# Composing Functions *

**F-BF.3, F-BF.2, F-IF.3:** Build and explore new functions that are made from existing functions, and explore graphs of the related functions using technology. Recognize sequences are functions and are sometimes defined recursively (CDE 2023, 57).

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 new functions from existing functions. [Linear, exponential, quadratic, and absolute value; for F-BF.4a, linear only] (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.3. Identify the effect on the graph of replacing
**Build a function that models a relationship between two quantities. [For F-BF.1–2, linear, exponential, and quadratic] (CDE 2013, 100)**- 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).

### Primary Standards

**Understand the concept of a function and use function notation. [Learn as general principle; focus on linear and exponential and on arithmetic and geometric sequences.] (CDE 2013, 128)**- F-IF.3. Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers.
*For example, the Fibonacci sequence is defined recursively by*

*f(0)=f(1)=1, f(n+1)=f(n)+f(n-1) for n ≥ 1*(CDE 2013, 91).

- F-IF.3. Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers.

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 Category1 of Composing Functions, students can express new functions created from existing functions using visual/graphic representations (icons) or card sorts to match the functions with their corresponding graph, and dataset.

For Category 2, students can use visual representations to express the sequences in the functions that occur recursively like using a visual depiction of the Fibonnaci sequence in a sunflower. For both categories, they can describe their results verbally or in writing and they can use a digital graphing program such as Desmos, Geogebra, or a hand-held graphing calculator to graph the functions.

## 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 Composing Functions (Teacher Guide)

This performance task evaluates students’ understanding of key concepts within the Integrated 1 Composing Functions 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.

Integrated Mathematics I Composing 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:

- work with patterns and to work out the nth term of a sequence.
- build a function that models a relationship between two quantities.

### 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:

- Articulate verbally the relationships between variables arising in everyday contexts.
- Translate between everyday situations and sketch graphs of relationships between variables.
- Interpret algebraic functions in terms of the contexts in which they arise.
- Reflect on the domains of everyday functions and in particular whether they should be discrete or continuous.

### 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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