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.
| Content | Skills | Learning Targets | Big Ideas | Essential Questions |
|---|---|---|---|---|
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Geometric Objects
Undefined Terms
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1 - Know (definitions angle, circle, perpendic-ular lines, parallel lines, line segment)
1 - Know (undefined terms or notions point, line, line distance, arc distance or length)
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1 - I can know (or state) the definitions of angle, circle, perpendic- ular lines, parallel lines, and line segment.
1 - I can know the undefined terms or notions of point, line, distance on a line, and distance around a circular arc.
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A. The undefined notions of point, line, distance along a line, and distance around a circular arc serve as the basis for precise definitions of geometric objects and shapes.
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A.1 How are undefined notions or terms in Geometry considered?
A.2 How can undefined notions serve as the basis for precise definitions of Geometric objects?
A.3 How do a Geometric object’s basic characteristics relate to its precise definition?
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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).
| Content | Skills | Learning Targets | Big Ideas | Essential Questions |
|---|---|---|---|---|
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Transformations in the -Plane
Preservation
Non-Preservation
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2 - Represent (transforma-tions in the plane)
2 - Describe (transformations as functions using points in plane as input and generating other points as output)
4 - Compare (transformations that preserve distance and angle to those that do not, including translations versus horizontal stretch)
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2 - I can represent transformations in the plane with transparencies and software.
2 - I can describe transformations as functions that use points in the plane as input and generate other points as output.
4 - I can compare transformations that preserve distance and angle to those that do not, including translations versus horizontal stretches.
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A. Transformations of geometric objects in the plane can be represented in various ways, including transparencies and geometry software.
B. Transformations can be considered as a function that takes points in the plane as inputs and generates other points as outputs.
C. Some transformations in the plane preserve distance and angle while others do not.
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A.1 By what methods can transformations of geometric objects in the plane be accomplished?
B.1 Why can transformations be considered functions?
C.1 What is a translation?
C.2 What is a horizontal stretch?
C.3 How do some transformations in the plane preserve distance and angle?
C.4 How are transformations that preserve distance and angle alike or different from transformations that do not?
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Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.
| Content | Skills | Learning Targets | Big Ideas | Essential Questions |
|---|---|---|---|---|
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Rotations
Reflections
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2 - Describe (rotations and reflections of a given rectangle, parallelogram, trapezoid, or regular polygon that carries it onto itself)
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2 - I can describe the rotations and reflections of a given rectangle, parallelogram, trapezoid, or regular polygon that carries it onto itself.
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A. For a given geometric object, certain rotations and reflections will carry (or map) the object onto itself.
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A.1 What is a rotation?
A.2 What is a reflection?
A.3 For a specific geometric object, how can the rotations and reflections that carry it onto itself be determined?
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Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.
| Content | Skills | Learning Targets | Big Ideas | Essential Questions |
|---|---|---|---|---|
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Geometric Concepts
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6 - Develop (definitions of rotations, reflections, translations, in terms of angles, circles, perpendicular lines, parallel lines, and line segments)
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6 - I can develop definitions of rotations, reflections, and translations as they relate to angles, circles, perpendicular lines, parallel lines, and line segments.
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A. Rotations, reflections, and translations of geometric objects can be defined in terms of angles, circles, perpendicular lines, and line segments.
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A. 1 What is the relationship among rotations, reflections, and translations of Geometric objects and the terms angles, circles, perpendicular lines, and line segments?
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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.
| Content | Skills | Learning Targets | Big Ideas | Essential Questions |
|---|---|---|---|---|
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Geometric figure
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3 - Draw (a transformed figure, given a geometric figure and a rotation, reflection, or translation)
3 - Use (graph paper, tracing paper, or geometry software)
6 - Specify (a sequence of transforma-tions that will carry a given figure onto another.)
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3 - I can draw a transformed figure when I am given a geometric figure, and a rotation, reflection, or translation with the use of graph paper, tracing paper, or geometry software.
6 - I can specify a sequence of transformations that will carry a given figure onto another.
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A. A rotation, reflection, or translation of a geometric figure into a transformed figure can be accomplished using graph paper, tracing paper, or geometry software.
B. The transformations that carry a given figure onto another can be specified by a sequence of transformations.
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A.1 Given a geometric figure, how can a transformed figure be drawn?
B.1 How can a sequence of transformations that carries a given figure onto another be specified?
