215 Krone Hughes Geom Transformation · ©"2012"Boise"State"University"...

© 2012 Boise State University The MTI project is sponsored by the Idaho Legislature

and the State Department of EducaDon

Transforma)ons and Congruence in the CCSS-‐M 6-‐10

Dr. Gwyneth Hughes Boise State University and University of Wisconsin

Keith Krone Boise State University

Initiative for Developing Mathematical Thinking

© 2012 Boise State University IDMT

So...Why change? Congruence and

Similarity

TransformaDons

TransformaDons

Congruence Similarity

1.  Finding coherence and structure across mathemaDcs 2.  SpaDal reasoning in STEM fields


Traditional Mathematics

Iceberg analogy stolen shamelessly from Jeff Frykholm

Rotate Reflect Translate Dilate y = mx + b

ASA k =1.5

Finding Structure in Mathematics

Relate Shapes and Maneuver in Space y1 = 3x

y2 = 3x + 2

Rotate Reflect Translate Dilate y = mx + b

ASA k =1.5


ConnecDng TransformaDons to CPCFC •  Okolica and Macrina (1992) •  TransformaDonal geometry unit

before deducDve geometry •  Students prove congruence both

by tradiDonal methods and by transformaDons

•  When students see transformaDon connecDon they more easily idenDfy corresponding parts.

HolisDc

AnalyDcal


Spatial reasoning in chemistry

“A significant spaDal ability main effect was found in this study when: (1) exam quesDons required students to mentally manipulate two-‐dimensional representaDons of molecules, and/or (2) exams focused on higher order cogniDve skills such as problem solving.”

Pribyl and Bodner 1987, Journal of Research in Science Teaching

Purdue spatial visualization

Example problem from Purdue Visualization of Rotations Test


Applications – Earth Science

“…Between one fourth and one third of the total variaDon in ESC scores can be accounted for by spaDal ability, an ability that has not been systemaDcally fostered in tradiDonal educaDon.” ESC = Earth Science Concept Test

Black, 2005, Journal of Geoscience Educa@on


Seeing ConnecDons Start with High School

Focus Area: “The concepts of congruence, similarity, and symmetry can be understood from the perspecDve of geometric transformaDon.”

Focus Area: “Geometric transformaDons of the graphs of equaDons correspond to algebraic changes in their equaDons.”

y = x2 y = x2 + 2


“The concepts of congruence, similarity, and symmetry can be understood from the perspecDve of geometric transformaDon.”

High School

8th Grade “use ideas about distance and angles, how they behave under translaDons, rotaDons, reflecDons, and dilaDons, and ideas about congruence and similarity to describe and analyze 2D figures and to solve problems.”

“find areas of right triangles, other triangles, and special quadrilaterals by decomposing these shapes, rearranging or removing pieces, and relaDng the shapes to rectangles.

6th Grade

“reason about relaDonships among two-‐dimensional figures using scale drawings and informal geometric construcDons”

7th Grade

TransformaDons across Geometry


TransformaDons across Geometry High School

8th Grade

Can we prove that the shaded triangles are congruent?

7th Grade

Adapted from IllustraDve MathemaDcs

Adapted from sample SBAC item

Describe transformaDons that take ∆ABC to ∆DEF

>

> 40 n:1cm scale

20 n:1cm scale

≤6th Grade

?

Find area that is shaded.


“Geometric transformaDons of the graphs of equaDons correspond to algebraic changes in their equaDons.”

High School

8th Grade

1-‐7th Grade Write equivalent mathemaDcal expressions based on properDes of operaDons.

“Understand the connecDons between proporDonal relaDonships, lines, and linear equaDons.”

ConnecDng Symbols with Geometry


ConnecDng Symbols with Geometry High School

8th Grade

1-‐7th Grade

14 32

7 64

AssociaDve Property (7 x 2) x 32 = 7 x (2 x 32)

n Dles

4(n + 1) 4n + 4

y1 = 3x y2 = 3x + 2

y = x2

y = x2 + 2


K-‐5: Decompose / Compose / Create / Compare / ParDDon / Reason with shapes*, Lines of symmetry (fold and draw), Coordinate plane movement

* Shaping gets overlooked * Visual ↔ Verbal (a lot) * "QuesDons surrounding idenDficaDon and definiDon can get in

the way of developing imagery."

(*NCTM Developing EssenDal Understanding of Geometry Grades 6-‐8, page 25, 27)

“The concepts of congruence, similarity, and symmetry can be understood from the perspecDve of geometric transformaDon.”


Practical Considerations •  SpaDal reasoning warm-‐ups •  Connect spaDal reasoning across curriculum

(e.g. area formulas, graphing, parallel lines) •  Move or combine congruence and

transformaDon chapters •  Encourage high school students to defend

congruence or similarity via transformaDons AND standard criteria, relaDng the two.


Contact Info

Dr. Gwyneth Hughes [email protected]

Keith Krone [email protected]

References

Black, A. A. (2005). SpaDal ability and earth science conceptual understanding. Journal of Geoscience Educa@on, 53(4), 402.

Boulter, D. R., & Kirby, J. R. (1994). IdenDficaDon of strategies used in solving transformaDonal geometry problems. The Journal of Educa@onal Research, 87(5), 298–303.

Okolica, S., & Macrina, G. (1992). IntegraDng TransformaDon Geometry into TradiDonal High School Geometry. Mathema@cs Teacher, 85(9), 716–719.

Pribyl, J. R., & Bodner, G. M. (1987). SpaDal ability and its role in organic chemistry: A study of four organic courses. Journal of research in science teaching, 24(3), 229–240.

Sinclair, N., Pimm, D., & Skelin, M. (2012). Developing essen@al understanding of geometry for teaching mathema@cs in grades 6-‐8. NCTM.

Webinars available at dmt.boisestate.edu -‐> Professional Development -‐> Online

215 Krone Hughes Geom Transformation · ©"2012"Boise"State"University"...

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