# Variability *

**S-ID.5, S-ID.6, S-ID.7, S-ID.1, S-ID.2, S-ID.3, A-SSE.1:** Summarize, represent, and interpret data. For quantitative data, use a scatter plot and describe how the variables are related. Summarize categorical data in two-way frequency tables and interpret the relative frequencies. (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

**Summarize, represent, and interpret data on a single count or measurement variable. (S-ID.1, S-ID.2, S-ID.3) (CDE 2013, 94)**- S-ID.1. Represent data with plots on the real number line (dot plots, histograms, and box plots) (CDE 2013, 94).
- S-ID.2. Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets (CDE 2013, 94).
- S-ID.3 Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers) (CDE 2013, 94).

**Summarize, represent, and interpret data on two categorical and quantitative variables.[Linear focus; discuss general principle.] [Emphasize linear, quadratic, and exponential models.] (S-ID.6) (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).

- S-ID.6. Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.
**Interpret linear models. (S-ID.7) (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).

**Interpret the structure of expressions. [Linear expressions and exponential expressions with integer exponents] (A-SSE.1) (CDE 2013, 98)**- 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, 98).

- A-SSE.1. Interpret expressions that represent a quantity in terms of its context.

### Primary Standards

**Summarize, represent, and interpret data on two categorical and quantitative variables.[Linear focus; discuss general principle.] [Emphasize linear, quadratic, and exponential models.] (S-ID.5) (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).

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 a student could use visual/graphic representations (icons) or physical manipulatives to organize the data, then use a digital graphing program to design a scatterplot, and conclude with verbal or written explanations to demonstrate how the variables are related.

For Category 2, a student could represent the categorical data using icons related to the category (e.g. green or blue eyes for eye color) on a frequency table and then use a presentation format or program to show the interpretation of the data with emphasis on what is more or less frequent.

## 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 Variability and Corr/Cause (Teacher Guide)

This performance task evaluates students’ understanding of key concepts within the Integrated 1 Variability and Correlation and Causation 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 Variability and Corr/Cause (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 use a scatter diagram to compare results of two school tests using bivariate data.

### Performance Task

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.

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 activity is intended to help you assess how well students are able to:

- select and use mathematical ideas to solve a problem and then compare and critique alternative approaches.
- investigate patterns of association in bivariate data.

### 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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