Coordinate Geometry: Transformations

Coordinate Geometry: Transformations
In this next unit you will study transformational geometry, a subset of coordinate geometry. Coordinate Geometry is another name for Analytical Geometry which is, “the use of algebra to study geometric properties; (and) operates on symbols defined in a coordinate system This type of geometry has a lot of applications, mostly seen in the areas of chemistry (including medicine), engineering, manufacturing, and design.

Y N Intro Activity 1.) Copy what’s below into your book.
2.) Across the line from each letter below, draw what you think the mirror image of that letter will look like, or each letter’s reflection. 3.) Describe the difference between each letter’s reflection? Y N

4.) Shake hands with a neighbor. Which hands did each of you use?
5.) Only one of you change the hand you used, and shake hands again, and describe the difference between the shaking of your hands from the first time. 6.) Definition: Chiral. A shape, object, is said to be “chiral” if it is distinct or unique compared to its reflection or mirror image. Why is this important? …………

Chemistry/Medical Application
Thalidomide is a sedative drug that was prescribed to pregnant women, from 1957 into the early 60's. It was present in at least 46 countries under different brand names. "When taken during the first trimester of pregnancy, Thalidomide prevented the proper growth of the fetus, resulting in horrific birth defects in thousands of children around the world" [1]. Why? The Thalidomide molecule is chiral. There are left and right-handed Thalidomides, just as there are left and right hands. The drug that was marketed was a 50/50 mixture. One of the molecules, say the left one, was a sedative, whereas the right one was found later to cause fetal abnormalities. "The tragedy is claimed to have been entirely avoidable had the physiological properties of the individual thalidomide [molecules] been tested prior to commercialization [2]."

POD: Coordinate plane review:
1.) On a coordinate plane, plot, and label A(2, 3), B(-2, 1), C(-4, -3), D(3, -2) 2.) How did you find point D? 3.) Why does each point have 2 coordinates? Why not just one, or why not 3? 4.) How do we refer to all the different parts of the coordinate plane? 5.) How are all the points that lie in the same quadrant, similar? How does this vary from quadrant to quadrant?

What is the coordinate plane?
a coordinate plane is a two-dimensional number line where the vertical line is called the y-axis and the horizontal is called the x-axis. These lines are perpendicular and intersect at their zero points. This point is called the origin. The axes divide the plane into four quadrants.

1st Investigation, Unit 1:
1.) clear your desks, and have your notebook and pencil handy, on your lap, or in your desk. 2.) Work with your group, but write your own work in your own notebook, to achieve the goal: Describe in steps, (step 1, step2, …)exactly what kinds of motions one figure must undergo such that it exactly overlaps the other shape (like an eclipse). Before each next step, sketch what the shapes look like so you also have a pictorial representation of your steps. Make sure what you write is specific enough, so anyone reading your steps, will do exactly what you did, and only what you did, in order for your shapes to overlap exactly. Check with your group. Discuss with your group how the two shapes might be related. Fill out the T-Chart in your notebook. Confer with your group about your findings. Ways my two shapes are the sames Ways my two shapes are different.

POD On a coordinate plane, plot the following points. A(-2, 3), B(-1, -1), C(-1, 0), D(-4, -5), E(2, 3), F(5, 3), G(5, -5), H(2, -5). Can you connect these points in a particular way so that you spell out a greeting?

POD On a coordinate plane, plot some points so if you connect them, you can form a capitol letter. Label each of your points with ordered pair coordinates, (x, y).

POD Suppose you took a capitol letter, and flipped it up, down, left or right. Try this with a letter of your choosing, and sketch the result. Which of the following categories describes what happened? See if you can find 3 letters that fit into each of the different categories below. It becomes itself again, It turns into a different letter in our alphabet, and It turns into a shape that is not any letter in our alphabet.

POD Draw a sketch of a capitol letter. (don’t bother plotting its exact location on the coordinate plane). Now redraw what you think the image of that letter will look like, after it has rotated 90 degrees. Then, draw what you think it will look like after its been rotated 180 degrees.

POD #3: Draw line segment AB, where the endpoints are A(-4, 3), and B(5, -2). Reflect this line segment over the X axis, to form line segment AIBI. #4: Lets do a sequence of two transformations. Take point T(2, -4), and reflect it over the Y axis, and then translate it left 5 units, and down 3 units. INCLUDE ARROW NOTATION to the side.

POD Substitute a number for x in each equation below which makes that equation true. How did you know which number would work? 1.) -x = 10 2.) -x = -10

POD #5: graph line segment GH, G(4, 3), H(-1, -2). Rotate GH 90 degrees clockwise, to form G l Hl. Notate this rotation using arrow notation. #6: graph line segment YZ, Y(3, 2), Z(-1, -3), and translate 1 unit up, and 4 units left, and then dilate this by a scale factor of 2.\

POD Plot triangle GHI, G(1, -3), H(2, -1), I(4, -2), and triangle JKL, J(-4, 3), K(-6, 4), L(-5, 6). Make a conjecture which states how, in two transformations, GHI  JKL occurred, and prove your conjecture. Look at pre and final image on slide 55, and write a statement that justifies why a sequence of two transformations changing ABCD, into PQRS, CAN NOT include a REFLECTION.

Lets all start off on the same page, by reviewing some coordinate plane basics.
How do we remember how we identify x vs. y axis? How do we remember where the negative and positive numbers go? How can we plot, or define exact coordinates, using an ordered pair? How can we use/identify the quadrants, each quadrant’s properties, to check our work, and communicate about the coordinate plane?

Some WWW resources Some basic animations (not on coordinate plane) for reflections, rotations, & translations Above site useful for gaining a general sense of what we can imagine A nice simple interactive game, for rotations, translations, & reflections

(4,4) (-4,4) (-1,1) (1,1) (-5,0) (-5,0) (5,0) (-1,-1) (-4,-4)
The Triangle ABC undergoes a series of two transformations, and becomes triangle AIIBIICII . 1.) Describe the two transformations triangle ABC undergoes. 2.) What different types of reflections do we see? 3.) How are the coordinates of each vertex changing after the 1st reflection over the Y axis? 4.) How do they change after the 2nd reflection over the X axis? (4,4) (-4,4) A AI (-1,1) (1,1) CI C (-5,0) BI B (-5,0) (5,0) BII CII (-1,-1) AII (-4,-4) 5.) Why, or why not, do you think that all three triangles have the same area?

