ALEX Classroom Resource

  

Geometry Module 1, Topic G: Axiomatic Systems

  Classroom Resource Information  

Title:

Geometry Module 1, Topic G: Axiomatic Systems

URL:

https://www.engageny.org/resource/geometry-module-1-topic-g-overview

Content Source:

EngageNY
Type: Lesson/Unit Plan

Overview:

In Module 1, Topic G, students review material covered throughout the module. Additionally, students discuss the structure of geometry as an axiomatic system.

Content Standard(s):
Mathematics
MA2015 (2016)
Grade: 9-12
Geometry
4 ) Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. [G-CO4]


Alabama Alternate Achievement Standards
AAS Standard:
M.G.AAS.HS.4- Given a geometric figure of a reflection or a translation of that figure, identify if the geometric figure is a reflection or translation.


Mathematics
MA2015 (2016)
Grade: 9-12
Geometry
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. [G-CO5]


Alabama Alternate Achievement Standards
AAS Standard:
M.G.AAS.HS.5- Given a figure and that figure after a vertical or horizontal translation, identify the vertical or horizontal translation.


Mathematics
MA2015 (2016)
Grade: 9-12
Geometry
8 ) Explain how the criteria for triangle congruence, angle-side-angle (ASA), side-angle-side (SAS), and side-side-side (SSS), follow from the definition of congruence in terms of rigid motions. [G-CO8]


Alabama Alternate Achievement Standards
AAS Standard:
M.G.AAS.HS.8- Given two congruent triangles and angle measures of one of the triangles, identify the angle measures of the other triangle.


Mathematics
MA2015 (2016)
Grade: 9-12
Geometry
10 ) Prove theorems about triangles. Theorems include measures of interior angles of a triangle sum to 180o, base angles of isosceles triangles are congruent, the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length, and the medians of a triangle meet at a point. [G-CO10]


Alabama Alternate Achievement Standards
AAS Standard:
M.G.AAS.HS.10- Given a measure of a leg or base angle of an isosceles triangle, identify the measure of the other leg or other base angle.


Mathematics
MA2015 (2016)
Grade: 9-12
Geometry
11 ) Prove theorems about parallelograms. Theorems include opposite sides are congruent, opposite angles are congruent; the diagonals of a parallelogram bisect each other; and conversely, rectangles are parallelograms with congruent diagonals. [G-CO11]


Alabama Alternate Achievement Standards
AAS Standard:
M.G.AAS.HS.11- Given the measure of one side or one angle of a parallelogram, identify the measure of the opposite side or opposite angle.


Mathematics
MA2015 (2016)
Grade: 9-12
Geometry
12 ) Make formal geometric constructions with a variety of tools and methods such as compass and straightedge, string, reflective devices, paper folding, and dynamic geometric software. Constructions include 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. [G-CO12]


Alabama Alternate Achievement Standards
AAS Standard:
M.G.AAS.HS.12- Given a drawing with angles and a protractor overlay, determine which angles are congruent. Sample image below.
Image


Mathematics
MA2015 (2016)
Grade: 9-12
Geometry
13 ) Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. [G-CO13]


Alabama Alternate Achievement Standards
AAS Standard:
M.G.AAS.HS.13- Identify an equilateral triangle from a set of triangles or identify a regular hexagon from a set of hexagons. Make sure sides/angles are marked so that students can identify congruence.


Mathematics
MA2019 (2019)
Grade: 9-12
Geometry with Data Analysis
21. Represent transformations and compositions of transformations in the plane (coordinate and otherwise) using tools such as tracing paper and geometry software.

a. Describe transformations and compositions of transformations as functions that take points in the plane as inputs and give other points as outputs, using informal and formal notation.

