Cutting Edge STEM

Professional Learning Day, 26 Nov, 2025

How to use free-online digital technologies to enhance student understanding of key concepts from the Specialist Maths syllabus

Juan Carlos Ponce Campuzano

Workshop

Gold Coast Campus

Linear regression: Residuals

👉 geogebra.org

Code:

GTXF J8AA

🔗 geogebra.org

🔗 geogebra.org/calculator

Let's get ready to explore maths in GeoGebra

GeoGebra calculator

1. Exploring systems of linear equations in 3D

Consider the system corresponding to three planes

\begin{array}{rcl} x+ 5y &=& 1+ 2z\\ x+z &=& 3y+3\\ 8y- \lambda &=& 3z \end{array}

\(\lambda\) is a parameter.

Use Gaussian elimination to determine the value of \(\lambda\) for which this system has infinitely many solution.
Use part a) to determine the infinitely many solutions. Express your answer in the form of a vector equation

1. Exploring systems of linear equations in 3D

\begin{array}{rcl} x+ 5y &=& 1+ 2z\\ x+z &=& 3y+3\\ 8y- \lambda &=& 3z \end{array}

\begin{array}{rrrrrrr} x&+& 5y &- &2z &=& 1\\ x&-&3y &+&z &=& 3\\ 0x &+& 8y&-& 3z &=& \lambda \end{array}

\left( \begin{array}{rrr|r} 1 & 5 & -2 & 1\\ 1 & -3 & 1 & 3\\ 0 & 8 & -3 & \lambda \end{array} \right)

\left( \begin{array}{rrr|r} 1 & 5 & -2 & 1\\ 0 & -8 & 3 & 2\\ 0 & 8 & -3 & \lambda \end{array} \right)

R_2-R_1

\lambda = -2

R_1\\ R_2

the system is consistent with \(\infty\) solutions

1. Exploring systems of linear equations in 3D

lambda = Slider(-10, 10, 0.1)
eq1: x + 5y = 1 + 2z
eq2: x + z= 3y + 3
eq3: 8y - lambda = 3z

Solve({eq1, eq2, eq3})

line: X = (9/4, -1/4, 0) + t * (1/8, 3/8, 1)

🔗 Solve

🔗 Slider

🔗 Equations of lines

🔗 Online demo

2. Differential equations and slope fields

The slope fied for the differential equation \(\dfrac{dy}{dx}=\dfrac{-0.5(y-4)}{x},\) with \(x\neq 0,\) where \(-6\leq x\leq 6\) and \(-6\leq y \leq 6\) is shown below.

Determine the value of the slope at point \(A=(-4,-2)\).
Use the slope field to sketch the solution curve given \(x=-6,\) \(y=3.5\)

2. Differential equations and slope fields

n = Slider(10, 30, 1)
s = Slider(0.1, 1, 0.1)

F(x, y) = -0.5 * (y - 4)/x

SlopeField(F, n, s, -6, -6, 6, 6)

A = (0, 0)

SolveODE(F, x(A), y(A), 6, 0.01)
SolveODE(F, x(A), y(A), -6, 0.01)

🔗 SlopeField

🔗 SolveODE

🔗 Online demo

3. Roots of complex numbers

Represent in the Argand diagram the solutions of

z^{1/n}

\text{with}\;k=0,1,\ldots, n-1

z^4= 16 \,\text{cis}\left(\dfrac{2\pi}{3}\right)

= 16 \cos\left(\dfrac{2\pi}{3}\right) + i\,16\sin\left(\dfrac{2\pi}{3}\right)

=\sqrt[n]{r}\left[\cos\left(\frac{\theta}{n}+\frac{2\pi k}{n}\right)+i\sin\left(\frac{\theta}{n}+\frac{2\pi k}{n}\right) \right]

3. Roots of complex numbers

Represent in the Argand diagram the solutions of

z^{1/n}

\text{with}\;k=0,1,\ldots, n-1

z^4= 16 \,\text{cis}\left(\dfrac{2\pi}{3}\right)

= 16 \cos\left(\dfrac{2\pi}{3}\right) + i\,16\sin\left(\dfrac{2\pi}{3}\right)

= r^{1/n} \exp\Bigg(\frac{i \left( \theta+2\pi k\right)}{n} \Bigg)

3. Roots of complex numbers

z_1 = i
u = Vector(z_1)
r = abs(z_1)
theta = arg(z_1)
n = Slider(2, 10, 1)

L = Sequence(r^(1/n) * exp(i * ( theta + 2pi * k )/n ), k, 0, n-1)

Sequence(Vector(Element(L, k)), k, 1, Length(L))

# Alternatively
Zip(Vector(Point), Point, L)

🔗 Sequence

🔗 Vector

🔗 Zip (advance)

🔗 Predefined functions and operators

🔗 Online demo