Juan Carlos Ponce Campuzano
Mathematics Educator
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
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Code:
GTXF J8AA
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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           Â
🔗 Slider           Â
🔗 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)
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Cutting Edge STEM 2025
By Juan Carlos Ponce Campuzano
Cutting Edge STEM 2025
Discover innovative ways to leverage free online digital tools, like GeoGebra, to deepen student comprehension of Specialist Maths concepts, including linear regression and differential equations. Join us in transforming maths education!
