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Transformations and Congruence *

G-CO.1, G-CO.2, G-CO.3, G-CO.4, G-CO.5, G-CO.12, G-CO.13, G-GPE.4, G-GPE.5, G.GPE.7, F-BF.3: Explore congruence of triangles, including quadrilaterals built from triangles, through geometric constructions. Investigate transformations in the plane. Use geometry software to study transformations, developing definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, and parallel lines. Express translations algebraically (CDE 2023, 58).

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.

Secondary Standards

  • Make geometric constructions. [Formalize and explain processes.] (G-CO.12, G-CO.13) (CDE 2013, 134)
    • G-CO.12. Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line (CDE 2013, 134).
    • G-CO.13. Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle (CDE 2013, 134).

Primary Standards

  • Experiment with transformations in the plane. (G-CO.1, G-CO.2, G-CO.3, G-CO.4, G-CO.5) (CDE 2013, 72)
    • G-CO.1. Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc (CDE 2013, 72).
    • G-CO.2. Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch) (CDE 2013, 72).
    • G-CO.3. Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself (CDE 2013, 72).
    • G-CO.4. Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments (CDE 2013, 72).
    • G-CO.5. Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another (CDE 2013, 72).
  • Use coordinates to prove simple geometric theorems algebraically. [Include distance formula; relate to Pythagorean Theorem.] (G-GPE.4, G-GPE.5, G.GPE.7) (CDE 2013, 74)
    • G-CO.4. Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments (CDE 2013, 72).
    • G-CO.5. Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another (CDE 2013, 72).
    • G-GPE.7. Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent (CDE 2013, 72).

Secondary Standards

  • Build new functions from existing functions. [Linear and exponential; focus on vertical translations for exponential.] (F-BF.3) (CDE 2013, 92)
    • 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, 92)

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 Transformations and Congruence, students may verbally or in writing explain the congruence of shapes using rigid transformations. Tools such as GeoGebra or Desmos can aid students in measuring lengths, recognizing slopes of line segments, and creating additional lines and line segments on a coordinate plane. Students can annotate on their graphs the effects of transformations on their shapes. They also can complete a simplified number of coordinate transformations that alter the shapes and express these changes algebraically, using either speech or text.

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 Shapes and Structures, Building with Triangles, Transformation and Congruence (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.

Shapes and Structures, Building with Triangles, Transformation and Congruence: PT Student Materials

Mathematics

Integrated Mathematics 1 *

These materials provide the student facing resources needed to deliver the option 1 performance task.

Comment

Integrated Mathematics I Shapes and Structures, Building with Triangles, Transformation and Congruence (Teacher Guide)

This performance task evaluates students’ understanding of key concepts within the Integrated 1 Shapes and Structures, Building with Triangles, and Transformations and Congruence Big Ideas. It is divided into parts, with each targeting specific overlapping components of these three Big Ideas related to plane geometry. 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 to promote an inclusive and relevant educational experience.

When preparing to administer this performance task, it is important for the teacher to distinguish between flexible and fixed elements to ensure students have multiple ways to demonstrate their knowledge without compromising the depth and the rigor of the standards. As always, educators should also consult the student’s Individualized Education Program (IEP) to ensure that all required accommodations and supplementary aids are provided during the assessment.

Shapes and Structures, Building with Triangles, Transformation and Congruence: PT Teacher Guide

Mathematics

Integrated Mathematics 1 *

This guide provides a sample performance task for this big idea created by the project team.

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

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 reflections and rotations of a given figure on a grid and to describe transformations needed to make a given pattern.

Transformation and Congruence: Public Performance Task

Mathematics

Integrated Mathematics 1 *

This task assesses drawing and describing reflections, rotations, and transformations.

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 you assess how well students are able to:

  • Describe in words the transformation that maps an object to a transformed image.
  • Given a geometric figure and a rotation, reflection or translation, draw the transformed figure (or the original figure if the image is given.)
  • Describe transformations as algebraic functions that take points in the plane as inputs and give other points as outputs.

Transformation and Congruence: Card Sort

Mathematics

Integrated Mathematics 1 *

This card sort assesses describing, drawing, and defining transformations as functions.

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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