Geometry Similar Polygons Assignment Satisfaction

Course Overview

Students are able to gain credit if they have previously completed this course but did not successfully earn credit. For each unit, students take a diagnostic test that assesses their current knowledge of fundamental content. The results of these tests help students create individualized study plans.

Students move at their own pace and then are assessed by computer-scored unit tests for a grade. Teacher-graded assignments are available as optional or for review only. Students review core geometric concepts as they develop sound ideas of inductive and deductive reasoning, logic, concepts, and techniques and applications of Euclidean plane and solid geometry. Students use visualizations, spatial reasoning, and geometric modeling to solve problems. Topics include points, lines, and angles; triangles, polygons, and circles; coordinate geometry; three-dimensional solids; geometric constructions; symmetry; and the use of transformations.

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Course Length

Two Semesters

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Prerequisites

Student completed the course or its equivalent, but did not receive credit; teacher/school counselor recommendation required

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Course Outline

SEMESTER ONE

Unit 1: An Introduction

Even the longest journey begins with a single step. Any journey into the world of geometry begins with the basics. Points, lines, segments, and angles are the foundation of geometric reasoning. This unit provides students with basic footing that will lead to an understanding of geometry.

  • Semester Introduction
  • Basic Geometric Terms and Definitions
  • Measuring Length
  • Measuring Angles
  • Bisectors and Line Relationships
  • Relationships between Triangles and Circles
  • Transformations
  • Using Algebra to Describe Geometry

Unit 2: Methods of Proof and Logic

Professionals use logical reasoning in a variety of ways. Just as lawyers use logical reasoning to formulate convincing arguments, mathematicians use logical reasoning to formulate and prove theorems. With definitions, assumptions, and previously proven theorems, mathematicians discover and prove new theorems. It’s like building a defense, one argument at a time. In this unit, students will learn how to build a defense from postulates, theorems, and sound reasoning.

  • Reasoning, Arguments, and Proof
  • Conditional Statements
  • Compound Statements and Indirect Proof
  • Algebraic Logic
  • Inductive and Deductive Reasoning

Unit 3: Polygon Basics

We can find polygons in many places: artwork, sporting events, architecture, and even in roads. In this unit, students will discover symmetry, work with special quadrilaterals, and work with parallel lines and slopes.

  • Polygons and Symmetry
  • Quadrilaterals and Their Properties
  • Parallel Lines and Transversals
  • Converses of Parallel Line Properties
  • The Triangle Sum Theorem
  • Angles in Polygons
  • Midsegments
  • Slope

Unit 4: Congruent Polygons and Special Quadrilaterals

If two algebraic expressions are equivalent, they represent the same value. What about geometric shapes? What does it mean for two figures to be equivalent? A pair of figures can be congruent the same way that a pair of algebraic expressions can be equivalent. Students will learn, use, and prove theorems about congruent geometric figures.

  • Congruent Polygons and Their Corresponding Parts
  • Triangle Congruence: SSS, SAS, and ASA
  • Isosceles Triangles and Corresponding Parts
  • Triangle Congruence: AAS and HL
  • Using Triangles to Understand Quadrilaterals
  • Types of Quadrilaterals
  • Constructions with Polygons
  • The Triangle Inequality Theorem

Unit 5: Perimeter, Area, and Right Triangles

If we have a figure, we can take many measurements and calculations. We can measure or calculate the distance around the figure (the perimeter or circumference), as well as the figure’s height and area. Even if we have just a set of points, we can measure or calculate the distance between two points.

  • Perimeter and Area
  • Areas of Triangles and Quadrilaterals
  • Circumference and Area of Circles
  • The Pythagorean Theorem
  • Areas of Special Triangles and Regular Polygons
  • Using the Distance Formula
  • Proofs and Coordinate Geometry

Unit 6: Semester Review and Test

  • Semester Review
  • Semester Test

SEMESTER TWO

Unit 1: Three-Dimensional Figures and Graphs

One-dimensional figures, such as line segments, have length. Two-dimensional figures, such as circles, have area. Objects we touch and feel in the real world are three-dimensional; they have volume.

