discovering geometry - high school math | prek 12 correlation of discovering geometry(©2004) key...

Minnesota correlation of Discovering Geometry (© 2004) Key Curriculum Press 1

Minnesota Correlation of Discovering Geometry

Standard

Apply skills of mathematicalrepresentation, communicationand reasoning throughout theremaining three content strands.

Benchmarks

1. Assess the reasonableness of a solution by comparing the solution to appropriate graphical or numericalestimates or by recognizing thefeasibility of solutions in a givencontext and rejecting extraneoussolutions.

2. Appropriately use examples andcounterexamples to make and testconjectures, justify solutions, andexplain results.

Discovering Geometry

Chapter 2: Reasoning in Geometry2.2: Deductive ReasoningProject: Best-Fit Lines2.5: Angle Relationships2.6: Special Angles on Parallel Lines

Chapter 4: Discovering and Proving Triangle Properties4.1: Triangle Sum Conjecture4.2: Properties of Special Triangles

Chapter 6: Discovering and Proving Circle Properties6.5: The Circumference/Diameter RatioProject: Needle Toss

Chapter 8: Area8.3: Area ProblemsExploration: Alternative Area Formulas

Throughout the text students make well over a hundred conjectures.While formulating each conjecture, students use examples to make andtest its validity.

For uses of counterexamples, see:

Chapter 1: Introducing Geometry1.3: What's a Widget?1.5: Triangles and Special QuadrilateralsChapter 1 Review

Chapter 2: Reasoning in Geometry2.1: Inductive Reasoning2.4: Mathematical Modeling

MATHEMATICAL REASONING

GeometryDiscovering

GeometryAn Investigative Approach

by Michael Serra


Standard Benchmarks

3. Translate a problem described verbally or by tables, diagrams orgraphs, into suitable mathematicallanguage, solve the problemmathematically and interpret the result in the original context.


Chapter 4: Discovering and Proving Triangle Properties4.8: Proving Isosceles Triangle Conjectures

Chapter 5: Discovering and Proving Triangle Properties5.6: Properties of Special Parallelograms

Chapter 6: Discovering and Proving Circle Properties6.4: Proving Circle Conjectures

Chapter 10: Volume10.2: Volume of Prisms and Cylinders

Chapter 11: SimilarityTake Another Look

Chapter 12: TrigonometryChapter 12 Review

Chapter 1: Introducing Geometry1.7: A Picture is Worth a Thousand Words

Chapter 2: Reasoning in Geometry2.1: Inductive Reasoning2.3: Finding the nth Term2.4: Mathematical Modeling2.5: Angle RelationshipsExploration: Patterns in FractalsChapter 2 Review

Chapter 4: Discovering and Proving Triangle PropertiesUsing Your Algebra Skills 4: Writing Linear EquationsProject: Polya's Problem4.8: Proving Isosceles Triangle Conjectures

GeometryDiscovering


by Michael Serra


Standard Benchmarks Discovering Geometry

Chapter 5: Discovering and Proving Polygon Properties5.1: Polygon Sum Conjecture

Chapter 6: Discovering and Proving Circle Properties6.5: The Circumference/Diameter Ratio6.6: Around the World

Chapter 8: Area8.3: Area ProblemsExploration: Pick's Formula for Area

Chapter 9: The Pythagorean Theorem9.1: The Theorem of Pythagoras9.4: Story ProblemsChapter 9 Review

Chapter 10: Volume Exploration: Euler's Formula for Polyhedrons 10.2: Volume of Prisms and Cylinders10.4: Volume Problems10.5: Displacement and DensityProject: Maximizing Volume10.7: Surface Area of a SphereChapter 10 Review

Chapter 11: Similarity11.3: Similar Triangles11.5: Proportions with Area and Volume

Chapter 12: Trigonometry12.2: Problem Solving with Right Triangles12.5: Problem Solving with Trigonometry

GeometryDiscovering


by Michael Serra


Standard Benchmarks

4. Support mathematical results byexplaining why the steps in a solutionare valid and why a particular solutionmethod is appropriate.


Chapter 2: Reasoning in Geometry2.2: Deductive Reasoning

Chapter 4: Discovering and Proving Triangle Properties4.7: Flowchart Thinking4.8: Proving Isosceles Triangle Conjectures

Chapter 5: Discovering and Proving Polygon Properties5.7: Proving Quadrilateral Properties


Chapter 9: The Pythagorean Theorem9.1: The Theorem of Pythagoras9.2: The Converse of the Pythagorean Theorem

Chapter 10: VolumeExploration: Sherlock Holmes and Forms of Valid Reasoning

Chapter 11: SimilarityExploration: Two More Forms of Valid Reasoning

Chapter 13: Geometry as a Mathematical System13.1: The Premises of Geometry13.2: Planning a Geometry Proof13.3: Triangle Proofs13.4: Quadrilateral Proofs13.5: Indirect Proof13.6: Circle Proofs13.7: Similarity ProofsUsing Your Algebra Skills 10: Coordinate ProofChapter 13 Review

GeometryDiscovering


by Michael Serra


Standard Benchmarks

5. Determine whether or not relevantinformation is missing from a problemand if so, decide how to best expressthe results that can be obtained withoutthat information.

