NEKSDC CCSSM HS

GEOMETRY

February

12, 2013

PRESENTATION WILL INCLUDE…

• Overview of K – 8 Geometry• Overarching Structure of HS Geometry

Content Standards• Closer Look at Several Key Content

Standards• Discussion and Activities around

Instructional Shifts and Tasks to engage students in Geometry Content Standards and reinforce Practice Standards

K – 6 GEOMETRYSTUDENTS BECOME FAMILIAR WITH

GEOMETRIC SHAPES • THEIR COMPONENTS (Sides, Angles, Faces)• THEIR PROPERTIES (e.g. number of sides,

shapes of faces)• THEIR CATEGORIZATION BASED ON

PROPERTIES (e.g. A square has four equal sides and four right angles.)

K – 6 GEOMETRY

COMPOSING AND DECOMPOSING GEOMETRIC SHAPES

The ability to describe, use, and visualize the effects of composing and decomposing geometric regions is significant in that the concepts and actions of creating and then iterating units and higher-order units in the context of construction patterns, measuring, and computing are established bases for mathematical understanding and analysis.

K-6 GEOMETRY PROGRESSIONS

SPATIAL STRUCTURING AND SPATIAL RELATIONS IN GRADE 3• Students are using abstraction when they

conceptually structure an array understand two dimensional objects and sets of objects in two dimensional space as truly two dimensional. • For two-dimensional arrays, students must see a

composition of squares (iterated units) and also as a composition of rows or columns (units of units)

SPATIAL STRUCTURING AND SPATIAL RELATIONS IN GRADE 5

• Students must visualize three-dimensional solids as being composed of cubic units (iterated units) and also as a composition of layers of the cubic units (units of units).

CLASSIFY TRIANGLES IN GRADE 4By Side Length Equilateral

Isosceles

Scalene

CLASSIFY TRIANGLES IN GRADE 4By Angle Size Acute

Obtuse

Right

ANGLES, IN GRADE 4, STUDENTS

Understand that angles are composed of two rays with a common endpoint

Understand that an angle is a rotation from a reference line and that the rotation is measured in degrees

PERPENDICULARITY, PARALLELISMIN GRADE 4, STUDENTS Distinguish between lines and line segments

Recognize and draw Parallel and perpendicular lines

COORDINATE PLANEPlotting points in Quadrant I is introduced in Grade 5By Grade 6, students understand the continuous nature of the

2-dimensional coordinate plane and are able to plot points inall four quadrants, given an ordered pair composed of rational numbers.

ALTITUDES OF TRIANGLESIn Grade 6, students recognize that there are three altitudes in

every triangle and that choice of the base determines the altitude.

Also, they understand that an altitude can lie…Outside the triangle On the triangle

Inside the triangle

POLYHEDRAL SOLIDSIn Grade 6, students analyze, compose, and decompose

polyhedral solidsThey describe the shapes of the faces and the number of faces,

edges, and vertices

VISUALIZING CROSS SECTIONS In Grade 7, students describe cross sections parallel to the

base of a polyhedron.

SCALE DRAWINGSIn Grade 7, students use their understanding of proportionality

to solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale.

Scale: ¼ inch = 3 feet

UNIQUE TRIANGLESIn Grade 7 students recognize when given conditions will result in a

UNIQUE TRIANGLE

They partake in discovery activities, and form conjectures, but do not formally prove until HS.

IMPOSSIBLE TRIANGLESIn Grade 7 students recognize when given side lengths will or will not result

in a triangle The triangle inequality theorem states that any side of a triangle is always shorter than the sum of the other two sides. If the sum of the lengths of A and B is less than the length of C, then the 3

lengths will not form a triangle.

If the sum of the lengths of A and B are equal to the length of C, then the 3 lengths will not form a triangle, since segments A and B will lie flat on side C when they are connected.

GRADE 7 FORMULAS FOR CIRCLESKnow the formulas for the area and

circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle.  

C = 2πr A = πr2

GRADE 7 ANGLE RELATIONSHIPSUse 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.

GRADE 7 PROBLEMS INVOLVING 2-D AND 3-D SHAPES

Solve real-world and mathematical problems involving area, volume and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms.  

Find the volume and surface area

GRADE 8 TRANSFORMATIONSUnderstand congruence and similarity using

physical models, transparencies, or geometry software.  

Verify experimentally the properties of rotations, reflections, and translations:

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

Angles are taken to angles of the same measure. Parallel lines are taken to parallel lines.

 

GRADE 8 TRANSFORMATIONSDescribe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.  

GRADE 8 CONGRUENCE VIA RIGID TRANSFORMATIONS

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.  

