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.

We want and need your ideas, suggestions, and observations. What would you like to know more about? What questions have your students asked? We invite you to share with us and other readers by posting your comments. Please check back often for our newest posts or download the RSS feed for this blog. Let us know what you think and tell us how we can serve you better. We appreciate your feedback on all of our Middle School Portal 2 publications. You can also email us at msp@msteacher.org. Post updated 11/08/2011.

Math Games – Part II

You probably already incorporate games in your teaching. Games are a great way to focus students’ attention as few other teaching strategies can. The ones selected here deal directly with the math content covered in the middle grades. Each has a learning objective; each could be embedded in a lesson plan. Here are a few more games that you can add to your store of games that teach.

Fraction Game
For work on fractions, this applet is a winner! It allows students to individually practice working with relationships among fractions and ways of combining fractions. It helps them visualize what is meant by equivalence of fractions. A link to an applet for two-person play is also given here.

Polygon Capture
This excellent lesson uses a game to review and stimulate conversation about properties of polygons. A player draws two cards, one about the sides of a polygon, such as “All sides are equal,” and one about the angles, such as “Two angles are acute.” The player then captures all the polygons on the table that fit both of the properties. Provided here are handouts of the game cards, the polygons, and the rules of the game.

The Factor Game
A two-player game that immerses students in factors! To play, one person circles a number from 1 to 30 on a gameboard. The second person circles (in a different color) all the proper factors of that number. The roles are switched and play continues until there are no numbers remaining with uncircled factors. The person with the largest total wins. A lesson plan outlines how to help students analyze the best first move in the game, which leads to class discussion of primes and squares as well as abundant and deficient numbers.

Planet Hop
In this online one-person computer game, four planets are shown on a coordinate grid. A player must pass through each on a journey through space. The player must find the coordinates of the four planets and, finally, the equation of the line connecting them. Three levels of difficulty are available. This is one of 12 interactive games created by the Maths File Games Show.

Towers of Hanoi: Algebra (Grades 6-8)
This online version of the Towers of Hanoi puzzle features three spindles and a graduated stack of two to eight discs, a number decided by the player, with the largest disc on the bottom. The player must move all discs from the original spindle to a new spindle in the smallest number of moves possible, while never placing a larger disc on a smaller one. The algebra learning occurs as the player observes the pattern of number of discs to number of moves needed. Generalizing from this pattern, students can answer the question: What if you had 100 discs? The final step is expressing the pattern as a function.

Traffic Jam Activity
Why the jam? There are seven stepping stones and six people. Three stand on the left-hand stones and three on the right-hand; all face center. Everyone must move so that the people on the right and the people on the left pass each other, eventually standing on the side opposite from where they started. But no two people may stand on the same stone at the same time! This problem requires reasoning, but its solution also reveals a pattern that leads to an algebraic expression. A lesson plan is provided.