Reflections Lets practice making our own. Use prime notation to label the vertices of the resulting figures. 1.) Reflect triangle ABC, over the x axis, if A(3, 2), B(1, 1), and C(-1, 4) 2.) Reflect quadrilateral PQRS, over the Y axis, if P(-3, 4), Q(-1, 5), R(-4, -4), and S(-2, -1). 3.) Reflect triangle ABC over the y axis, A(2, 4), B(2, -2), C(4, -6), and then reflect that result over the x axis.

Project #1) Group A) Group B) Group C) U(-5, 1), T(-1, 6)
1.) plot line segment UT. U(-5, 1), T(-1, 6) 2.) on the same coordinate plane, translate UT right 6 units, and down 3 units, to form U’T’. 3.) on the same coordinate plane, reflect U’T’ over the X axis to form U’’T’’. 4.) Describe in detail how you know that UT and U’’T’’ are congruent, by describing the exact sequence of two transformations which will overlap UT exactly onto U’’T’’. 5.) Next to your graph, represent this sequence of two transformations with arrow notation. Group B) 1.) plot triangle EFG. E(-5, 1), F(-1, 6), G(-1, 1) 2.) on the same coordinate plane, translate EFG right 7 units, and down 5 units, to form E’F’G’. 3.) on the same coordinate plane, reflect E’F’G’ over the X axis to form E’’F’’G’’. 4.) Describe in detail how you know that EFG and E’’F’’G’’ are congruent, by describing the exact sequence of two transformations which will overlap EFG exactly onto E’’F’’G’’. 5.) Next to your graph, represent this sequence of two transformations with arrow notation. Group C) 1.) plot figure PQRS. P(-5, 2), Q(-1, 2), R(-2, -3), S(-4, -3) 2.) on the same coordinate plane, translate PQRS right 4 units, and down 5 units, to form P’Q’R’S’. 3.) on the same coordinate plane, reflect P’Q’R’S’ over the X axis to form P’’Q’’R’’S’’. 4.) Describe in detail how you know that PQRS and P’’Q’’R’’S’’ are congruent, by describing the exact sequence of two transformations which will overlap PQRS exactly onto P’’Q’’R’’S’’. 5.) Next to your graph, represent this sequence of two transformations with arrow notation.

Project #1 Rubric. Translation (graph) Reflection (graph)
Arrow Notation Written Description of Transformation Sequence 4 Exceeding Standards All vertices neatly, correctly plotted, and labeled appropriately. All points neatly, correctly plotted, and labeled appropriately. All elements (Vertex letter names, any prime notation used on the graph, arrows, rules up top, correct x and y coordinate pairs) of arrow notation are clearly indicated in the appropriate placement, and exactly matched what is on the graph. Entirely clear, specific, and thorough description. The response conveys a completely correct sequence, proving the shapes over lap exactly, and are congruent. 3 Meeting Standards Most points neatly, correctly plotted, and labeled appropriately. Most elements of arrow notation are correct, included, and appropriately placed, matching the graph Mostly clear, specific, and thorough description. The response conveys a mostly correct sequence, proving the shapes over lap exactly, and are congruent. 2 Approaching Standards Some points neatly, correctly plotted, and labeled appropriately. Some points neatly, correctly plotted, and labeled appropriately Some elements of arrow notation are correct, included, and appropriately placed, matching the graph Only some of the response is clear, specific, and thorough. The response only partially attempts to describe the sequence that proves the shapes overlap exactly and are congruent. 1 Below Standards Few to no points neatly, correctly plotted, and labeled appropriately Few to no elements of arrow notation are correct, included, and appropriately placed, matching the graph Little to none of the response is clear, specific, and thorough. The response conveys little to no proof of the sequence of transformations which prove the shapes overlap exactly, and are congruent. 0 Incomplete

Reflections (looks like a mirror image)
Over Y axis: (x, y)  (-x, y) Over X axis: (x, y)  (x, -y) Over line y=x: Point A(x, y) becomes Al(y, x) Over line y=-x: Point A(x, y) becomes Al(-y, -x)

How do we perform

What do we mean when we say a rotation “about the origin”?
We mean that the origin is the “center of rotation”. Its like how the axel is the center of a wheel.

Example: Rotation, counter-clockwise, 90 degrees.

(1,5) (3,5) (-5,3) (-5,3) (1,3) (-5,1) (-3,1) (3,-1) (5,-1) (-3,-5) G
1.) How would we describe how ABC rotates to form GHI? How does QRS, form GHI? 2.) How can we determine a rule or pattern that a figure’s coordinates follow, when it is being rotated. 3.) How can we describe the pattern that vertex A, B, or C undergoes as it rotates 90, and 180 degrees counter clockwise? 4.) Does a rotation of 180 degrees clockwise follow the same pattern as a rotation of 180 degrees counter-clockwise? Discuss with your group and justify your response in writing. (1,5) G (3,5) A (-5,3) (-5,3) H I (1,3) (-5,1) (-3,1) Q S (3,-1) C B (5,-1) (-3,-5) R

Rotations about the origin (meaning the origin is the center of rotation)
Counter Clockwise 90o: (x, y)  (-y, x) 180o : (x, y)  (-x, -y) 270o : Same as clockwise 90o rotation. Clockwise 90o : (x, y) (y, -x) 180o : same as counter clockwise 180 rotation 270o : Same as counter clockwise 90o rotation.

Rotations Lets practice making our own. Use prime notation to label the vertices of the resulting figures. Also include arrow notation which accurately represents your transformations 1.) Rotate triangle ABC 90 degrees counterclockwise. A(3, 2), B(1, 1), C(-1, 4) 2.) Rotate quadrilateral PQRS 90 degrees clockwise. P(-3, 4), Q(-1, 5), R(-1, 1), S(-2, 1). 3.) Rotate triangle ABC 90 degrees counter clockwise, A(2, 4), B(5, 0), C(1, 1), and then reflect that result over the x axis.

