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Mathematics Department Geometry Curriculum Map Volusia County Schools May 2014

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Curriculum Map

2014 - 2015

Mathematics Florida Standards Volusia County Curriculum Maps are revised annually and updated throughout the year.

The learning goals are a work in progress and may be modified as needed.


Common Core State Standards Standards for Mathematical Practice

1. Make sense of problems and persevere in solving them. (MAFS.K12.MP.1)

Solving a mathematical problem involves making sense of what is known and applying a thoughtful and logical process

which sometimes requires perseverance, flexibility, and a bit of ingenuity.

2. Reason abstractly and quantitatively. (MAFS.K12.MP.2)

The concrete and the abstract can complement each other in the development of mathematical understanding:

representing a concrete situation with symbols can make the solution process more efficient, while reverting to a concrete

context can help make sense of abstract symbols.

3. Construct viable arguments and critique the reasoning of others. (MAFS.K12.MP.3)

A well-crafted argument/critique requires a thoughtful and logical progression of mathematically sound statements and

supporting evidence.

4. Model with mathematics. (MAFS.K12.MP.4)

Many everyday problems can be solved by modeling the situation with mathematics.

5. Use appropriate tools strategically. (MAFS.K12.MP.5)

Strategic choice and use of tools can increase reliability and precision of results, enhance arguments, and deepen

mathematical understanding.

6. Attend to precision. (MAFS.K12.MP.6)

Attending to precise detail increases reliability of mathematical results and minimizes miscommunication of mathematical

explanations.

7. Look for and make use of structure. (MAFS.K12.MP.7)

Recognizing a structure or pattern can be the key to solving a problem or making sense of a mathematical idea.

8. Look for and express regularity in repeated reasoning. (MAFS.K12.MP.8)

Recognizing repetition or regularity in the course of solving a problem (or series of similar problems) can lead to results

more quickly and efficiently.


Geometry, Geometry Honors: Florida Standards

The fundamental purpose of the course in Geometry is to formalize and extend students’ geometric experiences from the middle grades. Students explore more complex geometric situations and deepen their explanations of geometric relationships, moving towards formal mathematical arguments. Important differences exist between this Geometry course and the historical approach taken in Geometry classes. For example, transformations are emphasized early in this course. Close attention should be paid to the introductory content for the Geometry conceptual category found in the high school CCSS. The Mathematical Practice Standards apply throughout each course and, together with the content standards, prescribe that students experience mathematics as a coherent, useful, and logical subject that makes use of their ability to make sense of problem situations. The critical areas, organized into five units are as follows. Topic 1-Congruence, Proof, and Constructions: In previous grades, students were asked to draw triangles based on given measurements. They 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. They 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 triangles, quadrilaterals, and other polygons. They apply reasoning to complete geometric constructions and explain why they work. (Units 1, 2, and 3)

Topic 2-Similarity, Proof, and Trigonometry: Students apply their earlier experience with dilations and proportional reasoning to build a formal understanding of

similarity. They identify criteria for similarity of triangles, use similarity to solve problems, and apply similarity in right triangles to understand right triangle trigonometry, with particular attention to special right triangles and the Pythagorean theorem. Students develop the Laws of Sines and Cosines in order to find missing measures of general (not necessarily right) triangles, building on students’ work with quadratic equations done in the first course. They are able to distinguish whether three given measures (angles or sides) define 0, 1, 2, or infinitely many triangles. (Units 4 and 5)

Topic 3-Extending to Three Dimensions: Students’ experience with two-dimensional and three-dimensional objects is extended to include informal explanations

of circumference, area and volume formulas. Additionally, students apply their knowledge of two-dimensional shapes to consider the shapes of cross-sections and the result of rotating a two-dimensional object about a line. (Units 7 and 8)

Topic 4-Connecting Algebra and Geometry Through Coordinates: Building on their work with the Pythagorean theorem in 8th grade to find distances, students

use a rectangular coordinate system to verify geometric relationships, including properties of special triangles and quadrilaterals and slopes of parallel and perpendicular lines, which relates back to work done in the first course. Students continue their study of quadratics by connecting the geometric and algebraic definitions of the parabola. (Unit 7)

Topic 5-Circles With and Without Coordinates: In this unit students prove basic theorems about circles, such as a tangent line is perpendicular to a radius,

inscribed angle theorem, and theorems about chords, secants, and tangents dealing with segment lengths and angle measures. They study relationships among segments on chords, secants, and tangents as an application of similarity. In the Cartesian coordinate system, students use the distance formula to write the equation of a circle when given the radius and the coordinates of its center. Given an equation of a circle, they draw the graph in the coordinate plane, and apply techniques for solving quadratic equations, which relates back to work done in the first course, to determine intersections between lines and circles or parabolas and between two circles. (Unit 9)