6. Know and use the relationship thatexists among a logical implication ofthe form "if A, then B," its converse "if B, then A," its inverse "if not A, then not B," and its contrapositive "if not B, then not A."


Chapter 2: Reasoning in Geometry2.1: Inductive Reasoning2.2: Deductive Reasoning

Chapter 4: Discovering and Proving Triangle Properties4.4: Are There Congruence Shortcuts?4.5: Are There Other Congruence Shortcuts?4.6: Corresponding Parts of Congruent Triangles4.8: Proving Isosceles Triangle ConjecturesChapter 4 Review


Chapter 2: Reasoning in Geometry2.5: Angle Relationships2.6: Special Angles on Parallel Lines

Chapter 3: Using Tools of Geometry3.2: Constructing Perpendicular Bisectors

Chapter 4: Discovering and Proving Triangle Properties4.2: Properties of Special Triangles4.8: Proving Isosceles Triangle Conjectures

Chapter 5: Discovering and Proving Polygon Properties5.4: Properties of Midsegments

Chapter 9: The Pythagorean Theorem9.2: The Converse of the Pythagorean Theorem

Chapter 10: VolumeExploration: Sherlock Holmes and Forms of Valid Reasoning

Chapter 11: SimilarityExploration: Two More Forms of Valid Reasoning

GeometryDiscovering


by Michael Serra


Standard

Use real numbers, representedin a variety of ways, to quantify information and tosolve real-world andmathematical problems.

Benchmarks Discovering Algebra

Chapter 1: Introducing Geometry1.1: Using Your Algebra Skills1: Midpoint

Chapter 2: Reasoning in Geometry2.1: Inductive Reasoning2.3: Finding the nth Term2.4: Mathematical ModelingUsing Your Algebra Skills 2: Slope

Chapter 4: Discovering and Proving Triangle PropertiesUsing Your Algebra Skills 4: Writing Linear Equations

Chapter 5: Discovering and Proving Polygon Properties5.1: Polygon Sum ConjectureUsing Your Algebra Skills 5: Solving Systems of Equations

NUMBER SENSE, COMPUTATION AND OPERATIONSA. Number Sense

Standard

Appropriately use calculatorsand other technologies to solve algebraic, geometric,probabilistic and statisticalproblems.

Benchmarks

1. Apply the correct order of operationsand grouping symbols when usingcalculators and other technologies.

2. Know, use and translate calculatornotational conventions to mathematicalnotation.

Discovering Algebra

See Key Curriculum's Discovering Algebra and Discovering AdvancedAlgebra for coverage of this topic.

Chapter 4: Discovering and Proving Triangle PropertiesProject: Lines and Isosceles Triangles

Chapter 5: Discovering and Proving Triangle Properties Project: Drawing Regular Polygons

Chapter 10: VolumeProject: Maximizing Volume

Chapter 12: TrigonometryProject: Trigonometric Functions

B. Computation and Operation

GeometryDiscovering


by Michael Serra


Standard Benchmarks

3. Recognize the impact of units such asdegrees and radians on calculations.

4. Recognize that applying an inversefunction with a calculator may lead toextraneous or incomplete solutions.

5. Understand the limitations ofcalculators such as missing oradditional features on graphs due toviewing parameters or misleadingrepresentations of zero or very large numbers.

6. Understand that use of a calculatorrequires appropriate mathematicalreasoning and does not replace the need for mental computation.



See Key Curriculum's Discovering Advanced Algebra for coverage ofthis topic.


Chapter 2: Reasoning in GeometryProject: Line Designs

Chapter 4: Discovering and Proving Triangle PropertiesProject: Lines and Isosceles Triangles

Chapter 8: AreaProject: Maximizing Area


GeometryDiscovering


by Michael Serra


Standard

Represent and analyze real-world and mathematicalproblems using numeric,graphic and symbolic methodsfor a variety of functions.

Benchmarks

1. Know the numeric, graphic andsymbolic properties of linear, step,absolute value and quadratic functions.Graphic properties may include rates of change, intercepts, maxima andminima.

