# STUDENTS’ WAYS OF REASONING ABOUT NONLINEAR FUNCTIONS IN GUESS-MY-RULE GAMES

• John M. Francisco
• Markus Hähkiöniemi
Article

## Abstract

There is a growing movement toward the introduction of algebra in early grades. This is supported by an increasing number of research studies that have reported success in getting young students to “do” mathematics considered beyond their reach. Yet, the consensus is that more research is needed to provide insights into the processes by which students make sense of algebraic ideas. In particular, more studies are needed on how nonlinear functions can be introduced in early grades. This article reports an algebra research strand that introduced seventh grade students to quadratic functions using Guess-My-Rule games. The article describes several instances of the students engaging successfully with ideas and forms of reasoning involving quadratic functions. The purpose is to contribute to the debate on the introduction of algebra in early grades by providing further evidence of young students’ ability to engage with algebraic ideas usually considered to be beyond their reach.

## Key words

algebraic reasoning generalization Guess-My-Rule games nonlinear functions quadratic functions

## Supplementary material

10763_2011_9310_MOESM1_ESM.doc (420 kb)
Fig. 1 Hector notices and tries to prove that the graph of $$x \cdot x + y \cdot y = 25$$ is a circle. (DOC 419 kb)
10763_2011_9310_MOESM2_ESM.doc (153 kb)
Fig. 2 Ian notices symmetry in the function $$y = {(x - 1)^2}$$ and uses it to add points on the function table. (DOC 153 kb)
10763_2011_9310_MOESM3_ESM.doc (463 kb)
Fig. 3 Terry solves a quadratic equation by factoring. (DOC 463 kb)
10763_2011_9310_MOESM4_ESM.doc (119 kb)
Fig. 4 Ian notices a pattern in the rate of change. (DOC 119 kb)
10763_2011_9310_MOESM5_ESM.doc (92 kb)
Fig. 5 Ian’s family of functions [$$2 \times 2--3 = 1$$; $$3 \times 3--5 = 4$$; $$5 \times 5--9 = 16$$; $$4 \times 4--7 = 9$$]. (DOC 92 kb)
10763_2011_9310_MOESM6_ESM.doc (595 kb)
Fig. 6 Stefanie relied on a recursive heuristic strategy to come up with the rule $$x \cdot (x - 1) = y$$, which she described as “square times one number less than it is”. (DOC 595 kb)
10763_2011_9310_MOESM7_ESM.doc (62 kb)
Fig. 7 Illustration of Stefanie’s reasoning. (DOC 62 kb)
10763_2011_9310_MOESM8_ESM.doc (258 kb)
Fig. 8 Chris’s rule [left] and Jerel’s rule [right]. Jerel’s rule reads “$$x + 1 = nx + nx = y$$,” but, based on his explanation, Jerel meant nx × nx. (DOC 258 kb)

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