# Algebra: Teaching Concepts

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.

# Solving Equations: Lesson Ideas

Your own textbook has numerous pages on equations and their solutions. What I’ve pulled together here are scenarios that involve students in solving equations in unusual contexts.  A change of pace, a different approach. If you have found lesson ideas on working with equations that you’d like to share with colleagues, please share the wealth!

Algebra—Fun with Calendars
Using any calendar, tell a friend to choose four days that form a square, then tell you only the sum of the four days. You tell her the four numbers! The trick lies in setting up an equation and simplifying it to an algebraic expression.

The Yo-Yo Problem
In these activities, students explore patterns, write a pattern in algebraic language, and solve equations using algebra tiles, symbolic manipulation, and the graphing calculator.

Balance Beam Activity
Students explore the meaning of balance, a key concept in developing mathematical understanding of solving equations. Working with shapes of differing weights, the students must experiment to balance the virtual scale by adding shapes of unknown weights. They use basic equation-solving principles throughout the activity.

Algebra Balance Scales-Negatives
This applet presents an equation for students to illustrate by balancing the scale, using blue blocks for positive units and variables and red balloons for negative units and variables. Students then work with the arithmetic operations to solve the equation. A record of the steps taken by the student is shown on the screen and on the scale.

Amby’s Math Resources: Order of Operations
This resource is a tutorial and practice on a topic that often frustrates the younger middle school student. Immediate feedback is given when an incorrect answer is chosen, plus a full explanation of the correct solution.

Planet Hop
Here students concentrate on writing an equation. In an interactive online game, they find the coordinates of four planets shown on a grid or locate the planets when given the coordinates. Finally, they must find the slope and y-intercept of the line connecting the planets in order to write its equation. Tips for students are available as well as a full explanation of the key instructional ideas underlying the game.