Geometry/Geometry Honors: Common Core State Standards At A Glance

First Quarter Second Quarter Third Quarter Fourth Quarter

DSA

Unit 1- Definitions and Constructions MAFS.912.G-CO.1.1 MAFS.912.G-CO.4.12 MAFS.912.G-CO.3.9

Unit 2- Transformations MAFS.G-CO.1.2 MAFS.G-CO.1.3 MAFS.G-CO.1.4 MAFS.G-CO.1.5 MAFS.G-CO.2.6 MAFS.G-SRT.1.1

DIA #1 (Units 1 and 2)

Unit 3-Triangles and Congruence MAFS.912.G-CO.3.10 MAFS.912.G-CO.2.7 MAFS.912.G-CO.2.8 MAFS.912.G-SRT.2.5

DIA #2 (Units 3)

Unit 4-Similarity MAFS.912.G-SRT.1.1 MAFS.912.G-SRT.1.2 MAFS.912.G-SRT.1.3 MAFS.912.G-SRT.2.5 MAFS.912.G-SRT.2.4

DIA #3 (Unit 4) Unit 5-Right Triangles and Trigonometry MAFS.912.G-SRT.2.4 MAFS.912.G-SRT.3.6 MAFS.912.G-SRT.3.7 MAFS.912.G-SRT.3.8 *MAFS.912.G-SRT.4.10 *MAFS.912.G-SRT.4.11

SSA

Unit 6-Quadrilaterals and Coordinate Geometry MAFS.912.G-CO.3.11 MAFS.912.G-GPE.2.4 MAFS.912.G-GPE.2.5 MAFS.912.G-GPE.2.6

DIA #4 (Unit 6) Unit 7-Two-Dimensional Measurements MAFS.912.G-GMD.1.1 MAFS.912.G-GPE.2.7 MAFS.912.G-MG.1.1 MAFS.912.G-MG.1.2 MAFS.912.G-MG.1.3

DIA #5 (Unit 7) Unit 8- Three-Dimensional Measurements MAFS.912.G-GMD.2.4 MAFS.912.G-GMD.1.3 MAFS.912.G-GMD.1.1 *MAFS.912.G-GMD.1.2 MAFS.912.G-MG.1.1 MAFS.912.G-MG.1.2 MAFS.912.G-MG.1.3

DIA #6 (Unit 8)

Unit 9–Circles with and without Coordinates MAFS.912.G-C.1.1 MAFS.912.G-C.1.2 MAFS.912.G-C.1.3 MAFS.912.G-CO.4.13 MAFS.912.G-C.2.5 MAFS.912.G-GPE.1.1 MAFS.912.G-GPE.2.4

DIA #7 (Unit 9) *MAFS.912.G-C.1.4 *MAFS.912.G-GPE.1.2 *MAFS.912.G-GPE.1.3

PRACTICE EOC

Unit 10–Algebra 2 Preparation MAFS.912.F-IF.3.7a, b MAFS.912.F-IF.3.8a

* Standards highlighted are for HONORS only*


The following English Language Arts Florida Standards should be taught throughout the course: LAFS.910.RST.1.3: Follow precisely a complex multistep procedure when carrying out experiments, taking measurements or performing tasks, attending to special cases or exceptions defined in the text. LAFS.910.RST.2.4: Determine the meaning of symbols, key terms, and other domain-specific words and phrases as they are used in context and topics. LAFS.910.RST.3.7: Translate quantitative or technical information expressed in words in a text into visual form and translate information expressed visually or mathematically into words. LAFS.910.SL.1.1: Initiate and participate effectively in a range of collaborative discussions with diverse partners. LAFS.910.SL.1.2: Integrate multiple sources of information presented in diverse media or formats evaluating the credibility and accuracy of each source. LAFS.910.SL.1.3: Evaluate a speaker’s point of view, reasoning, and use of evidence and rhetoric, identifying any fallacious reasoning or exaggerated or distorted evidence. LAFS.910.SL.2.4: Present information, findings and supporting evidence clearly, concisely, and logically such that listeners can follow the line of reasoning. LAFS.910.WHST.1.1: Write arguments focused on discipline-specific content. LAFS.910.WHST.2.4: Produce clear and coherent writing in which the development, organization, and style are appropriate to task, purpose, and audience.

LAFS.910.WHST.3.9: Draw evidence from informational texts to support analysis, reflection, and research.


Course: Geometry/ Geometry Honors Unit 1-Defintions and Constructions

Essential Question(s):

How can I construct the basic tools of geometry?

Standard The students will:

Learning Goals I can:

Remarks

MAFS.912.G-CO.1.1 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. SMP #6

identify the undefined notions used in geometry (point, line, plane, distance) and describe their characteristics

identify angles, perpendicular lines, parallel lines, rays, and line segments.

define angles, perpendicular lines, parallel lines, rays, and line segments precisely using the undefined terms and “if-then” and “if-and-only-if” statements.

Vocabulary should include: point, line, plane, distance, angle, circle, perpendicular, parallel, line segment, ray, vertex, equidistant, intersect, right angle. A brief review of polygon names and properties could be done at this point.

