table of contents - swwmath2.pbworks.comswwmath2.pbworks.com/w/file/fetch/81827021/geo sco… ·...

Table of Contents

Geometry

Return to Page 1

Scope & Sequence

Table of Contents

Fall Semester

TSTransformations & Similarity

In high school, students formalize much of the geometric exploration from middle school. In this unit, students develop rigorous definitions of three familiar congruence transformations: reflections, translations, and rotations and use these transformations to discover and prove geometric properties. Throughout the course, students will use transformations as a tool to analyze and describe relationships between geometric figures. This unit moves away from rigid motion and focuses on dilations and similarity. Students prove theorems involving similarity and apply dilations and similarity to model situations in the real world.

CPConjectures & Proof

This unit gives students the foundational tools for developing viable geometric arguments using relationships students studied in middle school related to lines, transversals, and special angles associated with them. Students learn how to combine true statements within a mathematical system to deductively prove other statements. Students should begin to see the structure of a mathematical system as they make conjectures and then prove statements involving lines and angles.

TRTriangles

This unit explores basic theorems and conjectures about triangles, including the triangle inequality conjecture, the Triangle Sum Theorem, and theorems regarding centers of a triangle. Students explored some of these relationships in middle school but will build on their work in unit 2 with deductive reasoning and proof related to triangles in this unit. Students make and verify conjectures related to isosceles triangles and explore physical properties of the centroid of a triangle. In this unit, students also learn basic construction techniques and use these as they explore triangle properties. Throughout this unit, students will use the precise definitions developed in GMCO.A.1. This unit builds on students work with transformations in unit 1 and properties of triangles in unit 3 to formalize the definition of congruent triangles. Students reason to identify criteria for triangle congruence and use precise notation to describe the correspondence in congruent triangles.

QLQuadrilaterals

Prior units of this course have focused on triangles. This unit extends that work to the study of quadrilaterals. Students use triangle congruence as they prove theorems about parallelograms. This unit also provides an opportunity for students to become proficient with coordinate proofs.

Additional content if time permits*

CICircles*

This unit explores properties of circles. Students draw on geometric relationships involving lines, angles, triangles and quadrilaterals as they derive the equation of a circle and explore properties of chords, arcs, and angles on circles.

Spring Semester

TGTrigonometry

This unit extends the idea of similarity to indirect measurements. Students develop properties of special right triangles, and use properties of similar triangles to develop trigonometric ratios. Students apply these ideas as they model real-world situations and solve problems involving unknown side lengths and angle measures.

MGModeling with Geometry

In prior units of this course, students learned about many geometric relationships and developed a mathematical system. This unit provides the opportunity to bring together all of the relationships students have learned in this course and apply them to real-world situations. The unit should present in depth problems that require students to draw on their understanding of geometric figures and strategically use the tools they have been developing throughout the course. This unit explores three-dimensional geometry including representations of real-world situations with three-dimensional objects and calculating volume. Students make connections between two-dimensional and three-dimensional representations of objects. Students culminate the course with modeling problems involving three-dimensional objects, allowing them again to integrate their knowledge and apply complex geometric reasoning.

IDInterpreting Data

This unit reviews the univariate data representations students studied previously and then introduces statistical models for bivariate categorical and quantitative data. Students have already addressed in previous units many of the standards in this unit, and they should now be able to apply their understandings from that previous work in the new work with the statistics standards in this unit. This unit provides opportunities to reinforce students work from the previous unit with representing linear functions symbolically and graphically, as described in A-SSE.A.1a, A-CED.A.2, F-IF.A.2, F-IF.B.4, F-IF.B.5, F-FI.C.9, F-BF.B.3, and F-LE.A.2, and F-LE.B.5.

DCDrawing Conclusions from Data

Student learn to use probability, relative frequencies, and discrete distributions. Drawing correct conclusions from data is highly dependent on how the data are collected. In particular, "cause and effect" conclusions can only arise from properly conducted experiments, in which the researcher actively imposes a treatment. In this unit students study design of experiments based on three fundamental principles: control of outside variables, randomization, and replication within the experiment. This unit is an introduction to these and other key issues in experimental design.


PB Probability*

Students last formally studied probability in Grade 7, when they found probabilities of simple and compound events and designed and used simulations. This unit builds on these concepts, as well as fundamental counting principles and the notion of independence, to develop rules for probability and conditional probability.

