Seeing Similarity – From the classroom to the real world

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Teri Coles, Oklahoma City, OK,

Varied opinions indicate the jury is divided on whether explaining mathematics in the classroom using real world examples enables students to transfer concepts learned to new situations (Felton, Matthew D; Ksenija Simic-Muller and Menendez, Jose Maria. 2012; Science, 2008, Ford, Matt.) Personally, with 30 plus years of experience in the classroom, I have found that incorporating the real world as examples either as a steppingstone to introduce a concept or included as a part of a performance task at the end of the unit helps students relate the mathematics learned in the classroom to new or alternative situations.Within my high school geometry classroom, the unit on similarity provides me an opportunity to see if students are able to transfer skills taught on similarity, dilations and reductions, to a different environment. I developed this idea when preparing my Understanding by Design (UbD) for the unit. As I was reviewing the resources in CPM’s Core Connections Geometry text, specifically problem 3-106, I decided to expand upon the use of the landmark idea.

I asked a student who has significant talent (at least in my opinion) to improve upon my very rough sketches of landmarks: the Gateway Arch, the Washington Monument, the Statue of Liberty, the Christ Statue, the Eiffel Tower, and the Space Needle. She did a beautiful job (two of the sketches included).

I assigned each of the landmarks a number that corresponded to a number on a six-sided die students rolled to determine which landmark they would use to complete the work. After completing problem 3-106 in the text showing the Big Ben Clock Tower and being asked to find the height of the tower showing the use of similar triangles, they would then have an additional opportunity to use nested triangles to find lengths. Using the sketches of the additional landmarks, students again complete the same tasks set out in problem 3-106. Upon completion students’ confidence level is strong on the concepts of similarity, dilations/reductions, ratio and proportion. Comfortable now using nested triangles, they were ready for the performance task.

For the performance task, I asked students to locate a landmark within the Oklahoma City metro area. Working with a partner, pairs then submit their selection for approval. Once approved, students must visit the landmark and take a photo of each other in front of the landmark. Then, using estimation skills or actually measuring, they determine the approximate distance (exact if possible) between themselves and the landmark.

Students then create a sketch, showing the concept of nested triangles, the landmark, and themselves. Multiple techniques to solve the situation could be and were used: nested triangles, the concepts of similarity, dilation, ratio and proportion, as well as the Pythagorean Theorem. Approximated or actual measures were found for lengths on the ground. Combining their skills and measurements, they were able to determine the height of the local landmark.

I then retain these sketches for use as I introduce the trigonometry unit, specifically, the tangent ratio. It serves as a wonderful, real world image to use as a launch into the world of trigonometry.

In this case, I used the same the real-world application to serve as a steppingstone into trigonometry, and as a concluding piece for the similarity unit. Students are able to visualize similarity in the real world.

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Algebra Tiles Blue Icon

Algebra Tiles Session

  • Used throughout CPM middle and high school courses
  • Concrete, geometric representation of algebraic concepts.
  • Two-hour virtual session,
  •  Learn how students build their conceptual understanding of simplifying algebraic expressions
  • Solving equations using these tools.  
  • Determining perimeter,
  • Combining like terms,
  • Comparing expressions,
  • Solving equations
  • Use an area model to multiply polynomials,
  • Factor quadratics and other polynomials, and
  • Complete the square.
  • Support the transition from a concrete (manipulative) representation to an abstract model of mathematics..

Foundations for Implementation

This professional learning is designed for teachers as they begin their implementation of CPM. This series contains multiple components and is grounded in multiple active experiences delivered over the first year. This learning experience will encourage teachers to adjust their instructional practices, expand their content knowledge, and challenge their beliefs about teaching and learning. Teachers and leaders will gain first-hand experience with CPM with emphasis on what they will be teaching. Throughout this series educators will experience the mathematics, consider instructional practices, and learn about the classroom environment necessary for a successful implementation of CPM curriculum resources.

Page 2 of the Professional Learning Progression (PDF) describes all of the components of this learning event and the additional support available. Teachers new to a course, but have previously attended Foundations for Implementation, can choose to engage in the course Content Modules in the Professional Learning Portal rather than attending the entire series of learning events again.

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Building on Instructional Practice Series

The Building on Instructional Practice Series consists of three different events – Building on Discourse, Building on Assessment, Building on Equity – that are designed for teachers with a minimum of one year of experience teaching with CPM instructional materials and who have completed the Foundations for Implementation Series.

Building on Equity

In Building on Equity, participants will learn how to include equitable practices in their classroom and support traditionally underserved students in becoming leaders of their own learning. Essential questions include: How do I shift dependent learners into independent learners? How does my own math identity and cultural background impact my classroom? The focus of day one is equitable classroom culture. Participants will reflect on how their math identity and mindsets impact student learning. They will begin working on a plan for Chapter 1 that creates an equitable classroom culture. The focus of day two and three is implementing equitable tasks. Participants will develop their use of the 5 Practices for Orchestrating Meaningful Mathematical Discussions and curate strategies for supporting all students in becoming leaders of their own learning. Participants will use an equity lens to reflect on and revise their Chapter 1 lesson plans.

Building on Assessment

In Building on Assessment, participants will apply assessment research and develop methods to provide feedback to students and inform equitable assessment decisions. On day one, participants will align assessment practices with learning progressions and the principle of mastery over time as well as write assessment items. During day two, participants will develop rubrics, explore alternate types of assessment, and plan for implementation that supports student ownership. On the third day, participants will develop strategies to monitor progress and provide evidence of proficiency with identified mathematics content and practices. Participants will develop assessment action plans that will encourage continued collaboration within their learning community.

Building on Discourse

In Building on Discourse, participants will improve their ability to facilitate meaningful mathematical discourse. This learning experience will encourage participants to adjust their instructional practices in the areas of sharing math authority, developing independent learners, and the creation of equitable classroom environments. Participants will plan for student learning by using teaching practices such as posing purposeful questioning, supporting productive struggle, and facilitating meaningful mathematical discourse. In doing so, participants learn to support students collaboratively engaged with rich tasks with all elements of the Effective Mathematics Teaching Practices incorporated through intentional and reflective planning.