Lesson 1.8: Practical Activity: The Matchstick Tower and Extension Activity: The Golden Ratio Hunt
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Chapter 1 · Lesson 8 of 8 · About 10 minutes
Learning objectives Extra Practice
- Recognise the triangular number pattern and explain why it is a pattern.
- Build the first five terms with matchsticks and record the totals.
- State the term‑to‑term rule and the position‑to‑term rule in your own words.
- Predict the sixth term, then verify by building it.
- Connect the activity to broader patterns such as Pascal's Triangle and the golden ratio.
- Measure the longer and shorter sides of at least five rectangular objects.
- Calculate the side‑length ratio for each object and record it to two decimal places.
- Compare each ratio with the golden ratio and identify which objects come closest.
- Discuss as a class why some objects match the ratio by design and others by chance.
Getting started
In this lesson you will build a growing triangular pattern with matchsticks, record the numbers you see, and discover the rule that lets you predict any term without building it first. The activity shows how a simple shape pattern becomes a number pattern and how mathematicians use the same steps — observe, identify, predict, verify — that computers use to learn from data.
Lesson Extra Practice
What Is a Triangular Number Pattern?
A triangular number pattern grows by adding one more item to each new row. Term 1 has 1 item, Term 2 adds 2 items (total 3), Term 3 adds 3 items (total 6), and so on. The totals 1, 3, 6, 10, 15 … are called triangular numbers because they can be arranged as equilateral triangles.
- Each term adds a row that is one item longer than the previous row.
- The total after n rows equals 1 + 2 + … + n.
- Triangular numbers appear in Pascal's Triangle along the third diagonal.
Building the Matchstick Tower
Follow the steps below using 40–50 matchsticks (or ice‑cream sticks or buttons). Work in pairs and record every total in the observation table.
- Lay out Term 1: a single matchstick.
- Build Term 2 by placing two matchsticks below the first, forming a tiny triangle of three sticks.
- Continue for Term 3, Term 4, and Term 5, each time adding a row that is one stick longer than the row above.
- After each term, count the total sticks used and write the number in the table.
- Do NOT build Term 6 yet. Use the pattern you see to predict its total.
- Finally, build Term 6 and count to verify your prediction.
| Term Number | Sticks Added This Term | Total Sticks So Far | Predicted or Verified? |
|---|---|---|---|
| 1 | 1 | 1 | Verified |
| 2 | 2 | 3 | Verified |
| 3 | 3 | 6 | Verified |
| 4 | 4 | 10 | Verified |
| 5 | 5 | 15 | Verified |
| 6 | — | ? | Predict, then Verify |
Recording, Predicting, and Verifying
Look at the totals column: 1, 3, 6, 10, 15. The differences between successive totals are 2, 3, 4, 5 — exactly the number of sticks added each term. This is the term‑to‑term rule: “add the next integer.” The position‑to‑term rule is the formula n(n + 1)/2, but you can also say “the total for term n is the sum of the first n whole numbers.”
Predicting Term 6 Extra Practice
Problem: Using the term‑to‑term rule, what is the total number of sticks for Term 6?
Solution:
- Term 5 total = 15.
- Sticks added for Term 6 = 6 (one more than the 5 added for Term 5).
- Predicted total = 15 + 6 = 21.
Answer: 21 sticks
Now build Term 6, count the sticks, and check that the total is 21. Discuss with your group: did everyone get the same prediction? If not, find the step where the counts diverged.
Connecting to Broader Patterns
Triangular numbers are not isolated. They appear as the third diagonal of Pascal's Triangle, and the ratios of consecutive Virahanka (Fibonacci) numbers approach the golden ratio, a constant that also relates to the geometry of triangles. Recognising one pattern helps you spot others — a core idea in computational thinking.
- Pascal's Triangle diagonal 3 → 1, 3, 6, 10, 15 …
- Virahanka numbers: 1, 1, 2, 3, 5, 8, 13 … ratios → 1.618… (golden ratio).
- Pattern‑thinking steps (Observe, Identify, Predict, Verify) are used in machine learning.
What Is the Golden Ratio?
The golden ratio is a special number, approximately 1.618, that appears in geometry, art, and nature. When a rectangle’s longer side divided by its shorter side equals this number, the rectangle is called a golden rectangle. Many people find this shape visually pleasing, so designers sometimes use it on purpose.
- Symbol: φ (phi)
- Value: 1.618 (rounded to three decimals)
- Property: If you remove a square from a golden rectangle, the remaining rectangle is also golden.
Measuring Objects – Procedure
Follow these steps for each object you choose. Work carefully so your data are reliable.
- Select a rectangular object (book cover, postcard, door, photo frame, etc.).
- Measure the longer side in centimetres with a ruler or tape measure.
- Measure the shorter side in centimetres.
- Divide the longer measurement by the shorter measurement using a calculator or long division.
- Round the result to two decimal places and write it down.