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Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent.
| Content | Skills | Learning Targets | Big Ideas | Essential Questions |
|---|---|---|---|---|
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Rigid Motions
descriptions
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3 - Use (geometric descriptions of rigid motions)
3 - Transform (figures)
6 - Predict (effect of a rigid motion on a figure)
3 - Use (definition of congruence)
5 - Decide (if two figures are congruent)
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3 - I can use geometric descriptions of rigid motions to transform figures.
6 - I can use geometric descriptions of rigid motions to predict the effect of a given rigid motion on a given figure.
5 - I can decide if two figures are congruent using the definition of congruence in terms of rigid motion.
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A. Rigid motions on a figure transform the figure in the plane.
B. The effect of a rigid motion on a figure can be predicted by the geometric description of the motion.
C. The definition of congruence (in terms of rigid motion) can be used to decide if two given figures are congruent.
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A.1 How can the geometric descriptions of rigid motion produce a transformed figure?
B.1 How can the geometric descriptions of rigid motions suggest the effect on a given figure?
C.1 How can the definition of congruence (in terms of rigid motion) be used to decide if two given figures are congruent?
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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.
| Content | Skills | Learning Targets | Big Ideas | Essential Questions |
|---|---|---|---|---|
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Rigid motions
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3 - Use (definition of congruence in terms of rigid motion)
3 - Show (that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent)
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3 - I can show that two triangles are congruent if corresponding pairs of sides and corresponding pairs of angles are congruent.
3 - I can show that if corresponding pairs of sides and corresponding pairs of angles are congruent in two triangles, then the triangles are congruent.
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A. The definition of congruence (in terms of rigid motion) can be used to show that two triangles are congruent when corresponding pairs of sides and corresponding pairs of angles are congruent.
B. The definition of congruence (in terms of rigid motion) can be used to show that when corresponding pairs of sides and corresponding pairs of angles are congruent in two triangles, then the triangles are congruent.
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A.1 How can the definition of congruence in terms of rigid motion be used to show that two triangles are congruent?
A.2 What is the meaning of if and only if in math?
B.1 How can two triangles be shown to be congruent when it is given that corresponding pairs of sides and corresponding pairs of angles are congruent?
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Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.
| Content | Skills | Learning Targets | Big Ideas | Essential Questions |
|---|---|---|---|---|
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Triangle Congruence Criteria
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4 - Explain (how criteria for triangle congruence, ASA (Angle-Side-Angle)
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4 - I can explain how the ASA (Angle-Side-Angle) triangle congruence criteria follows from the definition of congruence in terms of rigid motions.
4 - I can explain how the SAS (Side-Angle-Side) triangle congruence criteria follows from the definition of congruence in terms of rigid motions.
4 - I can explain how the SSS (Side-Side-Side) triangle congruence criteria follows from the definition of congruence in terms of rigid motions.
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A. The three criteria for triangle congruence (ASA, SAS, and SSS) are based on the definition of congruence in terms of rigid motion.
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A.1 Why does the triangle congruence criterion ASA (Angle-Side-Angle) follow from the definition of congruence in terms of rigid motions?
A.2 Why does the triangle congruence criterion SAS (Side Angle-Side) follow from the definition of congruence in terms of rigid motions?
A.3 Why does the triangle congruence criterion SSS (Side-Side-Side) follow from the definition of congruence in terms of rigid motions?
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Prove theorems about lines and angles.
Theorems include: 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.
| Content | Skills | Learning Targets | Big Ideas | Essential Questions |
|---|---|---|---|---|
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3 - Prove (theorems about lines and angles)
3 - Prove (vertical angles are congruent)
3 - Prove (when parallel lines are crossed by a transversal, alternate interior angles are congruent)
3 - Prove (when parallel lines are crossed by a transversal, corresponding angles are congruent)
3 - Prove (points on a perpendicular bisector of a line segment are equidistant from the endpoints of the segment)
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3 - I can prove theorems about lines and angles.
3 - I can prove that vertical angles are congruent.
3 - I can prove that when parallel lines are crossed by a transversal, alternate interior angles are congruent.
3 - I can prove that when parallel lines are crossed by a transversal, corresponding angles are congruent.
3 - I can prove that points on the perpendicular bisector of a line segment are equidistant from the endpoints of the segment.
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A. Relationships among lines and angles expressed as theorems can be demonstrated through mathematical proofs.
B. Specific relationships related to vertical angles, angles formed by parallel lines and transversals, and points on a perpendicular bisector of a segment can be proven mathematically by a variety of methods.
