An applet allowing students to practice solving and simplifying equations through virtual algebra tiles can be found on NCTM’s Illuminations website. It is free to use for anyone who has an Internet browser. This technology addresses two primary topics, each with its own audience. Firstly, middle school students can practice solving multi-step equations or evaluating expressions at specific values. The applet also has an option for high school students to expand a product whose factors are polynomials up to degree 2; they can also factor these expressions. All four activities aim to emphasize the concept of zero pairs when working with various single-variable expressions.
Standards
Content Standards
MCC6.EE.2c: Evaluate expressions at specific values for their variables.
MCC6.EE.7: Solve real-world and mathematical problems by writing and solving equations of the form x+p=q and px=q for cases in which p,q, and x are all nonnegative rational numbers.
MCC7.NS.1a: Describe situations in which opposite quantities combine to make 0.
MCC9-12.A.REI.3: Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.
MCC9-12.A.APR.1: Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract and multiply polynomials.
MCC9-12.A.REI.4: Solve quadratic equations in one variable.
Practice Standards
1. Make sense of problems and persevere in solving them. The solutions for some of the problems, such as factoring the trinomials, may not initially be clear to students. Thus, they will need to persevere in solving these exercises that may require higher cognitive thinking.
2: Reason abstractly and quantitatively.
Students reason abstractly through the algebra tiles to determine which of them are necessary for the quantity provided.
4: Model with mathematics.
Students use the tools in the applet to create an area model of a given algebraic expression.
5: Use appropriate tools strategically.
Students must decide which of the virtual tiles are necessary to accurately complete the model.
8: Look for and express regularity in repeated reasoning.
After a few examples, students will likely begin to notice some patterns in the algebra tiles, such as the product of two green or yellow tiles is a red one.
Technology Critique
How well does it work?
This applet works extremely well in all aspects. Each of the buttons accomplishes its purpose, and I did not see any mistakes or glitches with them while I was testing the technology. I appreciate that the applet provides students with feedback on their work, either with green or orange shading. All of the answers I submitted gave a mathematically correct response from the program.
Are the written materials well organized and useful?
The “Instructions” tab provided thorough descriptions for each button in the applet. All of the actions were initially overwhelming, but these explanations helped me understand each one’s function. The small pictures and colors would help students recall part of the applet when they are reading the materials. My only suggestion would be to have them on the same screen as the applet. Some students may forget the purpose of a button, but they will get frustrated when the progress on their model is lost if they click on change to the appropriate page.
The website could have provided clearer instructions on the requirements of each activity. Under the “Exploration” tab, it only provided an explanation of what students needed for a “Solve” problem. Thus, when I first tried a “Substitute” exercise, I didn’t know you always need a green x tile on one half of the “Value of X” section. Even with a written description, the “Exploration” page could benefit from a pictorial example for each type of problem that shows student what should be on their screen at all times.
What are the purposes and goals for using this technology? Does the technology reach this goal?
This technology allows students to look at previously learned concepts in a different way, as well as introduce new ones. The “Zero Pair” button in the technology will help students who are having trouble solving single-variable equations. It provides a visual model of the “cancelling” step. I have also effectively transitioned students to the idea of factoring after using algebra tiles to expand polynomials. However, if students are being introduced to any of the concepts in the applet, scaffolding by the teacher is necessary. The technology does not provide any prompts to challenge students’ thinking of the concepts. Many students will not appreciate the significance of the zero pairs and area representations without guided questions from their instructor.
Is the technology relatively easy to use?
Once students understand each button’s purpose and the requirements for each type of problem, the technology should be easy to use. The buttons are very responsive and accomplish what they should. I also like that in the first two types of problems, students can place tiles anywhere in the corresponding region; since the applet only checks for a number, they do not have to be arranged in a particular way. One minor complaint regards the copy tool: I wish that the copies were not on top of each other so I would know how many I have made.
Does this technology enhance or extend the teaching and learning process for the intended mathematics concepts? How and why? This technology extends the learning process by encouraging students to create models for problems that may otherwise be procedural. A great aspect of the feedback system in the applet is that students must have all parts of a problem correct to be able to proceed. This feature prevents students from guessing or only solving the problems using algebra. Thus, students are forced to reach the answers using the abstract representation. When solving these problems, many students become lost in the procedures and forget why each step can occur. The tools in this technology allow them to connect those steps to an area representation.
Would you recommend this product to a school? Why or why not? I would highly recommend this product to middle or high school math teachers. I have used physical algebra tiles with students before, and I thought they were extremely useful in developing their understanding of the concept. This technology has several advantages over using traditional algebra tiles. It is free to use, eliminating the cost of a classroom set of algebra tiles. The copy feature prevents students from being limited to the number of tiles in their bag, and the zero pair tool encourages them to utilize this concept in their models. In the “Expand” and “Factor” section, the applet also helps students by automatically placing tiles next to each other. When I have observed students working with these manipulatives, many of them would leave spaces between the tiles, leading to incorrect results.
As I discuss in the activity, the only hesitation I have in recommending the applet is that all of the questions are built into the program. Teachers need to either trust the randomness of those problems, or design some of their own, provided the students could not check their work using the technology. However, each section contains many problems with varying degrees of difficulty; while testing the applet, I was never given the same exercise twice.
Activity
A teacher should begin a lesson using this applet by demonstrating the function of each button. He or she can lead the class through a few examples that encompass all of the sidebar tools. When the class seems ready to try some on their own, the teacher can begin the activity. As I mentioned earlier, the biggest drawback of this technology is that teachers cannot program their own problems into the applet. They could give students a separate worksheet, but then the class would not feedback within the applet. I would then recommend that teachers instruct student to try four problems (or as many as time allows) from each appropriate section (“Solve” and “Substitute" for middle school and Coordinate Algebra, “Expand” and “Factor” for Analytic Geometry). I suggest four because while students receive a random problem each time they click refresh, ideally at least one of those will vary in difficulty level. Students would be required to write down the original problem, their final answer, and make a sketch of each of their models. Before the end of class, the teacher should lead the class in a discussion to discuss the students’ work. These are some possible questions to ask: · What were the similarities and differences between each type of problem? · Can you think of any applications for representing algebraic expressions or equations in this way? · In what other contexts have you seen the concept of a zero pair? · How is a zero pair related to what we sometimes call “cancelling”? · Did you notice any patterns that made the other problems easier? For students who finish their problems early, teachers can ask them a few of these questions as an extension of the activity.