Navigating numerical methods on the fx-CG50

Using numerical methods to solve equations that have no obvious algebraic solution is a concept students will come across at GCSE, and need to fully understand and apply at A-level.

It can be a challenging aspect of mathematics, but it’s also an area where students can get a lot of help from their calculator.

In this blog, we highlight some of the ways you can use the fx-CG50 to teach and improve student understanding of numerical methods, with insights from our fx-CG50 training instructor and mathematician, Simon May.

Decimal search and change of sign

Our most advanced graphic calculator offers various ways to explore the decimal search method of equation solving.

A good place to start is Table mode. Here, you can experiment with entering a y = function and inputting a range of x values to approximate the point at which the corresponding y values change from negative to positive.

This helps you identify the range in which y equals zero, and therefore where the solution lies, to a required number of decimal places.

We can put this into context with the example of a quadratic equation such as y = x² – 2x – 2.

When you enter Table mode, you can input your function in the table relation list, then press F5 to open the settings menu. Here you can add your chosen start and end values for x, as well as the increments by which you want values to increase within this range.

For example, with x² – 2x – 2 as the saved function, you could enter your start and end points for x as -0.73 and -0.74, respectively, and choose to go up in one thousandths by inputting 0.001 in the Step field.

When you press F6 in the table relation list screen to view your table, you’ll see that the change of sign in the y column occurs when the x value is within the range of -0.732 to -0.733.

Depending on the level of accuracy you need, you could then go through the same process again to narrow down the range of values even further, using -0.732 and -0.733 as your start and end points.

The fx-CG50 also gives you the option to visualise what’s happening by graphing the data in your table.

“You can trace the points on a graph, which will give you one value at a time,” Simon pointed out. “It’s quite nice to do it in Table mode and scroll through the values, or if you wanted to you could even do the same thing in the calculator’s Spreadsheet mode.”

The Newton-Raphson method

Another fx-CG50 app worth looking into is Recursion. This offers an interesting way to explore the Newton-Raphson method, a numerical approach students will need to be familiar and comfortable with, particularly at A-level.

In an exam, they might be asked to use this method to find the roots of a given equation to a certain number of decimal places.

It’s possible to do this in Run-Matrix mode, by entering the Newton-Raphson formula and the relevant values, then using the ANS key to calculate repeated iterations until the root is found to the required degree of accuracy.

However, another way to approach this is by entering the formula in Recursion mode and choosing a starting point. After that, you only need to press EXE once to show a list of the next values in the sequence, making it a fairly simple task to find the value you need.

“Recursion is a really nice way to do the Newton-Raphson method and it’s a somewhat underused function on the fx-CG50 as well, so it’s definitely worth investigating,” Simon commented.

There are lots of resources available that can help you get to grips with Recursion mode, including this video on our YouTube channel, which looks at how iteration can be used as the basis of numerical methods of solving equations.

Fixed point iteration

Recursion can also come into play when exploring another numerical method: fixed point iteration.

If a student is presented with an equation in terms of y =, once they’ve rearranged it in terms of x =, they can enter the data into Recursion mode to create a sequence of values.

A key benefit of doing this in Recursion mode on the fx-CG50 is that they can then use the values to plot a spiral, or cobweb, graph.

“The functionality allowing you to actually draw the spiral is a particularly useful feature, because it gives students a nice way to see the graph gradually converging to a value,” Simon said.

If you’re interested in learning more about the fx-CG50 and how it can help you and your students get to grips with numerical methods, pay a visit to our online resources centre.

You’ll find information leaflets covering decimal search, the Newton-Raphson method and fixed point iteration, including producing a sequence and its graph, along with many other useful resources.

We’ll also be hosting an fx-CG50 Topic Deep Dive session on numerical methods on March 13th. You can find more information and sign up on our webinar page.