2014-15

Unit TS* Transformations & Similarity

Unit 2 Radicals & Right Triangles except for section 2.2 Special Right Triangles.Alg 1 Unit 9 Parallel LinesUnit 3 Triangle Properties & ProofUnit 4 Quadrilateral PropertiesUnit 5 Quadrilateral ProofsUnit 6 Coordinate Geometryand Proof

Unit MG* Modeling GeometryUnit

Unit ID* Interpreting Data


DC* Drawing Conclusions from Data(Surveys, Sampling)

Add to Geometry*

2015-16

TS Transformations & Similarity

CP Conjectures & Proof

TR Triangles

QL Quadrilaterals


CI Circles*

TG Trigonometry

MG Modeling with Geometry

ID Interpreting Data

DC Drawing Conclusions from Data


PB Probability*

Common Core State Standards - Mathematics

Content Emphases by Cluster

Geometry

Not all of the content in a given grade is emphasized equally in the standards. Some clusters require greater emphasis than the others based on the depth of the ideas, the time that they take to =

To say that some things have greater emphasis is not to say that anything in the standards can safely be neglected in instruction. Neglecting material will leave gaps in student skill and understanding and may leave students unprepared for the challenges of a later grade.

In addition to identifying the Major, Additional, and Supporting Clusters for each grade, suggestions are given to connect the Supporting to the Major Clusters of the grade. Thus, rather than suggesting even inadvertently that some material not be taught, there is direct advice for teaching it, in ways that foster greater focus and coherence.

The major clusters identified in this course represent a subset of the critical areas of focus from the Smarter Balanced 11th grade Career and College Readiness assessment. The assessment consortium, PARCC, has identified these clusters aspart of the end-of-course boundaries that will be used fortheir assessments. Washington State is not part of the PARCC consortium but anticipates future instructional materials will be aligned to PARCCs course outline configuration rather than the course sequence outlined in Appendix A from CCSS. Therefore, we encourage districts to consider the below information when determining scope and sequence for their courses.

Numerals in parentheses denote content standards within each cluster applicable for the given course.

OSPI Algebra 1 Cluster Document

From http://www.k12.wa.us/CoreStandards/pubdocs/HighschoolAlg1GeoAlg2.docx

KEY: ( Major Content ( Supporting Content ( Additional Content

Reported EOC 1

Mathematical Practices

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.

Quantities (N-Q)

( Reason quantitatively and use units to solve problems (1, 2, 3)

Seeing Structure in Expressions (A-SSE)

( Interpret the structure of expressions (1)

( Write expressions in equivalent forms to solve problems (3)

Creating Equations (A-CED)

( Create equations that describe numbers or relationships (1, 2, 3, 4)

Reasoning with Equations and Inequalities (A-REI)

( Solve equations and inequalities in one variable (3)

( Solve systems of equations (5, 6)

( Represent and solve equations and inequalities graphically (10, 11, 12)

Arithmetic with Polynomials & Rational Expressions (A-APR)

( Understand the relationship between zeros and factors of polynomials (3)

Interpreting Functions (F-IF)

( Understand the concept of a function and use function notation (1, 2, 3)

( Interpret functions that arise in applications in terms of the context (4, 5, 6)

( Analyze functions using different representations (7, 9)

Building Functions (F-BF)

( Build a function that models a relationship between two quantities (1)

Linear, Quadratic, and Exponential Models (F-LE)

( Construct and compare linear, quadratic, and exponential models and solve problems (1, 2, 3)

( Interpret expressions for functions in terms of the situation they model (5)

Interpreting categorical and quantitative data (S-ID)

( Summarize, represent, and interpret data on a single count or measurement variable (1, 2, 3)

( Summarize, represent, and interpret data on two categorical and quantitative variables (5, 6)

( Interpret linear models (7, 8, 9)

Reported EOC 1

In Geometry



In Algebra 2


( Create equations that describe numbers or relationships (3)

Not on EOC

In Algebra 2

The Real Number System (N-RN)

( Use properties of rational and irrational numbers (3)


( Interpret the structure of expressions (2)

Arithmetic with Polynomials and Rational Expressions (A-APR)

( Perform arithmetic operations on polynomials (1)



Not Reported EOC




( Analyze functions using different representations (8)


( Build new functions from existing functions (3)

OSPI Geometry Cluster Document



Reported EOC 2


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.

