Transformations Tamara Bonn Indian Springs High School San Bernardino Unified School District

Seago, Jacobs, Driscoll, Nikula, Matass, and Callahan (2013) make note of recommended priority areas from the McCallum draft report as follows: Geometry in the Common Core State Standards is based on transformations, an approach that is significantly different from previous state standards. This is a change for students, teachers, and teachers of teachers. Challenges include attention to precision and language about transformations…The transformational approach to congruence and similarity is likely unfamiliar to many middle grades teachers. (p.74) My focus!

Why do we as teachers avoid teaching transformations? “I think the lack of clear understanding of what is given and the order in which it’s presented. It is not a logical progression.” Transformations are complex and difficult to understand. If I do not truly understand them than I am not confident enough to teach them.” “Many times it is difficult to explain to students thte process of what we are thinking.” “I first need to understand it myself and see myself how they work in order to teach it to students.” “I don’t know the answer without having come across them while teaching 5, 6, and 7th grade math. I can see why a math teacher would struggle if they didn’t know how to do them themselves.” “I think we avoid them (transformations) because we are uncomfortable with them. We have not seen their importance so we don’t think they are.” General survey question to see how participant's responses lined up with the research.

CCSS The common cores state standards emphasis the importance of transformations. Euclid defined that two triangles are congruent if one triangle can be mapped onto another using a series of rigid transformations. The following are the standards that ask students to develop, use and apply transformations. 8.G.1, 2, 3, and 4. G.CO. 2, 3, 4, 5, 6, 7, and 8.

Definitions Transformations may be a challenge to some students if they have not been exposed to physically manipulating shapes. Having experiences with the physical manipulation of shapes will allow students to attain a better grasp of transformations and congruence. (Adding it Up, pp. 284-287) Take a moment to use the frayer models in your packet on pages 3-4 to define each of the rigid motions. Leave the notation box blank so we can fill that in together.

Definition A rotation is an isometry that moves each point about a fixed point through a given angle. A rotation also preserves orientation as well as distance. . Picture Concept: Rotation Explanation A rotation is an Isometry that takes a point or a set of points and rotates all of these points about a center of rotation. Function Notation

Definition A translation is an isometry that moves each point through a given distance in a given direction, more commonly known as a vector. Picture Concept: Translation Explanation This means that if two points A and B are a specific distance apart on the pre-image then their image points A’ and B; are that same distance apart after the transformation. Function Notation

Definition A reflection fixes every point on a given line (axis) and maps every other point to that point such that the segment between these points is perpendicular to the axis and bisected by it Picture Concept: Reflection Explanation A reflection preserves distance but not orientation. A reflection reverses orientation of an object. Function Notation

Composition of transformations Example: Since this is a composition of functions we Reflect over line a, then translate 3 units left and 2 units up.

Transformations with a grid Describe fully the single transformation that maps triangle ABC onto triangle A’B’C’. Describe fully the single transformation which maps triangle QSG onto triangle Q’S’G’.

Functions? Are transformations functions? Can we use some function notation to define the particular function that maps one figure onto another?

Now off the grid-the main idea of the CCSS Turn to page 6 in your packet. Place your shape to the right of line a. Label your figure clockwise (not required by helps everyone be on the same page). Use any tool on the table to aid in reflecting over the line. State reflection as a function Pick your favorite shape from the ones in the bag on the table. This will be your figure that you will work with during our explorations.

Translation Translate your figure 3 cm up and 2 cm to the right. How will you accomplish this by without allowing your figure to drift. Use any tool on the table to aid in your translation. State translation as a function Place your shape anywhere on page 7 and trace around it. Don’t forget to label your shape.

Now off the grid-the main idea of the CCSS Turn to page 8 in your packet. Place your shape to the left of line a. Reflect over line a and then again over line b. What do you notice about the final image? How can we make sure our shape doesn’t float up or down on the page? We are now going to reflect over two parallel lines.

How to find the center of rotation. Turn to page 9 in the packet. Using the patty paper on your table trace both figures on the page. How can you use the paper to verify the center of rotation? What can we do now that we have the center to find the degrees of rotation? This question was asked to increase participant understanding

Two reflections over perpendicular lines Turn to page 10. There is a pre-drawn shape on this page. Using patty paper-reflect over line a and then over line b. What kind of transformation did you perform? Trace both figures onto patty paper and mark the center of rotation. Using the 360 protractor on pg. 10 verify the angle of rotation.

Transformations to Congruence Using the properties of transformations students can no state whether two figures are congruent without having to rely on memorizing SAS, SSS, etc. As stated before Euclid stated that two figures ARE congruent if they can be mapped one onto another by a series of rigid transformations. Work together on the jigsaw activity that begins on page 12 to state how you KNOW that the figures are congruent using rigid motions and items from the 8th grade standards.

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