F-IF.1–2) (CDE 2013, 91)
- F-IF.1. 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)*(CDE 2013, 91). - F-IF.2. Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context (CDE 2013, 91).

- F-IF.1. 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
**Interpret functions that arise in applications in terms of the context. [Linear and exponential (linear domain)] (F-IF.4) (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.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.
**Interpret expressions for functions in terms of the situation they model. [Linear and exponential of form ] (F-LE.5) (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.5. Interpret the parameters in a linear or exponential function in terms of a context. [Linear and exponential of form
**Interpret linear models. (S-ID.7)**- S-ID.7. Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data (CDE 2013, 94)

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 both categories of Modeling with Functions, students can use a graphing program like Desmos, Geogebra, or a handheld graphing calculator to create the functions, tables, and graphs, paying specific attention to equalities and inequalities. Students can illustrate mathematical relationships between two quantities by stating a rule or equation as an equation, a written sentence, a verbal sentence, or through building an equation from given cards with parts to the equation.. To convey their understanding, students can demonstrate the relationships and explain their work through speech-to-text software or video presentations.

## 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.

Integrated Mathematics I Modeling with Functions, Comparing Models (Teacher Guide)

This performance task evaluates students’ understanding of key concepts within the Integrated 1 Modeling with Functions and Comparing Models Big Ideas. It is divided into parts, each targeting a specific component of the Big Ideas. 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 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.

Integrated Mathematics I Modeling with Functions, Comparing Models (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.

### 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 theMathematics 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 draw and interpret a graph of speed, distance, and time.

### 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 assess how well students are able to interpret distance–time graphs and, in particular, to help you identify students who:

- Interpret distance–time graphs as if they are pictures of situations rather than abstract representations of them.
- Have difficulty relating speeds to slopes of these graphs.

### 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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