b. Compare transformations which preserve distance and angle measure to those that do not.
Unpacked Content
Evidence Of Student Attainment:
Students:
Given a variety of transformations (translations, rotations, reflections, and dilations),
  • Represent the transformations and compositions of transformations in the plane using a variety of methods (e.g., technology, transparencies, semi-transparent mirrors (MIRAs), patty paper, compass).
  • Describe transformations and compositions of transformations functions that take points in the plane as inputs and give other points as outputs, explain why this satisfies the definition of a function, and adapt function notation to that of a mapping [e.g., f(x,y) → f(x+a, y+b)].
  • Compare transformations that preserve distance and angle to those that do not.
Teacher Vocabulary:
  • Transformation
  • Reflection
  • Translation
  • Rotation
  • Dilation
  • Isometry
  • Composition
  • Horizontal stretch
  • Vertical stretch
  • Horizontal shrink
  • Vertical shrink
  • Clockwise
  • Counterclockwise
  • Symmetry
  • Preimage
  • Image
Knowledge:
Students know:
  • Characteristics of transformations (translations, rotations, reflections, and dilations).
  • Methods for representing transformations.
  • Characteristics of functions.
  • Conventions of functions with mapping notation.
Skills:
Students are able to:
  • Accurately perform dilations, rotations, reflections, and translations on objects in the coordinate plane with and without technology.
  • Communicate the results of performing transformations on objects and their corresponding coordinates in the coordinate plane, including when the transformation preserves distance and angle.
  • Use the language and notation of functions as mappings to describe transformations.
Understanding:
Students understand that:
  • Mapping one point to another through a series of transformations can be recorded as a function.
  • Some transformations (translations, rotations, and reflections) preserve distance and angle measure, and the image is then congruent to the pre-image, while dilations preserve angle but not distance, and the pre-image is similar to the image.
  • Distortions, such as only a horizontal stretch, preserve neither.

Alabama Alternate Achievement Standards
AAS Standard:
M.G.AAS.10.21 Identify and/or model characteristics of a geometric figure that has undergone a transformation (reflection, rotation, translation) by drawing, explaining, or using manipulatives.


Mathematics
MA2019 (2019)
Grade: 9-12
Geometry with Data Analysis
22. Explore rotations, reflections, and translations using graph paper, tracing paper, and geometry software.

a. Given a geometric figure and a rotation, reflection, or translation, draw the image of the transformed figure using graph paper, tracing paper, or geometry software.

b. Specify a sequence of rotations, reflections, or translations that will carry a given figure onto another.

c. Draw figures with different types of symmetries and describe their attributes.
Unpacked Content
Evidence Of Student Attainment:
Students:
Given a geometric figure,
  • Explore rotations, reflections, and translations using graph paper, tracing paper, and geometry software.
  • Produce the image of the figure under a rotation, reflection, or translation using graph paper, tracing paper, or geometry software.
  • Describe and justify the sequence of transformations that will carry a given figure onto another.
  • Draw figures such as rectangles, parallelograms, trapezoids, or regular polygons.
  • Identify which figures that have rotations or reflections that carry the figure onto itself.
  • Perform and communicate rotations and reflections that map the object to itself.
  • Distinguish these transformations from those which do not carry the object back to itself.
  • Describe the relationship of these findings to symmetry.
Teacher Vocabulary:
  • Transformation
  • Reflection
  • Translation
  • Rotation
  • Dilation
  • Isometry
  • Composition
  • horizontal stretch
  • vertical stretch
  • horizontal shrink
  • vertical shrink
  • Clockwise
  • Counterclockwise
  • Symmetry
  • Trapezoid
  • Square
  • Rectangle
  • Regular polygon
  • parallelogram
  • Mapping
  • preimage
  • Image
Knowledge:
Students know:
  • Characteristics of transformations (translations, rotations, reflections, and dilations).
  • Techniques for producing images under transformations using graph paper, tracing paper, or geometry software.
  • Characteristics of rectangles, parallelograms, trapezoids, and regular polygons.
Skills:
Students are able to:
  • Accurately perform dilations, rotations, reflections, and translations on objects in the coordinate plane with and without technology.
  • Communicate the results of performing transformations on objects and their corresponding coordinates in the coordinate plane.
Understanding:
Students understand that:
  • Mapping one point to another through a series of transformations can be recorded as a function.
  • Since translations, rotations and reflections preserve distance and angle measure, the image is then congruent.
  • The same transformation may be produced using a variety of tools, but the geometric sequence of steps that describe the transformation is consistent.

Alabama Alternate Achievement Standards
AAS Standard:
M.G.AAS.10.21 Identify and/or model characteristics of a geometric figure that has undergone a transformation (reflection, rotation, translation) by drawing, explaining, or using manipulatives.


Mathematics
MA2019 (2019)
Grade: 9-12
Geometry with Data Analysis
24. Define congruence of two figures in terms of rigid motions (a sequence of translations, rotations, and reflections); show that two figures are congruent by finding a sequence of rigid motions that maps one figure to the other.
Example: △ABC is congruent to △XYZ since a reflection followed by a translation maps △ABC onto △XYZ.

