Park City Mathematics Institute Lesson Study Project Abstract
Factoring Polynomials by Grouping
Subject: Integrated II
Authors:
Group A: Chris Bolognese, Sam Hilkey, Traci Jackson, Cristina Jimenez-Shawcroft, Brian Shay, Roy Snyder, Hemang Srikishan
Group B: John Carter, Kate Carter, Jasper DeAntonio, Kimberly Elicker, Siaka Kone, Jennifer Outzs, Kayleigh Rose

The purpose of our lesson study was to collaboratively craft, teach, and revise a lesson for high school students in a 10th grade Integrated II classroom. We met an average of ten hours per week for two and a half weeks. The lesson was piloted to the other lesson study group in the Park City Mathematics Institute Teacher Leadership Program which followed with a feedback session for revisions. The second time, the lesson was taught by group A to 27 high school students attending the Winter Sports School in Park City, Utah and by group B to 16 high school students who attended the Park City Mathematics Institute Summer Math Camp. The overarching goal of the lesson was for students to look for and make use of the structure of polynomials to write equivalent expressions depending on the context/purpose. The explicit content in the lesson includes identifying common factor(s) among the terms of a polynomial, writing a polynomial as a product of factors, including by grouping, and using the distributive property to justify the equivalence of expanded and factored expressions.

Group A started Act 1 with a "what do you notice/wonder?" activity where students analyzed a pictorial multiplication model and made connections to the distributive property. In Act 2, the lesson moved to students applying two representations (the pictorial multiplication model and an algebraic model) to factor polynomials multiple ways: common factors and grouping. In Act 3, students discussed relationships between these two models and individually completed an exit ticket to demonstrate all the ways they could factor a given polynomial.

Group B started Act 1 with a card sort to help students represent equivalent expressions and discuss the connection between the geometric model and factors. In Act 2, the lesson moved to having students use a geometric rectangle model to break apart polynomials and group them so they can be represented as one product of factors. In Act 3 the students describe a strategy for writing polynomials as factors that is useful to them and then apply strategies to rewrite polynomials as products of factors and justify the equivalence of expressions