MAFS.912.G-CO.4.12 Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc. This includes: 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.) SMP #5, #6

identify the tools used in formal constructions.

use tools and methods to precisely copy a segment.

use tools and methods to precisely copy an angle.

use tools and methods to precisely bisect a segment.

use tools and methods to precisely bisect an angle.

construct perpendicular lines and bisectors.

construct a line parallel to a given line through a point not on the line.

Students should be able to informally perform the constructions listed using string, reflective devices, paper folding, and/or dynamic geometric software. Emphasize the need for precision and accuracy when doing constructions (i.e. a sketch is not the same as a construction).

MAFS.912.G-CO.3.9 Prove theorems about lines and angles; use theorems about lines and angles to solve problems. (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.) SMP #2, #3

identify and use the properties of congruence and equality (reflexive, symmetric, transitive) in my proofs.

order statements based on the Law of Syllogism when constructing my proof.

interpret geometric diagrams and identifying what can and cannot be assumed.

apply theorems, postulates, or definitions to prove theorems about lines and angles.

With this standard, students should also learn definitions of complementary/supplementary angles, as well as the angle addition postulate.


Course: Geometry/ Geometry Honors

Unit 2-Transformations

Essential Question(s): In what ways can I precisely move a figure in space?



Remarks

MAFS.912.G-CO.1.2 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). SMP #6

draw transformations of reflections, rotations, translations, and combinations of these using graph paper, transparencies, and/or geometry software.

determine the coordinates for the image (output) of a figure when a transformation rule is applied to the pre-image (input).

distinguish between transformations that are rigid and those that are not.

-This unit will draw upon student’s prior knowledge of polygon names and properties. -Rigid transformations preserve distance and angle measure (reflections, rotations, translations, or combinations of those).

MAFS.912.G-CO.1.3 Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself. SMP #7

describe and illustrate how a rectangle, parallelogram, and isosceles trapezoid are mapped onto themselves using transformations.

calculate the number of lines of reflection symmetry and the degree of rotational symmetry of any regular polygon.

This is a discussion of symmetry.



Unit 2-Transformations




Remarks

MAFS.912.G-CO.1.4 Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. SMP #6

construct the definition of reflection, translation, and rotation.

construct the reflection definition by connecting any point on the pre-image to its corresponding point on the reflected image and describing the line segment’s relationship to the line of reflection.

construct the translation definition by connecting any point on the pre-image to its corresponding point on the translated image, and connecting a second point on the pre-image to its corresponding point on the translated image, and describing how the two segments are equal in length, point in the same direction, and are parallel.

construct the rotation definition by connecting the center of rotation to any point on the pre-image and to its corresponding point on the rotated image, and describing the measure of the angle formed and the equal measures of the segments that formed the angle as part of the definition.

The terms “mapping” and “under” are

used in special ways when studying

transformations. Students sometimes

confuse the terms “transformation”

and “translation.” Remind students

that that corresponding vertices have

to be listed in order so that

corresponding sides and angles can

be easily identified and that included

sides or angles are apparent.

MAFS.912.G-CO.1.5 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. SMP #5

draw a specific transformation when given a geometric figure and a rotation, reflection, or translation.

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

Students may confuse rotations and reflections and be unable to differentiate the two. Allowing them the opportunity to physically manipulate the shapes (such as with cut-outs or patty paper) can clear up misconceptions.



Unit 2-Transformations (cont)




Remarks

MAFS.912.G-CO.2.6 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. SMP #3

define rigid motions as reflections, rotations, translations and combinations of these, all of which preserve distance and angle measure.

define congruent figures as figures that have the same shape and size and state that a composition of rigid motions will map one congruent figure onto the other.

predict the composition of transformations that will map a figure onto a congruent figure.

determine if two figures are congruent by determining if rigid motions will turn one figure into the other.

Students may believe that all transformations, including dilations, are rigid motions or that any two figures that have the same area represent a rigid transformation. Provide counterexamples.

MAFS.912.G-SRT.1.1 Verify experimentally the

properties of dilations given

by a center and a scale

factor:

a. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged.

b. The dilation of a line segment is longer or shorter in the ratio given by the scale factor.

SMP #6, #8

define dilation.

perform a dilation with a given center and scale factor on a figure in the coordinate plane.

verify that when a side passes through the center of dilation, the side and its image lie on the same line.

verify that corresponding sides of the pre-image and images are parallel.

verify that a side length of the image is equal to the scale factor multiplied by the corresponding side length of the pre-image.

Relate dilations to things students might already know: dilated pupils, scale drawings, etc.



Unit 3-Congruence

Essential Question(s): How can I prove that two polygons are the exact same shape?