2. Model exponential growth and decay,numerically, graphically andsymbolically, using exponentialfunctions with integer inputs

3. Analyze the effects of coefficientchanges on linear and quadraticfunctions and their graphs.

Discovering Algebra

Chapter 2: Reasoning in GeometryUsing Your Algebra Skills 2: Slope

Chapter 3: Using Tools of GeometryUsing Your Algebra Skills 3: Slopes of Parallel andPerpendicular Lines


Chapter 8: AreaProject: Maximizing Area

See Key Curriculum's Discovering Algebra and Discovering AdvancedAlgebra for a more complete coverage of this topic.




PATTERNS, FUNCTIONS AND ALGEBRAA. Patterns and Functions

GeometryDiscovering


by Michael Serra


Standard Benchmarks

4. Apply basic concepts of linear,quadratic and exponential expressionsor equations in real-world problemssuch as loans, investments and the path of a projectile.

5. Distinguish functions from otherrelations using graphic and symbolicmethods.


Chapter 2: Reasoning in Geometry 2.4: Mathematical Modeling Exploration: Patterns in Fractals


Chapter 5: Discovering and Proving Polygon Properties Using Your Algebra Skills 5: Solving Systems of Linear Equations

See Key Curriculum's Discovering Algebra and Discovering AdvancedAlgebra for coverage of quadratic and exponential models.


GeometryDiscovering


by Michael Serra


Standard

Solve simple equations andinequalities numerically,graphically, and symbolically.Use recursion to model andsolve real-world andmathematical problems.

Benchmarks

1. Translate among equivalent forms ofexpressions, such as, simplify algebraicexpressions involving nested pairs ofparentheses and brackets, simplifyrational expressions, factor a commonterm from an expression and applyassociative, commutative anddistributive laws.

2. Understand the relationship betweenabsolute value and distance on thenumber line and graph simpleexpressions involving absolute valuesuch as, |x – 3| = 6 or |x + 2| <5.

3. Find equations of a line given twopoints on the line, a point and the slope of the line or the slope and the y-intercept of the line.

4. Translate among equivalent forms oflinear equations and inequalities.

Discovering Algebra

Chapter 13: Geometry as a Mathematical System13.1: The Premises of Geometry



Chapter 4: Discovering and Proving Triangle PropertiesUsing Your Algebra Skills 4: Writing Linear Equations4.4: Are There Congruence Shortcuts?4.6: Corresponding Parts of Congruent Triangles

Chapter 6: Discovering and Proving Circle PropertiesUsing Your Algebra Skills 6: Finding the Circumcenter

Chapter 7: Transformations and TessellationsUsing Your Algebra Skills 7: Finding the Orthocenter and the Centroid

Chapter 8: AreaExploration: Pick's Formula for Area



B. Algebra (Algebraic Thinking)

GeometryDiscovering


by Michael Serra


Standard Benchmarks

5. Use a variety of models such asequations, inequalities, algebraicformulas, written statements, tables and graphs or spreadsheets to representfunctions and patterns in real-world and mathematical problems.

6. Apply the laws of exponents to performoperations on expressions with integerexponents.

7. Solve linear equations and inequalitiesin one variable with numeric, graphicand symbolic methods.


Chapter 2: Reasoning in Geometry2.3: Finding the nth Term2.4: Mathematical ModelingExploration: Patterns in FractalsChapter 2 Review


Chapter 5: Discovering and Proving Polygon PropertiesUsing Your Algebra Skills 5: Solving Systems of Linear Equations



Chapter 2: Reasoning in Geometry2.2: Deductive Reasoning

Chapter 11: SimilarityUsing Your Algebra Skills 9: Proportion and Reasoning

Chapter 12: Trigonometry12.2: Problem Solving with Right Triangles



GeometryDiscovering


by Michael Serra


Standard Benchmarks

8. Find real solutions to quadraticequations in one variable with numeric,graphic and symbolic methods.

9. Use appropriate terminology andmathematical notation to define andrepresent recursion.

10. Create and use recursive formulas to model and solve real-world andmathematical problems.

11. Solve systems of two linear equationsand inequalities with two variablesusing numeric, graphic and symbolicmethods.




Chapter 2: Reasoning in GeometryExploration: Patterns in Fractals


Chapter 4: Discovering and Proving Triangle Properties4.4: Are There Congruence Shortcuts?4.6: Corresponding Parts of Congruent Triangles

Chapter 5: Discovering and Proving Polygon PropertiesUsing Your Algebra Skills 5: Solving Systems of Linear Equations


Chapter 7: Transformations and TessellationsUsing Your Algebra Skills 7: Finding the Orthocenter and Centroid

See Key Curriculum's Discovering Algebra and Discovering AdvancedAlgebra for coverage of systems of inequalities.