From Algebra 1



From Algebra 2


( Summarize, represent, and interpret data on a single count or measurement variable (4)

( Summarize, represent, and interpret data on two categorical and quantitative variables (6)

Making Inferences and Justifying Conclusions (S-IC)

( Understand and evaluate random processes underlying statistical experiments (1, 2)

( Make inferences and justify conclusions from sample surveys, experiments and observational studies (3, 4, 5, 6)

Optional Unit

Conditional Probability and Rules of Probability (S-CP)

( Understand independence and conditional probability and use them to interpret data (1, 2, 3, 4, 5)

( Use the rules of probability to compute probabilities of compound events in a uniform probability model (6, 7)

Similarity, Right Triangles, and Trigonometry (G-SRT)

( Understand similarity in terms of similarity transformations (1, 2, 3)

( Prove theorems using similarity (4, 5)

( Define trigonometric ratios and solve problems involving right triangles (6, 7, 8)

Geometric measurement and dimension

(G-GMD)

( Explain volume formulas and use them to solve problems (1, 3)

( Visualize relationships between two-dimensional and three-dimensional objects (4)

Modeling with Geometry (G-MG)

( Apply geometric concepts in modeling situations (1, 2, 3)

Not Reported Geometry

Congruence (G-CO)

( Experiment with transformations in the plane (1, 2, 3, 4, 5)

( Understand congruence in terms of rigid motions (6, 7, 8)

( Prove geometric theorems (9, 10, 11)

( Make geometric constructions (12, 13)

Expressing Geometric Properties with Equations (G-GPE)

( Translate between the geometric description and the equation of a conic section (1)

( Use coordinates to prove simple geometric theorems algebraically (4, 5, 6, 7)


( Summarize, represent, and interpret data on two categorical and quantitative variables (6)

Optional Unit

Circles (G-C)

( Understand and apply theorems about circles (1, 2, 3)

( Find arc lengths and areas of sectors of circles (5)

OSPI Algebra 2 Cluster Document



Quantities (N-Q)

( Reason quantitatively and use units to solve problems (1, 2, 3)

The Real Number System (N-RN)

( Extend the properties of exponents to rational exponents (1, 2)

( Use properties of rational and irrational numbers (3) - From Algebra 1

The Complex Number System (N-CN)

( Perform arithmetic operations with complex numbers (1, 2)

( Use complex numbers in polynomial identities and equations (7)


( Interpret the structure of expressions (1,2)

( Write expressions in equivalent forms to solve problems (3.4)

Arithmetic with Polynomials and Rational Expressions (A-APR)

( Perform arithmetic operations on polynomials (1) - From Algebra 1

( Understand the relationship between zeros and factors of polynomials (2, 3)

( Use polynomial identities to solve problems (4)

( Rewrite rational expressions (6)





( Understand solving equations as a process of reasoning and explain the reasoning (1, 2)


( Solve systems of equations (6, 7)

( Represent and solve equations and inequalities graphically (11)


( Understand the concept of a function and use function notation (3)

( Interpret functions that arise in applications in terms of the context (4, 6)

( Analyze functions using different representations (7, 8, 9)


( Build a function that models a relationship between two quantities (1, 2)

( Build new functions from existing functions (3, 4a)


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.

In Geometry

Making Inferences and Justifying Conclusions (S-IC)

( Understand and evaluate random processes underlying statistical experiments (1, 2)

( Make inferences and justify conclusions from sample surveys, experiments and observational studies (3, 4, 5, 6)

Conditional Probability and the Rules of Probability (S-CP)

( Understand independence and conditional probability and use them to interpret data (1, 2, 3, 4, 5)

( Use the rules of probability to compute probabilities of compound events in a uniform probability model (6, 7)

Transformations and Similarity (TS) Target

Practice 4: Model with mathematics

Practice 5: Use appropriate tools strategically Practice 6. Attend to precision.

Major Content

Supporting Content

Learning Target

Unit Emphasis

Learning Target

Unit Emphasis

G.COb I can explain congruence in terms of rigid motions.

Determine whether figures are congruent using rigid transformation

If two figures are congruent, then the corresponding pairs of angles are congruent and the corresponding sides are congruent. G-CO.6

G.SRTa I understand similarity in terms of similarity transformations.

NOT LIMITED TO TRIANGLES

Corresponding segments are parallel.

Special Case: If center of dilation of collinear to the vertices on one side of the figure then the corresponding vertices are also collinear.

The ratio of the distance from the vertex of the image to the center of dilation to the distance from the vertex of the preimage to the center of dilation is proportional; and conversely. Level 3 = area is not proportional

Level 4 = area is square of proportion.

Verify that a side length of the image is equal to the scale factor multiplied by the corresponding side length of the preimage.

The center of dilation and the corresponding vertices of the image and preimage are collinear. G-SRT.1

Determine whether figures are similar using

rigid transformations, and

dilations.

If two triangles are similar, then the corresponding pairs of angles are congruent and the corresponding sides are proportional. G-SRT.2

Justify that two triangles are similar if two pairs of angles are congruent. G-SRT.3

G.SRTb I can prove theorems involving similarity.

Prove two triangles are similar by using triangle similarity postulates SSS~, SAS~, and AA. Tasks

Apply triangle similarity to solve problem situations (e.g., indirect measurement, missing sides/angle measures).

G.SRT.5

G.GPE I can express properties with equations.