Remarks

MAFS.912.G-CO.3.10 Prove theorems about triangles; use theorems about triangles to solve problems. (Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; 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.) SMP #2, #3

interpret geometric diagrams (what can and cannot be assumed).

apply theorems, postulates, or definitions to prove theorems about triangles, including:

a) Measures of interior angles of a triangle sum

to 180; b) Base angles of isosceles triangles are

congruent; c) An exterior angle of a triangle is equal to the

sum of the measures of the remote interior angles.

d) The medians of a triangle meet at a point.

Properties and classifications of triangles should be discussed with this standard.

MAFS.912.G-CO.2.7 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. SMP #3

identify corresponding sides and corresponding angles of congruent triangles.

explain that in a pair of congruent triangles, corresponding sides are congruent (distance is preserved) and corresponding angles are congruent (angle measure is preserved).

demonstrate that when distance is preserved (corresponding sides are congruent) and angle measure is preserved (corresponding angles are congruent) the triangles must also be congruent.

Remind students of the transformations covered in the previous unit to demonstrate congruence. Use those transformations to remind students why AAA would not prove triangles congruent.



Unit 3-Congruence (cont) Essential Question(s):

How can I prove that two polygons are the exact same shape?



Remarks

MAFS.912.G-CO.2.8 Explain how the criteria for triangle congruence (ASA, SAS, SSS, and Hypotenuse-Leg) follow from the definition of congruence in terms of rigid motions. SMP #2, #3

list the sufficient conditions to prove triangles are congruent.

map a triangle with one of the sufficient conditions (e.g., SSS) onto the original triangle and show that corresponding sides and corresponding angles are congruent.

-Include AAS as a criterion for congruence. -Proofs (both formal and informal) showing triangle congruence should be included in this standard. -Students should be able to identify the transformations used to create congruent triangles AFTER triangles are proved congruent.

MAFS.912.G-SRT.2.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. SMP #1

apply triangle congruence and triangle similarity to solve problems (e.g., indirect measure, missing sides/angle measures, side splitting).

apply triangle congruence and triangle similarity to prove relationships in geometric figures.

-Use CPCTC (Corresponding Parts of Congruent Triangles are Congruent) to find missing angles/side lengths AFTER triangles are proved congruent. -Remind students not to assume pieces of triangles are congruent unless the information is given.


Course: Geometry/ Geometry Honors Unit 4-Similarity


How might the features of one figure be useful when solving problems about a similar figure?



Remarks

MAFS.912.G-SRT.1.1 Verify experimentally the properties of

dilations given by a center and a scale

factor:

c. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged.

d. The dilation of a line segment is longer or shorter in the ratio given by the scale factor.

SMP #6, #8

define dilation.

perform a dilation with a given center and scale factor on a figure in the coordinate plane.

verify that when a side passes through the center of dilation, the side and its image lie on the same line.

verify that corresponding sides of the pre-image and images are parallel.

verify that a side length of the image is equal to the scale factor multiplied by the corresponding side length of the pre-image.

In this unit, dilations should be brought up in the context of similarity.

MAFS.912.G-SRT.1.2 Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. SMP #3

define similarity as a composition of rigid motions followed by dilations in which angle measure is preserved and side length is proportional.

identify corresponding sides and corresponding angles of similar triangles.

demonstrate that in a pair of similar triangles, corresponding angles are congruent (angle measure is preserved) and corresponding sides are proportional.

determine that two figures are similar by verifying that angle measure is preserved and corresponding sides are proportional.

-Students think of similarity and congruence as separate and distinct categories. Remind students that congruent figures are just similar figures with a scale factor of 1:1. -Students should be able to identify transformations that occurred to create the similar figures (including the dilation).


Course: Geometry/ Geometry Honors Unit 4-Similarity (cont)


How might the features of one figure be useful when solving problems about a similar figure?



Remarks

MAFS.912.G-SRT.1.3 Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. SMP #3

show and explain that when two angle measures are known (AA), the third angle measure is also known (Third Angle Theorem).

conclude and explain that AA similarity is a sufficient condition for two triangles to be similar.

Students often confuse the triangle congruence theorems with the triangle similarity theorems. It may be necessary to go back and review SSS, SAS, ASA and HL theorems to distinguish them from AA~, SAS~ and SSS~.

MAFS.912.G-SRT.2.4 Prove theorems about triangles. (Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity.) SMP #3

apply theorems, postulates, or definitions to prove theorems about triangles, including: a) A line parallel to one side of a triangle divides the other two

proportionally. b) If a line divides two sides of a triangle proportionally, then it is

parallel to the third side. c) The Pythagorean Theorem proved using triangle similarity.

Students often cannot visualize corresponding parts of similar overlapping triangles. Have them separate and redraw the triangles.


Course: Geometry/ Geometry Honors Unit 5- Right Triangles and Trigonometry


How can right triangles be used to solve application problems?



Remarks

MAFS.912.G-SRT.2.4 Prove theorems about triangles. (Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity.) SMP #3

apply theorems, postulates, or definitions to prove theorems about triangles, including: ( c ) The Pythagorean Theorem proved using triangle similarity.

