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Gr. HS Unit: Congruence, Proof and Constructions

Lesson Unit

Lesson Plans

Lesson Seeds

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Unit Overview

In previous grades, students were asked to draw triangles based on given measurements. Students also have prior experience with rigid motions: translations, reflections, and rotations and have used these to develop notions about what it means for two objects to be congruent. In this unit, students establish triangle congruence criteria, based on analyses of rigid motions and formal constructions. Students use triangle congruence as a familiar foundation for the development of formal proof. Students prove theorems—using a variety of formats—and solve problems about lines, angles, triangles, quadrilaterals, and other polygons. Students also apply reasoning to complete geometric constructions and explain why constructions work.

Essential Questions:

  • How is visualization essential to the study of geometry?
    • How does the concept of rigid motion connect to the concept of congruence?
  • How does geometry explain or describe the structure of our world?
    • How do geometric constructions enhance understanding of the geometric properties of objects?
  • How can reasoning be used to establish or refute conjectures?
    • What are the characteristics of a valid argument?
    • What is the role of deductive or inductive reasoning in validating a conjecture?
    • What facts need to be verified in order to establish that two figures are congruent?

A question is essential when it stimulates multi-layered inquiry, provokes deep thought and lively discussion, requires students to consider alternatives and justify their reasoning, encourges re-thinking of big ideas, makes meaningful connections with prior learning, and provides students with opportunities to apply problem-solving skills to authentic situations.

Unit Lesson

Additional information such as Teachers Notes, Enduring Understandings,Content Emphasis by Cluster, Focus Standards, Possible Student Outcomes, Essential Skills and Knowledge Statements and Clarifications, and Interdisciplinary Connections can be found in this Lesson Unit.

Available Model Lesson Plans

The lesson plan(s) have been written with specific standards in mind. Each model lesson plan is only a MODEL - one way the lesson could be developed. We have NOT included any references to the timing associated with delivering this model. Each teacher will need to make decisions related ot the timing of the lesson plan based on the learning needs of students in the class. The model lesson plans are designed to generate evidence of student understanding.

This chart indicates one or more lesson plans which have been developed for this unit. Lesson plans are being written and posted on the Curriculum Management System as they are completed. Please check back periodically for additional postings.

CCSC Alignment: G.CO.6, G.CO.7, G.CO.8

This lesson plan focuses on the concept of congruence and its connection to rigid motion. (i.e. Objects in space which can be transformed in an infinite number of ways and describing/analyzing those transformations.)

Available Model Lesson Seeds

The lesson seed(s) have been written with specific standards in mind. These suggested activity/activities are not intended to be prescriptive, exhaustive, or sequential; they simply demonstrate how specific content can be used to help students learn the skills described in the standards. Seeds are designed to give teachers ideas for developing their own activities in order to generate evidence of student understanding.

This chart indicates one or more lesson seeds which have been developed for this unit. Lesson seeds are being written and posted on the Curriculum Management System as they are completed. Please check back periodically for additional postings.

CCSC Alignnment: G.CO.2

Practice Activity: Students who learn best through kinesthetic means will have the opportunity to move while reviewing what they know about various transformations.

CCSC Alignnment: G.CO.2

Developmental Activity: Students will compare and contrast rigid transformations and transformations that are not rigid as well as the various specific types of transformations.

CCSC Alignnment: G.CO.5, G.CO.6

Practice Activity: This is a computer game that could be used to provide practice on transformations to students who respond well to using technology.

CCSC Alignnment: G.CO.6

Investigation: This investigation can be used in direct instruction or with students in small groups or partners. Students answer the questions provided, while investigating rotational transformations on the computer.

CCSC Alignnment: G.CO.9

Warm-Up/Intervention: In 8th grade students use informal arguments to establish facts about the angles created when parallel lines are cut by a transversal. This activity helps to bring this knowledge back to the forefront. This information is needed to prove theorems about lines and angles.

Triangle Congruence

CCSC Alignnment: G.CO.10

Practice Activity: The problems found in this document should not be used all at one time. The problems should be used one or two at a time to provide distributed practice over time.

Using Triangle Congruence in Proofs

CCSC Alignnment: G.CO.10

Practice Activity: This activity will provide students with practice on completing proofs using a variety of formats. It is highly recommended that these problems be used one or two at a time for distributed practice over time to help student maintain skills.

Triangle Sum Theorem

CCSC Alignnment: G.CO.10

Practice Activity: This lesson seed describes an activity which requires students to complete a flow chart proof and a two-column proof while proving the Triangle Sum Theorem and the Exterior Angle Theorem.

CCSC Alignnment: G.CO.10

Investigation: Introduce this activity after proof of the interior angle sum theroem for triangles. Students will use sum of the interior angles of a triangle to determine the sum of the interior angles of any convex polygon.

CCSC Alignnment: G.CO.10

Investigation/Carousel Practice Activity: The beginning of this lesson seeds provides an investigation which helps students develop conjectures about midsegments. The activity can be followed with a class discussion to solidify these geometric relationships.

CCSC Alignnment: G.CO.9, G.CO.10, G.CO.11

Developmental Activity: This activity could be used to help students begin thinking about how the logical order of a sequence of statements is necessary for a process to make sense. Completing this activity could serve as motivation for a lesson which targets proofs.

Constructions Manual

CCSC Alignnment: G.CO.12

Resource:This lesson seed serves as a resource rather than an activity. It provides step-by-step directions to make geometric constructions using a compass and straightedge. Students may elect to refer to this document as needed when required to make formal geometric constructions.

CCSC Alignnment: G.CO.9, G.CO.12

Investigation: This lesson seed describes three different activities which allow students to first construct a perpendicular bisector of a segment and then discover that all points on the perpendicular bisector of a segment are equidistant from the endpoints of the segment.

CCSC Alignnment: G.CO.12

Application: This lesson seed could be used as an application after students have studied the Perpendicular Bisector Theorem.