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A.1 What is a theorem?
A.2 What is a proof?
A.3 How can theorems about lines and angles be proven?
B.1 How can specific theorems related to vertical angles, angles formed by parallel lines and transversals, and points on a perpendicular bisector of a segment be proven?
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Prove theorems about triangles.
Theorems include: measures of interior angles of a triangle sum to 180°; 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.
| Content | Skills | Learning Targets | Big Ideas | Essential Questions |
|---|---|---|---|---|
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Triangle Theorems
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3 - Prove
3 - Prove (the measures of the interior angles of a triangle sum to 180°)
3 - Prove (base angles of an isosceles triangle are congruent)
3 - Prove (the segment that joins the midpoints of two sides of a triangle is parallel to and half the length of the third side)
3 - Prove (the medians of a triangle meet at a point)
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3 - I can prove theorems about triangles.
3 - I can prove the measures of the interior angles of a triangle sum to 180°.
3 - I can prove base angles of an isosceles triangle are congruent.
3 - I can prove that the segment that joins the midpoints of two sides of a triangle is parallel to and half the length of the third side.
3 - I can prove the medians of a triangle meet at a point.
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A. Specific properties of triangles, isosceles triangles, and segments of triangles can be proven mathematically by a variety of methods.
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A.1 How can specific theorems related to triangles, isosceles triangles, and segments of triangles be proven?
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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.
| Content | Skills | Learning Targets | Big Ideas | Essential Questions |
|---|---|---|---|---|
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Parallelogram Theorems
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3 - Prove (parallelogram theorems)
3 - Prove (opposite sides of parallelograms are congruent)
3 - Prove (opposite angles of a parallelogram are congruent)
3 - Prove (diagonals of a parallelogram are congruent, and the converse)
3 - Prove (that rectangles are parallelograms with congruent diagonals)
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3 - I can prove theorems about parallelograms.
3 - I can prove opposite sides of parallelograms are congruent.
3 - I can prove opposite angles of a parallelogram are congruent.
3 - I can prove diagonals of a parallelogram are congruent, and the converse.
3 - I can prove (that rectangles are parallelograms with congruent diagonals.
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A. Properties of parallelograms expressed as theorems can be demonstrated by mathematical proofs.
B. Some proofs require that the converse of a theorem be proven in addition to the theorem itself.
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A.1 How can theorems related to parallelograms be proven?
B.1 What is the converse of theorem?
B.2 How can the converse of a theorem be proven?
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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.
| Content | Skills | Learning Targets | Big Ideas | Essential Questions |
|---|---|---|---|---|
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Constructions
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3 - Make (formal geometric constructions by using compass and straightedge, string, reflective devices, paper folding, and geometric software)
3 - Copy (a segment)
3 - Copy (an angle)
3 - Bisect (a segment)
3 - Bisect (an angle)
3 - Construct (perpendicular lines)
3 - Construct (perpendicular bisector of a line segment)
3 - Construct (a line parallel to a given line through a point not on the line)
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3 - I can make formal geometric constructions using compass and straightedge, string, reflective devices, paper folding, and geometric software.
3 - I can copy a segment.
3 - I can copy an angle.
3 - I can bisect a segment.
3 - I can bisect an angle.
3 - I can construct perpendic-ular lines.
3 - I can construct a perpendic-ular bisector of a line segment.
3 - I can construct a line parallel to a given line through a point not on the line.
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A. Basic formal geometric constructions can be accomplished by using a variety of tools.
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A.1 How can formal geometric constructions be accomplished?
A.2 What tools are available to perform formal geometric constructions?
A.3 How are various constructions related to one another?
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Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.
| Content | Skills | Learning Targets | Big Ideas | Essential Questions |
|---|---|---|---|---|
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Constructions
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4 - Construct (an equilateral triangle inscribed in a circle)
4 - Construct (a square inscribed in a circle)
4 - Construct (a regular hexagon inscribed in a circle)
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4 - I can construct an equilateral triangle inscribed in a circle.
4 - I can construct a square inscribed in a circle.
4 - I can construct a regular hexagon inscribed in a circle.
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A. Certain figures can be constructed as inscribed figures in a circle by composing a sequence of basic constructions.
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A.1 What is the meaning of inscribed?
A.2 Why can certain figures be constructed as inscribed figures in a circle while other figures cannot?
A.3 How can an equilateral triangle, a square, and a regular hexagon inscribed in a circle be constructed?
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