Students often cannot visualize corresponding parts of similar overlapping triangles. Have them separate and redraw the triangles.

MAFS.912.G-SRT.3.6 Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. SMP #2, 7

demonstrate that within a right triangle, line segments parallel to a leg create similar triangles by angle-angle similarity.

use characteristics of similar figures to justify the trigonometric ratios.

define the following trigonometric ratios for acute angles in a right triangle: sine, cosine, and tangent.

-Extension: Use division and the Pythagorean Theorem (a2 + b2 = c2) to prove that sin2A + cos2A = 1. -Some students believe that right triangles must be oriented a particular way or they do not realize that opposite and adjacent sides need to be identified with reference to a particular acute angle in a right triangle.

MAFS.912.G-SRT.3.7 Explain and use the relationship between the sine and cosine of complementary angles. SMP #2

define complementary angles.

calculate sine and cosine ratios for acute angles in a right triangle when given two side lengths.

use a diagram of a right triangle to explain that for a pair of complementary angles A and B, the sine of angle A is equal to the cosine of angle B and the cosine of angle A is equal to the sine of angle B.


Course: Geometry/ Geometry Honors Unit 5- Right Triangles and Trigonometry (cont)


How can right triangles be used to solve application problems?



Remarks

MAFS.912.G-SRT.3.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. SMP #1, #4

use angle measures to estimate side lengths and vice versa.

solve right triangles by finding the measures of all sides and angles in the triangles using Pythagorean Theorem and/or trigonometric ratios and their inverses.

draw right triangles that describe real world problems and label the sides and angles with their given measures.

solve application problems involving right triangles, including angle of elevation and depression, navigation, and surveying.

-Include Special Right Triangles in this standard (30-60-90 and 45-45-90) -Some students believe that the trigonometric ratios defined in this cluster apply to all triangles, but they are only defined for acute angles in right triangles.

MAFS.912.G-SRT.4.10 (Honors Only) Prove the Law of Sines and Cosines and use them to solve problems. SMP #2, 7

derive the Law of Sines by drawing an altitude in a triangle, using the sine function to find two expressions for the length of the altitude, and simplifying the equation that results from setting these expressions equal.

draw an altitude to create two right triangles and can establish the relationships of the sides in each right triangle using the sine and cosine functions of a single angle in the original triangle.

derive the Law of Cosines using the Pythagorean Theorem, two right triangles formed by drawing an altitude, and substitution.

generalize the Law of Cosines to apply to each included angle (a2=b2+c2-2bccosA)

use the Law of Sines and Law of Cosines to solve real world problems.

This standard involves deriving the Law of Sines and the Law of Cosines.


Course: Geometry/ Geometry Honors Unit 5- Right Triangles and Trigonometry (cont)

Essential Question(s): How can right triangles be used to solve application problems?



Remarks

MAFS.912.G-SRT.4.11 (Honors Only) Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g., surveying problems, resultant forces). SMP #1, 4

use the triangle inequality and side/angle relationships (e.g. largest angle is opposite the largest side) to estimate the measures of unknown sides and angles.

distinguish between situations that require the Law of Sines (ASA, AAS, SSA) and situations that require the Law of Cosines (SASS, SSS).

apply the Law of Sines to find unknown side lengths and unknown angle measures in right and non-right triangles.

use the Law of Sines to determine if two given side lengths and a given non-adjacent angle measures (SSA) make two triangles, one triangle, or no triangle.

apply the Law of Cosines to find unknown side lengths and unknown angle measures in right and non-right triangles.

represent real world problems with diagrams of right and non-right triangles and use them to solve for unknown side lengths and angle measures.

This standard involves using the Law of Sines and the Law of Cosines to find missing sides/angle measures in non-right triangles.


Course: Geometry/ Geometry Honors Unit 6-Quadrilaterals and Coordinate Geometry


How does geometry relate to algebra?



Remarks

MAFS.912.G-CO.3.11 Prove theorems about parallelograms; use theorems about parallelograms to solve problems. (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.) SMP #2, #3

apply theorems, postulates, or definitions to prove theorems about parallelograms, including: a) Prove opposite sides of a parallelogram are congruent; b) Prove opposite angles of a parallelogram are

congruent; c) Prove the diagonals of a parallelogram bisect each

other; d) Prove that rectangles are parallelograms with

congruent diagonals.

Rectangles, squares, and rhombi are special parallelograms. Use the definition of parallelogram to distinguish their defining characteristics.

MAFS.912.G-GPE.2.4 Use coordinates to prove simple geometric theorems algebraically. SMP #3, #7

connect a property of a figure to the tool needed to verify the property.

use coordinates and the right tool to prove or disprove a claim about a figure.

Example: use the distance formula to determine if sides are congruent; use the midpoint formula or the distance formula to decide if a side has been bisected; use slope to determine if sides are parallel, intersecting, or perpendicular.