GeometryDiscovering


by Michael Serra


Standard Benchmarks

12. Understand how slopes can be used todetermine whether lines are parallel orperpendicular. Given a line and a pointnot on the line, find the equations forthe lines passing through that point and parallel or perpendicular to thegiven line.



Chapter 4: Discovering and Proving Triangle Properties4.2: Properties of Special Triangles Using Your Algebra Skills 4: Writing Linear Equations4.5: Are There Other Congruence Shortcuts?4.8: Proving Isosceles Triangle Conjectures

Chapter 5: Discovering and Proving Polygon Properties5.1: Polygon Sum Conjecture5.4: Properties of Midsegments5.6: Properties of Special Parallelograms

Chapter 6: Discovering and Proving Circle Properties6.2: Tangent PropertiesUsing Your Algebra Skills 6: Finding the CircumcenterChapter 6 Review


Chapter 13: Geometry as a Mathematical SystemUsing Your Algebra Skills 10: Coordinate Proof

GeometryDiscovering


by Michael Serra


Standard

Represent data and use variousmeasures associated with data to draw conclusions and identifytrends. Understand the effects of display distortion andmeasurement error on theinterpretation of data.

Benchmarks

1. Construct and analyze circle graphs, bar graphs, histograms, box-and-whisker plots, scatter plots and tables,and demonstrate the strengths andweaknesses of each format by choosing appropriately among them for a given situation.

2. Use measures of central tendency andvariability, such as, mean, median,maximum, minimum, range, standarddeviation, quartile and percentile, to describe, compare and drawconclusions about sets of data.

3. Determine an approximate best-fit line from a given scatter plot and use the line to draw conclusions.

Discovering Algebra

Chapter 2: Reasoning in Geometry2.3: Finding the nth Term2.4: Mathematical Modeling

Chapter 6: Discovering and Proving Circle PropertiesTake Another Look

Chapter 8: AreaProject: Random RectanglesProject: Different Dice

Chapter 11: SimilarityProject: In Search of the Perfect Rectangle


Chapter 2: Reasoning in GeometryProject: Best Fit lines

Chapter 6: Discovering and Proving Circle PropertiesTake Another Look

Chapter 11: SimilarityProject: In Search of the Perfect Rectangle


DATA ANALYSIS, STATISTICS AND PROBABILITYA. Data and Statistics

GeometryDiscovering


by Michael Serra


Standard Benchmarks

4. Know the influence of outliers onvarious measures and representations ofdata about real-world and mathematicalproblems.

5. Understand the relationship betweencorrelation and causation.

6. Interpret data credibility in the context of measurement error anddisplay distortion.

7. Compare outcomes of voting methodssuch as majority, plurality, ranked bypreference, run-off and pair-wisecomparison.





This topic is not covered.

GeometryDiscovering


by Michael Serra


Standard

Use appropriate countingprocedures, calculateprobabilities in various waysand apply theoretical probabilityconcepts to solve real-world and mathematical problems.

Benchmarks

1. Select and apply appropriate countingprocedures to solve real-world andmathematical problems, includingprobability problems.

2. Use area, trees, unions and intersectionsto calculate probabilities and relate the results to mutual exclusiveness,independence and conditionalprobabilities, in real-world andmathematical problems.

3. Use probability models, including areaand binomial models, in real-world and mathematical problems.

Discovering Algebra

Chapter 1: Introducing GeometryChapter 1 Review

Chapter 2: Reasoning in Geometry2.4: Mathematical Modeling2.6: Special Angles on Parallel LinesExploration: Patterns in Fractals

See Key Curriculum's Discovering Advanced Algebra for completecoverage of this topic.

Chapter 1: Introducing GeometryExploration: Geometric Probability I

Chapter 8: Area Exploration: Geometric Probability II

Chapter 1: Introducing GeometryExploration: Geometric Probability 1

Chapter 6: Discovering and Proving Circle Properties6.5: The Circumference/Diameter Ratio


Chapter 9: The Pythagorean Theorem9.2: The Converse of the Pythagorean TheoremChapter 9 Review

B. Probability

GeometryDiscovering


by Michael Serra


Standard Benchmarks

4. For simple probability models,determine the expected values ofrandom variables.

5. Know the effect of sample size onexperimental and simulationprobabilities.

6. Use a variety of experimental,simulation and theoretical methods to calculate probabilities.




Chapter 9: The Pythagorean Theorem9.2: The Converse of the Pythagorean TheoremChapter 9 Review

Chapter 8: Area Project: Different Dice

See Key Curriculum's Discovering Advanced Algebra for a morecomplete coverage of this topic.


Chapter 6: Discovering and Proving Circle PropertiesProject: Needle Toss


GeometryDiscovering


by Michael Serra


Standard

Use models to represent andunderstand two- and three-dimensional shapes and howvarious motions affect them.Recognize the relationshipbetween different representa-tions of the same shape.