Find the coordinate that divides a line segment into two proportional segments. G-GPE.6

G.COa I can perform transformations in the plane.

Know precise definitions of angle, perpendicular lines, parallel lines, line segment based on the undefined notions of point, line, distance along a line. G-CO.1

Represent transformatons using

Patty paper or other transparencies

Geometry software (geogebra)

Distinguish between transformations that use rigid motions from those that do not.

Use and understand mapping notation (Translate ABC using the rule (x, y) (x 6, y 5) for transformations. G-CO.2

Describe the rotations and reflections of rectangle, parallelogram, trapezoid, or regular polygon, that carry it onto itself.

Calculate the number of lines of reflection symmetry

Calculate the degree of rotational symmetry of any regular polygon. G-CO.3

Use academic language to describe rigid motion.

Use terms from G-CO.1

Concept.

Rigid motions preserve size and shape. G-CO.4

Perform and predict rigid motion transformations.

G-CO.5

G.COd I can make geometric constructions.

Skills G-CO.12

Copying a segment; copying an angle;

bisecting a segment;

bisecting an angle;

constructing perpendicular lines, including the perpendicular bisector of a line segment;

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

Conceptual understanding

Arcs maintain distance

Straightedges maintain direction

Geometric properties and theorems behind the construction steps.

Construct an G-CO.13

equilateral triangle,

a square,

Foundational Content

Understand congruence and similarity using physical models, transparencies, or geometry software.

8.G.1- Verify experimentally the properties of rotations, reflections, and translations:

1.a Lines are taken to lines, and line segments to line segments of the same length.

1.b Angles are taken to angles of the same measure.

1.c Parallel lines are taken to parallel lines.

8.G.2 Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them.

8.G.3 Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.

8.G.7 Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.

8.G.8 Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.

7.G.2 Draw freehand, with ruler and protractor, and with technology) geometric shapes with given conditions. Focus on constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle.

6.NS.c Find and position integers and other rational numbers on a horizontal or vertical number line diagram; find and position pairs of integers and other rational numbers on a coordinate plane.

6.G.3 Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to find the length of a side joining points with the same first coordinate or the same second coordinate. Apply these techniques in the context of solving real-world and mathematical problems.

Geometric measurement: understand concepts of angle and measure angles.

4.MD.5- Recognize angles as geometric shapes that are formed wherever two rays share a common endpoint, and understand concepts of angle measurement:

5.a An angle is measured with reference to a circle with itscenter at the common endpoint of the rays, by considering the fraction of the circular arc between the points where the two rays intersect the circle. An angle that turns through 1/360 of a circle is called a one-degree angle, and can be used to measure angles.

5.b An angle that turns through n one-degree angles is said to have an angle measure of n degrees.

4.MD.6 Measure angles in whole-number degrees using a protractor. Sketch angles of specified measure.

4.MD.7 Recognize angle measure as additive. When an angle is decomposed into non-overlapping parts, the angle measure of the whole is the sum of the angle measures of the parts. Solve addition and subtraction problems to find unknown angles on a diagram in real world and mathematical problems, e.g., by using an equation with a symbol for the unknown angle measure.

Draw and identify lines and angles, and classify shapes by properties of their lines and angles.

4.G.1 Draw points, lines, line segments, rays, angles right, acute, obtuse), and perpendicular and parallel lines. Identify these in two-dimensional figures.

4.G.2 Classify two-dimensional figures based on the presence or absence of parallel or perpendicular lines, or the presence or absence of angles of a specified size. Recognize right triangles as a category, and identify right triangles.

4.G.3 Recognize a line of symmetry for a two-dimensional figure as a line across the figure such that the figure can be folded along the line into matching parts. Identify line-symmetric figures and draw lines of symmetry.

Transformations and Similarity Standards

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

Practice 4. Model with mathematics. Practice 5. Use appropriate tools strategically.

Practice 6. Attend to precision.

Major Content

Congruence CO)

( Understand congruence in terms of rigid motions

G- CO.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. [Focuses on the effect of a given rigid motion.]

Similarity, Right Triangles & Trigonometry SRT)

( Understand similarity in terms of similarity transformations

G.SRT.1 Verify experimentally the properties of dilations given by a center and a scale factor.

G.SRT.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.

G.SRT.3 Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.

( Prove theorems involving similarity

G.SRT.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.

Expressing Geometric Properties with Equations GPE)

( Use coordinates to prove simple geometric theorems algebraically

G.GPE.6 Find the point on a directed line segment between two given points that partitions the segment in a given ratio.

Supporting Content

Congruence CO)

( Experiment with transformations in the plane

G-CO.1 Know precise definitions of angle, circle, perpendicular lines, parallel lines, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. [Focuses on definitions not related to a circle.]

