TECHNOLOGY INTEGRATION IN PROBLEM SOLVING
Introducing lessons using tools to create sequences and analyzing growing patterns is a powerful way for students to learn about the connections between growing patterns and algebra. Using manipulatives, students can be motivated to model the evolution of the pattern using multiple representations including a table or an algebraic formula or a graph. Having a tool to help them visualize a pattern as they build it is helpful for them to understand the growing nature of the pattern. The following technology tool helps students to not only create any pattern they want but helps them to organize their data in different ways. Consider the following technology tool that includes a dynamic visualization of growing patterns.
The tool allows students to enter any data that they identify from any source such as a textbook or something they have discovered through a hands-on experience. For instance, if the students saw the following pattern (1, 2), (2, 3), (3, 4), (4, 5), etc., then they can enter these numbers into the table provided as
and when they click “get formula,” the tool would give them the following.
Note that the pattern entered is now described algebraically as (x + 1) as well as the associated graph y = x + 1 is shown in the graph. This now helps for the students to make the appropriate connections to the growing pattern being linear.
Another example that can be motivated for example using toothpicks is to have them build a growing pattern using toothpicks with different stages indicating as follows with stage 1 having one square, stage 2 with four, and so on.
A challenging question to ask is to ask them the number of toothpicks required to build stage x. This allows them to formulate a table of growing numbers as (1, 4), (2, 12), (3, 24), and so on. These can then be entered into the technology tool that will yield:
The students are now able to interpret the nonlinearity in the graph and recognize that the function is y = 2x2 +2x which may be interpreted many ways. For example, in stage 1, there are two rows of one toothpick and two columns of one tooth yielding (1 x 2) + (2 x 1) = 2 (1 x 2). Similarly for stage 2, there are three rows of two toothpicks and three columns of two toothpicks to give (2 x 3) + (3 x 2) = 2 (2 x 3). Note that the result in each case start to motivate an algebraic pattern given at any stage x but, (x (x + 1 )) + (( x + 1)x) = 2(x ( x + 1)) = 2x2 + 2x. Such technology-based learning discoveries not only help the students to discover new patterns but also help them to understand these discoveries in a conceptual way being able to make sense of the algebraic formula or the graphical interpretations. Students can try more patterns at: http://completecenter.gmu.edu/java/functionpattern/index.html.
Describe the advantages of using multiple representations, including technology, to develop algebraic thinking.
Modeling Math Ideas with Patterns and Algebraic Reasoning