Lesson 1.4: 1.2 Patterns in Numbers (part 2)
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Chapter 1 · Lesson 4 of 8 · About 14 minutes
Learning objectives Extra Practice
- Identify and extend cube numbers, the doubling pattern, and Virahanka numbers.
- Explain the term‑to‑term and position‑to‑term rules for each pattern.
- Represent patterns with dots, cubes, tables, and simple charts.
- Discover the connection between Virahanka numbers and the golden ratio.
- Apply the four‑step pattern‑thinking process (Observe, Identify, Predict, Verify) to new situations.
Getting started
In this lesson we explore three fascinating number patterns that grow in very different ways: cube numbers, the doubling pattern (powers of two), and the Virahanka (Fibonacci) numbers. By building models, calculating terms, and observing ratios, you will learn to recognise a pattern, describe its rule, and see how these patterns appear in nature, technology, and even in the way computers learn.
Lesson Extra Practice
1. Cube Numbers – Building Solids
Square numbers arrange dots in a flat grid. Cube numbers take the same idea into three dimensions: we stack unit cubes to form a larger cube that is n cubes long, n cubes wide, and n cubes tall. The first four cube numbers are 1, 8, 27, and 64.
- 1³ = 1 (a single cube)
- 2³ = 8 (a 2 × 2 × 2 block)
- 3³ = 27 (a 3 × 3 × 3 block)
- 4³ = 64 (a 4 × 4 × 4 block)
- 5³ = 125 (a 5 × 5 × 5 block)
| n (position) | n³ (cube number) |
|---|---|
| 1 | 1 |
| 2 | 8 |
| 3 | 27 |
| 4 | 64 |
| 5 | 125 |
| 6 | 216 |
Finding the 7th cube number Extra Practice
Problem: What is the 7th cube number? Show the position‑to‑term rule you used.
Solution:
- The position‑to‑term rule for cube numbers is: term = n × n × n (or n³).
- Substitute n = 7: 7 × 7 × 7 = 343.
- Therefore the 7th cube number is 343.
Answer: 343
Image prompt 1: Section 1 – Cube Numbers
2. The Doubling Pattern – Powers of Two
Not every pattern grows by addition. The doubling pattern starts at 1 and multiplies by 2 each step: 1, 2, 4, 8, 16, 32, … This is the sequence of powers of two (2⁰, 2¹, 2², …). A famous story tells of a king who agreed to place one grain of rice on the first square of a chessboard, two on the second, four on the third, and so on. By the 64th square the total exceeds all rice ever produced.
- Term‑to‑term rule: multiply the previous term by 2.
- Position‑to‑term rule: term = 2^(n‑1) where n is the position (starting at 1).
- The pattern grows slowly at first, then extremely quickly.
- Write the first term: 1.
- To get the next term, multiply the current term by 2.
- Repeat step 2 for as many terms as you need.
Rice on the first 10 squares Extra Practice
Problem: How many grains of rice are on the 10th square? What is the total number of grains on squares 1 through 10?
Solution:
- 10th square: term = 2^(10‑1) = 2⁹ = 512 grains.
- Total grains = sum of first 10 terms = 2¹⁰ – 1 = 1024 – 1 = 1023 grains.
Answer: 10th square: 512 grains; total on squares 1‑10: 1023 grains
Image prompt 2: Section 2 – Doubling Pattern
3. Virahanka Numbers – The Fibonacci Sequence
Around the sixth century the Indian scholar Virahanka studied Sanskrit poetry and found a sequence where each term is the sum of the two preceding terms: 1, 1, 2, 3, 5, 8, 13, 21, … This sequence appears in flower petals, sunflower seed spirals, and nautilus shells, showing that mathematical patterns can describe natural growth.
- Term‑to‑term rule: add the two previous terms to get the next term.
- Position‑to‑term rule: no simple closed form at this level; we generate terms step by step.
- The ratio of consecutive terms approaches the golden ratio (≈ 1.618).
| Position |
Term | Ratio (term ÷ previous) |
|---|---|---|
| 1 | 1 | — |
| 2 | 1 | 1.000 |
| 3 | 2 | 2.000 |
| 4 | 3 | 1.500 |
| 5 | 5 | 1.667 |
| 6 | 8 | 1.600 |
| 7 | 13 | 1.625 |
| 8 | 21 | 1.615 |
| 9 | 34 | 1.619 |
| 10 | 55 | 1.618 |
Extending the Virahanka sequence Extra Practice
Problem: Find the 11th and 12th Virahanka numbers and their ratios to the previous term.
Solution:
- 11th term = 55 + 34 = 89.
- 12th term = 89 + 55 = 144.
- Ratio 11th/10th = 89 ÷ 55 ≈ 1.618.
- Ratio 12th/11th = 144 ÷ 89 ≈ 1.618.