G-CO.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.

G-CO.3 Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.

G-CO.4 Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.

G-CO.5 Given a geometric figure and a rotation, reflection, and 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.[focuses on performing transformations.]

Show Solution( Make geometric constructions

G-CO.12 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.

G-CO.13 Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. [Focus on construction of an equilateral triangle.]

Transformations and Similarity Language

angle

congruence

distance along a line

line

line segment

parallel lines

perpendicular lines

point

quadrilaterals

rectangles

parallelograms

trapezoids

rigid motions

similarity transformations

similar triangles and polygons

transformations

translation

of a polygon

rotation of a polygon

reflection

of a point

of a line segment

of a polygon

dilation of a polygon

Conjecture vs. Proof (CP) Targets

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

Practice 6. Attend to precision. Practice 7. Look for and make use of structure.

Major Content

Supporting Content

Learning Target

Unit Emphasis

Learning Target

Unit Emphasis


Determine whether figures are congruent using rigid transformation.

If two figures are congruent, then the corresponding pairs of angles are congruent and the corresponding sides are congruent. G- CO.6

G-COc I can prove geometric theorems.

linear pair postulate

Prove and apply the

vertical angle theorem

alternate interior angles are congruent (what about the others)

corresponding angles are congruent

points on a perpendicular bisector of a line segment are exactly those equidistant from the segments endpoints. G.CO.9

Students prove and apply

measures of interior angles of a triangle sum to 180

the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length. G.CO.10

G-SRTa I understand similarity in terms of similarity transformations.

Prove and apply

a line parallel to one side of a triangle divides the other two proportionally, and conversely;

Pythagorean Theorem proved using triangle similarity.

The altitude from the hypotenuse of a right triangle divides the triangle into two triangles that are similar to the largers right triangle.

The altitude of a triangle is the geometric mean of the two segments into which its foot divides the hypotenuse. G.SRT.4

G-GPE I can express geometric properties with equations.

Prove and solve problems using the slope criteria for parallel

Parallel lines have the same slope

Prove and solve problems using the slope criteria for perpendicular lines

The slopes of perpendicular lines are

negative reciprocals. G.GPE.5

COa I can perform transformation in the plane.

Know precise definitions of

angle

perpendicular lines

parallel lines

line segment

based on the undefined notions of point, line, distance along a line. G-CO.1


G-CO.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).

G-CO.4 Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.

G-CO.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.

A-CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

A-REI.6 Solve systems of linear equations exactly and approximately e.g., with graphs), focusing on pairs of linear equations in two variables. [Would include discussion of parallel lines having the same slope, to recognize a system of equations of parallel lines which has no solution.]

F-LE.2 Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs include reading these from a table).

8.F.4 -Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.

8.G.5 Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles. For example, arrange three copies of the same triangle so that the sum of the three angles appears to form a line, and give an argument in terms of transversals why this is so.

8.EE.8- Analyze and solve pairs of simultaneous linear equations. [Would include discussion of parallel lines having the same slope, to recognize a system of equations of parallel lines which has no solution.]

8.G.7 Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.

8.G.8 Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.

7.G.2 Draw freehand, with ruler and protractor, and with technology) geometric shapes with given conditions. Focus on constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle.

7.G.5 Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure.

Geometric measurement: understand concepts of angle and measure angles.

4.MD.5- Recognize angles as geometric shapes that are formed wherever two rays share a common endpoint, and understand concepts of angle measurement:

5.a An angle is measured with reference to a circle with itscenter at the common endpoint of the rays, by considering the fraction of the circular arc between the points where the two rays intersect the circle. An angle that turns through 1/360 of a circle is called a one-degree angle, and can be used to measure angles.

5.b An angle that turns through n one-degree angles is said to have an angle measure of n degrees.

4.MD.6 Measure angles in whole-number degrees using a protractor. Sketch angles of specified measure.

4.MD.7 Recognize angle measure as additive. When an angle is decomposed into non-overlapping parts, the angle measure of the whole is the sum of the angle measures of the parts. Solve addition and subtraction problems to find unknown angles on a diagram in real world and mathematical problems, e.g., by using an equation with a symbol for the unknown angle measure.

Draw and identify lines and angles, and classify shapes by properties of their lines and angles.

4.G.1 Draw points, lines, line segments, rays, angles right, acute, obtuse), and perpendicular and parallel lines. Identify these in two-dimensional figures.

Conjecture vs. Proof Standards

Major Content

Congruence CO)

( Prove geometric theorems

G.CO.9 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 segments endpoints.

G.CO.10 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. [Focus on midsegments]


( Prove theorems involving similarity

G.SRT.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 similarityExpressing Geometric Properties with Equations GPE)


G.GPE.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).