Rotations, Further Practice
Lets practice making our own. Use prime notation to label the vertices of the resulting figures. Also include arrow notation which accurately represents your transformations 1.) Rotate triangle ABC 180 degrees counterclockwise. A(3, 2), B(1, 1), C(-1, 4). 2.) Rotate quadrilateral PQRS 180 degrees clockwise. P(-3, 4), Q(-1, 5), R(-1, 1), S(-2, 1). 3.) Rotate triangle ABC 180 degrees counter clockwise, A(2, 4), B(5, 0), C(1, 1), and then reflect that result over the x axis.

Practice on Rotations. Choose either #1, 2, or 3 with your group
Practice on Rotations. Choose either #1, 2, or 3 with your group. Be sure and use arrow notation to notate any transformations that occur. If you finish early, try another example on a new coordinate plane. 1.) Rotate line segment AB 90 degrees counter-clockwise to form line segment AB prime, from A(1, 4), B(-4, 2). 2.) Rotate triangle GHL 90 degrees clockwise, to form GHL prime, if GHL begins at G(-3, -1), H(-2, 2), L(3, 4). 3.) Rotate quadrilateral JKLM, from J(1, 3), K(2, -1), L(5, 2), M(3, 4) 90 degrees counter clockwise, to form JKLM prime. Investigation Task #1: A.) On a new coordinate plane, plot your original figure again (from #1, 2, or 3) and rotate it 180 degrees. Use prime notation. B.) On another different coordinate plane, plot the same original figure yet again, but this time reflect it over the X axis, and then reflect that result over the Y axis. Use prime notation to label your figures. C.) Write the arrow notation for both part A, and B, as well as label all of the vertices on each coordinate plane using prime notation. D.) Compare the results of part A and B. Two of your friends, have a disagreement about the results of part A and B. Giovani says that the position the shape after the two reflections is the same as the position after the 180 degree rotation, but Helima says that they shouldn’t be in the same position. Write a viable argument for who you think is correct, and justify your decision.

Extra… In Need of A Challenge? Plot and label your own irregular polygon with at least five vertices on three different coordinate planes, and show an example of a clockwise 90 degree, counter clockwise 90 degree, and a 180 degree rotation of your personally designed polygon. Want an Easier Example? Plot line segment AB, where A(1, 2), and B(3, 6). Do all of the following on the same coordinate plane: A.) Rotate AB, 90 degrees clockwise B)Rotate AB 90 degrees counter-clockwise C.) Rotate AB, 180 degrees. (Note: for each part, you go back and apply the given rotation to the original line segment AB, not the result of one of the other parts…

B A Transformation Analysis #1
2.) Describe in words the sequence of transformations which will transform triangle ABC onto triangle PQR, or point G onto GII. Be sure to include enough detail, so your statement has only ONE interpretation. 3.) Use arrow notation, to notate where and how each original point is transformed, in order to overlap its final image. Transformation Analysis #1 1.) Choose coordinate plane A or B below, and plot all of the points/figures into your notebook. B A R GII GI Q P A B G C

1.) How can we communicate the specific type of transformations we see happening, such that we all have the exact same interpretation? 2.) How exactly did vertex A transform, to overlap AI? 3.) How exactly did vertex CI transform to overlap CII? The rectangle ABCD undergoes a sequence of transformations to form AIIBIICIIDII A B AI BI All Bll D C (x, y)  (x – 9, y – 2)  (x + 5, y – 3) A(3, 5)  AI(-6, 3)  AII (-1, 0) B(5, 5)  BI(-4, 3)  BII(1, 0) C(5, -1)  CI(-4, -3)  CII(1, -6) D(3, -1)  DI(-6, -3)  DII(-1, -6) DI CI DII CII

Translations If point A is translated, it will shift (slide), a certain “C” number of spaces, to the left or right, and then a certain “D” number of spaces up or down. So, point (x, y)  (x ± C, y ± D) The symbol ± means “plus or minus”. Adding to the X coordinate represents a shift right Subtracting from the X coordinate represents a shift left Adding to the Y coordinate represents a shift upward Subtracting from the Y coordinate represents a shift downward Example: If a point was translated left 5 and up 3, the rule in arrow notation would look like… (x, y)  (x – 5, y + 3)

Lets check our understanding of translations
Lets check our understanding of translations. Turn and talk to your group about how we would write the arrow notation rule that represents the translation performed to overlap Triangle ABC onto Triangle AIBICI (x, y)  ( ?, ?)

Practice Translating: For each bullet, plot the original image, its translated result, and notate each using arrow notation 1.) Translate triangle ABC, down 4 spaces, and to the right 6 spaces. A(-3, 1), B(-1, -1), C(-4, 3). 2.) Translate triangle QRS, up 1 space, and to the left 3 spaces. Q(1, -2), R(4, 3), S(5, -1) 3.) Based on this arrow notation, write a rule for how the line segment WX was translated. W(3, 4)  Wl (6, 0) X(-1, 2)  Xl (2, -2) 4.) Translate triangle WRX, by the rule (X-1, Y+7), if it starts at W(-6, -6), R(-4, -2), X(1, -7)

Translation practice continued
1.) Translate triangle PQR, right 4 spaces, and to the up 3 spaces. P(-2, 1), Q(1, -1), R(-3, 3). 2.) Translate triangle JKL, up 2 spaces, and to the left 5 spaces. J(5, -2), K(4, -6), L(5, -4) 3.) Based on this arrow notation, write a rule for how the line segment WX was translated. W(2, -1)  Wl (6, -5) X(-2, 2)  Xl (2, -2) 4.) Translate triangle EFG, by the rule (X+3, Y-1), if it starts at E(-7, -5), F(-5, -1), G(4, 1)

Transformation Task #2 1.) On a coordinate plane, plot any polygon of your choice that has any where from 3 to 6 vertices, and label each vertex with a capitol letter. (choose letters in an alphabetical sequence, like DEFG). Don’t label the points with the ordered pairs. 2.) Translate your figure either left or right, and either up or down, a number of spaces of your choosing. Label the vertices of the resulting figure using prime notation (like DIEIFIGI). 3.) Take the result of your translation, and reflect that either over the x or y axis, and label the resulting figure as double prime (like DIIEIIFIIGII). 4.) Near your coordinate plane, represent the two transformations by using arrow notation. Don’t forget the rule!! 5.) In a complete sentence or two, narrate a description of how your starting image turned into your second image and then your final image.