MAFS.912.G-GPE.2.5 Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point). SMP #3, #8

state that parallel lines have the same slope.

determine if lines are parallel using their slopes.

state that parallel lines have the opposite reciprocal slopes.

determine if lines are perpendicular using their slopes.

May need to review graphing coordinates and finding slope before determining if lines are parallel or perpendicular.

MAFS.912.G-GPE.2.6 Find the point on a directed line segment between two given points that partitions the segment in a given ratio. SMP #7

calculate the point(s) on a directed line segment whose endpoints are (x1, y1) and (x2, y2) that partitions the line segment into a given ratio.

- In addition to using the formula, students may find the midpoint graphically using slope. -This includes the midpoint formula, as well as dividing a segments into thirds, fourths, etc. -This includes finding a missing endpoint given the midpoint and the other endpoint of a segment.



Unit 7-Two-Dimensional Measurements

Essential Question(s): In what ways can geometric figures be used to understand real-world situations?



Remarks

MAFS.912.G-GMD.1.1 Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments. SMP #3, #7

Circumference of a Circle

define π (pi) as the ratio of a circle’s circumference to its diameter.

use algebra to demonstrate that because π is the ratio of a circle’s circumference to its diameter that the formula for a circles’ circumference must

be C = π d. Area of a Circle

inscribe a regular polygon in a circle and break it into many congruent triangles to find its area.

explain and use the dissection method on regular polygons to generate an area formula for regular polygons

A = ½ apothem perimeter (A =

aP).

calculate the area of a regular polygon A = ½ apothem perimeter (A =

aP).

use pictures to explain that a regular polygon with many sides is nearly a circle, its perimeter is nearly the circumference of a circle, and that its apothem is nearly the radius of a circle.

substitute the “nearly” values of a many sided regular polygon into A = ½

apothem perimeter (A =

aP) to show that the formula for the area of a circle is

A = π r2.

-Focus on the two-dimensional measures in this standard. -This includes finding area of regular polygons. -An informal survey of students from elementary school through college showed the number pi to be the mathematical idea about which more students were curious than any other. There are at least three facets to this curiosity: the symbol π itself, the number 3.14159…, and the formula for the area of a circle. All of these facets can be addressed here, at least briefly.

MAFS.912.G-GPE.2.7 Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. SMP #1

apply the distance formula to compute the perimeter and area given the coordinates of vertices of a polygon.

Graphing the given coordinates of the vertices may help students visualize the polygon in order to find the perimeter and area.


Course: Geometry/ Geometry Honors Unit 7-Two-Dimensional Measurements (cont)


In what ways can geometric figures be used to understand real-world situations?



Remarks

MAFS.912.G-MG.1.1 Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder). SMP #4

represent real-world objects as geometric figures.

estimate measures (circumference, area, perimeter, volume) of real-world objects using comparable geometric shapes or three-dimensional figures.

apply the properties of geometric figures to comparable real-world objects.

Focus on two-dimensional modeling. Example: The spokes of a wheel of a bicycle are equal lengths because they represent the radii of a circle. Students may have issues with estimating (rounding, conceptual, not being exact, etc.).

MAFS.912.G-MG.1.2 Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot). SMP #1, #4

decide whether it is best to calculate or estimate the area or volume of a geometric figure and perform the calculation or estimation.

break composite geometric figures into manageable pieces.

convert units of measure.

apply area and volume to situations involving density.

Focus on two-dimensional modeling. Example: Determine the population in an area. Students have difficulty converting units of area and volume due to the difference in scale factors.

MAFS.912.G-MG.1.3 Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios). SMP #1, #4

create a visual representation of a design problem.

solve design problems using a geometric model (graph, equation, table, formula).

interpret the results and make conclusions based on the geometric model.

Focus on two-dimensional modeling. Mathematical modeling involves solving problems in which the path to the solution is not obvious. A challenge for teaching modeling is finding problems that are interesting and relevant to high school students and, at the same time, solvable with the mathematical tools at the students’ disposal.



Unit 8-Three-Dimensional Measurements

Essential Question(s): In what ways can geometric solids be used to understand real-world situations?



Remarks

MAFS.912.G-GMD.1.1 Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. (Use dissection arguments, Cavalieri’s principle, and informal limit arguments.) SMP #3, #7

Volumes:

identify the base for prisms, cylinders, pyramids, and cones.

calculate the area of the base for prisms, cylinders, pyramids, and cones.

calculate the volume of a prism using the formula V =

B h and the volume of a cylinder V = πr2h.

defend the statement, “The formula for the volume of a cylinder (or cone) is basically the same as the formula for the volume of a prism (or pyramid).”

explain that the volume of a pyramid (or cone) is 1/3 the volume of a prism (or cylinder) with the same base area and height.