Answer: 11th = 89 (ratio ≈ 1.618); 12th = 144 (ratio ≈ 1.618)
4. Connecting Patterns – From Numbers to Computational Thinking
All three patterns can be represented visually (cubes, chessboard, spiral of squares) and each follows a clear rule that a computer could follow. The four‑step pattern‑thinking process — Observe, Identify, Predict, Verify — is exactly how machine‑learning algorithms learn to spot patterns in data.
- Observe: look at the given terms or shapes.
- Identify: state the rule (term‑to‑term or position‑to‑term).
- Predict: use the rule to find the next terms.
- Verify: check the prediction against known terms or a model.
These patterns also link to each other: triangular numbers appear inside Pascal’s Triangle, square numbers can be built from odd numbers, and Virahanka ratios converge to the golden ratio. Recognising such connections deepens mathematical insight.
Teacher–student conversation Extra Practice
Teacher: Look at the cube numbers table. What do you notice about how fast they grow compared with square numbers?
Student: They grow much faster because each step multiplies in three directions instead of two.
Teacher: Exactly. If a square of side 4 has area 16, a cube of side 4 has volume 64 — four times larger in each dimension.
Teacher: If we continue the doubling pattern, what will the 15th term be? Can you use the position‑to‑term rule?
Student: The rule is 2^(n‑1). So for n = 15 it is 2^14 = 16384.
Teacher: Well done. Notice how quickly the numbers become huge — that is why the chessboard story surprises the king.
Real-world connections Extra Practice
- Cube numbers model the volume of cubic containers (e.g., shipping boxes, water tanks).
- The doubling pattern explains viral spread, compound interest, and limits of paper folding.
- Virahanka numbers appear in plant phyllotaxis (leaf arrangement), pinecone spirals, and the proportions of many natural forms.
Build Cube Numbers with Blocks Extra Practice
- Gather unit cubes (e.g., centimetre cubes or sugar cubes).
- Build the first four cube numbers: 1, 8, 27, 64.
- Record the number of cubes used for each model in a table.
- Predict how many cubes are needed for the 5th cube number and verify by building.
Materials: Unit cubes (at least 125), table template, pencil
Paper‑Folding Doubling Experiment Extra Practice
- Take a standard A4 sheet of paper.
- Fold it in half, count the layers (2).
- Continue folding and recording layers after each fold until you cannot fold further.
- Compare the recorded layers with the doubling pattern (1, 2, 4, 8, …).
Materials: A4 paper, ruler, notebook
Draw a Virahanka Spiral Extra Practice
- On graph paper, draw squares with side lengths equal to the Virahanka numbers: 1, 1, 2, 3, 5, 8, 13 …
- Arrange each new square adjacent to the previous two so they form a growing rectangle.
- Draw a quarter‑circle arc inside each square to create a spiral.
- Observe how the spiral approximates the golden spiral.
Materials: Graph paper, ruler, compass, coloured pencils
Important points to remember Extra Practice
- Cube numbers are n³; they represent three‑dimensional growth.
- The doubling pattern follows the rule term = 2^(n‑1); it grows exponentially.
- Virahanka numbers use the term‑to‑term rule: add the two previous terms.
- Ratios of consecutive Virahanka numbers converge to the golden ratio (≈ 1.618).
- The four‑step pattern‑thinking process (Observe, Identify, Predict, Verify) is a foundation for computational thinking.
Lesson summary Extra Practice
We have explored three distinct number patterns: cube numbers that fill space, the doubling pattern that multiplies by two each step, and the Virahanka sequence that adds the two previous terms. Each pattern can be described with a clear rule, represented visually, and connected to real‑world phenomena. Mastering the Observe‑Identify‑Predict‑Verify cycle equips you to recognise and extend patterns in mathematics, nature, and technology.
Illustrations retained from the textbook From Your Textbook
Fig. 1.6: The first four cube numbers, shown as isometric stacks of unit cubes: 1, 8, 27, and 64 From Your Textbook
Four-panel isometric illustration, n×n×n stacked unit cubes per term, one accent colour per term (navy, gold, maroon, green), title states the multiplication and total above each stack
Fig. 1.7: The doubling pattern grows slowly at first, then extremely quickly From Your Textbook
Vertical bar chart, six bars (terms 1 –6), navy fill, value label above each bar, x -axis term number, y-axis value
Fig. 1.8: The Virahanka numbers: each term is the sum of the two terms before it From Your Textbook
Vertical bar chart, eight bars (terms 1 –8), gold fill, value label above each bar, x -axis term number, yaxis value
Fig. 1.9: The ratio of consecutive Virahanka numbers oscillates ever closer to the golden ratio, approximately 1.618 From Your Textbook
Line chart with markers, x -axis term position 3–9, y-axis ratio value, gold dashed horizontal reference line at 1.618, data labels above each marker
Teacher review note
All numerical facts and descriptions are taken directly from the provided source text. No additional historical dates or formulas have been invented. The activities use only materials listed and follow the patterns described in the source.
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