Transformation Rubric
4.) Exceeding Standards: All points are plotted and labeled accurately/clearly on the coordinate plane, and exactly match the ordered pairs listed in your arrow notation. All of the arrow notation includes where necessary, arrows, letters representing the points on the graph, prime notation, and the rules which describe accurately each transformation occurring on your graph. The written statement conveys clearly, accurately, and unambiguously how the image(s) was transformed, the sequence (order) in which the transformations were applied, and which is the starting image, and resulting image(s). 3.) Meeting Standards: Mostly… all of the above is present with few or minor exceptions. No major conceptual errors. 2.) Approaching Standards: Somewhat… all of the above is only somewhat demonstrated, included, or correct. About a little more than half of the above requirements have been met. May include limited conceptual errors. 1.) Mostly does not…Student work mostly does not meet the requirements listed above. Major conceptual errors are made or major requirements are not demonstrated.

Writing an analysis of geometric transformations.
I. Identify the figures you are describing. Name figures by both type, and vertices (ex. Quadrilateral ABCD) Identify the original starting shape, and the new resulting shape. II. Describe the type, and specific kind of transformation. III. Cite the rule for this transformation, as evidence of your conclusion (what you state in I. and II.) IV. Cite a specific vertex to prove that the above rule is effect. Example: Triangle ABC is the result of Triangle QRS being reflected across the X axis. I know this because a reflection across the x axis follows the rule (x,y)  (x, -y), and you can see B(1,3) becomes BI(1, -3)

Isometric transformations (iso = same, metric=measurement, size)
Rotation, translation, reflection Resulting figures are congruent Same shape Side lengths same Same angles Parallel lines remain (Overlaps exactly through a series of rotations, reflections, translations)

How can we apply our “clues” to help determine which trans -formations occurred?
B Cl Al Bl

? Al Bl Cl C B A

G P I H Q R

G G’ P I H H’ I’ Q R

B C 1.) Write a statement proving EFG is congruent to TUV
2.) Create the ARROW NOTATION which represents the sequence of 2 transformations which maps EFG onto TUV. 3.) Repeat the above 2 steps, proving EFG is congruent to PQR B C F E G V T T V P U U R P Q Q R

Congruent? 1.) Analyze the following two coordinate planes, to determine whether or not the two shapes are congruent. If you don’t think so, give as much supporting evidence/reasoning to support/justify your assertion. If they are, write a congruence statement supporting your conclusion with the transformation sequence that will map one shape exactly onto the other. Include arrow notation as supporting evidence for your statement. Finished early? What other sequences of transformations, could have also supported your conclusion

Dilations A dilation is a type of transformation, which applies a scale factor that either enlarges, or shrinks the original figure, resulting in a shape similar to the original. Similar: Two shapes are said to be similar, if one can become (overlap exactly) the other, by a series of dilations, and/or rotations, reflections, and translations.

Dilations; Through the Origin.
When a figure is dilated, it expands in size or shrinks, but always does so proportionally through a specific point. Think of when we say our pupils in our eye dilate, the size of our pupil expands or reduces to let in different amounts of light. We will learn to dilate through the origin. The point of dilation

Congruent Figures 13 12 13 5 5 12

Similar Figures 5 4 7.5 6 3 4.5

Definitions of Congruent and Similar
Congruent: Two shapes are congruent when one can exactly overlap the other by undergoing a sequence of one or more of the following, and nothing other than: rotation, reflection, AND/OR translation. Similar: Two shapes are similar when one can exactly overlap the other by undergoing a sequence which MUST INCLUDE A DILATION, but also may include one or more of the following: Rotation, reflection, AND/OR translation.

Dilations Dilations expand or shrink the size of a shape, based upon a common “scale factor”, we’ll call scale factor “k”. Rule: (x,y)  (kx, ky) EX: (x,y)  (3x, 3y), (x,y)  (0.5x, 0.5y) kx means x is multiplied by k. Every coordinate, x, and y, are multiplied by the scale factor. Different dilations, have different scale factors. An enlargement occurs when k is greater than 1 A reduction occurs when k is greater than 0, but less than 1. The scale factor is never 1, and never less than zero.

Dilations How do we understand how a dilation transforms a figure on the coordinate plane? How do we interpret/apply the “scale factor” How can we generalize about the ranges of scale factors, 0 < k < 1, and k > 1, and why k ≠ 0. (and why we won’t be concerned with k = 1, and will not use k<0) How can we check our work after we have performed a dilation? How can we use dilations to understand similarity relationships between figures?

For the dilation occurring when ABC  A’B’C’ there would be a scale factor of 2, which is an ENLARGEMENT. The formula that transforms ABC onto A’B’C’ is (x, y)  (2x, 2y) If we went in reverse, and dilated A’B’C’ to map onto ABC, this would be a REDUCTION and the formula would have a different scale factor. (x, y)  ( x, y) 1 1 2 2

(x, y)  (?x,?y) A(1 , 4)W(1.5, 6) B(6, 2)X( 9, 3) (x, y)  (?x,?y)
Using Arrow Notation: Lets determine the rule for the dilations we see here. 1.) How did □ ABCD become □ WXYZ? 2.) How did □ WXYZ become □ ABCD? W X (x, y)  (?x,?y) A(1 , 4)W(1.5, 6) B(6, 2)X( 9, 3) A B D C Z (x, y)  (?x,?y) Y(0, -6)C(0, -4) Z( -6, -3)D( -4, -2) Y

Determine the scale factor of the dilation used below for the following…
1.) For ABC  A’B’C’ (x, y)  (?x, ?y) 2.) For A’B’C’  ABC (x, y)  (?x, ?y)

A C B Plot and label triangle ABC, and then dilate it by a scale factor of 3, to form A’B’C’

A’ C’ B’ A C B

A B Plot and label triangle ABC, and then dilate it by a scale factor of 1/2, to form A’B’C’ C

A A’ B B’ Plot and label triangle ABC, and then dilate it by a scale factor of 1/2, to form A’B’C’ C’ C

Practice finding the scale factor
1.) (x, y)  (?x,?y) Y( 3 , 4 )Yl( 12 , 16 ) 2.) (x, y)  (?x,?y) Y( 2 ,6 )Yl( 5 , 15 ) 3.) (x, y)  (?x,?y) Y( 2 , 3 )Yl( 3 , 4.5 )