Supporting Content

Congruence CO)

( Experiment with transformations in the plane

G-CO.1 Know precise definitions of angle, circle, perpendicular lines, parallel lines, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. [Focuses on definitions not related to a circle.]

Triangles (TR) Targets

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

Practice 5. Use appropriate tools strategically.

Practice 6. Attend to precision. Practice 7. Look for and make use of structure.

Major Content

Learning Target

Unit Emphasis

Supporting Content

Learning Target

Unit Emphasis


Determine whether figures are congruent using rigid transformations.

If two triangles are congruent, then the corresponding angles are congruent and the corresponding sides are congruent.

I can identify corresponding parts of triangles.

I can create triangles given AAS, SAS, ASA, SSS

I can show how examples of non congruent triangles AAA, SSA


Isosceles triangles theorem base angels of isosceles triangles are congruent

The medians of a triangle meet at a point.

G-COd I can make geometric constructions.

I can construct an equilateral triangle.


Triangles Standards

Major Content

Congruence CO)

( Understand congruence in terms of rigid motions

G- CO.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. [Focuses on the effect of a given rigid motion.]

G-CO.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.

G-CO.8 Explain how the criteria for triangle congruence ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions .


G.CO.10 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. [Leave for later the medians of a triangle meeting at a point and midsegments.]

Supporting Content

Congruence CO)

( Make geometric constructions 12, 13)

G-CO.13 Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. [Focus on construction of an equilateral triangle.]

Additional Content

Circles C)

( Understand and apply theorems about circles

G-C.3 Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. [Focus on inscribed and circumscribed circles of a triangle.]

Quadrilaterals (QL) Targets


Practice 5. Use appropriate tools strategically. Practice 6. Attend to precision.

Practice 7. Look for and make use of structure.

Advancing Content

Learning Target

Unit Emphasis

G-C I understand and can apply theorems about circles.

G-C.3 on next page

Major Content

Learning Target

Unit Emphasis

Supporting Content

Learning Target

Unit Emphasis


Students can use geometric simulations (computer software or graphing calculator) to explore theorems about parallelograms.

Use triangle congruence to prove theorems about parallelograms

Know definition of parallelogram

Know how to create a proof using either flow chart or 2 column proof structure

Given a parallelogram, prove the characteristics of the parallelogram ex. opposite angles are congruent, opposite sides are congruent, diagonals bisect each other.

Given a quadrilateral, prove that it is a parallelogram

Know which quadrilaterals are parallelograms and be able to prove, ex. Rectangle, square, rhombus. G.CO.11

G-COd I can make geometric constructions.

Understand that the compass preserves distance

Understand that the method for constructing an equilateral triangle is also a form of constructing a 60 degree angle

Know how to construct perpendicular segments

Be able to construct a square given only a perpendicular and a length

Understand that a hexagon is composed of equilateral triangles.

The radius of a circle is the same as the side length of a hexagon. Select a point on the circle and mark off the radius length six times as you go around the circle. The intersections of the arcs and the circle are the vertices of the hexagon. G-CO.13

G-GPE I can express geometric properties with equations.

Know all of the properties of parallelogram; opp sides are congruent, opp angles are congruent, diagonals bisect each other

Use slope formula to prove sides are parallel

Use distance formula to prove sides are congruent

Use distance formula and midpoint to show that diagonals bisect each other

Know the properties of a rectangle; all 4 angles are 90 degrees (use slope formula to prove perpendicular), diagonals are congruent (use distance formula to prove)

Know the properties of a square; sides are congruent (distance), angles are 90 (slope), diagonals are perpendicular (use slope formula to show perpendicular)

Know properties of rhombus; all sides congruent (distance formula), diagonals are perpendicular (slope), can show that sides are not perpendicular (slope)

Given a circle and the radius prove or disprove if a given point lies on the circle (Pythagorean theorem) G.GPE.4

G-C I understand and can apply theorems about circles.

Define the terms inscribed, circumscribed, angle bisector and perpendicular bisector

Know how to construct angle bisectors

The intersection of angle bisectors is the incenter

The incenter is the center of the circle inscribed in a triangle

The incenter is an equal distance from each side of the triangle

Know how to construct perpendicular bisectors

The intersection of perpendicular bisectors is the circumcenter

The circumcenter is the center of the circumscribed circle of the triangle

The circumcenter is an equal distance from the vertices of the triangle

Prove that opposite angles of a quadrilateral inscribed in a circle are supplementary


Quadrilaterals Standards

Major Content

Congruence CO)


G.CO.11 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.



G.GPE.4 Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point 1, 3) lies on the circle centered at the origin and containing the point 0, 2).