-Focus on the three-dimensional measures in this standard. -The inclusion of the coefficient 1/3 in the formulas for the volume of a pyramid or cone and 4/3 in the formula for the volume of a sphere remains a mystery for many students. In high school, students should attain a conceptual understanding of where these coefficients come from. Concrete demonstrations, such as pouring water from one shape into another should be followed by more formal reasoning.

MAFS.912.G-GMD.1.3 Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. SMP #4

calculate the volume of a cylinder, pyramid, cone, and sphere and use the volume formulas to solve problems.

Students may not understand when volume would be needed to solve a problem versus area, perimeter, or surface area. Students have difficulty with “B” in volume/surface area formulas (using the length of a side of the base instead of area of the base).

MAFS.912.G-GMD.2.4 Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. SMP #4, #7

identify the shapes of the two-dimensional cross-sections of three-dimensional objects.

rotate a two-dimensional figure and identify the three-dimensional object that is formed.

Hands-on models of three-dimensional figures will help students when they are first determining cross sections. Rubber bands may also be stretched around a solid to show a cross section.



Unit 8-Three-Dimensional Measurements (cont)

Essential Question(s): In what ways can geometric solids be used to understand real-world situations?



Remarks

MAFS.912.G-MG.1.1 Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder). SMP #4

represent real-world objects as geometric figures.

estimate measures (circumference, area, perimeter, volume) of real-world objects using comparable geometric shapes or three-dimensional figures.

apply the properties of geometric figures to comparable real-world objects.

Focus on three-dimensional models.

MAFS.912.G-MG.1.2 Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot). SMP #1, #4

decide whether it is best to calculate or estimate the area or volume of a geometric figure and perform the calculation or estimation.

break composite geometric figures into manageable pieces.

convert units of measure.

apply area and volume to situations involving density.

-Focus on three-dimensional models. -Students have difficulty converting units of area and volume due to the difference in scale factors.

MAFS.912.G-MG.1.3 Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios). SMP #1, #4

create a visual representation of a design problem.

solve design problems using a geometric model (graph, equation, table, formula).

interpret the results and make conclusions based on the geometric model.

Mathematical modeling involves solving problems in which the path to the solution is not obvious. A challenge for teaching modeling is finding problems that are interesting and relevant to high school students and, at the same time, solvable with the mathematical tools at the students’ disposal.


Course: Geometry/ Geometry Honors Unit 8-Three-Dimensional Measurements (cont)


In what ways can geometric solids be used to understand real-world situations?



Remarks

MAFS.912.G-GMD.1.2 (Honors Only) Give an informal argument using Cavalieri’s principle for the formulas for the volume of a sphere and other solid figures. SMP #2, 3, 7

state that, if two solid figures have the same total height and their cross-sectional areas are identical are every level, the figures have the same volume (Cavalieri’s Principle).

use objects such as a deck of cards or stack of coins to demonstrate Cavalieri’s Principle

use Cavalieri’s Principle to demonstrate that a right prism (or cylinder) and a slanted prism (or cylinder) have the same volume when the base areas and heights are the same.

use the Pythagorean Theorem and formula for the area of a circle to find the cross-sectional area of each level of a hemisphere as a function of the radius; to find the cross-sectional area of each level of a right cone when the height and radius are equal.

find the cross-sectional area of a cylinder when the height and radius are equal.

use Cavalieri’s Principle, a hemisphere, a right cone, and a right cylinder to derive the formula for the area of a sphere.

Example:

Same height and cross-sectional area, so these two would have the same volume.


Course: Geometry/ Geometry Honors Unit 9- Circle with and without Coordinates


How can the properties of circles, polygons, lines, and angles be useful when solving geometric problems?



Remarks

MAFS.912.G-C.1.1 Prove that all circles are similar. SMP #3

prove that all circles are similar by showing that for a dilation centered at the center of a circle, the preimage and the image have equal central angle measures.

The definition of similarity and dilation will need to be reviewed with students. Online applets can be helpful in seeing this relationship.

MAFS.912.G-C.1.2 Identify and describe relationships among inscribed angles, radii, and chords. (Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle.) SMP #1, #6

identify central angles, inscribed angles, circumscribed angles, diameters, radii, chords, and tangents.

describe the relationship between a central angle, inscribed angle, or circumscribed angle and the arc it intercepts.

recognize that an inscribed angle whose sides intersect the endpoints of the diameter of a circle is a right angle and that the radius of a circle is perpendicular to the tangent where the radius intersects the circle.

Students may think they can tell by inspection whether a line intersects a circle in exactly one point. It may be beneficial to formally define a tangent line as the line perpendicular to a radius at the point where the radius intersects the circle.

MAFS.912.G-C.1.3 Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. SMP #5

define the terms inscribed, circumscribed, angle bisector, and perpendicular bisector.

construct the inscribed circle whose center is the point of intersection of the angle bisectors (the incenter)and circumscribed circle whose center is the point of intersection of the perpendicular bisectors of each side of the triangle (the circumcenter).

apply the Arc Addition Postulate to solve for missing arc measures.

prove that opposite angles in an inscribed quadrilateral are supplementary.