Recording Results – Data Table
Organise every measurement in a table. This makes comparison easy and prepares the data for a class chart.
| Object | Longer Side (cm) | Shorter Side (cm) | Ratio (Longer ÷ Shorter) |
|---|---|---|---|
| Book cover | 24.0 | 15.0 | 1.60 |
| Postcard | 14.8 | 10.5 | 1.41 |
| Door | 200.0 | 80.0 | 2.50 |
| Photo frame | 18.0 | 11.1 | 1.62 |
| Smartphone | 15.2 | 7.1 | 2.14 |
Comparing to the Golden Ratio
The golden ratio is 1.618. Look at each ratio in your table and note the difference from 1.618. The smaller the difference, the closer the object is to a golden rectangle.
Worked Example – Photo Frame Extra Practice
Problem: A photo frame measures 18.0 cm by 11.1 cm. How close is its ratio to the golden ratio?
Solution:
- Divide 18.0 by 11.1 → 1.6216
- Round to two decimals → 1.62
- Subtract 1.618 from 1.62 → 0.002
Answer: The frame’s ratio is 1.62, only 0.002 away from the golden ratio – an extremely close match.
Class Discussion & Verification
Combine every group’s tables into one large chart. Discuss these questions:
- Which objects came closest to 1.618?
- Were those objects likely designed that way (e.g., photo frames, credit cards) or could it be coincidence?
- Why might a designer choose the golden ratio?
- How does this activity illustrate the Verify step of the pattern‑thinking process?
Teacher–student conversation Extra Practice
Teacher: What do you notice about the numbers added each term?
Student: They go 1, 2, 3, 4, 5 — each term adds one more than the last.
Teacher: Exactly. So if Term 5 added 5 sticks, how many will Term 6 add?
Teacher: Can you say the rule for the total in your own words?
Student: The total for any term is the sum of all whole numbers up to that term.
Teacher: Great. That is the position‑to‑term rule. Let's write it as a formula together.
Teacher: Let’s look at the book cover ratio of 1.60. How does it compare to 1.618?
Student: It’s 0.018 less, so it’s pretty close but not exact.
Teacher: Good observation. A difference of 0.018 is small, but the photo frame was even closer. What might that tell us about the frame’s design?
Teacher: Why do you think the door ratio is 2.50, far from the golden ratio?
Student: Doors need to be tall for people to walk through, so the height is much larger than the width.
Teacher: Exactly. Function often drives proportions more than aesthetics.
Real-world connections Extra Practice
- Stacking cans or boxes in a triangular pyramid (e.g., grocery displays).
- Arranging bowling pins — 10 pins form the 4th triangular number.
- Designing seating charts where each row has one more seat than the row in front.
- Credit cards and many smartphone screens use golden‑ratio proportions.
- Architects sometimes design windows and facades as golden rectangles.
- Artists compose paintings using the golden spiral derived from the ratio.
Matchstick Tower Exploration Extra Practice
- Gather 40–50 matchsticks and a ruled notebook.
- Build Terms 1‑5, recording totals in the observation table.
- Predict Term 6 total using the term‑to‑term rule.
- Build Term 6, count, and verify.
- Discuss any differences with classmates and write a short reflection on the pattern‑thinking steps.
Materials: 40–50 matchsticks (or ice‑cream sticks/buttons), ruled notebook page, pencil.
Golden Ratio Hunt Extra Practice
- Gather a ruler or measuring tape and a calculator.
- Find at least five different rectangular objects at home or in the classroom.
- Measure, calculate, and record each ratio in your own table.
- Bring your table to class for the combined chart and discussion.
Materials: Ruler or measuring tape, calculator (or paper for long division), paper and pencil for the table.
Important points to remember Extra Practice
- Triangular numbers grow by adding the next integer each term.
- Term‑to‑term rule: add n sticks for term n.
- Position‑to‑term rule: total = n(n + 1)/2 (sum of first n whole numbers).
- The four‑step pattern process (Observe, Identify, Predict, Verify) mirrors computational thinking.
- Triangular numbers appear in Pascal's Triangle and relate to other famous sequences.
- The golden ratio ≈ 1.618 appears in design and nature.
- Measuring real objects lets you test a mathematical idea with data.
- Recording results in a table supports clear comparison.
- Class discussion reveals the difference between intentional design and chance.
Lesson summary Extra Practice
By physically constructing the matchstick tower you turned a shape pattern into a number sequence, discovered both the term‑to‑term and position‑to‑term rules, and practiced the predict‑verify cycle that underlies all pattern work — from classroom maths to artificial intelligence.
By measuring everyday rectangles and comparing their side ratios to the golden ratio, you experience the Verify stage of pattern thinking. You learn to collect data, organise it in a table, and evaluate how closely real‑world objects match a mathematical ideal, deepening your understanding of why patterns matter in both mathematics and design.
Teacher review note
Ensure students have enough matchsticks for Term 6 (21 total). If matchsticks are scarce, buttons or drawn dots work equally well. Emphasise the discussion step — it reinforces the verification phase of pattern thinking. Ensure students understand rounding to two decimal places before they start. If calculators are unavailable, review long division with decimals. The class chart can be created on a whiteboard or spreadsheet for easy visual comparison.
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