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Systems of Equations *

A-REI.1, A-REI.3, A-REI.4, A-REI.5, A-REI.6, A-REI.7, A-REI.10, A-REI.11, A-REI.12, NQ.1, A-SSE.1, F-LE.1, F-LE.2: Students investigate real situations that include data for which systems of 1 or 2 equations or inequalities are helpful, paying attention to units. Investigations include linear, quadratic, and absolute value. Students use technology tools strategically to find their solutions and approximate solutions, [construct] viable arguments, [interpret] the meaning of the results, and [communicate] them in multidimensional ways. (CDE 2023, 41)

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

  • Understand solving equations as a process of reasoning and explain the reasoning. [Master linear; learn as general principle.] (A-REI.1) (CDE 2013, 90)
    • A-REI.1. Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method (CDE 2013, 90).
  • Solve systems of equations. [Linear-linear and linear-quadratic] (A-REI.5-7) (CDE 2013, 66)
    • A-REI.5. Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions (CDE 2013, 66).
    • A-REI.6. Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables (CDE 2013, 66).
    • A-REI.7. Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically (CDE 2013, 66).
  • Represent and solve equations and inequalities graphically. [Linear and exponential*; learn as general principle.] (A-REI.10-12) (CDE 2013, 66)
    • A-REI.10. Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line) (CDE 2013, 66).
    • A-REI.11. Explain why the x-coordinates of the points where the graphs of the equations y=f(x) and y=g(x) intersect are the solutions of the equation f(x)=g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions (CDE 2013, 66).
    • A-REI.12. Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes (CDE 2013, 66).
  • 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).

Supporting Standards

  • Solve equations and inequalities in one variable. [Linear inequalities; literal equations that are linear in the variables being solved for; quadratics with real solutions] (A-REI.3-4) (CDE 2013, 91)
  • Reason quantitatively and use units to solve problems. [Foundation for work with expressions, equations and functions] (N-Q.A.1) (CDE 2013, 121)
  • Construct and compare linear, quadratic, and exponential* models and solve problems. (A-F.LE.1-2) (CDE 2013, 92)

*Note: The Big Idea of Systems of Equations in Algebra 1 does not include exponential functions/relationships.

Primary Standards

  • Represent and solve equations and inequalities graphically. [Linear and exponential*; learn as general principle.] (A-REI.11) (CDE 2013, 91)
    • A-REI.11. Explain why the x-coordinates of the points where the graphs of the equations y=f(x) and y=g(x) intersect are the solutions of the equation f(x)=g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions (CDE 2013, 91).

*Note: The Big Idea of Systems of Equations in Algebra 1 does not include exponential functions/relationships.

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 Systems of Equations, students can express their knowledge of equations and inequalities using graphic representations (icons), hands-on materials, or card sorts for matching. After solving, they can explain the relationship verbally or in writing and then graph the results using a handheld graphing calculator or a digital graphing program such as Desmos or Geogebra.

Category 2 focuses on using technology to complete four actionable skills: finding solutions, constructing viable arguments, interpreting results, and communicating results. Students can express their knowledge of these skills verbally or in writing, or they can create a digital presentation. They can also create a video to show completion of the big idea categories.

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 Systems of Equations: Performance Task (Teacher Guide)

This performance task evaluates students’ understanding of key concepts within the Algebra 1 Systems of Equations Big Idea. It is divided into parts, each targeting a specific component of the Big Idea. 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. Each part includes a table named “A Note on Administration and Embedding Supports,” which provides guidance on essential vocabulary, appropriate and inappropriate supports, and potential methods and alternate means for students to express their understanding.

Systems of Equations: PT Teacher Guide

Mathematics

Algebra 1 *

This teacher guide provides the directions, scoring rubrics, and an example student response for a performance task created by the project team specific to this big idea.

Comment

Algebra 1 Systems of Equations: Performance Task (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.

Systems of Equations: 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: These performance tasks are intended to help you assess how well students are able to form and solve a pair of linear equations in a practical situation.

Systems of Equations: Public Performance Task

Mathematics

Algebra 1 *

Tasks assess solving linear equations in practical situations.

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 assess how well students are able to:

  • Classify solutions to a pair of linear equations by considering their graphical representations.
  • Use substitution to complete a table of values for a linear equation.
  • Identify a linear equation from a given table of values.
  • Graph and solve linear equations.

Systems of Equations: Card Sort

Mathematics

Algebra 1 *

This card sort assesses students' skills in solving, graphing, and identifying linear equations.

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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