# Model with Functions *

**F-IF.1, F-IF.2, F-IF.4, F-IF.5, F-IF.6, F-IF.7, F-IF.8, F-IF.9, F-BF.1, F-BF.2, F-BF.4, F-LE.1, F-LE.2, S-ID.5, S-ID.6, S-ID.7, S-ID.8, S-ID.9**: Investigate data sets by table and graph and using technology; fit and interpret functions** to model the data between two quantities. Interpret information from the functions, noticing key features* and symmetries. Develop understanding of the meaning of the function and how it represents the data that it is modeling; recognizing possible associations and trends in the data – including consideration of the correlation coefficients of linear models.

- Students can disaggregate data by different characteristics of interest (populations for example), and compare slopes to examine questions of fairness and bias among groups.
- Students have opportunities to consider how to communicate relevant concerns to stakeholders and/or community members.
- Students can identify both extreme values (true outliers) and data errors, and how the inclusion or exclusion of these observations may change the function that would most appropriately model the data.

*intercepts, slope, increasing or decreasing, positive or negative

** functions include linear, quadratic and exponential (CDE 2023, 40)

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**

**Build a function that models a relationship between two quantities. (F-BF. 1-2) (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.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).

- F-BF.1. Write a function that describes a relationship between two quantities.
**Construct and compare linear, quadratic, and exponential models and solve problems. (F-LE 1–2) (CDE 2013, 92)**- F-LE.1. Distinguish between situations that can be modeled with linear functions and with exponential functions.
- Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals.
- Recognize situations in which one quantity changes at a constant rate per unit interval relative to another.
- 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.1. Distinguish between situations that can be modeled with linear functions and with exponential functions.

**Primary Standards**

**Understand the concept of a function and use function notation. (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
**Analyze functions using different representations. (F-IF 7–9) (CDE 2013, 92)**- F-IF.7. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.
- Graph linear and quadratic functions and show intercepts, maxima, and minima.
- Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.
- Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude (CDE 2013, 92).

- F-IF.8. Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.
- Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context.
- Use the properties of exponents to interpret expressions for exponential functions.
*For example, identify percent rate of change in functions such as y = (1.02)t, y = (0.97)t, y = (1.01)12t, and y = (1.2)t/10, and classify them as representing exponential growth or decay*(CDE 2013, 100).

- 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, 100).

- F-IF.7. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.

**Primary Standards**

**Interpret functions that arise in applications in terms of the context. (F-IF 4–6) (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). - F-IF.6. 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 (CDE 2013, 92).

- 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.
**Build new functions from existing functions (F-BF 4) (CDE 2013, 92)**- F-BF.4. Find inverse functions.
- 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).

- Solve an equation of the form

- F-BF.4. Find inverse functions.
**Summarize, represent, and interpret data on two categorical and quantitative variables. (S-ID 5–6) (CDE 2013, 94)**- S-ID.5. Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data (CDE 2013, 94).
- S-ID.6. Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.
- Fit a function to the data; use functions fitted to data to solve problems in the context of the data.
*Use given functions or choose a function suggested by the context. Emphasize linear, quadratic, and exponential models.* - Informally assess the fit of a function by plotting and analyzing residuals.
- Fit a linear function for a scatter plot that suggests a linear association (CDE 2013, 94).

- Fit a function to the data; use functions fitted to data to solve problems in the context of the data.

**Interpret linear models. (S-ID 7–9) (CDE 2013, 94)**- 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).
- S-ID.8. Compute (using technology) and interpret the correlation coefficient of a linear fit (CDE 2013, 94).
- S-ID.9. Distinguish between correlation and causation (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 Category 1 of Modeling with Functions, students can use graphing tools such as Desmos, Geogebra, or a graphing calculator to create tables, graph data, and organize results. After working out the math formulas related to their data, they can explain their findings and how they modeled the data relationship either verbally, in writing, or using a text-to-speech program.

For Category 2, students can do a card matching activity where they pair the function with its graph and explain the slope and intercepts out loud or in writing. As well, students can complete a simplified dataset on each graph to deduce the function type it represents and then present the main features and their calculations verbally or in a presentation (e.g. Google Slides or PowerPoint).

For Category 3, students can create a video on a device (e.g. tablet, phone, or laptop) to express the meaning of the results, including an explanation of the trends they found in the data and how a linear model might represent the relationships.

## 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 Model with Functions, Function Investigations, Features of Functions (Teacher Guide)

This performance task evaluates students’ understanding of key concepts within the Algebra 1 Model with Functions, Features of Functions, and Function Investigations Big Ideas. 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.

Algebra 1 Model with Functions, Function Investigations, Features of Functions (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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