  • Semester Introduction
  • Solid Shapes and Three-Dimensional Drawing
  • Lines, Planes, and Polyhedra
  • Prisms
  • Coordinates in Three Dimensions
  • Equations of Lines and Planes in Space

Unit 2: Surface Area and Volume

Every three-dimensional figure has surface area and volume. Some figures are more common and useful than others. Students probably see pyramids, prisms, cylinders, cones, and spheres every day. In this unit, students will learn how to calculate the surface area and volume of several common and useful three-dimensional figures.

  • Surface Area and Volume
  • Surface Area and Volume of Prisms
  • Surface Area and Volume of Pyramids
  • Surface Area and Volume of Cylinders
  • Surface Area and Volume of Cones
  • Surface Area and Volume of Spheres
  • Three-Dimensional Symmetry

Unit 3: Similar Shapes

A map of a city has the same shape as the original city, but the map is much, much smaller. A mathematician would say that the map and the city are similar. They have the same shape but are different sizes.

  • Dilations and Scale Factors
  • Similar Polygons
  • Triangle Similarity
  • Side-Splitting Theorem
  • Indirect Measurement and Additional Similarity Theorems
  • Area and Volume Ratios

Unit 4: Circles

You probably know what a circle is and what the radius and diameter of a circle represent. However, a circle can have many more figures associated with it. Arcs, chords, secants, and tangents all provide a rich set of figures to draw, measure, and understand.

  • Chords and Arcs
  • Tangents to Circles
  • Inscribed Angles and Arcs
  • Angles Formed by Secants and Tangents
  • Segments of Tangents, Secants, and Chords
  • Circles in the Coordinate Plane

Unit 5: Trigonometry

Who uses trigonometry? Architects, engineers, surveyors, and many other professionals use trigonometric ratios such as sine, cosine, and tangent to compute distances and understand relationships in the real world.

  • Tangents
  • Sines and Cosines
  • Special Right Triangles
  • The Laws of Sines and Cosines

Unit 6: Beyond Euclidian Geometry

Some people break rules, but mathematicians are usually very good at playing by them. Creative problem-solvers, including mathematicians, create new rules, and then play by their new rules to solve many kinds of problems.

  • The Golden Rectangle
  • Taxicab Geometry
  • Graph Theory
  • Topology
  • Spherical Geometry
  • Fractal Geometry
  • Projective Geometry
  • Computer Logic

Unit 7: Semester Review and Test

  • Semester Review
  • Semester Test
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Lesson Scheduling

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In this section we first describe the data set. We then present the helical curve fitting algorithm and the definitions and computations of helix score, helix axis angle and Cα RMSD as well as the definitions of three standard protein helices. Finally we detail the assignment algorithm itself.

