COMBINING LIKE TERMS IN ALGEBRAIC EXPRESSIONS WORKSHEETS
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Teachers

What are learning tiles?

Let's use the term learning tiles to describe learning objects that are little "tiles" that can be pieced together like a puzzle. The pieces could be the shapes the states in the United States--they would be moved to form the entire map of the USA. Instead of "state tiles" we use "number tiles" in the examples on these web pages. Number tiles are tiles representing constants.

Note: Algebra tiles are manipulatives used to represent variables and constants. Product names may vary according to the retailer or manufacturer of these manipulatives--for instance, Algebra Tiles, AlgeBlocks, Number Tiles. These manipulatives look similar to Diene's Base-Ten Blocks. Perhaps Diene's Blocks provided the impetus for the creation of the algebra tiles.

How can you obtain a set of these manipulatives?

You can easily and cheaply create your own manipulatives out of colored construction paper or poster board paper. Or you can check with commercial resource suppliers (e.g., Cuisenaire). A third way is to access Internet sites that have "virtual" manipulatives. for number tiles designed to help teach integer arithmetic to students who comprehend, are accurate, and are already fluent at whole number arithmetic.

My advice on using these manipulatives:

"Manipulatives, such as algebra tiles, should be used to bridge the gap from the concrete to the abstract" (Tangible math correlations, 2001). I wholeheartedly recommend you use number tiles for very basic operations. For example, the tiles can be used when students are first introduced to integer addition. Also, students who have an understanding of the basic field properties can "manipulate" the tiles to combine like terms.

Make certain the manipulatives complement, not replace, other instructional approaches (use of theorems and postulates, real-world applications that illustrate uses of the various operations, etc.) to help students comprehend basic operations essential for understanding algebra.

Don't let manipulatives interfere with learning mathematics. Overuse of manipulatives at the expense of other strategies won't move the student toward understanding abstract concepts. It might even lead students to avoid learning abstract concepts. Fifth through seventh-graders have to be provided increasing levels of abstraction if they are to be prepared for algebra (Wu, 1999). Although teachers in the United States often use concrete objects to represent abstract concepts, they typically don't follow up with class discussions (Ma, 1999). We should gain the knowledge of mathematics to the breadth and depth required to present the level of abstraction the students need to learn.

Frequently and regularly state justifications of the steps used to expand or simplify mathematical expressions. Verbalizing these reasons provides students more opportunity to build mathematical logic and the foundations for proof. Encourage your students to use the proper terms associated with each operation (for the example below: associative, inverse, and identity).

 (+1) + (-2) = [(+1)] + [(-1) + (-1)] = [(+1) + (-1)] + [(-1)] = [0] + [(-1)] = -1

Practice before you demonstrate to students how manipulatives are used. Overhead viewgraph projector surfaces are so small. It is difficult to show students a problem without placing mixing in extra tiles that you mean to leave at the base of the viewgraph surface. I have observed students who looking at a hodgepodge of tiles that doesn't accurately portray the intended problem.

Many of you promote the use of algebra tiles because students enjoy working with them. Some studies have been conducted that indicate the use of these manipulatives positively impacts students' attitudes (Sharp, 1995).

Fade out the use of manipulatives as students master both understanding and fluency. Manipulatives are good for reinforcing concepts students understand and practice. But, just as children must move beyond doing whole number operations with their fingers, at some stage the tiles must be removed. Students must be able to do integer arithmetic mentally or using paper-and-pencil. Pre-algebra students can use tiles before they develop fluency with integer arithmetic. They can also use algebra tiles for the most basic level of algebraic expressions or equations. My experiences, observations, and readings suggest that using algebra tiles beyond those stages doesn't improve the students' understanding.

Please read and contribute to research on this topic. Most of the studies cover manipulatives in general. The two studies below were about students using algebra tiles. If you have a reference you want me to add to this list, and some pertinent entries from that resource, please send it to me at .

Five high school algebra classes used algebra tiles to study operations with algebraic expressions. Results suggested no differences between groups who used or did not use manipulatives when tested with traditional chapter tests. However, the majority of students entered diary narrative data that stated the tiles added a mental imagery that made learning "easier." Students indicated that they found it easy to think about algebraic manipulations when they visualized the tiles. (Sharp, 1995)

Two West Virginia high school Algebra I classes (24 and 25 students) participated. One experienced a traditional teaching method of lecture, homework, and in-class worksheets. The other used a traditional teaching method of lecture and homework, but instead of in-class worksheets, students worked with the manipulative Algeblocks. The only difference in the instructional strategies was the use of the Algeblocks. A pretest was administered. No significant difference was identified in the achievement levels of the two groups. A posttest identical to the pretest was given to both groups. Mean scores were compared. Posttest results indicated that there was a significant difference in achievement levels: the students taught using the traditional method of lecture, homework, and in-class worksheets outperformed the students taught using the manipulatives. (McClung, 1998)

Here's a discussion between two mathematics teachers. Algeblocks and Algebra Tiles are mentioned in the conversation. I didn't notice any comments saying to study the research on specific manipulatives. Let's encourage one another to use research-based best practices. I believe the research indicates that algebra manipulatives are effective if used in the right way at the right time!

References:

Ma, L. (1999). Knowing and teaching elementary mathematics: Teachers' understanding of fundamental mathematics inChina and the U.S. Mahwah, NJ: Lawrence EribaumAssociates.

McClung, L. W. (1998, January). A Study on the use of manipulatives and their effect on student achievement in a high school Algebra I class. Master of Arts Thesis, Salem International University (formerly named Salem-Teikyo University) Salem, WV. (ERIC Document No. ED425077)

Sharp, J. M. (1995, October 13). Results of using algebra tiles as meaningful representations of algebra concepts. Paper presented at the Annual Meeting of the Mid-Western Education Research Association, Chicago, IL. (ERIC Document No. ED398080)

Tangible math correlations: Correlations for West Virginia Content Standards in Mathematics. (2001). Riverdeep Interactive Learning Limited. 125 Cambridge Park Drive, Cambridge, MA 02140. (Phone: 617-995-1000. Fax: 617-491-5855. .) Retrieved August 11, 2001, from

Wu, H. (1999). Basic skills versus conceptual understanding: A bogus dichotomy in mathematics education. American Educator, 23(3),14-19,50-52. Retrieved August 11, 2001 from