Benchmarks

1. Use models and visualization tounderstand and represent three-dimensional objects and their crosssections from different perspectives.

Discovering Algebra

Chapter 1: Introducing Geometry1.1: Building Blocks of Geometry1.2: Poolroom Math1.3: What's a Widget?1.4: Polygons1.5: Triangles and Special Quadrilaterals1.6: Circles1.7: A Picture is Worth a Thousand Words 1.8: Space GeometryChapter 1 Review

Chapter 2: Reasoning in Geometry2.1: Inductive Reasoning

Chapter 3: Using Tools of Geometry3.1: Duplicating Segments and Angles3.2: Constructing Perpendicular Bisectors3.3: Constructing Perpendiculars to a Line3.4: Constructing Angle Bisectors3.5: Constructing Parallel Lines3.6: Construction ProblemsExploration: Perspective Drawing3.7: Constructing Points of Concurrency3.8: The CentroidChapter 3 Review

Chapter 5: Discovering and Proving Polygon Properties5.5: Properties of Parallelograms

SPATIAL SENSE, GEOMETRY AND MEASUREMENTA. Spatial Sense

GeometryDiscovering


by Michael Serra



Chapter 8: Area8.1: Areas of Rectangles and Parallelograms8.2: Areas of Triangles, Trapezoids, and Kites8.4: Areas of Regular Polygons8.5: Areas of Circles8.7: Surface AreaChapter 8 Review

Chapter 10: Volume10.1: The Geometry of Solids10.2: Volume of Prisms and CylindersProject: The Soma Cube10.3: Volume of Pyramids and ConesProject: The World's Largest Pyramid10.4: Volume Problems 10.5: Displacement and Density10.6: Volume of a Sphere10.7: Surface Area of a SphereChapter 10 ReviewTake Another Look

Chapter 11: Similarity11.1: Similar PolygonsExploration: Constructing a Dilation Design11.4: Corresponding Parts of Similar Triangles11.5: Proportions with Area and VolumeExploration: Why Elephants Have Big EarsChapter 11 Review

GeometryDiscovering


by Michael Serra


Standard

Apply basic theorems of plane geometry, right triangletrigonometry, coordinategeometry and a variety ofvisualization tools to solve real-world and mathematicalproblems.

Benchmarks

1. Know and use theorems about trianglesand parallel lines in elementarygeometry to justify facts about variousgeometrical figures and solve real-world and mathematical problems.These theorems include criteria for two triangles to be congruent or similarand facts about parallel lines cut by a transversal.

Discovering Algebra

Chapter 2: Reasoning in Geometry2.6: Special Angles on Parallel Lines

Chapter 4: Discovering and Proving Triangle Properties4.1: Triangle Sum Conjecture4.2: Properties of Special Triangles4.3: Triangle Inequalities4.4: Are There Congruence Shortcuts?4.5: Are There Other Congruence Shortcuts?4.6: Corresponding Parts of congruent Triangles4.7: Flowchart Thinking4.8: Proving Isosceles Triangle ConjecturesExploration: Napoleon's TheoremChapter 4 Review

Chapter 11: Similarity11.2: Similar Triangles11.3: Indirect Measurement with Similar Triangles11.4: Corresponding Parts of Similar Triangles11.6: Proportional Segments Between Parallel LinesChapter 11 Review

Chapter 13: Geometry as a Mathematical System13.2: Planning a Geometry Proof13.3: Triangle Proofs13.7: Similarity Proofs

B. Geometry

GeometryDiscovering


by Michael Serra


Standard Benchmarks

2. Know and use theorems about circles tojustify geometrical facts and solve real-world and mathematical problems.These theorems include therelationships involving tangent linesand radii, the relationship betweeninscribed and central angles and therelationship between the measure of acentral angle and arc length.

3. Know and use properties of two- andthree-dimensional figures to solve real-world and mathematical problemssuch as: finding area, perimeter, volumeand surface area; applying direct orindirect methods of measurement; thePythagorean theorem and its converse;and properties of 45o –45o –90o and 30o –60o –90o triangles


Chapter 6: Discovering and Proving Circle Properties6.1: Chord Properties6.2: Tangent Properties6.3: Arcs and Angles6.4: Proving Circle Conjectures6.5: The Circumference/Diameter Ratio6.6: Around the World6.7: Arc LengthChapter 6 Review

Chapter 9: The Pythagorean Theorem9.6: Distance in Coordinate Geometry

Chapter 13: Geometry as a Mathematical System13.6: Circle Proofs

Chapter 8: Area8.1: Areas of Rectangles and Parallelograms8.2: Areas of Triangles, Trapezoids and Kites8.3: Area Problems8.4: Areas of Regular Polygons8.5: Areas of circles8.6: Any Way You Slice It?8.7: Surface AreaExploration: Alternative Area FormulasChapter 8 Review

Chapter 9: The Pythagorean Theorem9.1: The Theorem of Pythagoras9.2: The Converse of the Pythagorean Theorem9.3: Two Special Right Triangles9.4: Story Problems9.5: Distance in Coordinate GeometryChapter 9 ReviewTake Another Look

GeometryDiscovering


by Michael Serra


Standard Benchmarks

4. Apply the basic concepts of righttriangle trigonometry including sine,cosine and tangent to solve real-worldand mathematical problems.

