Subject

Geometry

Nine Week

1st Nine Weeks (1st Quarter)

Standard

G.CO.A.1, 2, 3, 4, 5 Experiment with transformations in the plane.

G.CO.B.6, 7 Understand congruence in terms of rigid motions

G.CO.C.9, 10 Prove geometric theorems

G.CO.D.12 Make geometric constructions.

G.MG.A.1, 2 Apply geometric concepts in modeling situations.

G.GPE.B.2, 3, 4, 5 Use coordinates to prove simple geometric theorems algebraically.

The following practice standards will be used throughout the quarter:

1. Make sense of problems and persevere in solving them.

2. Reason abstractly and quantitatively.

3. Construct viable arguments and critique the reasoning of others.

4. Model with mathematics.

5. Use appropriate tools strategically.

6. Attend to precision.

7. Look for and make use of structure.

8. Look for and express regularity in repeated reasoning.

Objectives

(Student Friendly “I can” statements)

The learner will...

-Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, plane, distance along a line, and distance along a line, and distance around a circular arc

(I can describe and name the undefined notions of points, lines, and planes. I can precisely define line segments, rays, parallel lines, perpendicular lines, and skew lines and describe their characteristics.)

-Represent transformations in the plane in multiple ways, including technology. Describe transformations as functions that take points in the plane (pre-image) as inputs and give other points (image) as outputs. Compare transformations that preserve distance and angle measure to those that do not (e.g., translation versus horizontal stretch).

(I can draw transformations of reflections, rotations, translations, and combinations of these using graph paper, transparencies, and patty paper, both on and off the coordinate plane. I can determine the coordinates for the image of a figure when a transformation rule is applied to the pre-image. I can explain rigid motion as motion that preserves distance and angle measure. I can distinguish between congruence transformations that are rigid (reflections, rotations, translations) and those that are not (dilations or rigid motions followed by dilations)).

-Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.

(I can determine if a figure has rotational symmetry (maps on to

itself), and if so, determine the angle of rotation.

I can determine if a figure has line symmetry, and if so, find all the

lines of symmetry.)

- Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.

(I can define and describe transformations in terms of angles, circles, lines, and line segments (i.e. reflecting a figure over a line or parallel lines, rotating a figure 180°, etc.). I can use a rule to define reflections, rotations, and translations on the coordinate plane (i.e. (𝑥, 𝑦) → (−𝑦, 𝑥) when rotating 90° counterclockwise about the origin, etc.))

- Given a geometric figure and a rigid motion, draw the image of the figure in multiple ways, including technology. Specify a sequence of rigid motions that will carry a given figure onto another.

(I can predict and verify the sequence of transformations (a composition) that will map a figure onto another.)

- 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 determine informally if they are congruent.

(I can define congruent figures as figures that have the same shape and size and state the composition of rigid motions (reflections, rotations, translations, and combinations of these) that will map one congruent figure onto the other.)

- 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.

(I can determine if two figures are congruent by determining if rigid motions will turn one figure onto the other (preserving distance and angle measure). I can explain and prove that in a pair of congruent triangles, corresponding sides are congruent and corresponding angles are congruent.

- Prove theorems about lines and angles. Proving includes, but is not limited to, completing partial proofs; constructing two-column or paragraph proofs; using transformations to prove theorems; analyzing proofs; and critiquing completed proofs. Theorems include but are not limited to: 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. Theorems include but are not limited to: 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.

- Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometry 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. Constructions include but are not limited to: Copying a segment; copying an angle; 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 and constructing the following objects inscribed in a circle: an equilateral triangle, square, and a regular hexagon.

(I can prove vertical angles are congruent. I can prove and apply theorems about the angles formed by parallel lines and a transversal (corresponding, alternate interior, same-side interior). (*ACT) I can prove points on a perpendicular bisector of a line segment are exactly equidistant from the segment’s endpoints.)

-Use coordinates to prove simple geometric theorems algebraically.

(I can use coordinate geometry to prove theorems algebraically)

-Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems.

(I can use slope to prove lines are parallel or perpendicular. I can find the equation of a line parallel or perpendicular to a given line that passes through a given point.)

-Find the point on a directed line segment between two points that partitions the segment in a given ratio.

(I can find the point on a line segment, given two endpoints, that divides the segment into a given ratio.)

-Know and use coordinates to compute perimeters of polygons and areas of triangles and rectangles.

(I can use coordinate geometry and the distance formula to find the area

and perimeters of polygons on the coordinate plane.)

-Use geometric shapes, their measures, and their properties to describe objects.

(I can use geometric shapes, their measures and their properties to describe objects.)

-Apply geometric methods to solve real-world problems.

(I can apply geometric methods to solve design problems.)

Honors Addendum

Embed the Honors Addendum within the regular Scope & Sequence.

-Understand and apply the rules of logic as they relate to proving geometric theorems.

-Investigate forms of non-Euclidean geometry.

-Understand and apply the rules of logic as they relate to proving geometric theorems.

-Construct truth tables to determine the truth-value of logical statements.

(I can find counterexamples to disprove conjectures.

I can write and analyze biconditional statements.

I can write the inverse, converse, and contrapositive of a conditional

statement.

I can order statements based on logic when constructing my proof.

I can research and explain the other types of geometry besides

Euclidean such as spherical, hyperbolic, and elliptical.

I can identify, write, and analyze the truth-value of conditional statements.

I can apply the Law of Detachment and Law of Syllogism in logical reasoning.

I can follow logical steps to write a simple indirect proof.

I can construct truth tables to determine the truth-value of logical statements.)

Topics

•

Slope and parallel lines

•

Slope and perpendicular lines

•

Coordinate proof using distance with segments and triangles

•

Coordinate proofs using distance with quadrilaterals

•

Perimeter and area on the coordinate plane

•

Proving figures are similar using transformations

•

transversals and parallel lines

•

proving lines are parallel

•

perpendicular line

•

equations of parallel and perpendicular lines

•

segment length and midpoints

•

angle measures and angle bisectors

•

representing and describing transformations

•

reasoning and proof

•

translations, reflections, rotations

•

investigating symmetry

•

sequences of transformations

•

proving figures are congruent using rigid motions

•

corresponding parts of congruent figures are congruent

•

angles formed by intersecting lines

•

exploring what makes triangles congruent

Major Assignment/s

Pre-assessments; ongoing module assessments to periodically measure understanding; unit assessment

Instructional Materials

Computer; manipulatives; geometers sketch pad; projection screen; laser printer; KUTA software; HRM AGA resources; dry erase boards; HRM textbook; protractor; compass; ruler, Promethean Board and clickers

Field Trip/s

NONE