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MATHEMATICS CARNIVAL:

A V A VEHI HICLE LE FOR OR AL ALL TO L TO LE LEAR ARN N CCSS SSM- GE GEOM OMETRI TRIC CON ONCEPTS TS!*

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• Prof. Vivian La Ferla, ED.D

Professor of Mathematics and Computer Science and Educational Studies

Rhode Island College Providence, Rhode Island

National Council of Teachers of Mathematics (NCTM) 2016 Annual Meeting and Exhibition San Francisco, California April 14, 2016 Moscone 2016

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OUTLINE FOR TODAY

• 1. Introduction
• 2. Carnival Miras and Funny Mirrors



Understanding reflections

• 3. Spinning for Transformations



Coordinate Geometry

• reflection
• translation
• rotation



Spin the wheels for transformations

• 4. Ferris Wheel Math—rotation and other geometric understandings
• 5. Enlarge the Carousel-dilation
• 6. Analyze Three-Dimensional Solids Using K’nex
• 7. Using the TI-inspire to study transformational geometry

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RATIONALE

Understand congruence and similarity using physical models, transparencies,

• r geometry software.

 CCSS.MATH.CONTENT

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

 CCSS.MATH.CONTENT

.8.G.A.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.

 CCSS.MATH.CONTENT

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

 CCSS.MATH.CONTENT

.8.G.A.4 Understand 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.

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Experiment with transformations in the plane



CCSS.MATH.CONTENT .HSG.CO.A.1 Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.



CCSS.MATH.CONTENT .HSG.CO.A.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).



CCSS.MATH.CONTENT .HSG.CO.A.3 Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.



CCSS.MATH.CONTENT .HSG.CO.A.4 Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.



CCSS.MATH.CONTENT .HSG.CO.A.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.

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Understand congruence in terms of rigid motions

 CCSS.MATH.CONTENT

.HSG.CO.B.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.

 CCSS.MATH.CONTENT

.HSG.CO.B.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.

 CCSS.MATH.CONTENT

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

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 Make geometric constructions

 CCSS.MATH.CONTENT

.HSG.CO.D.12 Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.).

 Apply geometric concepts in modeling situations

 CCSS.MATH.CONTENT

.HSG.MG.A.1 Use geometric shapes, their measures, and their properties to describe

• bjects

 CCSS.MATH.CONTENT

.HSG.MG.A.3 Apply geometric methods to solve design problems

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Which Standards for Mathematical Practice are addressed?

 CCSS.MATH.PRACTICE.MP1 Make sense of problems and persevere in solving

them.

 CCSS.MATH.PRACTICE.MP2 Reason abstractly and quantitatively.  CCSS.MATH.PRACTICE.MP3 Construct viable arguments and critique the

reasoning of others.

 CCSS.MATH.PRACTICE.MP4 Model with mathematics.  CCSS.MATH.PRACTICE.MP5 Use appropriate tools strategically.  CCSS.MATH.PRACTICE.MP6 Attend to precision.  CCSS.MATH.PRACTICE.MP7 Look for and make use of structure.  CCSS.MATH.PRACTICE.MP8 Look for and express regularity in repeated

reasoning.

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CARNIVAL MIRAS AND FUNNY MIRRORS

PART I

• Use the Mira to reflect several shapes below over one reflection line.
• Place the Mira on the line of reflection with the beveled edge facing the

shape and look through the Mira at an angle and draw its reflection.

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2.Label the pre-image A,B,C,D and their image A’,B’C’,D’

• 3. What do you notice about the reflection in relation to the pre-image?
• 4. Connect AA’, BB’, CC’
• a. Is there a relationship between AA’ and the line of reflection?
• b. Is there a relationship between BB’ and the line of reflection? Is it different that AA’ or

the same? Explain.

• c. Is there a relationship between AA’, BB’, CC’?

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5.When a shape is reflected over a line, some properties hold true for any shape.

Analyze these properties by answering the following questions:

 Does the shape change into another shape when it is reflected or does it stay the same?  Is the shape the same size?  Examine the orientation of the shape. Did it stay the same or change?

• 6. What happens when a point is on the line of reflection?
• 7. What happens when a point is not on the line of reflection?
• 8. Can you write a definition for reflections?

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Note: the notation for a reflection of a shape over line, m, is 𝑠

𝑛 𝑡ℎ𝑏𝑞𝑓 𝐵 =

𝐵′ PART II 1.Use the Mira to perform the series of reflections: 𝑠

𝑛(∆𝐵𝐶𝐷) and label ΔA’B’C’ then 𝑠 𝑜 ∆𝐵′𝐶′𝐷′ = ∆𝐵"B"𝐷" where m ll n

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• 2. What do you notice about the relationship between the pre-image and the image when the

lines of reflection are parallel?

• 3. Connect AA”, BB” and CC”. Measure the distances and angle measures and record in the

chart below.