Unpacked Content
Evidence Of Student Attainment:
Students:
  • Given two geometric figures, determine if a sequence of rotations, reflections, and translations will carry the first to the second, and if so justify their congruence by the definition of congruence in terms of rigid motions.
Teacher Vocabulary:
  • Rigid motions
  • Congruence
Knowledge:
Students know:
  • Characteristics of translations, rotations, and reflections including the definition of congruence.
  • Techniques for producing images under transformations using graph paper, tracing paper, compass, or geometry software.
  • Geometric terminology (e.g., angles, circles, perpendicular lines, parallel lines, and line segments) which describes the series of steps necessary to produce a rotation, reflection, or translation.
Skills:
Students are able to:
  • Use geometric descriptions of rigid motions to accurately perform these transformations on objects.
  • Communicate the results of performing transformations on objects.
Understanding:
Students understand that:
  • Any distance preserving transformation is a combination of rotations, reflections, and translations.
  • If a series of translations, rotations, and reflections can be described that transforms one object exactly to a second object, the objects are congruent.

Alabama Alternate Achievement Standards
AAS Standard:
M.G.AAS.10.24 When given two congruent triangles that have been transformed (limit to a translation), determine the congruent parts. (Ex: Determine which leg on Triangle A is congruent to which leg on Triangle B.)


Mathematics
MA2019 (2019)
Grade: 9-12
Geometry with Data Analysis
25. Verify criteria for showing triangles are congruent using a sequence of rigid motions that map one triangle to another.

a. Verify that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.

b. Verify that two triangles are congruent if (but not only if) the following groups of corresponding parts are congruent: angle-side-angle (ASA), side-angle-side (SAS), side-side-side (SSS), and angle-angle-side (AAS).

Example: Given two triangles with two pairs of congruent corresponding sides and a pair of congruent included angles, show that there must be a sequence of rigid motions will map one onto the other.
Unpacked Content
Evidence Of Student Attainment:
Students:
  • Given a triangle and its image under a sequence of rigid motions (translations, reflections, and translations), verify that corresponding sides and corresponding angles are congruent.
  • Given two triangles that have the same side lengths and angle measures, find a sequence of rigid motions that will map one onto the other.
  • Use rigid motions and the basic properties of rigid motions (that they preserve distance and angle), which are assumed without proof to establish that the usual triangle congruence criteria make sense and can then be used to prove other theorems.
Teacher Vocabulary:
  • Corresponding sides and angles
  • Rigid motions
  • If and only if
  • Triangle congruence
  • Angle-Side-Angle (ASA)
  • Side-Angle-Side (SAS)
  • Side-Side->Side (SSS)
Knowledge:
Students know:
  • Characteristics of translations, rotations, and reflections including the definition of congruence.
  • Techniques for producing images under transformations.
  • Geometric terminology which describes the series of steps necessary to produce a rotation, reflection, or translation.
  • Basic properties of rigid motions (that they preserve distance and angle).
  • Methods for presenting logical reasoning using assumed understandings to justify subsequent results.
Skills:
Students are able to:
  • Use geometric descriptions of rigid motions to accurately perform these transformations on objects.
  • Communicate the results of performing transformations on objects.
  • Use logical reasoning to connect geometric ideas to justify other results.
  • Perform rigid motions of geometric figures.
  • Determine whether two plane figures are congruent by showing whether they coincide when superimposed by means of a sequence of rigid motions (translation, reflection, or rotation).
  • Identify two triangles as congruent if the lengths of corresponding sides are equal (SSS criterion), if the lengths of two pairs of corresponding sides and the measures of the corresponding angles between them are equal (SAS criterion), or if two pairs of corresponding angles are congruent and the lengths of the corresponding sides between them are equal (ASA criterion).
  • Apply the SSS, SAS, and ASA criteria to verify whether or not two triangles are congruent.
Understanding:
Students understand that:
  • If a series of translations, rotations, and reflections can be described that transforms one object exactly to a second object, the objects are congruent.
  • It is beneficial to have minimal sets of requirements to justify geometric results (e.g., use ASA, SAS, or SSS instead of all sides and all angles for congruence).

Alabama Alternate Achievement Standards
AAS Standard:
M.G.AAS.10.24 When given two congruent triangles that have been transformed (limit to a translation), determine the congruent parts. (Ex: Determine which leg on Triangle A is congruent to which leg on Triangle B.)


