Equation App (Pt 2 in series) – Solving Equations – Why Use a Calculator?

Solving equations is a large part of the mathematics curriculum as students move into those upper-level concepts. If we look at the Common Core Standards, students start solving one-step equations for one variable in grade 6, adding on to the complexity as they move into higher mathematics where they have multiple variables and simultaneous equations and complex functions. It is important to help students understand what solving equations really represents – i.e. determining the values of unknown quantities and to help them solve them in a variety of ways (i.e. graphically, using a table, using symbolic manipulation, and yes….using technology such as a graphing calculator). And connecting those unknown quantities to real-world contexts is a big part of this as well. Students should solve in multiple ways and express their solutions in multiple ways so that they really understand the inter-connectedness of the multiple representations (graphs, tables, symbolic) and what all these quantities mean in context.

That said, many teachers are reluctant to use the equation solver that is often part of a graphing calculator because, as I have heard multiple times, it does the work for the students and just gives them the answer. True. But – there are ways to utilize the equation solver so that it supports the learning, not just ‘gives the solution’. The obvious way, and probably the most frequent way, is to have students solve the equation (s) by hand, showing all their inverse operations/work, maybe even sketching a graph of the solutions, and then using the graphing calculator to check their solution. Very valid way for students to both do the work, show their steps, and verify their solutions. But – the reverse is also a great way to try to help students learn HOW to solve equations. Working backwards, so to speak.

By this, I mean, use the equation solver to give students the answer first, and then see if they can figure out how to use symbolic manipulation and inverse operations to reach that outcome. As an example, start with a simple linear equation, such as 2x – 5 = 31. Have students plug this into the equation solver and get the solution of 18. Then, in pairs or small groups, have students look at the original problem and try to figure out how they can manipulate the coefficients and constants using inverse operations to get to that solution of 18. So maybe, plug the 18 in for the x.  What would they have to do to the other numbers in order to isolate that 18?  This forces students to use inverse operations to try to ‘undo’ the problem and end up with 18. In doing so, they are discovering the idea that to isolate a variable, you have to undo all the things that happened to it.  Give them a harder problem. Same process….and let them get to a point where they try to solve using their ‘understanding’ of inverse, and then they use the calculator to ‘check’.  The idea here is students are figuring it out by starting with the solution and working backwards to understand the process for solving equations. And they develop the process themselves versus memorizing it.

Rather than thinking of the calculator as a solution tool, think of it as another way to help students discover where those solutions come from.

Here’s a quick video on using the Equation App (solver) on the CG50. The process is the same on Casio’s other graphing calculators. This is another installment in the app exploration series, started last week with the Physium App.

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Multiple Representations on the Casio Graphing Calculators

One of the key things we try to help students with when studying functions is the idea of multiple representations – i.e. graphical, symbolic (equation) and table.  Ideally, we want students to be able to discern what the function represents or looks at no matter what representation they are given, and to be able to find patterns and important components about that functions from all representations.  Students should never learn about functions just through graphing, or just through symbolic manipulations or just through looking at data points in a table – they should be able to go back and forth and determine which representation is the most useful for the situation.

Unfortunately, too often, the emphasis is on one representation at a time, or at most 2. Let’s look at the graph and find the minimum, maximum, or intersection. Or, let’s find the roots of a quadratic by factoring, or symbolic manipulation. Or, here’s a table of points, where are the x-intercepts or the y-intercepts? Ideally, we want students to be able to look at all of these representations simultaneously so that they see the relationships between the representations and come to understand what the points represent in the table, in the equation, or in the graph.

Technology is one way to show all these representations at the same time, and then quickly manipulate and explore. There are obviously many technology tools out there, but as I have stated in previous posts, the most accessible technology tool for most students and teachers is the graphing calculator, not only because of it’s affordability, but because it is a tool most students have readily available.  It would be nice if all students had computers or tablets for daily classroom use, but that is still NOT the reality.

I have put together a quick video showing Casio’s three graphing calculators – the fx-9750GII, the fx-9860GII, and the CG10/20 or Casio Prizm, and how they can display the equation, graph and table representations of a function on one screen. No matter which model you have, you can achieve the same functionality, allowing students to work with multiple representations and explore relationships quickly and efficiently.

Check it out: