Solving equations on calculators: The benefits to students
Solving equations on calculators: The benefits to students
When teaching students to solve equations, you might start with something straightforward, like 2x+3=5. But then you’ll build on it and make it harder. You might make the right-hand side big or very small, and you might make it a fraction or decimal. You may even make it an integer or a fraction that doesn’t easily convert to a decimal solution.
We spoke with Casio Education Manager Gerard Dummett to understand the benefits calculators and graphing technology can bring to students.
As you build it up, both scientific and graphing calculators can help reinforce students’ understanding because the calculators allow students to change one thing at a time. They start with 2x+3=5. Then they change the equation to equal six, which introduces a fraction. Now onto seven, and the solution has increased. But what happens if we go down? What happens if we make it negative? What happens if we make the right-hand side a fraction rather than an integer? What happens if, higher up in school, we make it a surd? When the examples get harder, the principal issue of rearranging remains the same. Creative use of calculators helps students comprehend the concept and develop it further.
You may be interested to know that we provide free training for teachers to build confidence when using the fx-CG50 graphing calculator, showing how you can use calculators to reinforce teaching. We also have a huge resource centre to support you with videos and guides that help with student revision and exam preparations.
How do calculators help with solving equations?
Introducing a calculator enables fast computation so students can start seeing something unusual when facing these questions. That unusual might be a fraction, a negative number or a recurring decimal, maybe a surd. Using the calculator means they can start exploring the concept of variables and understand that the variable x is changing value each time.
You can take this further in the classroom by representing it visually. Students can start to see a picture of how these gradually increasing solutions look. For example, increasing the right-hand side by one each time and showing students that it produces a pattern, then drawing it.
We could plot y=2x+3. Then, when we say 2x+3=5, students start to grasp that two things can vary. There’s the value of x and the number on the right-hand side, and we can begin to link to coordinates. Maybe introduce the line y=5.
Connecting to other principles:
Introducing the unknown on both sides
With the unknown on both sides, we traditionally consider rearranging. But using graphing technology, students could use a table of values and then progress to the intersection of graphs, reinforcing that x has the same value for each graph, and so does y.
For example, we could try 2x+3=x+5. Create a table of values for the function on each side and see when they are equal. Change a coefficient. This obviously needs careful introduction, but it’s powerful for understanding concepts.
Using calculators (particularly graphing calculators) helps embed the principle that equality does not mean finding the answer; it means the same value. This concept can be difficult for some students because they see equals as supplying the solution. Moving students away from this way of thinking, particularly in Y7 and Y8, can be really hard work. The calculator does the heavy lifting of computation; the students explore the concept.
Then plot the two functions. When are the two sides equal in value? We have an intersection of graphs. Having the visual representation helps students to comprehend this relationship.
Building on the foundations
Other concepts, which are particularly valuable at A-Level, are in quadratic functions. Students quickly get used to some of the GCSE work they’ve done. However, they then have to relate the various forms of the function (factorised, expanded, completed square) and understand the relationship between the different forms and the properties of the discriminant. The discriminant is something that is frequently asked about in the A-Level examinations. The technologies help students link the three different forms and discriminant together, so they can get into harder questions and solve not just equations but inequalities. The calculator helps to connect all those different concepts, which are easier to see graphically.
It can be done using either a scientific or graphing calculator. However, one of the key distinctions between scientific and graphing calculators is that the scientific calculators can only show or find one solution at a time. In contrast, all of the solutions appear simultaneously on the graphing calculator.
This feature becomes important when discussing periodic functions. Here, the problem is that scientific calculators can’t find all of the roots when restricting the interval and focussing on a specific part of the axis. But a graphing calculator can, and it can show them as an image. This visual representation helps students understand the intersection of functions and understand those as solutions of equations.
Any graphing technology can perform this visual representation. But the benefit of the calculator is it’s immediately available. You don’t have to go to a computer room, get the laptop set up in the classroom or ask students to use any software on their phones.
What about the exam?
The short answer is calculators are now expected in exams, and exam boards know about calculator functionality and write questions accordingly.
The advice for students in an exam is to show their working because if they write an incorrect solution from a calculator and show no working, they will get zero marks. Whereas if they offer some working, they may get a method mark. As well, sometimes questions have an instruction to show working. Or maybe sometimes the question is such that students have to use working because they can’t do it directly on a calculator. Examiners are getting quite good at setting those sorts of questions, especially higher up the chain at A-Level.
What benefit is a calculator when solving equations?
It comes down to linking multiple concepts and embedding understanding. The calculator’s value here is that it gives students a foundation to tackle harder questions. Without that, tackling harder questions (with or without a calculator) is more difficult.
If students have the structures to solve the problems, even if it’s a problem they’ve not necessarily faced before, then it’s not going to faze them in the exam hall. Whereas if they don’t have that foundation, it’s much harder to understand what’s required. If they know the principles, then they stand a good chance of being able to do it.