2.1 The data set

To evaluate the performance of our algorithm, we have downloaded from the current version of the PDB a non-redundant set of x-ray structures (29,093 in total) with at most 70% sequence identity and each one has at least one helix according to the PDB. Out of them we have selected a set of 3,287 high-resolution structures (set 𝕊) each one has at least three helices, a resolution ≤ 2.0Å and a R-factor ≤ 25.0% to obtain the statistics for four helical parameters and two RMSDs. The set of the remaining 25,806 structures (set ℕ) is used to evaluate the performance of our algorithm and two previous programs dssp and stride. To compare our algorithm with the five programs (kaksi, plasse, stick, stlsstr and sst) that we are not able to obtain a local copy we have uploaded to a web server [23] a set of 100 x-ray protein structures (set 𝕋) with the first 50 structures having resolutions between 1.0Å–2.0Å and the rest having resolutions ≥ 2.5Å(1AKG,1BGF,1EZW,1GSU,1I1N,1K1B,1NTE,1O98,1PVM,1SJD,1UKF,1VZY,1XGW,1YQD,2ASC, 2BJI, 2CWH,2FD5,2GB2, 2GG6,2H1V,2I2C,2NSF, 2POK,2VBA,2W6K,2WRA,2X7H,2YSK,2ZJ3,3C9U,3EDF, 3GG7,3HG7,3IDV,3LCC,3LFJ,3P4H,3PUA, 3TOU,3V7Q,3ZOO;1AZ2,1BTN,1CAX,1F1F,1FXA,1HMY,1KX8, 1MJ9,1MSC,1NJ1,1PYP,1RIN,2CND,2GAE,2HAF,2HXB,2QQV, 2RJQ,2SPT,2WO7,2X2B,2Y5Q,2ZIY,3AT0, 3G0A,3IIA,3L1G,3M3T,3NPE,3O5K,3PGK,3PMQ,3QNT,3QYB,3T2Y,3UUF,3VGE, 3WMF,4AYH,4BOY, 4EWS,4GB0,4JVC,4LIF,4MBR,4N3G,4O1S,4PUT,4QGL,4TW8) to obtain their helix assignments. Set 𝕋 is also used for a comparison with the program disicl.

2.2 The curve fitting algorithm and helix assignment

Our solution to the helix assignment problem consists of two steps. The first step is to solve a minimization problem by a helical curve fitting algorithm that searches for a series of genuine helical curves each one best fits the coordinates of four successive Cα atoms. A helix model composed of a series of helical curves has been previously called a polyhelix [24]. We then define a standard protein helix for each helix type. The algorithmic solution to the minimization problem makes it possible to compute a helix axis angle ai, three helix scores hi, gi and πi, and two Cα RMSDs for residue i. A helix score quantifies the deviation from a standard protein helix of the helical curve that begins with this particular residue. The score, axis angle and Cα RMSD are used in the second step as input to make helix assignment.

2.2.1 The helical curve fitting algorithm.

A general helical curve in three dimensional space could be represented as: (1) where r = {x, y, z} is a point on the curve, r0 = {x0, y0, z0} its origin, and R the rotation matrix that specifies its helical axis n with respect to a coordinate system. The first three helical parameters, radius (r), pitch (p) and turn angle (t), define a standard helical curve, x = r sin t, y = r cos t, z = pt with its origin at {1.0,0.0,0.0} and its axis along the +Z axis. Together with n and r0 these five parameters completely define a general helical curve. Though r, p, t could be computed directly from the virtual bond length, bond angle and dihedral angle of a quadruple of Cαs [25], no simple analytic expression has been derived for the computation of a helical curve that best fits the coordinates of a quadruple of Cαs, that is, a helical curve that has the minimum RMSD (Δi) between the four Cαs and their closest points on the curve. In fact, this minimization (or curve fitting) problem is equivalent to finding the solutions to a high-degree monomial. In the following we describe briefly an algorithmic solution to this minimization problem.

We begin with the computations of r, p and t using previously-derived analytic expressions [25], and denote their values as rm, pm and tm. Then we proceed as follows to search discretely and exhaustively over two intervals, [rmdr, rm + dr] and [pmdp, pm + dp], for the r and p values of a helical curve such that it best fits the coordinates of a quadruple of Cαs of residue i, i + 1, i + 2 and i + 3. Both dr and dp are user-specified constants.

1. Δi = ∞       {the initial RMSD}

2. For each r in [rmdr, rm + dr]

  For each p in [pmdp, pm + dp]

   Computet   {the turn angle}

   Generate a helical curve   {byEq 1}

   Best-fit the curve to the four Cα coordinates using singular-value decomposition(SVD) to compute R

   IfRms < Δi

    Δi = Rms

ri = r, pi = p, ti = t, Ri = R

where Rms is the RMSD between the quadruple of Cαs and their closest points on the helical curve; ri, pi, ti and Ri are, respectively, the helical parameters and rotation matrix for the helical curve that best-fits the quadruple. Given both r and p and the distance di,i+1 between two consecutive Cαs, t could be computed as follows: . Singular-value decomposition (SVD) is applied to compute Δi and rotation matrix Ri; and from Ri, the helical axis ni for this helical curve could be calculated. In fact, the SVD step guarantees that the computed helical curve best fits the coordinates of the quadruple of Cαs. A set of six helical parameters (r, p, t, Δ, R and n) is computed for a protein chain by sliding over its sequence a window of four Cα atoms.

2.2.2 The computation of three helix scores and helix axis angle and Cα RMSD.

Except for the last three residues at the C-terminus of a protein chain three helix scores, hi, gi and πi, are computed for each residue i. They are used respectively for the assignment of α,310 and π helices. (2) where ri, pi, ti, Δi are computed as above using a quadruple of Cαs of residue i, i + 1, i + 2 and i + 3. The constants μr, σr; μp, σp; μt, σt and σΔ are respectively the normal distribution parameters for r, p, t, Δ that are determined as follows over the respective data sets for r, p, t, Δ calculated on the non-redundant set 𝕊. We first apply the program dssp [4] to assign 31,383 α-helices, 11,926 310-helices and 1,156 π-helices for the structures in 𝕊, and then for each helix type we calculate its r, p, t and Δ values. The r, p, t data sets for π-helix are calculated differently than those for either α or 310-helices. If dssp assigns a π-helix say composed of residues i, i + 1, i + 2, i + 3, i + 4, then the final value for each r, p, t is the average over the three values for the first three residues, that is, , , . Each triple of parameters μr, μp and μt defines a standard helical curve for a helix type that represents an average over all the helices of that particular type in 𝕊. For ease of reference we call them respectively the standard protein helix for α,310, and π helices. The helix scores h, g and π are computed with respect to the respective α,310 and π standard protein helices. The score measures the local deviation of the helical curve from the standard protein helix for that particular helix type: the higher the score the larger deviation from the standard protein helix. The term quantifies the spatial difference between a Cα atom and its closest point on the helical curve. In fact, Δ together with the r, p, t terms in Eq 2 and the helical axis n provide a pure geometrical definition for a helix in a protein, that is, as long as a segment of Cα coordinates conform to a genuine helical curve, it is assigned as a helix.

The minimum RMSD Δi for residue i is computed over the quadruple of residues i, i + 1, i + 2 and i + 3 and thus is useful for the determination of the N-terminus of a helix. For the determination of the C-terminus of a helix, we have computed a Cα RMSD δi for residue i using up to four helical curves best fit to four successive quadruples of Cαs starting with the quadruple of residues i − 2, i − 1, i and i + 1. The RMSD δ measures the goodness of fitting of up to four consecutive helical curves to seven successive Cα atoms. In the current implementation δ is used for the extension of the C-terminus of a 310-helix and the possible merge of two adjacent α-helices. In addition to the helix scores and δ, we have also calculated the angle between two successive helical axes ni−1 and ni and use it as input to the assignment algorithm. This angle measures the bending of the current helical curve starting at residue i relative to the previous helical curve starting at residue i − 1. For ease of reference we call this angle helix axis angle and call the four variables ri, pi, ti, aithe four helical parameters for residue i. For the first residue at the C-terminus we set ai = 0.0° and for a genuine helical curve all of its ais are zero. The set of four helical parameters for all the quadruples of consecutive Cαs in a protein chain are computed by sliding over its sequence a window of four Cα atoms.

Four thresholds hT, hmax, gT and πT for the three helix scores, four thresholds aT, amax, aG and aI for the helix axis angle and two thresholds, δG and δmax, for δ are required by our assignment algorithm. These ten thresholds are determined as detailed late by the analyses of the statistics for both helical parameters and RMSDs obtained on all the helices in 𝕊 assigned by the program dssp.

