Algebra+Tiles

Algebra Tiles



This resource enables users to work with algebra tiles to model factoring, the distributive property, evaluating expressions, and solving equations. When factoring and distributing there is a row and a column for students with which enter the two monomials/binomials. Once these are represented with tiles a box is formed where the factors need to be placed. Students then use the tiles to represent the answer and type the answer in the solution box. Immediate feedback is given based on if the factor or distribution is correct. The substitution feature of the applet allows students to represent an expression with tiles, substitute numbered tiles for the variable and then simplify the expression. Again, immediate feedback is given on the accuracy of the evaluation. Solving equations revolves around a similar process. Students represent the equation with variables and number tiles. Students are then expected to solve by adding or removing the same tiles from each side of the workspace. The process of adding tiles, removing tiles, and changing from positive to negative tiles is a bit complex. There are detailed directions provided above the applet.


 * Grade Level:** 7-11
 * PSSM Content Standard:** Algebra
 * Math Content:** Solving Linear Equations, Evaluating Expressions, Using the Distributive Property, Factoring

Evaluation & Annotations

 * What is being learned? What mathematics is the focus of the activity/technology? Is relational or instrumental understanding emphasized?**

This resource offers practice with a number of different mathematical processes. Student can use the applet to gain an understanding of solving equations, evaluating expressions, distributing, and factoring. With all of these processes students are required to use algebra tiles to represent the mathematical situation. The answer must be obtained symbolically as well as with the tiles. If the tiles are correctly distributed and the answer is correctly typed, students will proceed to a new problem. This resource exceeds instrumental understanding because students are not performing operation blindly. Rather, this resource fosters relational understanding through its use of multiple representations. Students are able to gain a deeper understanding of solving, evaluating, distributing, and factoring through the use of the algebra tiles. These visual aids deepen student understanding of the properties of equality, as students to add and subtract tiles from both sides of an equation. Furthermore, this resource demonstrates the necessity of the distributive property and F.O.I.L. by having students build models and multiple each component piece by piece.


 * How does learning take place? What are the underlying assumptions (explicit or implicit) about the nature of learning?**

With this resource, learning takes place through access to multiple representations. Allowing students to see the connection between solving an equation with symbols and with tiles ingrains the process in their mind. This connection should make it clear to students that items cannot be subtracted from one side of an equation without subtraction the same amount from the other side. In terms of distributing and factoring, the tiles enable students to visualize how each term must be multiplied by each other term. The tiles are able to bring the dry properties of equality and processes of distributing and factoring to life. The underlying assumption is that students need to learn mathematics in a hands on (albeit computer) environment where mathematics can be seen and understood at a less abstract level.


 * What role does technology play? What advantages or disadvantages does the technology hold for this role? What unique contribution does the technology make in facilitating learning?**

Technology is essential to this activity. The dynamic software present in the applet allows students to create a mathematical model of the situation with tiles. There are templates for each type of problem to assist students in this modeling. For solving equation problems, the applet is divided into two columns: one for each side of the equation. For problems involving substitution, there is one box to represent the value of x and another to demonstrate the substitution. Finally, for problems involving the distributive property and factoring there is a top and side border to place each term/quantity with the product being multiplied in the middle. Granted, this could be accomplished without the technology; however using tiles by hand would not provide the feedback which the applet offers. This technology provides a large quantity of mathematical problems and provides students with immediate feedback. The applet will not let students progress unless the tiles appropriately model the problem. Although the applet does not verify each step along the way, the students will not receive a correct response unless the answer is correct in symbol and tile form. This encourages students to find the answers through both methods. With the ability to add positive tiles, add negative tiles, and remove tiles from all parts of the applet, students have the ability to model the problem step by step.


 * How does it fit within existing school curriculum? (e.g., is it intended to supplement or supplant existing curriculum? Is it intended to enhance the learning of something already central to the curriculum or some new set of understandings or competencies?)**

This applet would best supplement an existing school curriculum. Solving equations, evaluating expressions, distributing, and factoring are all integral components already present in a algebra curriculum. This applet would enhance student understanding of these processes and the adjoining properties that support them. Prior to the use of this applet students would need to be introduced to the four processes with which the resource focuses. This tool would best be implemented in the middle of a lesson on solving equation, evaluating expressions, distributing, or factoring. It could assist struggling students and deepen the understanding of all students by providing visual representations of the rather abstract processes.


 * How does the technology fit or interact with the social context of learning? (e.g., Are computers used by individuals or groups? Does the technology/activity support collaboration or individual work? What sorts of interaction does the technology facilitate or hinder?)**

This resource could be used individually or in a collaborative group setting. While this resource aims to broaden student understanding, it is not centered on constructivism, reflection, and debate, where collaboration is essential. Still, students could work together with one solving the problem symbolically and the other using the tiles, while eventually rotating. This would enable students to see the connection between the two mediums. Additionally, having more than one student work together could be beneficial, as it is a bit complicated to work the algebra tiles when first introduced to them. The applet could also be used individually. Students with a sound foundation of the four processes of the applet could use the tiles to deepen their understanding while performing the operations by hand as well as with the tiles.


 * How are important differences among learners taken into account?**

Differences among learners are taken into account with the use of algebra tiles. Visual learners are being serviced with the implementation of algebra tiles and the feedback from the applet on the accuracy of the representation. However, there is no guidance for the symbolic representation of the mathematics. The applet would be very beneficial if the symbolic and visual representations of the algebra were used in conjunction, yet the applet does not foster complete symbolic solving. The applet encourages students to manipulate the tiles, yet does not provide any direction on solving by hand. While the resource does not play into the hands of learners with strengths in the symbolic form of mathematics, teachers can easily use this resource to join the two. Also, this resource does little to differentiate the problems for students of different abilities. The problems are random and cannot be set to various levels. This is a drawback that a teacher will need to take into account when implementing the applet.


 * What do teachers and learners need to know? What demands are placed on teachers and other "users"? What knowledge is needed? What knowledge supports does the innovation provide (e.g., skills in using particular kinds of technology)?**

Entering into the implementation of this tool, students need to have exposure to evaluating expressions, solving equations, distributing, and factoring. The power of this resource is that it can further student understanding of these processes through the use of a visual aide in algebra tiles. Without a base of understanding in these processes, students will be overwhelmed with learning the process and manipulating the tiles. Students also need to have an understanding of how to use the applet. There are directions on adding tiles, removing tiles, and the other feature of the applet, yet the directions may prove a bit confusing for students. It would be wise for the teacher to perform a practice problem with the students before sending them off individually or in groups. The demand placed upon the teacher is to create guidelines the narrow the focus of this applet. This is a powerful resource, and an activity which reinforces simplifying symbols with the use of tiles could foster a strong relational understanding for students. Teachers need to carefully construct a lesson for pairs of students or individual students that requires students to find solutions through multiple means and which forces students to review the properties in the process.