© 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
© 2012 Boise State University IDMT
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
© 2012 Boise State University IDMT
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
© 2012 Boise State University IDMT
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
© 2012 Boise State University IDMT
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
© 2012 Boise State University IDMT
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
© 2012 Boise State University IDMT
“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
© 2012 Boise State University IDMT
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.
© 2012 Boise State University IDMT
“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
© 2012 Boise State University IDMT
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
© 2012 Boise State University IDMT
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.”
© 2012 Boise State University IDMT
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.
© 2012 Boise State University IDMT
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