1.) A figure, Triangle ABC has been dilated to form Triangle ABC prime. How would we determine what scale factor was used to dilate, by interpreting the arrow notation below? Express the scale factor in a rule, using arrow notation. A(2,8)  A'(3,12) B(10,4)  B'(15,6) C(6,2)  C'(9,3) 2.) Plot figure EFGH and dilate it by a scale factor of 3, if E(-1, 1), F(-1, 2), G(-2, 3), H(-3, 2) 3.) Quadrilateral QRTS, Q(3, 4), R(4, -1), T(-2, 6), S(-4, 8), is dilated by a scale factor of one half. Before graphing your dilation, use arrow notation first to determine, where each Q‘, R‘, T‘, S‘ will be after the dilation, and then plot both QRTS and the resulting Q‘R‘T‘S‘ 4.) Dilate parallelogram WXYZ, by a scale factor of 1/3 and then rotate that result by 180 degrees. WXYZ begins at W(9,9), X(6, 6), Y(0,6), Z(3, 9) 5.) Rotate the irregular pentagon, QRSTU, 90 degrees counter clockwise, and the dilate it by a scale factor of QRSTU begins at Q(4, -2), R(6, -4), S(6, -6), T(2, -6), U(2, -4).

Definitions of Congruent and Similar
Congruent: Two shapes are congruent when one can exactly overlap the other by undergoing a sequence of one or more of the following, and nothing other than: rotation, reflection, AND/OR translation. Similar: Two shapes are similar when one can exactly overlap the other by undergoing a sequence which MUST INCLUDE A DILATION, but also may include one or more of the following: Rotation, reflection, AND/OR translation.

Example: Sequence of two transformations (with out showing 1st transformation)
1.) In a written statement, prove whether EFG is congruent or similar to ABC, by showing how DEF maps exactly onto ABC. 2.) Represent this sequence of 2 transformations with ARROW NOTATION. Rotate EDF, by 180 degrees, and then dilate that result by a scale factor, k=0.5.

Practice Proving Shapes Are Congruent or Similar
1.) In a written statement, prove whether ABCD is congruent or similar to PQRS, by showing how ABCD maps exactly onto PQRS. 2.) Represent this sequence of 2 transformations with ARROW NOTATION. (x, y)  ( , )  ( , ) A(-6, -2)  Al( , )  P( -3, 1 ) B(-2, -2)  Bl( , )  Q( -1 , 1) C(-4, -4)  Cl( , )  R( -2, 0) D (-8, -4)D’( , )  S( - 4, 0)

1.) Write a statement proving if EFG is congruent or similar to TUV
2.) Create the ARROW NOTATION which represents the sequence of 2 transformations which maps EFG onto TUV. 3.) Repeat the above 2 steps, proving EFG is congruent to PQR B C F G U P E T V Q R P Q U R T V R

TEST on TUESDAY 10/7/14 You must be able to…
1.) Know from memory all of the rules which determine new coordinate pairs after any type of rotation, reflection, translation, and dilation we’ve studied. 2.) Graph a figure and the result of a figure undergoing any sequence of any of the above types of transformations, on the coordinate plane. 3.) Recognize, and prove when necessary, whether or not two shapes are either congruent, similar, or neither. 4.) Define congruent, and similar figures, using transformations 5.) Analyze from a given starting and resulting figure, which sequence of transformations could map the starting figure onto the resulting figure. 6.) Be able to notate any sequence of transformations using arrow notation, including the rules.

More Practicing Performing Sequences of Transformations
1.) Draw triangle ABC, A(-4, 6), B(2, -6), C(-6, -6). Dilate this figure by a scale factor of 1.5, then rotate it 90 degrees clockwise. Draw the resulting triangle and label its new vertices. Explain how you found the new vertices. 2.) Draw the same triangle ABC from #1 above. Translate this 2 units right, and 3 units down, then rotate it 180 degrees. Draw the resulting triangle and label its new vertices. Explain how you found the new vertices. 3.) Draw the same triangle ABC from #1 above. Rotate it 270 degrees clockwise, and then dilate it by a scale factor of Draw the resulting parallelogram and label its new vertices. Explain how you

Example continued from previous slide.
Now, applying our rules for each transformation, we will finish filling out our arrow notation, determining where each new coordinate will be. (We still have not graphed any transformed shapes, just the original). (x,y)  ( -x, -y)(0.5x, 0.5y) E(-4, 6)  El(4, -6)C(2, -3) D(2, 6)  Dl(-2, -6)A(-1, -3) F(-2, -2)  Fl(2, 2) B( 1, 1)

Example from previous slide continued

Performing Sequences of (two or more) Transformations
1.) Draw quadrilateral ABCD, A(-2, 6), B(4, 6), C(-4, 2), D(2, -2). Dilate this figure by a scale factor of 0.5, then rotate it 90 degrees counter clockwise. Draw the resulting parallelogram and label its new vertices. Represent what you did with arrow notation, including your rules. 2.) Draw the quadrilateral ABCD from #1 above. Translate this 4 units right, and 2 units down, then rotate it 180 degrees. 3.) Draw the same quadrilateral ABCD from #1 above. Rotate it 90 degree clockwise, and then reflect it across the Y axis.

POD What must “x” equal for the following statement to be true. “3 times a number, x, is the same as negative 45”. Show work that supports your answer. (Bonus) If I can prove that triangle JKL can map exactly onto another triangle, WXY, what else have I proven about those two triangles? (Bonus Bonus) How could I prove that the way to map JKL onto WXY I came up with actually works?

Now, how do we analyze transformations that have already been produced?
This is how we prove that two figures are congruent, similar, or neither, by determining that one can map exactly onto the other, by some sequence of rotations, reflections, and/or translations. First, visually, you want to determine if you have a reflection, rotation, or translation (or dilation). Sometimes some of us will instantly recognize the type of transformation, but for when that doesn’t happen, lets go through a list of clues we can look for.