GR. 8 SIMILARITY VIA NON-RIGID AND RIGID TRANSFORMATIONSUnderstand that a two-dimensional figure is similar to

another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar two-dimensional figures, describe a sequence that exhibits the similarity between them. Enlarge PQR by a factor of 2.

GRADE 8 ANGLESUse informal arguments* to establish facts about:• the angle sum and exterior angle of triangles, • the angles created when parallel lines are cut by a

transversal• 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.  

bc a

GRADE 8 PYTHAGOREAN THEOREM

Understand and apply the Pythagorean Theorem.  

Explain a proof of the Pythagorean Theorem and its converse.  

 Here is one ofmany proofs of the PythagoreanTheorem.

How does this prove the Pythagorean Theorem?

GRADE 8 PYTHAGOREAN THEOREM Apply the Pythagorean Theorem to determine unknown

side lengths in right triangles in real-world and mathematical problems in two and three dimensions.

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

 FromKahnAcademy

GRADE 8 VOLUMESolve real-world and mathematical problems involving volume of cylinders, cones, and spheres.

Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.  

http://www.math.com

TURN AND TALK TO YOUR NEIGHBOR

What concepts and skills that HS Geometry have traditionally spent a lot of time on are now being introduced in middle school?

How does that change your ideas for focus in HS Geometry?

What concepts and skills do you predict will be areas of major focus in HS Geometry?

STRUCTURE OF THE HS GEOMETRY CONTENT STANDARDSCongruence (G-CO)Similarity, Right Triangles, and

Trigonometry (G-SRT)Circles (G-C)Expressing Geometric Properties with

Equations (G-GPE)Geometric Measurement and Dimension

(G-GMD)Modeling with Geometry (G-MG)

STRUCTURE OF THE HS GEOMETRY CONTENT STANDARDSCongruence (G-CO)• Experiment with transformations in the plane• Understand congruence in terms of rigid motions• Prove geometric theorems (required theorems

listed)• Theorems about Lines and Angles• Theorems about Triangles• Theorems about Parallelograms

Make geometric constructions (variety of tools and methods…by hand and using technology) (required constructions listed)

STRUCTURE OF THE HS GEOMETRY CONTENT STANDARDSSimilarity, Right Triangles, and Trigonometry

(G-SRT)• Understand Similarity in terms of similarity

transformations• Prove theorems involving similarity• Define trigonometric ratios and solve

problems involving right triangles• (+) Apply trigonometry to general triangles

• Law of Sines• Law of Cosines

STRUCTURE OF THE HS GEOMETRY CONTENT STANDARDSCircles (G-C)Understand and apply theorems about circles • All circle are similar• Identify and describe relationships among inscribed angles, radii, and chords.• Relationship between central, inscribed, and circumscribed angles• Inscribe angles on a diameter are right angles• The radius of a circle is perpendicular to the tangent where the radius intersects the circle

Find arc lengths and sectors of circles

STRUCTURE OF THE HS GEOMETRY CONTENT STANDARDSExpressing Geometric Properties with

Equations (G-GPE)• Translate between the geometric

description and the equation for a conic section

• Use coordinates to prove simple geometric theorems algebraically

STRUCTURE OF THE HS GEOMETRY CONTENT STANDARDSGeometric Measurement and Dimension (G-

GMD)• Explain volume formulas and use them to

solve problems• Visualize relationships between two-

dimensional and three-dimensional objects

Modeling with Geometry (G-MG)• Apply geometric concepts in modeling

situations

HS GEOMETRY CONTENT STANDARDSPrimarily Focused on Plane Euclidean GeometryShapes are studied Synthetically & Analytically• Synthetic Geometry is the branch of geometry

which makes use of axioms, theorems, and logical arguments to draw conclusions about shapes and solve problems

• Analytical Geometry places shapes on the coordinate plane, allowing shapes to defined by algebraic equations, which can be manipulated to draw conclusions about shapes and solve problems.

FINDING ANGLES

Work through this “synthetic” geometry problem. What definitions, axioms, and theorems do students need to know? What algebraic skills?

FINDING ANGLES

The next three shapes and the previous one were taken from a site filled with rich Geometry problems. http://donsteward.blogspot.com/In addition to being used to find angles, students can be asked to create a copy of each shape using GeoGebra, which reinforces many of the Practice Standards as well as knowledge of transformations.

FINDING ANGLES

FINDING ANGLES

FORMAL DEFINITIONS AND PROOF

HS Students begin to formalize the experiences with geometric shapes introduced in K – 8 by

• Using more precise definitions• Developing careful proofs

When you hear the word “proof”, what do you envision?

FORMAL DEFINITIONS AND PROOFIn a triangle, the segment connecting the midpoints

of two sides is parallel to the third side and has a length that is half the length of the third side.

Given the verbal statement of a

theorem, what are the steps that students need to take in order to prove the theorem?

