When we teach algebra, most teachers find that getting across the manipulation of expressions is far easier than teaching the big ideas that underlie algebra. Lately I’ve run across sites that help middle school students grasp those concepts. I’d like to share them with you in this post and ask for your ideas in return.

First, two excellent ideas on helping students walk the bridge from arithmetic to algebra:

**Building Bridges** In this lesson, students move from arithmetical to algebraic thinking by exploring problems that are not limited to single-solution responses. These are common, not complex, problems that are developed through questioning to a higher level. Within real-world contexts, students organize values into tables and graphs, then note the patterns, and finally express them symbolically.

**Difference of Squares** uses a series of related arithmetic experiences to prompt students to generalize into more abstract ideas. In particular, students explore arithmetic statements leading to a result that is the factoring pattern for the difference of two squares. Very well done.

Equivalence is one of those underlying concepts that make algebraic reasoning possible. **Everything Balances Out in the End** offers a unit in which students use online pan balances to study key aspects of equivalence. The three lessons focus on balancing shapes to study equality, then balancing algebraic statements in order to explore simplifying expressions, order of operations, and determining if algebraic expressions are equal.

The next lesson, **Equations of Attack**, is a game but developed to uncover the algebra beneath the strategies. The two players each plot points on a coordinate grid to represent their ships and points along the y-axis to represent cannons. Slopes are chosen randomly (from a deck of prepared cards) to determine the line and its equation of attack. Students use their algebraic skills to sink their opponent’s ships and win the game. Afterwards, the algebraic approach to the game is investigated.

**Walk the Plank** is also a game. You need to place one end of a wooden board on a bathroom scale and the other end on a textbook. Students can “walk the plank” and record the weight measurement as their distance from the scale changes. This investigation leads to a real world occurrence of negative slope.

A final teaching idea develops students’ understanding of algebraic symbols: **Extending to Symbols**. As students begin to use symbolic representations, they use variables as unknowns. To help their concept of symbolic representation to grow, they need to explore questions such as: What is an identity? and When are two symbolic representations equal? This activity engages students in work with an online algebraic balance.

Each of these lessons comes from NCTM’s **Illuminations** site, a rich source for K-12 teaching.

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msp@msteacher.org. Post updated 11/08/2011.