To anchor our work in differentiation and mathematical flexibility, we use NCTM’s 5 Practices for Orchestrating Productive Mathematics Discussions by Margaret Smith and Mary Kay Stein.
Kristi Story, Becky Holden, and I worked together during our professional learning time to meet the goals for the session shown below.
From NCTM’s 5 Practices, we know that we should do the math ourselves, predict (anticipate) what students will produce, and brainstorm what will help students most when in productive struggle and when in destructive struggle.
The learning goals for students include:
I can use ratio reasoning to solve problems and understand ratio concepts.
I can make sense of tasks and persevere in solving them.
I can look for and make use of structure.
I can notice and note to make my thinking visible.
Kristi selected Illustrative Math’s 6.RP Overlapping Squares task for students. Here are the ways we anticipated how students would approach and engage with the task.
Our plan for helping students who are stuck includes providing and encouraging the use of a graphing tool such as graph paper or TI-Nspire software installed on their MacBooks. We also intend to use the following learning progressions.
Finally, we also want our learners to work on how they show their work.
When mathematics classrooms focus on numbers, status differences between students often emerge, to the detriment of classroom culture and learning, with some students stating that work is “easy” or “hard” or announcing they have “finished” after racing through a worksheet. But when the same content is taught visually, it is our experience that the status differences that so often beleaguer mathematics classrooms, disappear. – Jo Boaler
Boaler, Jo, Lang Chen, Cathy Williams, and Montserrat Cordero. “Seeing as Understanding: The Importance of Visual Mathematics for Our Brain and Learning.” Journal of Applied & Computational Mathematics 05.05 (2016): n. pag. Youcubed. Standford University, 12 May. 2016. Web. 18 Mar. 2017.
Stein, Mary Kay., and Margaret Smith. 5 Practices for Orchestrating Productive Mathematics Discussions. N.p.: n.p., n.d. Print.
Experiments in Learning by Doing 
The 5 Practices are gorgeous guidelines for facilitating productive discourse around rich math tasks. On our website, we’ve woven these practices into some of Graham Fletcher’s 3-Act Tasks.
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