The five parameters, r, p, t, r0, n, completely defines a right-handed helical curve. By inverting just one component of every Cα coordinate, say from {x, y, z} to {−x, y, z}, the same five parameters together with the axis angle and two RMSDs could be computed similarly and used for the assignment of the left-handed helices.

2.2.3 The helix assignment algorithm.

The assignment proceeds as follows using helix score, axis angle and Cα RMSD as well as the ten thresholds as input. The assignment for each protein chain starts with π-helix.

letbi = 0, i = 0, …, n

whilei < n AND bi == 0  {residue i has NOT been assigned}

Ifπi < πT AND ai < aI AND ai+1 < aI AND ai+2 < aI {the N-terminus of a π helix}

iπ-helix  {assign residue i to π helix}

bi = 1

i + +

 while πi < πT AND i < (n − 3) AND bi == 0

iπ-helix

bi = 1

i + +

forj = i, i + 1, i + 2, i + 3

jπ-helix

j + +

bj = 1

i = j

i + +

where n is the number of residues in a protein chain, i residue index, and πT a threshold for helix score π and aI a threshold for helix axis angle used only in π-helix assignment.

Next the algorithm assigns 310-helices for the remaining residues of the same protein chain.

whilei < n AND bi == 0  {residue i has NOT been assigned}

 Ifgi < gT AND ai < aG  {the N-terminus of a 310helix}

i ∈ 310-helix  {assign residue i to 310helix}

bi = 1

i + +

whilegi < gT AND i < n AND bi == 0

i ∈ 310-helix

bi = 1

i + +

ifδi < δG AND δi+1 < δG  {assign two more residues to 310helix}

i ∈ 310-helix

bi = 1

i + +

i ∈ 310-helix

bi = 1

i + +

where gT, aG and δG are respectively the thresholds for helix score gi, axis angle and RMSD δ used only in 310-helix assignment.

Finally the algorithm assigns α-helices for the remaining residues of the same protein chain.

whilei < n AND bi == 0  {residue i has NOT been assigned}

 Ifhi < hT AND ai < aT AND hi+1 < hT  {the N-terminus of a helix}

iα-helix  {assign residue i to a helix}

bi = 1

i + +

whilehi < hT AND i < n AND bi == 0

iα-helix

bi = 1

i + +

i + +

where hT a threshold for helix score hi. For the assignment of α-helices only, additional steps may be needed to merge two α-helices adjacent in protein sequence and to extend their C-termini. The merge of two adjacent α-helices proceeds as follows. For every pair of adjacent helices less than four residues apart, if the axis angle aiamax, helix score hi < hmax and δi < δmax for every intermediate residue i, then the two helices are merged into a single helix. The thresholds hmax, amax and δmax are the respective thresholds for axis angle, helix score and δ. The threshold δmax is used only in the merge step.

After the merge step the C-terminus of an α-helix may be extended as follows.

letj − 1 be the last residue of an α-helix assigned above

ifj < n AND bj == 0  {residue j has NOT been assigned}

  Ifhj < hmax AND aj < amax

bj = 1

jα-helix

   j + +

bj = 1

jα-helix

where hmax and amax are the same thresholds in the merge step.

Though the three types of helices are assigned similarly at individual helix level, no merge step is necessary for either 310 or π helix since there exist rarely 310 or π helices with more than eight residues. In our current implementation, no extension is made for π helix.

A left handed helix is assigned similarly except that the helical parameters and RMSDs used in Eq 2 are those computed from a quadruple of inverted Cα coordinates as described above.

2.3 Helix classification

To better compare the assigned helices made by our and previous programs and to characterize their structures we have classified them using our geometric clustering algorithm [22]. The clustering is performed on sets of helices that have the same length. The RMSD threshold for clustering is 1.5Å.

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