Transformation Analysis (visual) Clues
Does the resulting shape enlarge, or shrink? Does the resulting shape seem like it is a “mirror” image, or that it has been flipped over on its side? Has the orientation of the shape changed? (is the top still the top? Is the bottom still the bottom? Is the left still the left? Does the right, still look like the right?) - Yes? - No? Rotation, but how far? -180 is always in the diagonally opposing quadrant. (Will show example on board if needed) - 90 degree rotations will result in a figure in an adjacent quadrant, directly, above, below, or to the left, or right. (Will show example on board if needed)

Transformation Analysis (visual) Clues
Does the resulting shape enlarge, or shrink? Yes?  DILATION!!!! Does the resulting shape seem like it is a “mirror” image, or that it has been flipped over on its side? Yes?  REFLECTION!!!!! Has the orientation of the shape changed? (is the top still the top? Is the bottom still the bottom? Is the left still the left? Does the right, still look like the right?) - Yes? REFLECTION, or ROTATION!!!!! - No? TRANSLATION!!!!!! (Unless of course the shape changed size, then it’s a dilation)

How do we proceed, once we have analyzed our transformation visually?
Now we need to verify our conjectures by testing our rule(s) on some of the coordinates or vertices that transformed. If we still are lost analyzing visually, this “test, and verify” method should help narrow your focus. The “old” and “new” vertex’s coordinates are also important clues. They are the MOST IMPORTANT clues we have. Continued…

6.) How can we check to see if our dilations were graphed correctly?
1.)How would you describe how triangle ABC changed after the dilation? How exactly do we mean “bigger”? Area? side lengths? Angles? 2.) How can these different ideas of “bigger” inform how we understand “similarity” or “similar figures” in geometry? 3.) How would you describe how the specific coordinates change from each vertex of triangle ABC , after it is dilated? 4.) How can we observe a pattern that each coordinate followed to get from ABC to ABC prime? 5.) How can we formalize our observations of a pattern, into a rule for all dilations? Observe the transformation that was applied to triangle ABC, that resulted in the new triangle ABC prime. This is an example of a dilation through the origin. All of the dilations we will study, will have a center of (0,0) 6.) How can we check to see if our dilations were graphed correctly?

Using the “old” and “new” coordinates to test and verify, which transformations occurred, by testing out different rules. Lets take two points from the original figure in the reflection we looked at, in arrow notation. (x, y)  (?, ?) } A(3, -3)  Al(-3, 3) B(4, -2)  Bl(-4, 2) This is what we do NOT know, and what it is we’re trying to determine. } This is what we DO know, from looking at where the points are on the coordinate plane Test #1) (x, y)  (x, -y)  “I’ll see if this was a reflection over the x axis.” A(3, -3)  Al(-3, 3) B(4, -2)  Bl(-4, 2) } “Wait, this can’t work, because (x, -y) would give me Bl(4, 2), but I need a rule to get me to Bl(-4, 2) Test #2) (x, y)  (-x, y)  “I’ll see if this was a reflection over the y axis.” A(3, -3)  Al(-3, 3) B(4, -2)  Bl(-4, 2) } “Wait, this can’t work, because (x, -y) would give me Bl(-4, -2), but I need a rule to get me to Bl(-4, 2) Test #3) (x, y)  (-x, -y)  “I’ll see if this was a rotation of 180 degrees.” A(3, -3)  Al(-3, 3) B(4, -2)  Bl(-4, 2) } “YES!! This works, because (-x, -y) will get me to both Al(-3, 3), and Bl(-4, 2)

1.) Prove whether or not the images below are congruent.
a.) Write a congruence statement which communicates the specific sequence of transformations which would exactly map ABC onto ABC double prime. If they are congruent, the sequence of transformations will be some combination of reflections, rotations, and/or translations. b.) Support your statement by attaching arrow notation which thoroughly shows how your transformation sequence works.

How can we use arrow notation templates to distinguish between when we ANALYZE a transformation sequence, and when we PERFORM a sequence of transformations? (x, y)  (-x, y)  (1.5x, 1.5y) A(3, -3)  Al( , )  D( , ) B(4, -2)  Bl( , )  E( , ) C(1, -2)  Cl( , )  F( , ) Why do we understand that one of these templates is for ANALYZING a sequence of two transformations, and the other is for when you are asked to PERFORM a sequence of two transformations? (x, y)  ( , )  ( , ) A(3, -3)  Al( , )  D( 3, 7 ) B(4, -2)  Bl( , )  E( 2 , 8) C(1, -2)  Cl( , )  F( 2, 5)

When we analyze for congruence, we have to find, in our arrow notation, the missing rules, and the missing coordinates of the “middle” or prime shape. (x, y)  ( , )  ( , ) A(3, -3)  Al( , )  D( 3, 7 ) B(4, -2)  Bl( , )  E( 2 , 8) C(1, -2)  Cl( , )  F( 2, 5)

POD Simplify by combining like terms: X X - 10

All the rules, for all the transformations we need to know for 8th grade.

Dilations Dilations expand or shrink the size of a shape, based upon a common “scale factor”, we’ll call scale factor “k”. Rule: (x,y)  (kx, ky) EX: (x,y)  (3x, 3y), (x,y)  (0.5x, 0.5y) kx means x is multiplied by k. Every coordinate, x, and y, are multiplied by the scale factor. Different dilations, have different scale factors. An enlargement occurs when k is greater than 1 A reduction occurs when k is greater than 0, but less than 1. The scale factor is never 1, and never less than zero.

Translations If point A is translated, it will shift (slide), a certain “C” number of spaces, to the left or right, and then a certain “D” number of spaces up or down. So, point A(x, y)  Al (x ± C, y ± D) The symbol ± means “plus or minus”.

Rotations (about the origin)
Counter Clockwise 90o – point (x, y)  (-y, x) 180o – point (x, y)  (-x, -y) 270o – point (x, y)  (y, -x) Same as clockwise 90o rotation. Clockwise 90o – point (x, y) (y, -x) 180o – same as counter clockwise (see above) 270o – point (x, y)  (-y, x) Same as counter clockwise 90o rotation.

Reflections (looks like a mirror image)
Over Y axis: Point (x, y)  (-x, y) Over X axis: Point (x, y)  (x, -y)

Congruence Analysis Task
Determine if the following two quadrilaterals are congruent. Quadrilateral ABCD: A(-4, -1), B(-3, -1), C(-2, -3), D(-4, -3). Quadrilateral TUVW: T(0,3), U(0, 2), V(-2, 1), W(-2, 3) Write a “congruence statement” which asserts whether or not the two figures are congruent, and supports your assertion with a congruence statement: a detailed description of how quadrilateral ABCD, through a sequence of either reflections, rotations, and/or translations, can map exactly onto quadrilateral TUVW. Fully notate your sequence of transformations as you describe in #1, using arrow notation. (Don’t forget the rule for each transformation!)