Students sometimes confuse inscribed angles and central angles. For example they will assume that the inscribed angle is equal to the arc like a central angle.



Unit 9- Circle with and without Coordinates (cont)

Essential Question(s): How can the properties of circles, polygons, lines, and angles be useful when solving geometric problems?



Remarks

MAFS.912.G-CO.4.13 Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. SMP #5, #6

define inscribed polygons.

construct an equilateral triangle, a square, a hexagon inscribed in a circle.

explain the steps to constructing and equilateral triangle, a square, and a regular hexagon inscribed in a circle.

MAFS.912.G-C.2.5 Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. SMP #6, #7

define similarity as rigid motions with dilations, which preserves angle measures and makes lengths proportional.

apply similarity to calculate the length of an arc.

define and calculate the radian measure of an angle as the ratio of an arc length to its radius.

convert degrees to radians using the constant of proportionality.

calculate the area of a circle.

define a sector of a circle.

calculate the area of a sector using the ratio of the intercepted

arc measure and 360 multiplied by the area of the circle.

Constant of proportionality for radian

measures: 2pi*angle measure / 360. The formulas for converting radians to degrees and vice versa are easily confused. Knowing that the degree measure of given angle is always a number larger than the radian measure can help students use the correct unit. Sectors and segments are often used interchangeably in everyday conversation. Care should be taken to distinguish these two geometric concepts.

MAFS.912.G-GPE.1.1 Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. SMP #2, #3, #7

identify the center and radius of a circle given its equation.

draw a right triangle with a horizontal leg, a vertical leg, and the radius of a circle as its hypotenuse.

use the distance formula (Pythagorean Theorem), the coordinates of a circle’s center, and the circle’s radius to write the equation of a circle.

convert an equation of a circle in general (quadratic) form to standard form by completing the square.

-Completing the square will need to taught this year (2014-2015). -Students forget to change the sign of (h,k) to find the coordinates of the center.



Unit 9- Circle with and without Coordinates (cont)

Essential Question(s): How can the properties of circles, polygons, lines, and angles be useful when solving geometric problems?



Remarks

MAFS.912.G-GPE.2.4 Use coordinates to prove simple geometric theorems algebraically. SMP #3, #7

connect a property of a figure to the tool needed to verify the property.

use coordinates and the right tool to prove or disprove a claim about a figure.

Example: prove or disprove that the point (1, √3) lies on the circle centered at the origin and containing the point (0, 2); use the distance formula to determine to decide if a point is inside a circle, outside a circle, or on the circle;

MAFS.912.G-C.1.4 (Honors Only) Construct a tangent line from a point outside a given circle to the circle. SMP #5

define and identify a tangent line.

construct a tangent line from a point outside the circle to the circle using construction tools or computer software.

MAFS.912.G-GPE.1.2 (Honors Only) Derive the equation of a parabola given a focus and directrix. SMP #2, 3, 7

define a parabola.

find the distance from a point on the parabola (x,y) to the directrix.

find the distance from a point on the parabola (x,y) to the focus using the distance formula.

equate the two distance expressions for a parabola to write its equation.

identify the focus and directrix of a parabola when given its equation.

MAFS.912.G-GPE.1.3 (Honors Only) Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances from the foci is constant. SMP #2, 3, 7

define an ellipse and a hyperbola.

define and identify the foci of an ellipse and a hyperbola.

use the distance formula to write an expression for the sum of the distances from a point (x,y) on the ellipse to each focus and equate it to the given constant sum.

use algebra to convert the derived equation for an ellipse to standard form.

identify the center, foci, and axes of an ellipse when given the standard form equation.

use the distance formula to write an expression for the difference of the distances from a point (s,y) on the hyperbola to each focus and equate it to the given constant sum.

use algebra to convert the derived equation for a hyperbola to standard form.

identify the center, foci, axes, and asymptotes of a hyperbola when given the standard form equation.



Unit 10-Algebra 2 Preparations

Essential Question(s): How can relationships between quantities best be represented?



Remarks

MAFS.912.F-IF.3.7a, b Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.

a. Graph linear and quadratic functions and show intercepts, maxima, and minima.

b. Graph absolute value functions.

SMP #7, 8

graph a line in point-slope form, standard form, and slope-intercept form.

graph a quadratic function using evaluated points

identify the maxima/minima, vertex, axis of symmetry, and transformations from the parent function y=x2 of a quadratic function

graph an absolute value function using evaluated points

identify the maxima/minima, vertex, axis of symmetry,

and transformations from the parent function y= of a quadratic function

MAFS.912.F-IF.3.8a Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.

a. Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph and interpret these in terms of a constant.

SMP #2, 7

find the x-intercepts of a quadratic written in factored form

convert a standard form quadratic to factored form by factoring.

convert a standard form quadratic to vertex form by completing the square

-Focus on reviewing factoring quadratics (including difference of squares) and completing the square

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