5. Use coordinate geometry to representand examine geometric concepts suchas the distance between two points, themidpoint of a line segment, the slope ofa line and the slopes of parallel andperpendicular lines.


Chapter 11: Similarity11.3: Indirect Measurement with Similar Triangles

Chapter 12: Trigonometry12.1: Trigonometric Ratios12.2: Problem Solving with Right TrianglesExploration: Indirect MeasurementChapter 12 Review

Chapter 1: Introducing GeometryUsing Your Algebra Skills 1: Midpoint

Chapter 2: Reasoning in GeometryUsing Your Algebra Skills 2: Slope



Chapter 7: Transformations and Tessellations Using Your Algebra Skills 7: Finding the Orthocenter and Centroid

Chapter 9: The Pythagorean Theorem9.5: Distance in Coordinate Geometry

Chapter 11: SimilarityTake Another Look

Chapter 13: Geometry as a Mathematical SystemUsing Your Algebra Skills 10: Coordinate Proof

GeometryDiscovering


by Michael Serra


Standard Benchmarks

6. Use numeric, graphic and symbolicrepresentations of transformations suchas reflections, translations and changeof scale in one, two and threedimensions to solve real-world andmathematical problems.


Chapter 1: Introducing Geometry1.4: Polygons1.5: Triangles and Special Quadrilaterals1.6: Circles1.7: A Picture is Worth a Thousand Words1.8: Space GeometryChapter 1 Review

Chapter 2: Reasoning in Geometry2.6: Special Angles on Parallel Lines

Chapter 3: Using Tools of Geometry3.1: Duplicating Segments and Angles3.6: Construction Problems

Chapter 4: Discovering and Proving Triangle Properties4.2: Properties of Special Triangles4.7: Flowchart ThinkingProject: Lines and Isosceles Triangles

Chapter 7: Transformations and Tessellations7.1: Transformations and Symmetry7.2: Properties of Isometries7.3: Compositions of transformations7.4: Tessellations with Regular Polygons7.5: Tessellations with Nonregular Polygons7.6: Tessellations Using Only Translations7.7: Tessellations That Use Rotations7.8: Tessellations That Use Glide ReflectionsChapter 7 Review

Chapter 11: Similarity11.1: Similar Polygons11.2: Similar Triangles11.4: Corresponding Parts of Similar TrianglesTake Another Look

GeometryDiscovering


by Michael Serra


Standard Benchmarks

7. Perform basic constructions with astraightedge and compass.


Chapter 3: Using Tools of Geometry3.1: Duplicating Segments and Angles3.2: Constructing Perpendicular Bisectors3.3: Constructing Perpendiculars to a Line3.4: Constructing Angle Bisectors3.5: Constructing Parallel Lines3.6: Construction Problems3.7: Constructing Points of Concurrency3.8: The Centroid Exploration: The Euler lineChapter 3 Review

Chapter 4: Discovering and Proving Triangle Properties4.2: Properties of Special Triangles4.4: Are There Congruence Shortcuts?4.5: Are There Other Congruence Shortcuts? 4.6: Corresponding Parts of Congruent Triangles4.8: Proving Isosceles Triangle ConjecturesChapter 4 Review

Chapter 5: Discovering and Proving Polygon Properties5.3: Kite and Trapezoid Properties5.4: Properties of Midsegments5.5: Properties of Parallelograms 5.6: Properties of Special ParallelogramsChapter 5 Review

Chapter 6: Discovering and Proving Circle Properties6.1: Chord Properties6.2: Tangent Properties6.3: Arcs and Angles6.7: Arc LengthChapter 6 Review

GeometryDiscovering


by Michael Serra


Standard Benchmarks

8. Draw accurate representations of planarfigures using a variety of tools.


Chapter 7: Transformations and Tessellations7.1: Transformations and Symmetry 7.2: Properties of Isometries7.5: Tessellations with Nonregular Polygons 7.8: Tessellations That Use Glide Reflections

Chapter 8: Area8.4: Areas of Regular Polygons8.7: Surface Area

Chapter 9: The Pythagorean Theorem9.3: Two Special Right Triangles9.6: Circles and the Pythagorean TheoremChapter 9 Review

Chapter 10: Volume10.4: Volume Problems10.6: Volume of a Sphere

Chapter 11: Similarity11.1: Similar Polygons11.3: Indirect Measurement with Similar Triangles11.4: Corresponding Parts of Similar Triangles11.6: Proportional Segments Between Parallel LinesChapter 11 Review

Chapter 0: Geometric Art0.1: Geometry in Nature and Art0.2: Line Designs0.3: Circle Designs0.4: Op Art0.5: Knot Designs0.6: Islamic Tile DesignsChapter 0 Review

GeometryDiscovering


by Michael Serra



Chapter 1: Introducing Geometry1.1: Building Blocks of Geometry1.2: Poolroom Math1.7: A Picture is Worth a Thousand Words

Chapter 3: Using Tools of Geometry3.1: Duplicating Segments

Chapter 5: Discovering and Proving polygon PropertiesProject: Drawing Regular Polygons

Chapter 7: Transformations and Tessellations7.3: Compositions of TransformationsProject: Penrose Tilings 7.6: Tessellations Using Only Translations7.7: Tessellations That Use Rotations

Chapter 11: SimilarityProject: Making a Mural

Chapter 12: Trigonometry12.5: Problem Solving with Trigonometry

GeometryDiscovering


by Michael Serra


Standard

Use the interconnectedness of geometry, algebra andmeasurement to explore real-world and mathematicalproblems.

Benchmarks Discovering Algebra

Chapter 1: Introducing GeometryUsing Your Algebra Skills 1: Midpoint

Chapter 2: Reasoning in Geometry2.3: Finding the nth Term2.4: Mathematical ModelingUsing Your Algebra Skills 2: SlopeChapter 2 Review


Chapter 4: Discovering and Proving Triangle Properties4.2: Properties of Special TrianglesUsing Your Algebra Skills 4: Writing Linear Equations4.4: Are There Congruence Shortcuts?4.6: Corresponding Parts of Congruent Triangles4.8: Proving Isosceles Triangle Conjectures

Chapter 5: Discovering and Proving Polygon Properties5.1: Polygon Sum Conjecture5.4: Properties of MidsegmentsUsing Your Algebra Skills 5: Solving Systems of Equations5.6: Properties of Special Quadrilaterals

Chapter 6: Discovering and Proving Circle Properties6.1: Chord PropertiesUsing Your Algebra Skills 6: Finding the CircumcenterChapter 6 Review

Chapter 7: Transformations and Tessellations7.1: Transformations and Symmetry7.2: Properties of Isometries7.3: Compositions of TransformationsUsing Your Algebra Skills 7: Finding the Orthocenter and Centroid

C. Measurement

GeometryDiscovering


by Michael Serra



Chapter 8: Area8.1: Areas of Rectangles and Parallelograms8.2: Areas of Triangles, Trapezoids, and Kites8.3: Area Problems8.4: Areas of Regular Polygons8.5: Areas of Circles8.6: Any Way You Slice It8.7: Surface AreaChapter 8 ReviewTake Another Look

Chapter 9: The Pythagorean Theorem9.1: The Theorem of Pythagoras9.2: The Converse of the Pythagorean Theorem9.3: Two Special Right Triangles9.5: Distance in Coordinate GeometryChapter 9 Review

Chapter 10: Volume10.1: The Geometry of Solids10.2: Volume of Prisms and Cylinders10.3: Volume of Pyramids and Cones 10.6: Volume of a Sphere10.7: Surface Area of a SphereTake Another Look

Chapter 11: Similarity11.2: Similar Triangles11.3: Indirect Measurement with Similar TrianglesExploration; Why Elephants Have Big Ears11.6: Proportional Segments Between Parallel Lines

GeometryDiscovering


by Michael Serra


Standard

Use tables of the normaldistribution and properties of that distribution to makejudgments about populationsbased on random samples from these populations.

Benchmarks

1. Use the concept of normal distributionand its properties to answer questionsabout sets of data.

2. Describe and use sampling distributionsand the central limit theorem. Calculateconfidence intervals when appropriate.

3. Understand the importance ofappropriate sampling methods. Forinstance, the time of day of a surveycould lead to inaccuracies in theoutcome.

Discovering Algebra




STATISTICS

GeometryDiscovering


by Michael Serra


Standard

Demonstrate facility with a wide range of algebraicoperations and use therelationship between coordinategeometry and algebraicequations to solve real-worldand mathematical problems.

Benchmarks

1. Solve systems of two, three or moresimultaneous linear equations orinequalities, in particular, decidingwhether a given system of equationshas one solution, no solution orinfinitely many solutions and, in this latter case, describing themparametrically.