Geometry, Proofs, and the Common Core Standards, Sue Olson, Ed.D, UCLA Curtis Center Mathematics Conference March 3, 2012

How has the proof of the theorem already been scaffolded at this step?

SCAFFOLDING PROOFS

WAYS TO SCAFFOLD THIS SYNTHETIC* PROOFEasiest to Most Challenging:• Provide a list of statements and a list of reasons

to choose from and work together as a class• The above, but no reasons provided• The above, but done individually• No list of statements or reasons and done

individually

*As opposed to Analytic (using coordinates)

CHANGE IT TO AN ANALYTIC APPROACHEasiest to HardestUse the methods of coordinate geometry to prove

that the segment connecting the midpoints of a triangle with vertices

A (8, 10), B (14, 0), and C (0, 0)is parallel to the third side and has a length that is

one-half the length ofthe third side. Start by drawing a diagram.

Would this method result in a proof? Why or why not?

CHANGE IT TO AN ANALYTIC APPROACHHarder:Use the methods of coordinate geometry to prove

that the segment connecting the midpoints of a triangle with vertices

A (2b, 2c), B (2a, 0), and C (0, 0)is parallel to the third side and has a length that is

one-half the length of the third side.

Would this method result in a proof? Why or why not?

CHANGE IT TO AN ANALYTIC APPROACHMost ChallengingUse the methods of coordinate geometry to prove

that the segment connecting the midpoints of any triangle is parallel to the third side and has a length that is one-half the length of the third side.

What could help make this less challenging?

INSTRUCTIONAL SHIFT: MORE FOCUS ON TRANSFORMATIONAL PERSPECTIVE

Congruence, Similarity, and Symmetry are understood

from the perspective of

Geometric Transformation

extending the work that was started in Grade 8

INSTRUCTIONAL SHIFT: MORE FOCUS ON TRANSFORMATIONAL PERSPECTIVERigid Transformations (translations, rotations,

reflections) preserve distance and angle and therefore result in images that are congruent to the original shape.

G-C0 Cluster Headings Revisited• Experiment with transformations in the plane• Understand congruence in terms of rigid

motions• Prove geometric theorems• Make geometric constructions

TRANSFORMATIONS AS FUNCTIONSUsing an Analytical Geometry lens,

transformations can be described as functions that take points on the plane as inputs and give other points on the plane as outputs.

What transformations do these functions imply?Will they result in congruent shapes?(x,y) (x + 3, y) (x, y) (y, x)(x,y) (x,-y) (x, y) (-y, x)(x,y) (2x, 2y) (x, y) (3x + 2, 3y + 2)(x, y) (.5x, y) (x, y) (x – 1, y – 1)

TRANSFORMATIONS AS FUNCTIONS(x,y) (x + 3, y)*

Turn and talk to your neighbor:*Compare and contrast the notation above that communicates a right shift of 3 and the function notation f(x – 3) used to indicate the function f(x) is shifted 3 to the right.

INSTRUCTIONAL SHIFT: MORE FOCUS ON TRANSFORMATIONAL PERSPECTIVE

Two shapes are defined to be congruent to each other if there

is a sequence of rigid motions that carries one onto the other.

Prove these triangles are congruentby writing the sequence of rigidtransformations

CONGRUENCY BY TRANSFORMATION

Prove these shapes are congruentby describing the sequence of rigid transformations

PROVING SIMILARITY VIA TRANSFORMATIONSDilation is a Non-Rigid Transformation that

preserves angle, but involves a scaling factor that affects the distance, which results in images that are similar to the original shape.

G-SRT Cluster Headings dealing with Similarity:

• Understand Similarity in terms of similarity transformations

• Prove theorems involving similarity

PROVING SIMILARITY VIA TRANSFORMATIONSFrom a transformational perspective…Two shapes are defined to be similar to

each other if there is a sequence of rigid motions followed by a non-rigid dilation that carries one onto the other.

A dilation formalizes the idea of scale factor studied in Middle School.

ANIMATION SHOWING DILATIONS OF LINES AND CIRCLES

Link to Charles A. Dana Center Mathematics Common Core ToolboxClick on the linkGo to Standards for Mathematical ContentGo to Key VisualizationsGo to Geometry

Discuss how this visualization could be used in the classroom.

What would be a good follow-up activity?

PROVE SIMILARITY BY TRANSFORMATIONSWhat non-rigid transformationproves that these trianglesare similar?What is the center of dilation?What is the scale factor of theDilation?

FIND SCALE FACTORS GIVEN A TRANSFORMATION

www.ck12.org Similarity Transformations Created by: Jacelyn O'Roark

TOOLS FOR CREATING TRANSFORMATIONSUsing• Compass• Ruler• Protractor • Transparencies

Task: Leaping Lizards

TOOLS FOR CREATING TRANSFORMATIONSUsing manipulatives such as a set of TangramsWhat shapes do you see?How are they related?Can you compose theshapes to form othercongruent or similar shapes?

