ALEX Classroom Resource

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.

Students know:

• Characteristics of transformations (translations, rotations, reflections, and dilations).
• Methods for representing transformations.
• Characteristics of functions.
• Conventions of functions with mapping notation.

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.

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.

Learning Objectives:
GEO.21.1: Define distance, angle, input, output, plane, translation, reflection, rotation, and dilation.
GEO.21.2: Compare transformation that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch).
GEO.21.3: Describe transformations as functions that take points in a plane as inputs and give other points as outputs.
GEO.21.4: Represent transformation in the plane.
GEO.21.5: Generate an input output table.
GEO.21.6: Compare the distance and angles of the figures from the pre-image to the image.
GEO.21.7: Measure distance and angle(s) of an image.

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.

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.

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.

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.

Learning Objectives:
GEO.22.1: Define rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.
GEO.22.2: Describe the effects of rotations, reflection, and translations on two dimensional figures using coordinates.
GEO.22.3: Illustrate figures transformed by a rotation, reflection or translation.
GEO.22.4: Describe the process of transforming a given figure.
GEO.22.5: Graph a figure on a coordinate plane.

Students:

• Use geometric terminology (angles, circles, perpendicular lines, parallel lines, and line segments) to describe the series of steps necessary to produce a rotation, reflection, or translation.
• Use these descriptions to communicate precise definitions of rotation, reflection, and translation.

Students know:

• Characteristics of transformations (translations, rotations, reflections, and dilations).
• -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).

Students are able to:

• Accurately perform rotations, reflections, and translations on objects with and without technology.
• Communicate the results of performing transformations on objects.
• Use known and developed definitions and logical connections to develop new definitions.

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.

Learning Objectives:
GEO.23.1: Define rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.
GEO.23.2: Describe the effects of rotations, reflection, and translations on two dimensional figures using coordinates.
GEO.23.3: Describe the effects of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.
GEO.23.4: Describe the process of transforming a given figure.
GEO.23.5: Illustrate figures transformed by a rotation, reflection or translation.
GEO.23.6: Recognize the type of transformation from a pre-image to an image.

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.

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.

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.

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.

Learning Objectives:
GEO.24.1: Define congruence.
GEO.24.2: Applying the definition of congruence determine if two figures are congruent.
GEO.24.3: Illustrate a sequence of rigid motions on a coordinate plane that maps one figure to another.
GEO.24.4: Illustrate a vertical and horizontal shift on a coordinate plane.
Example: Rectangle PQRS has vertices P(-3,5), Q(-4,2), R (3,0), 5(4,3). Translate PQRS vertically 3 units.
GEO.24.5: Recognize composition of transformations.
GEO.24.6: Graph a figure on a coordinate plane.

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.

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.

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.

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).

Learning Objectives:
GEO.25.1: Define congruent, corresponding, triangles, angles, and the concept of if and only if.
GEO.25.2: Compare angles and sides of two triangles to determine congruency.
GEO.25.3: Determine the lengths of sides and the measures of angles in triangles.
GEO.25.4: Identify corresponding parts of triangles.

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.

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).

Students are able to:

• Use known and developed definitions and logical connections to develop new definitions.

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.

Learning Objectives:
GEO.30.1: Define angle, circle, perpendicular line, parallel line, line segment, and distance.
GEO.30.2: Describe angle, circle, perpendicular line, parallel line, line segment, and distance.
GEO.30.3: Illustrate a point, line, distance along a line, and distance around a circular arc.
GEO.30.4: Identify angle, circle, perpendicular line, parallel line, line segment, and distance.

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.

Students know:

• Requirements for a mathematical proof.
• Techniques for presenting a proof of geometric theorems.

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.

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.

Learning Objectives:
GEO.31.1: Define vertical angle, transversal, parallel lines, alternate interior angles, corresponding angles, perpendicular bisector, line segment, equidistant, endpoints, interior angles of a triangle, base angles of isosceles triangles, isosceles triangles, midpoint, median, intersection, opposite sides, opposite angles, diagonals, parallelogram, bisector, and converse.
GEO.31.2: Develop a process that demonstrates the logical order of properties to form a proof.
GEO.31.3: Arrange statements to form a logical order.
GEO.31.4: Identify measures of vertical angles, alternate interior angles, corresponding angles, measures of interior angles of a triangle, base angles of isosceles triangles, isosceles triangles, midpoint, and median.
GEO.31.5: Illustrate vertical angle, transversal, parallel lines, alternate interior angles, corresponding angles, perpendicular bisector, line segment, equidistant, endpoints, interior angles of a triangle, base angles of isosceles triangles, isosceles triangles, midpoint, median, intersection, opposite sides, opposite angles, diagonals, parallelograms, bisectors, and their properties.
GEO.31.6: Find the measure of the third interior angle of a triangle when given the measure of the other two interior angles.