2. Solve problems with quadraticfunctions and equations, where some of the coefficients may be expressed in terms of parameters.

3. Perform the four arithmetic operationswith polynomials, except that divisionis restricted to division by monomialsand linear binomials.

4. Simplify a wide variety of algebraicexpressions, including those in whichnumerator or denominator needs to be rationalized.

Discovering Algebra

Chapter 4: Discovering and Proving Triangle Properties4.4: Are There Congruence Shortcuts?4.6: Corresponding Parts of Congruent Triangles

Chapter 5: Discovering and Proving Polygon Properties Using Your Algebra Skills 5: Solving Systems of Linear Equations



See Key Curriculum's Discovering Advanced Algebra for a morecomplete coverage of this topic.




ALGEBRA

GeometryDiscovering


by Michael Serra


Standard Benchmarks

5. Apply the laws of exponents to performoperations on expressions withfractional exponents.

6. Know the numeric, graphic andsymbolic properties of power,logarithmic and exponential functions.

7. Solve a wide variety of mathematicaland real-world problems involvingpower, exponential and logarithmicfunctions and equations, discardextraneous solutions and present results graphically.

8. Know the numeric, graphic andsymbolic properties of rationalfunctions.

9. Solve a wide variety of mathematicaland real-world problems involvingrational functions, discard extraneoussolutions and present results graphically.

10. Factor polynomials representing thedifference of squares, perfect squaretrinomials and quadratics with rationalfactors.




See Key Curriculum's Discovering Algebra and Discovering AdvancedAlgebra for a complete coverage of this topic.


See Key Curriculum's Discovering Algebra and Discovering AdvancedAlgebra for a complete coverage of this topic.




GeometryDiscovering


by Michael Serra


Standard Benchmarks

11. Make sketches including axes, centers,asymptotes, vertices of parabola,ellipses (including circles) andhyperbolas with axes parallel to thecoordinate axes, given their equations,and completing the square if necessary.

12. Find equations of parabolas, ellipsesand hyperbolas when presented withtheir graphs having axes parallel to the coordinate axes.

13. Add, subtract, multiply and dividecomplex numbers, interpret sumsgeometrically, and find complexsolutions of quadratic equations.

14. Know and use the Factor andRemainder Theorems.

15. Find the inverse of a function and thecomposition of functions by numericand symbolic methods. Know therelationship between the graphs of afunction and its inverse.

16. Know and use formal notation forsequences and series to solve relatedproblems.








GeometryDiscovering


by Michael Serra


Standard

Understand the properties of the standard trigonometricfunctions and apply them toreal-world and mathematicalproblems, especially geo-metrical problems. Developincreased mastery of geometricproof methodology.

Benchmarks

1. Know the six trigonometric functionsdefined for an angle in a right triangle.

2. Given the coordinates of a point on theterminal side of an angle in standardposition in the xy-plane, find the valuesof the trigonometric functions.

3. Convert between degrees and radianmeasures.

4. Solve applied problems about trianglesusing the law of sines including theambiguous case.

5. Solve applied problems about trianglesusing the law of cosines.

6. Graph the functions of the formAsin(Bt +C), Acos(Bt + C), andAtan(Bt + C) and know the meaning ofthe terms frequency, amplitude, phaseshift and period.

Discovering Algebra

Chapter 12: Trigonometry12.1: Trigonometric Ratios (3 functions only)

See Key Curriculum's Discovering Advanced Algebra for a completecoverage of this topic.

Chapter 12: TrigonometryExploration: Trigonometric Ratios and the Unit Circle



Chapter 12: Trigonometry12.3: The Law of Sines12.5: Problem Solving with Trigonometry Chapter 12 ReviewTake Another Look

Chapter 12: Trigonometry12.4: The Law of Cosines12.5: Problem Solving with TrigonometryChapter 12 Review


TRIGONOMETRY & GEOMETRY

GeometryDiscovering


by Michael Serra


Standard Benchmarks

7. Simplify trigonometric expressionsusing identities and verify simpletrigonometric identities including sin2x + cos2x = 1, sum, difference,double angle and half-angle formulasfor sine and cosine.

8. Find all the solutions of a trigonometricequation on various intervals.

9. Know and be able to use the definitionsof the inverse trigonometric functionsand related methods to solve problemssuch as find cos(x) and tan(x) given thevalue of sin x and the quadrantcontaining the terminal side.




Chapter 12: Trigonometry12.1: Trigonometric Ratios12.2: Problem Solving with Right TrianglesChapter 12 Review

GeometryDiscovering


by Michael Serra

discovering geometry - high school math | prek 12 correlation of discovering geometry(©2004) key...

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