Using Technology to Explore and Represent Patterns and Relationships

Overview

Technology — particularly calculators, spreadsheets, and graphing tools — extends what is possible when exploring patterns and relationships. In Foundation Mathematics, technology is a tool for investigation, not a shortcut that replaces understanding.

KEY TAKEAWAY: Technology speeds up calculations and reveals patterns in large datasets, but you still need to understand what the numbers mean and whether results are reasonable.

Scientific and CAS Calculators

Arithmetic sequences: Use the replay/answer key to generate terms.
- Enter the first term, then repeatedly add the common difference: $5$ [=] [+4=] [=] [=] …

Evaluating formulas: Substitute values directly.
- $t_n = 3n + 2$: For $n = 15$: enter \$3 \times 15 + 2 = 47$

Solving equations: Use solver or trial-and-error with calculator support.

EXAM TIP: Show the expression you entered and the result displayed. “I used a calculator” is not sufficient — write \$3 \times 15 + 2 = 47$.

Spreadsheets (e.g. Microsoft Excel, Google Sheets)

Spreadsheets are ideal for:
- Generating sequences automatically
- Calculating totals and running sums
- Modelling cost/income relationships
- Creating graphs from data

Generating an Arithmetic Sequence

Or use a direct formula:

Creating a Cost Model

A spreadsheet can model the phone plan example from the previous section:

Making Charts from Data

Select the data columns → Insert → Chart.

• Line chart: For sequences that change over time or step number
• Scatter plot: For relationships between two numerical variables
• Column chart: For comparing values across categories

EXAM TIP: When describing a graph created by technology, always include: the axes labels, the overall trend (increasing/decreasing/constant), and any notable features (e.g. where two lines intersect).

Graphing Applications

Graphing tools (Desmos, GeoGebra, CAS calculator) allow you to:
- Plot $y = mx + c$ and see the effect of changing $m$ (gradient) and $c$ (y-intercept)
- Find where two lines intersect (solving simultaneous equations graphically)
- Explore how a pattern changes when parameters are varied

Worked Example — Graphical Solution:

Plot $y = 60x + 80$ (Plumber A) and $y = 70x + 50$ (Plumber B) on the same axes. Where do they intersect?

Using Desmos or a CAS calculator, the intersection is at $(3, 260)$:
- After $3$ hours, both plumbers cost $\$260$
- For $x > 3$: Plumber A’s line is lower → cheaper
- For $x < 3$: Plumber B’s line is lower → cheaper

This matches the algebraic solution from the previous section.

Using Technology Responsibly

VCAA FOCUS: Foundation Mathematics assessments permit calculator use. Technology is expected in longer problems. However, you must record your working: write the formula, show substituted values, and state the final answer with units. A screen shot or “calculator says 47” earns no marks without mathematical reasoning.

STUDY HINT: Practice building spreadsheet models for the types of problems you study — wages, costs, savings plans, tile patterns. Seeing the numbers update automatically as you change inputs builds strong intuition about how relationships work.