Mathematics
MA2019 (2019)
Grade: 9-12
Geometry with Data Analysis
30. Develop and use precise definitions of figures such as angle, circle, perpendicular lines, parallel lines, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.
Unpacked Content
Evidence Of Student Attainment:
Students:
Given undefined notions of point, line, distance along a line, and distance around a circular arc,
  • Develop precise definitions of angle, circle, perpendicular line, parallel line, and line segment.
  • Identify examples and non-examples of angles, circles, perpendicular lines, parallel lines, and line segments.
Teacher Vocabulary:
  • Point
  • Line
  • Segment
  • Angle
  • Perpendicular line
  • Parallel line
  • Distance
  • Arc length
  • Ray
  • Vertex
  • Endpoint
  • Plane
  • Collinear
  • Coplanar
  • Skew
Knowledge:
Students know:
  • Undefined notions of point, line, distance along a line, and distance around a circular arc.
  • Properties of a mathematical definition, i.e., the smallest amount of information and properties that are enough to determine the concept. (Note: may not include all information related to concept).
Skills:
Students are able to:
  • Use known and developed definitions and logical connections to develop new definitions.
Understanding:
Students understand that:
  • Geometric definitions are developed from a few undefined notions by a logical sequence of connections that lead to a precise definition, A precise definition should allow for the inclusion of all examples of the concept, and require the exclusion of all non-examples.

Alabama Alternate Achievement Standards
AAS Standard:
M.G.AAS.10.30 Demonstrate perpendicular lines, parallel lines, line segments, angles, and circles by drawing, modeling, identifying or creating.


Mathematics
MA2019 (2019)
Grade: 9-12
Geometry with Data Analysis
31. Justify whether conjectures are true or false in order to prove theorems and then apply those theorems in solving problems, communicating proofs in a variety of ways, including flow chart, two-column, and paragraph formats.

a. Investigate, prove, and apply theorems about lines and angles, including but not limited to: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; the points on the perpendicular bisector of a line segment are those equidistant from the segment's endpoints.

b. Investigate, prove, and apply theorems about triangles, including but not limited to: the sum of the measures of the interior angles of a triangle is 180?; the base angles of isosceles triangles are congruent; the segment joining the midpoints of two sides of a triangle is parallel to the third side and half the length; a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem using triangle similarity.

c. Investigate, prove, and apply theorems about parallelograms and other quadrilaterals, including but not limited to both necessary and sufficient conditions for parallelograms and other quadrilaterals, as well as relationships among kinds of quadrilaterals.

Example: Prove that rectangles are parallelograms with congruent diagonals.

Unpacked Content
Evidence Of Student Attainment:
Students:
  • Make, explain, and justify (or refute) conjectures about geometric relationships with and without technology.
  • Explain the requirements of a mathematical proof.
    1. Present a complete mathematical proof of geometry theorems including the following: vertical angles are congruent. when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent. points on a perpendicular bisector of a line segment are exactly those equidistant from the segment's endpoints.
      Critique proposed proofs made by others.
    2. Present a complete mathematical proof of geometry theorems about triangles, including the following: measures of interior angles of a triangle sum to 180o. base angles of isosceles triangles are congruent. the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length. the medians of a triangle meet at a point.
      Critique proposed proofs made by others.
    3. Present a complete mathematical proof of geometry theorems about parallelograms, including the following: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals.
      Critique proposed proofs made by others.
Teacher Vocabulary:
  • Same side interior angle
  • Consecutive interior angle
  • Vertical angles
  • Linear pair
  • Adjacent angles
  • Complementary angles
  • Supplementary angles
  • Perpendicular bisector
  • Equidistant
  • Theorem Proof
  • Prove
  • Transversal
  • Alternate interior angles
  • Corresponding angles
  • Interior angles of a triangle
  • Isosceles triangles
  • Equilateral triangles
  • Base angles
  • Median
  • Exterior angles
  • Remote interior angles
  • Centroid
  • Parallelograms
  • Diagonals
  • Bisect
Knowledge:
Students know:
  • Requirements for a mathematical proof.
  • Techniques for presenting a proof of geometric theorems.
Skills:
Students are able to:
  • Communicate logical reasoning in a systematic way to present a mathematical proof of geometric theorems.
  • Generate a conjecture about geometric relationships that calls for proof.
Understanding:
Students understand that:
  • Proof is necessary to establish that a conjecture about a relationship in mathematics is always true, and may also provide insight into the mathematics being addressed.

Alabama Alternate Achievement Standards
AAS Standard:
M.G.AAS.10.31a When given an isosceles triangle and a measure of a leg or base angle, identify the measure of the other leg or base angle.
M.G.AAS.10.31b When given a parallelogram and the measure of one side or one angle, identify the measure of the opposite side or angle.


Tags: angle, arc, circle, compass, constructions, distance, equilateral triangle, geometry, hexagon, inscribe, line segment, parallel line, parallelogram, perpendicular line, polygon, proofs, reflections, rotations, square, straightedge, theorems, transformations, translations, trapezoid
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Author: Hannah Bradley