Length Length measurement Angle Angle measure AA’ m∠CGH A’A”

m∠HGI

BB’

m∠GIJ

B’B” CC’ C’C” Distance between m and n

• 4. By comparing values, can you make any conjectures?

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5.Use the Mira to perform the series of reflections when the lines of reflection s and t intersect. 𝑠

𝑇(∆𝐸𝐹𝐺) and label ΔD’E’F’ then 𝑠 𝑢 ∆𝐸′𝐹′𝐺′ = ∆𝐸"E"𝐺"

• 6. What do you notice about the relationship between the pre-image and the image when

the lines of reflection intersect?

7.Connect DD”, EE” and FF” and complete the chart below by measuring the listed angles.

ANGLE ANGLE MEASURE ANGLE m∠MOK

m∠ EKO

m∠KOL

m∠ OKE’ m∠ EKE’

• 8. Do you notice any relationships between the angles you measured and the angles between the

interesting lines?

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PART III: FUNNY MIRRORS

Use the mylar paper, place is over the curved line below and view the reflection. What do you notice?

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What happens when the curve gets narrower?

What happens when the curve gets wider?

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4.What happens if you place the “funny mirror” on the convex surface?

• 5. Play around and make your own curve and explains what happens.
• 6. How can you curve the mylar mirror so that you obtain many faces?

How many faces can you get?

• 7. Do the properties of reflection of preservation of distance, angle measure

and shape hold for the mylar mirror?

• 8. Can you draw any conclusions?

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SPIN FOR TRANSFORMATIONS

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DIRECTIONS

1.

COMPLETE THE PRELIMINARY ACTIVITIES ON RECOGNIZING PATTERNS THAT ARISE IN THE COORDINATES UNDER A TRANSFORMATION. YOU SHOULD BE ABLE TO RESPOND TO THE FOLLOWING QUESTIONS.

 HOW DOES A SHAPE MOVE UNDER A TRANSLATION? AND WHAT HAPPENS TO THE

CORRESPONDING COORDINATES AFTER A TRANSLATION?

 HOW DOES A SHAPE MOVE UNDER A REFLECTION? AND WHAT HAPPENS TO THE

CORRESPONDING COORDINATES AFTER A REFLECTION?

 HOW DOES A SHAPE MOVE UNDER A ROTATION OF 90°? AND WHAT HAPPENS TO THE

CORRESPONDING COORDINATES AFTER A ROTATION ABOUT (0,0)?

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SPINNING FOR TRANSFORMATIONS

1.

SELECT A GRID WITH A SHAPE ON IT .

2.

SPIN THE LEFT HAND SPINNER FOR A LETTER.

3.

FIND THE CARD WITH THE LETTER AND PERFORM THE TRANSFORMATION. MAKE SURE YOU LABEL THE COORDINATES OF THE VERTICES (A’,B’,C’, etc.).

4.

THEN PERFORM THE TRANSFORMATION ON THE TRANSFORMED SHAPE OBTAINED BY SPINNING THE SECOND SPINNER. THIS IS CALLED A SEQUENCE OF TRANSFORMATIONS. MAKE SURE YOU LABEL THE COORDINATES (A”,B”,C”, etc).

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A reflection is shown on the coordinate grid below of ΔABC to ΔA’B’C’. The reflection over the line m is given below. rm∆ABC=ΔA'B'C' is the notation for a reflection

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Perform a translation in the coordinate grid below of ABCD to A’B’C’D’. The translation vector shown moves the triangle horizontally, h units, and vertically, k units, in a direction given by the vector. T (h,k) (x+h, y+k) is the notation Complete the chart below and draw the translation and write the appropriate notation for this translation.

Coordinates of pre- image Coordinates of image (1,2) (4,2) (2,5) (5,5) 1. Do you notice a pattern between the coordinates of the pre-image and the coordinates of its image?________________ 2. Describe in your own words the relationship between the translation vector and the image? ______________________________________ 1. Label the parallelogram ABCD and its image A’B’C’D’. 2. Draw AA’, BB’,CC’,DD’ and measure the lengths. What do you notice?______________________

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POINT COORDINATES IMAGE COORDINATES A (0,0) A’ B (3,0) B ‘ C (3,4) C’

ON THE GRID BELOW IS AND EXAMPLE OF A ROTATION OF 90° ABOUT THE CENTER (0,0) OF ΔABC TO ΔA’B’C’. COMPLETE THE TABLE WITH COORDINATES FOR EACH VERTICE AND ITS ROTATIONS BELOW.

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CHALLENGE

COORDINATE GEOMETRY PROOFS

1.

TO DO THESE PROOFS, YOU WILL NEED SOME ALGEBRA

 MIDPOINT FORMULA-FOR MEDIAN  DISTANCE FORMULA-FOR LENGTHS OF SIDES  SLOPE FORMULA-TO SHOW PARALLEL LINES

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PROOF #1: DIAGONALS OF A RECTANGLE ARE EQUAL

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Thank you for your attention! Vivian La Ferla vlaferla@ric.edu Rhode Island College Providence, RI

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