Walking the Line

Seventh Grade Poster Problem
The Number System

The Common Core Mathematics Standard 7.NS.2a contains several key ideas. One major conceptual milestone students must pass in meeting this standard is understanding why (-1)(-1) = 1. The proof of this identity relies on the distributive property of multiplication over addition (see the Teacher Tune-up links for more explanation of the proof).

While the proof shows that (-1)(-1) = 1, it might not help your students develop a feeling for why this fact is true. This activity uses a familiar model, the number line, to develop an intuitive feeling for why it makes sense to define (-1)(-1) = 1.

Learning Objectives:

  • Understand that the rules for multiplication can be extended from whole numbers to include negative numbers.
  • Use a model to develop an intuitive understanding of why (-1)(-1) = 1.

Common Core State Standards for Mathematics:

  • About Models of Multiplication

    This lesson uses a number-line model of multiplication. You take a number of steps, and each step is so many units long. The total distance is the product of those two numbers. We model negative numbers in this situation by facing in the negative direction, by walking backwards, or both (see the slides for details). We hope this model will make sense to students, and help them see that if they face backwards, and step backwards, they move forwards: a negative times a negative is positive.

  • Number Lines

    Number lines in Grade 7? You bet. They are a clear model of the entire number system, not just a tool for multiplication. They also help students prepare for further math: first, they’re the one-dimensional analog of the all-important Cartesian plane. If you can’t navigate a number line, you are sure to get lost on the plane. Secondly, our model for addition (and multiplication as repeated addition), often using arrows, mimics what students will see later when they study vectors. This leads, eventually, to understanding complex numbers and linear algebra.

The Lesson Plan:

Project funding provided by The William and Flora Hewlett Foundation and S.D. Bechtel Jr. Foundation.

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