Transformation Congruence Analysis Rubric
4.) Exceeding Standards: The conclusion clearly, correctly asserts if the figures are congruent or not. All of the arrow notation includes where necessary, arrows, letters representing the points on the graph, prime notation, and the rules which describe accurately each transformation occurring on your graph, and represents accurately the sequence of transformations described in the congruence statement. The written congruence statement conveys clearly, accurately, and unambiguously how the initial image(s) can map exactly onto the other, through a correct sequence of reflections, rotations , and/or translations (no dilations). 3.) Meeting Standards: Mostly… all of the above is present with few or minor exceptions. No major conceptual errors. 2.) Approaching Standards: Somewhat… all of the above is only somewhat demonstrated, included, or correct. About a little more than half of the above requirements have been met. May include limited conceptual errors. 1.) Mostly does not…Student work mostly does not meet the requirements listed above. Major conceptual errors are made or major requirements are not demonstrated.

Use the transformations we’ve learned to support whether or not parallelogram PQRS is congruent, or similar to parallelogram ABCD. _________________________________________________________________________ 1.) Declare in writing if PQRS is congruent or similar to ABCD. Support this conclusion by… …Describing in complete sentences the exact transformation sequence necessary to prove your conclusion, which will map PQRS onto ABCD exactly, in a similarity or congruent statement. 2.) Show detailed arrow notation representing the transformations that allow ABCD to map exactly onto PQRS.

Rubric (transformation analysis task)
Arrow Notation Similarity/Congruence Statement 4 All elements (Vertex letter names, any prime notation used on the graph, arrows, rules up top, correct x and y coordinate pairs) of arrow notation are clearly indicated in the appropriate placement, and exactly matched what is on the graph. The correct claim has been indicated. Each transformation is indicated as specifically as possible, clearly communicating the sequence in which they occur. Precisely how the original figure becomes, or overlaps the final image exactly, is clearly communicated. Lots of appropriate math vocabulary is used. 3 Most elements of arrow notation are correct, included, and appropriately placed, matching the graph The correct claim has been indicated. The specific sequence of transformations which overlap the shapes exactly is indicated showing a core conceptual understanding, but lacks 1 or 2 minor details, or contains few careless errors. Some appropriate math vocabulary is used. 2 Some elements of arrow notation are correct, included, and appropriately placed, matching the graph Some understanding of how we can prove shapes are similar or congruent is shown, but lacking important aspects, such as parts of the sequence of transformations which maps the original figure onto the final image. Only some evidence of conceptual understanding is shown, as well as a lack of important math vocabulary (such as the names of the transformations, or how we name figures on a coordinate plane). Portions of writing may be too vague, or difficult to understand. 1 Few to no elements of arrow notation are correct, included, and appropriately placed, matching the graph Little to no understanding of how we can prove shapes are similar or congruent is shown, lacking most of the important aspects, such as how the sequence of transformations maps the original figure onto the final image. Little to no evidence of conceptual understanding is shown. Appropriate use of math vocabulary is not present.(such as the names of the transformations, or how we name figures on a coordinate plane). Much of the writing may be too vague, or difficult to understand

1.) after investigating the sequence of 2 transformations, write an appropriate congruence or similarity statement. 2.) Support this statement with thorough arrow notation.

Graded assignment: Based upon the rubrics we have discussed.
Each student will be responsible for… …plotting on a single coordinate plane, a pre-image, and resulting final figure. …a thoroughly written conjecture, which has been proven true. …thoroughly constructed proof of your conjecture using arrow notation.

…continued Group A: triangle ABC  triangle DEF. A(4, 6), B(6, 0), C(2, -4); D(-2, 3), E(-3, 0), F(-1, -2) Group B: trapezoid STUV  trapezoid WXYZ. S(-5, 2), T(-4, 4), U(-2, 4), V(-1, 2); W(3, 0), X(5, -1), Y(5, -3), Z(3, -4) Group C: Find two different sequences of two transformations for which pentagon ABCDE  pentagon FGHIJ. A(-4, 7), B(-2, 4), C(-2, 2), D(-6, 2), E(-6, 4); F(3, -4.5), G(4, -6), H(4, -7), I(2, -7), J(2, -6) Group C: If time allows, find two different sequences of two transformations such that pentagon ABCDE  pentagon FGHIJ If time still remains, write an analytical comparison of the two different successful sequences you came up with, comparing and contrasting each of the similarities and differences. How is it, that even though these are two different sequences of transformations, they produce the same result from the same pre-image. Us specific details to support your conjectures/claims. Group A done early? Analyze this sequence of two transformations: Quadrilateral ABCD  Quadrilateral WXYZ: A(1, 2), B(3, 2), C(2, -1), D(0, -1); W(6, -3), X(6 ,-9), Y(-3, -6), Z(-3, 0). Group B done early? Analyze this sequence of two transformations: Quadrilateral QRST  Quadrilateral EFGH: Q(2, 1), R(0, -3), S(-4, -3), T(-2, 1); E(-2, 4), F(6, 0), G(6, -8), H(-2, -4).

Congruence (produced by isometric transformations) & Similarity
How does our prior knowledge about Congruent and Similar figures inform how we think about these vocabulary terms? Congruent figures? How can we be more specific about what we mean by a shape’s size? Similar figures? How can understand exactly how similar figures are not of the same size?

Properties, and characteristics of Geometric Relationships we see when we apply the transformations we have learned about. Similar Figures Congruent Figures exact same____________ Different_______________ The ratio of the side lengths are always___________________ Similar figures are produced by_______________________ exact same____________ Exact same _______________ Similar figures are produced by one or more…. _______________________

Properties, and characteristics of Geometric Relationships we see when we apply the transformations we have learned about. Similar Figures Congruent Figures

Congruent Figures 13 12 13 5 5 12

Similar Figures 5 4 7.5 6 3 4.5

Congruence (produced by isometric transformations) & Similarity
If one shape can become another by a series of translations, reflections, and/or rotations, they are said to be congruent. Related vocabulary: ISOMETRY: A specific type of transformation sequence that results in a figure which is CONGRUENT to its starting pre-image, is an ISOMETRIC (adj.) transformation sequence, or an ISOMETRY (noun). Properties: When superimposed, they are coincide exactly. Angle measurements are preserved, identical. Side lengths arte preserved, identical. If one shape can become the other by a series of DILATIONS, or a series of translations, reflections, or rotations that also includes DILATIONS, they are said to be similar. Properties: Ratios of sides remain equal (proportional) but side lengths different. Angles still congruent.

