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Making sense of structure through patterns

This resource explores how patterns can enhance reasoning in mathematics through various activities. It discusses generalized arithmetic and how algebra captures structure, linking algebraic representations with geometry. Through engaging exercises like Toothpick Patterns, students learn to identify relationships, understand linearity versus quadratic patterns, and interpret data. The resource also highlights the importance of connecting various representations back to contextual problems and emphasizes the reasoning process in algebra classrooms. Teachers are encouraged to incorporate pattern-finding activities to promote deeper student understanding.

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Making sense of structure through patterns

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  1. Making sense of structure through patterns Tami S. Martin Illinois State University Roger Day Glencoe McGraw-Hill School Mathematics

  2. Patterns as a tool for reasoning • Generalized arithmetic: From counting to symbols. • Algebra as a way to concisely capture structure. • Connecting geometry or physical objects with algebraic representations. • Situations that lead into multiple, connected representations and interpretation of those representations; connections back to context.

  3. Toothpick Patterns

  4. Making Sense of Toothpick Patterns • How many toothpicks are needed for the 4th step? The 10th step? The nth step? • How many 1-by-1 squares are contained in the 4th step? The 10th step? The nth step? • Is the pattern describing the number of toothpicks per step a linear relationship? Quadratic? Something else? How about the number of 1-by-1 squares per step?

  5. Making Sense of Toothpick Patterns • How could these data be organized to help determine the patterns? • How does each representation connect back to the context? • How is slope revealed within your representations? How about intercepts? • In a table showing (step number, number of toothpicks), what is revealed when you calculate the differences between consecutive rows?

  6. MTWI: Stack Attack • Stack Attack: Rules and requirements • Try it in your small group! • Watch these videos: • A Traditional Stack Attack • A Young Stack Attack • A Fast Attack!! • What are some mathematical questions that could be asked about this scenario? • What representations could be used to model the scenario?

  7. Discussion • What are the key elements of reasoning and sense making that you can drawn upon or emphasize with the use of pattern-finding activities in the algebra classroom? • What actions will you take as a teacher to ensure that pattern-finding activities engender student thinking and sense-making?

  8. Discussion • What challenges to teaching and learning emerge when we engage students in pattern-finding activities such as these? • What are two or three actions you can take to begin (or increase!) implementation of pattern-finding activities in the courses you teach?

  9. Contact Info Tami S. Martin: tsmartin@ilstu.edu Roger Day: dayr@pontiac.k12.il.us

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