Rachel McAnallen's Tangram Activities

TANGRAM PARTNER ACTIVITYSwitch partner roles between “creator” and “maker”Place a file folder between the partners so theycan’t see each other’s shape.Each partner has a white sheet of paper markedN, S, E, W on the appropriate edges.1st couple of rounds: The creator creates a shape using all 7 pieces. Then stands up and gives directions while watching the “maker” create the shape. 2nd couple of rounds: Creator doesn’t watch the maker.

What Practice Standards are being used?

TANGRAM PARTNER ACTIVITY

Using two sets of tangrams, show an illustrationof the Pythagorean Theorem. What Practice

Standards are being used?

M.C. ESCHER HTTP://WWW.MCESCHER.COM/

GROUP ACTIVITYGo to the M.C Escher website and choose Picture Gallery and Symmetry. Choose a picture. Describe the transformations as clearly as you can. What transformations do you see. Are there more than one?

What Practice Standards did you use?

TOOLS FOR CREATING TRANSFORMATIONS• GeoGebra• Geometer’s Sketchpad• Other Dynamic Geometric Software• Roman MosaicWork with a partner or a group to createthis mosaic using GeoGebra.Discuss the Practice Standards and ContentStandards that were used.

C-C 5. ARC LENGTHS AND SIMILARITYDerive 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.

http://www.themathpage.com/atrig/arc-length.htm

C-C 5. ARC LENGTHS AND SIMILARITY

http://www.themathpage.com/atrig/arc-length.htm

C-C 5. ARC LENGTHS AND SIMILARITY

http://www.themathpage.com/atrig/arc-length.htm

The arc length s is proportional to the radius r. The radian measure θ is the constant of proportionality

C-C 5. ARC LENGTHS AND SIMILARITY

http://www.themathpage.com/atrig/arc-length.htm

The arc length s is proportional to the radius r. The radian measure θ is the constant of proportionality

RIGHT TRIANGLE TRIGONOMETRYUnderstand that by similarity, side ratios in right triangles are properties of angles in the triangle, leading to definitions of trigonometric ratios for acute angles.Explain and use the relationships between the sine and cosine of complementary angles.

Relationship between sine and cosine in complementary angles

CIRCLES IN ANALYTIC GEOMETRYG-GPE (Expressing Geometric Properties with Equations) Derive the equation of a circle given center (3,-2) and radius 6 using the

Pythagorean Theorem

Complete the square to find the center and radius of a circle with equation x2 + y2 – 6x – 2y = 26

Think of the time spent in Algebra I on factoringVersus completing the square to solve quadraticEquations. What % of quadratics can be solvedby factoring? What % of quadratics can be Solved by completing the square?Is completing the square using the area modelmore intuitive for students?

CONIC SECTIONS – CIRCLES AND PARABOLAS

• Translate between the geometric description and the equation for a conic section • Derive the equation of a parabola given a focus and directrix• Parabola – Note: completing the square to find the vertex of a parabola is in

the Functions Standards(+) Ellipses and Hyperbolas in Honors or Year 4

Sketch and derive the equation for the parabola withFocus at (0,2) and directrix at y = -2

Find the vertex of the parabola with equationY = x2 + 5x + 7

VISUALIZE RELATIONSHIPS BETWEEN 2-D AND 3-D OBJECTS

• Identify the shapes of 2-dimensional cross sections of 3-dimensional objects

VISUALIZE RELATIONSHIPS BETWEEN 2-D AND 3-D OBJECTS

• Identify 3-dimensional shapes generated by rotations of 2-dimensional objects

http://www.math.wpi.edu/Course_Materials/MA1022C11/volrev/node1.html

LINKS FOR TASKS ON CONSTRUCTION

CCSSI Math Tools Part 1

CCSSI Math Tools Part 2

http://ccssimath.blogspot.fr/2012/12/mathematical-tools-part-3.html

RICH HS GEOMETRY TASKhttp://www.illustrativemathematics.org/illustrations/607

This modeling task involves several different types of geometric knowledge and problem-solving: finding areas of sectors of circles (G-C.5), using trigonometric ratios to solve right triangles (G-SRT.8), and decomposing a complicated figure involving multiple circular arcs into parts whose areas can be found (MP.7).

Teachers who wish to use this problem as a classroom task may wish to have students work on the task in cooperative learning groups due to the high technical demand of the task. If time is an issue, teachers may wish to use the Jigsaw cooperative learning strategy to divide the computational demands of the task among students while requiring all students to process the mathematics in each part of the problem.