Supporting Content

Congruence CO)

( Make geometric constructions

G.CO.13 Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. [Focus on construction of a square]

Additional Content

Circles C)


G-C.3 Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. [Focus on inscribed and circumscribed circles of a triangle.]

Circles CI) Targets




Practice 8. Look for and express regularity in repeated reasoning.

Advancing Content

Learning Target

Unit Emphasis

Major Content

Learning Target

Unit Emphasis

Supporting Content

Learning Target

Unit Emphasis


Circles Standards (CI)

Additional Content

Circles C)


G-C.1 Prove that all circles are similar.

G-C.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.

G-C.4 +) Construct a tangent line from a point outside a given circle to the circle.

( Find arc lengths and areas of sectors of circles

G-C.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.


( Translate between the geometric description and the equation for a conic sectionG-GPE.1Derive 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.

Geometric Measurement and Dimension GMD)

( Explain volume formulas and use them to solve problems

G-GMD.1Give 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, Cavalieris principle, and informal limit arguments.

Trigonometric Ratios (TG) Targets

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

Practice 2 Reason abstractly and quantitatively.

Practice 4. Model with mathematics.

Advancing Content

Learning Target

Unit Emphasis

Major Content

Learning Target

Unit Emphasis

Supporting Content

Learning Target

Unit Emphasis


Trigonometric Ratios Standards

Major Content


( Define trigonometric ratios and solve problems involving right triangles

G-SRT.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.

G-SRT.7 Explain and use the relationship between the sine and cosine of complementary angles.

G-SRT.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.(

Additional Content


( Apply trigonometry to general triangles

G-SRT.11 +) 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).

Modeling with Geometry (MG) Targets





Advancing Content

Learning Target

Unit Emphasis

Major Content

Learning Target

Unit Emphasis

Supporting Content

Learning Target

Unit Emphasis


Modeling with Geometry - Standards

Major Content

Modeling with Geometry G-MG)

G-MG.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).(

G-MG.2 Apply concepts of density based on area and volume in modeling situations e.g., persons per square mile, BTUs per cubic foot). (

G-MG.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). (

Interpreting Data (ID) Target


Practice 3. Construct viable arguments and critique the reasoning of others.Practice 4. Model with mathematics.


Major Content

Supporting Content

Learning Target

Unit Emphasis

S-IDa I can summarize, represent, and interpret data on a single count or measurement variable.

Represent data with plots on the real number line dot plots, histograms, and box plots). S-ID.1

Use statistics appropriate to the shape of the data distribution to compare center median, mean) and spread interquartile range, standard deviation) of two or more different data sets. S-ID.2

Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points outliers.) S-ID.3

S-IDb I can summarize, represent and interpret data on two categorical and quantitative variables.

Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data including joint, marginal, and conditional relative frequencies). S-ID.5

N-Q I can reason quantitatively and use units to solve problems.

Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays. N-Q.1

Define appropriate quantities for the purpose of descriptive modeling. N-Q.2

Choose a level of accuracy appropriate to limitations on measurement when reporting quantities. N-Q.3

Foundational Understanding

Informally assess the degree of visual overlap of two numerical data distributions with similar variabilities, measuring the difference between the centers by expressing it as a multiple of a measure of variability. 7.SP.3

Use measures of center and measures of variability for numerical data from random samples to draw informal comparative inferences about two populations. 7.SP.4

Display numerical data in plots on a number line, including dot plots, histograms, and box plots. 6.SP.4

Summarize numerical data sets in relation to their context 6.SP.5

Interpreting Data

Supporting Content

QUANTITIES

( Reason quantitatively and use units to solve problems

N-Q.1 Use units as a way to understand problems and to guide the solution of multistep problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.

N-Q.2Define appropriate quantities for the purpose of descriptive modeling.

N-Q.3 Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.

INTERPRETING CATEGORICAL AND QUANTITATIVE DATA

( Summarize, represent and interpret data on two categorical and quantitative variables

S-ID.5Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data.

Additional Content

INTERPRETING CATEGORICAL AND QUANTITATIVE DATA

( Summarize, represent and interpret data on a single count or measurement variable

S-ID.1Represent data with plots on the real number line dot plots, histograms, and box plots).

S-ID.2Use statistics appropriate to the shape of the data distribution to compare center median, mean) and spread interquartile range, standard deviation) of two or more different data sets.

S-ID.3Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points outliers).

Drawing Conclusions from Data (DC) Targets


Practice 3. Construct viable arguments and critique the reasoning of others.Practice 4. Model with mathematics.


Advancing Content

Learning Target

Unit Emphasis

Major Content

Learning Target

Unit Emphasis

Supporting Content

Learning Target

Unit Emphasis


Drawing Conclusions from Data Standards

Major Content

Making Inferences & Justifying Conclusions IC)

( Make inferences and justify conclusions from sample surveys, experiments, and observational studies

S-IC.3 Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each.

