1. RATIONALE and CONCEPTUAL FRAME Students enter middle school at different levels of reasoning about multiplicative relationships. These levels affect the way they understand quantitative relationships. This in turn affects the way they work with and understand unknown quantities. Students who have interiorized three levels of units can work flexibly with 5x as a quantity containing 5 units, each of which contains the same unknown number of units, x. Students who have interiorized two levels of units (MC2 students) may be able to create this structure, but do not maintain it in further operating. (Hackenberg, Jones, Eker, Creager & Timmons, 2015) Robin Jones & Ayfer Eker, Indiana University, Bloomington Pieces of the Puzzle:Pieces of the Puzzle: Learning from differences in student drawing, notating and explaining fractional relationshipsLearning from differences in student drawing, notating and explaining fractional relationships THE TASK Students were posed the following problem: There is a tomato plant and stalk of corn growing in the garden, each of unknown height. The height of the stalk of corn is 5 times the height of the tomato plant. a.Draw a picture of this situation and describe what your picture represents. b.Write an equation for this situation that relates the two heights. Explain what your equation means in terms of your picture. After students wrote equations, they were given a numerical example (the stalk of corn’s height is 150 cm) and asked to find the height of the tomato plant and use the example to check their equations. ANDREA Andrea was an MC2 student in a regular math class. She tended to rely on equations and computation in her reasoning, drawing only when prompted. We saw two different ways she worked with this task:. TIM Tim was an MC2 student in a regular math class. He tended to prefer drawings (especially on computer) and verbal explanations to computation and equations. Initial drawing: maintains “unknownness”, loses 5x relationship. Once quantities are known, Andrea partitions cornstalk height to show 5x relationship. 5x relationship maintained, unknownness lost 5x x CONTEXT Our analysis is part of a larger 5-year project studying differentiated instruction in middle school The first two years of the project (2013 – 2015) included three 18- session, 9-week design experiments, each with 6-9 students, focusing on rational number knowledge and algebraic reasoning. Design experiments included students at different levels of multiplicative reasoning. OUR FOCUS Our question: for students at the same level of multiplicative reasoning, what can we learn about the different ways they reason about relationships between unknown quantities? What implications do these differences have for teaching Algebra? Our students: Our discussion highlights the work of 2 students from the fall, 2013 design experiment. Both were 7 th graders in regular math classes. Both were assessed in a selection interview as having interiorized two levels of units Both were working in small homogenous groups during the sessions included in this analysis. Our data: Video recordings (handheld camera and Screenflow software) and written work from sessions where students worked on representing multiplicative relationships of unknown quantities. IMPLICATIONS Students need to work explicitly with unknowns as potential measurement of quantity. Students who struggle to structure relationships between unknowns still need to work with them. Asking them to work only with known quantities (e.g. always rely on numerical examples) may not offer enough challenge to help them create meaningful relationships between algebraic representations of quantities. References: Hackenberg, A. J., Jones, R., Eker, A., Creager, M., & Timmons, R. (2015, April). “Approximate” multiplicative relationships between quantitative unknowns. Paper presentation at AERA in Chicago, IL. Steffe, L. P., & Olive, J. (2010). Children’s fractional knowledge. New York: Springer. Acknowledgement: This material is based on work supported by the National Science Foundation (grant no. DRL-1252575) In trying to work with three levels of units, Andrea ‘dropped’ one at a time, working in turn with each of the structures below: Unknownness maintained, 5x relationship lost Initial drawings: appeared to show 5x relationship and maintain “unknownness” of heights. However, the 5x relationship seemed hypothetical and imprecise. When asked to work with the numerical example, he said the tomato plant “supposedly fits 5 times…it could fit more…it could fit three times and each one would be fifty inches tall.” In trying to work with both unknown quantities, he lost the relationship. Both unknowns became disconnected and changeable. X ? =