Vocabulary Comprehension Practice:
Group A: Triangle QRS , Q(4, 3), R(3, 1), S(1, 2) needs to go through a series of isometric transformations. Name at least three different transformations that could fulfill this requirement. Why would a dilation, not fulfill this requirement. Group B: Triangle QRS, goes through a series of non-isometric transformation sequences. Justify how these transformations can still include reflections, and rotations, but still not be isometric Group C: A student in your class firmly believes that when he takes the final result of an isometric transformation sequence, which starts with triangle QRS, Q(4, 3), R(3, 1), S(1, 2) and then dilates it, that he has a shape that is neither congruent nor similar to the original starting shape. Conclude whether or not this final figure is either congruent or similar, and devise a method to prove this to the skeptical student.

Math Journal Entry #1 You encounter two fellow students having a debate about the word isometric, and this is what you hear them saying… Jack says: “Jillian, my sequence of a rotation and a dilation is isometric, because the interior angles of my starting figure compared to the final figure are still the same.” Jillian says: “No Jack, I think my sequence of a translation, rotation, and then reflection, is isometric, because each of these transformations produce a figure which is entirely congruent to the pre-mage.” Your teacher explains that only one student is correct about their transformation sequence being isometric. Write a response to this situation helping to clear up the confusion. You response should… Conclude which student is correctly understanding whether their transformation sequence is isometric. Justify your conclusion citing at least two specific reasons, facts, or observations that support your conclusion. (extension, if time allows: Why do you think the student who was incorrect was confused?)

(x, y)(-x, y) (x, y) (y, -x) (x, y) (5x, 5y) (x, y) (-x, -y)
POD: Without using our notes, from YOUR memory only, list the name of the transformation which is produced by each of the following rules, next to its rule. Analyze you responses in terms of their specificity, before finishing. (x, y)(-x, y) (x, y) (y, -x) (x, y) (5x, 5y) (x, y) (-x, -y) (x, y)(- y, x) (x, y) (x, -y) (x, y)(x – 3, y + 9) (x, y)(x + 7, y -1) (x, y) (0.5x, 0.5y)

How can we best benefit from our mock quiz?
Before we sit down to study, we should have a clear picture of what our strengths and weaknesses are; what we know and what still needs to be memorized. Understand the level of specificity required when identifying types of transformations

Analyze a sequence of two transformations
Unit test this Friday, on Transformational Geometry. (How do we understand the different expectations between the BOLD verbs below?) This test will be given in one period, and will assess the following skills and objectives: On this test students will be expected to... Perform a given sequence of two transformations, given a starting image notate their performed transformation sequence, with the rules, in arrow notation Analyze a sequence of two transformations Write a formal conject describing the sequence that successfully transforms the given pre-image into the given final image Notate, and prove their conjecture through the use of arrow notation Demonstrate an understanding of... Congruence Similarity Isometry(isometric transformations) The effect all of the transformations we have studied have on congruence, similarity, and isometry. Have memorized every transformation rule we have used. These rules will be posted under the resource seciton of the class website.

Lets analyze what we saw on the previous two slides…
How can we determine… If two shapes are congruent? If two shapes are similar? If a transformation sequence is isometric, or an isometry?

For the following sequences of transformations, start from the same pre-image, parallelogram NOPQ, N(0, 4), O(6, 4), P(8, 0), Q(2, 0) 1.) Perform a sequence of a 90 degree counter clockwise rotation, and a dilation by a scale factor of 0.5 Written response: Write a statement claiming how you know if this sequence is isometric or not, using at least 2 facts, observations, or reasons that justify your claim. 2.) Perform a sequence of first, a translation 5 units down, and 3 units to the left, and then reflect that result over the X axis. Show arrow notation. Written response: Write a statement claiming how you know if the original and final figures are congruent or not, and justify your claim. 3.) Graph parallelogram WXYZ, W(

Everyone for this slide will use the pre-image, triangle ABC, A(2, 4), B(2, -2), C(4, -6)
Perform two different sequences (each sequence, graphed on its own coordinate plane) of transformations on this starting figure above. You decide which transformations you want to apply (choose challenging ones for you), but must keep in mind… 1 sequence must be isometric 1 sequence should not be isometric Written response: Compare the different relationships between the pre-image and final image in each of these different sequences of transformations, making sure to address the following Identify which sequence produces similar figures, and which sequence produces congruent figures Justify how you know this by identifying at least two facts about congruent relationships Justify how you know this by identifying at least two facts about similar relationships

Homework #1) p261, copy the coordinate plane on top right of page. Label each point with the appropriate (x, y) ordered pair, label each quadrant of the coordinate plane, #2) p , read explanations, examples, definitions. Do quickchecks #1, and #2.

Homework #3) p #7, 8, 16, 17 #4) p.256 #9, 11, 13 #5) (reflections) p.259 – 260, read, explanations, examples, definitions, and do quick checks #1, #2. #6) p.261 #8-13 #7) p #14, 15, 23, 24 #8) (rotations) p.266, look at example #1, do quick checks #2a, 2b, according to the rules from class.

Homework #9) p.267, #8, 9 #10) p267, #12 #11) p , quick-checks #1, #2 (note: read examples for help) #12) p271, #5, 6 (before writing your statement, copy original and final figure onto a graph) #13) p272, #7, 8 (before writing your statement, copy original and final figure onto a graph)

Homework #14) (dilations) p.277-278 quickchecks #1, 2, 3
#16) (similarity) look over all of p283, then do p284, #8, 9

Geometry Transformations Unit Test
This Thursday, for your unit exam, you will have 2 periods, and be expected to… Identify, memorize the rules for, and plot figures which undergo 1 or more of the following… Rotation (90, 180, 270, clockwise & counter-clockwise) Translation (including direction, and distance) Reflection (over x-axis, or y-axis) Dilations (any scale factor, reduction or enlargement) Prove whether two figures on a graph are congruent or similar by describing a sequence of transformations which maps one exactly onto the other. Represent any sequence of transformations using arrow notation

Coordinate Geometry: Transformations

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