S-IC.4 Use data from a sample survey to estimate a population mean or proportion; develop a margin of error through the use of simulation models for random sampling.

S-IC.5 Use data from a randomized experiment to compare two treatments; use simulations to decide if differences between parameters are significant.

S-IC.6 Evaluate reports based on data.

Supporting Content


( Understand and evaluate random processes underlying statistical experiments

S-IC.1 Understand statistics as a process for making inferences about population parameters based on a random sample from that population.

Normal Distributions (ND) Targets



Practice 8. Look for and express regularity in repeated reasoning.

Advancing Content

Learning Target

Unit Emphasis

Major Content

Learning Target

Unit Emphasis

Supporting Content

Learning Target

Unit Emphasis


Normal Distributions Standards

Students investigate the normal distribution and estimate percentages based on the normal curve. Students also learn to evaluate whether the normal distribution is an appropriate model for a set of data.

Major Content


( Make inferences and justify conclusions from sample surveys, experiments, and observational studies

S-IC.4 Use data from a sample survey to estimate a population mean or proportion; develop a margin of error through the use of simulation models for random sampling.

Supporting Content


( Understand and evaluate random processes underlying statistical experiments

S-IC.2 Decide if a specified model is consistent with results from a given data-generating process, e.g., using simulation. For example, a model says a spinning coin falls heads up with probability 0.5. Would a result of 5 tails in a row cause you to question the model?

Additional Content

Interpreting Categorical & Quantitative Data ID)

( Summarize, represent, and interpret data on a single count or measurement variable

S-ID.4 Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages. Recognize that there are data sets for which such a procedure is not appropriate. Use calculators, spreadsheets, and tables to estimate areas under the normal curve.

Using Probability to Make Decisions MD)

( Use probability to evaluate outcomes of decisions

S-MD.6 +) Use probabilities to make fair decisions e.g., drawing by lots, using a random number generator).

S-MD.7 +) Analyze decisions and strategies using probability concepts e.g., product testing, medical testing, pulling a hockey goalie at the end of a game).

Probability (PB) Targets*

*Low priority according to OSPI resources: CI, 3D and PB




Advancing Content

Learning Target

Unit Emphasis

Major Content

Learning Target

Unit Emphasis

Supporting Content

Learning Target

Unit Emphasis


Probability Standards*

*Low priority according to OSPI resources: CI, 3D and PB

Additional Content

Conditional Probability & the Rules of Probability CP)

( Understand independence and conditional probability and use them to interpret data

S-CP.1 Describe events as subsets of a sample space the set of outcomes) using characteristics or categories) of the outcomes, or as unions, intersections, or complements of other events or, and, not).

S-CP.2 Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities, and use this characterization to determine if they are independent.

S-CP.3 Understand the conditional probability of A given B as P A and B)/P B), and interpret independence of A and B as saying that the conditional probability of A given B is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B.

S-CP.4 Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities. For example, collect data from a random sample of students in your school on their favorite subject among math, science, and English. Estimate the probability that a randomly selected student from your school will favor science given that the student is in tenth grade. Do the same for other subjects and compare the results.

S-CP.5 Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations. For example, compare the chance of having lung cancer if you are a smoker with the chance of being a smoker if you have lung cancer.

( Use the rules of probability to compute probabilities of compound events in a uniform probability model

S-CP.6 Find the conditional probability of A given B as the fraction of Bs outcomes that also belong to A, and interpret the answer in terms of the model.

S-CP.7 Apply the Addition Rule, P A or B) = P A) + P B) P A and B), and interpret the answer in terms of the model.

S-CP.8 +) Apply the general Multiplication Rule in a uniform probability model, P A and B) = P A)P B|A) = P B)P A|B), and interpret the answer in terms of the model.

S-CP.9 +) Use permutations and combinations to compute probabilities of compound events and solve problems.

Using Probability to Make Decisions (MD)

( Use probability to evaluate outcomes of decisions

S-MD.6 +) Use probabilities to make fair decisions e.g., drawing by lots, using a random number generator).

S-MD.7 +) Analyze decisions and strategies using probability concepts e.g., product testing, medical testing, pulling a hockey goalie at the end of a game).

HYPERLINK "http://www.parcconline.org/parcc-model-content-frameworks"http://www.parcconline.org/parcc-model-content-frameworks









In descriptive modeling, a model simply describes the phenomena or summarizes them in a compact form. Graphs of observations are a familiar descriptive model for example, graphs of global temperature and atmospheric CO2 over time.




table of contents - swwmath2.pbworks.comswwmath2.pbworks.com/w/file/fetch/81827021/geo sco… ·...

Documents