Patterns in Mathematics · Lesson 1 of 7
Mathematics as a Search for Patterns
“Learn to notice, describe, and extend number patterns, and ask why their rules work.”
• Explain how finding and explaining patterns helps in everyday life and scientific work. • Distinguish whole numbers from the counting sequence used in this chapter. • Recognise and describe the ten main number sequences. • Extend a sequence using addition, multiplication, or the preceding two terms. • Check a proposed rule against every displayed term.
A pattern invites an explanation
Think about a calendar: the days of the week repeat in the same order. A tiled floor may repeat a shape, and the Moon changes its appearance in a recurring cycle. Mathematics helps us describe such regularities clearly. It also asks a deeper question: what makes the pattern happen?
Finding a pattern can be creative, because you choose what to compare and how to represent it. Explaining the pattern requires careful reasoning. This combination is why mathematics has features of both an art and a science: imagination helps us discover a possibility, while reasoning helps us decide whether it works.
Patterns are useful beyond the place where they are first noticed. Studying the motions of planets and satellites helped people understand gravity and plan space travel. Patterns in genetic information help researchers study diseases. These are large applications of the same habit you can practise with a short list of numbers: observe, make a rule, check it, and seek an explanation.
Problem
A family buys 3 notebooks costing 40 rupees each and pays with 200 rupees. What mathematical decisions are involved?
- 1.The equal price repeats once for each notebook, so the total is 40 + 40 + 40 = 120 rupees.
- 2.Subtract the total from the payment: 200 − 120 = 80 rupees of change.
- 3.The family uses a repeated quantity to find a total, then checks the difference. Mathematics supports a practical decision.
Look for mathematics in cooking quantities, travel times, building designs, clocks, voting totals, or weather records. Describe what is counted or measured and what pattern might be useful. Discuss how repeated measurements in a scientific investigation can help reveal a relationship.
Numbers arranged in a sequence
A number sequence is an ordered list of numbers. The position of a number matters: 2, 4, 6 is different from 6, 4, 2. Each entry is called a term. In this chapter, we look for a rule that tells us how the terms are produced.
The numbers 0, 1, 2, 3, 4, and so on. The counting sequence used here begins at 1: 1, 2, 3, 4, and so on.
The study of patterns and relationships in whole numbers.
The three dots at the end of a sequence mean that the list continues. They do not tell us the rule by themselves. We must inspect the displayed terms and state which rule we intend to use. A good rule explains all the shown terms, not just the final change.
| Sequence | First terms | Rule to use |
|---|---|---|
| All 1s | 1, 1, 1, 1, 1, … | Keep writing 1. |
| Counting | 1, 2, 3, 4, 5, … | Add 1 each time. |
| Odd | 1, 3, 5, 7, 9, … | Start at 1 and add 2. |
| Even | 2, 4, 6, 8, 10, … | Start at 2 and add 2. |
| Triangular | 1, 3, 6, 10, 15, … | Add 2, then 3, then 4, then 5, and so on. |
| Square | 1, 4, 9, 16, 25, … | Multiply each counting number by itself. |
| Cube | 1, 8, 27, 64, 125, … | Multiply each counting number by itself three times. |
| Virahānka | 1, 2, 3, 5, 8, … | Start with 1 and 2; add the preceding two terms. |
| Powers of 2 | 1, 2, 4, 8, 16, … | Start at 1 and multiply by 2. |
| Powers of 3 | 1, 3, 9, 27, 81, … | Start at 1 and multiply by 3. |
Compare rules, not just appearances
Some sequences grow by a fixed addition; others grow by a changing addition or by multiplication. Odd and even numbers both increase by 2, but their starting terms differ. Triangular numbers also grow by addition, yet the amount added increases. Recognising this distinction prevents us from treating every growing list in the same way.
An even number can be shared into pairs with nothing left over. An odd number leaves one unpaired object. This explains why the two sequences alternate among the counting numbers. For squares, the products are 1 × 1, 2 × 2, 3 × 3, and so on. For cubes, they are 1 × 1 × 1, 2 × 2 × 2, 3 × 3 × 3, and so on. Their geometric names will become clearer when we draw them.
Problem
Find the next three triangular numbers after 1, 3, 6, 10, 15, 21, 28.
- 1.Compare the differences: 2, 3, 4, 5, 6, 7. Each new difference is 1 larger.
- 2.The next difference is 8, giving 28 + 8 = 36.
- 3.Then add 9 and 10: 36 + 9 = 45 and 45 + 10 = 55. The next terms are 36, 45, 55.
Growth by multiplication and by two earlier terms
Powers of 2 and powers of 3 use repeated multiplication. Powers of 2 begin with 1 and double each time; powers of 3 begin with 1 and triple each time. The word power describes repeated multiplication by the same number. For instance, 8 = 2 × 2 × 2 and 27 = 3 × 3 × 3.
Virahānka numbers use a different rule. You need to remember two terms, because the next term is their sum. Starting from 1 and 2 gives 3; then 2 and 3 give 5; then 3 and 5 give 8. The process always uses the most recent two terms.
Problem
Continue 1, 2, 3, 5, 8, 13, 21 by three terms.
- 1.Use 13 and 21: 13 + 21 = 34.
- 2.Move the pair forward to 21 and 34: 21 + 34 = 55.
- 3.Move it forward again to 34 and 55: 34 + 55 = 89. The next three terms are 34, 55, 89.
Problem
Extend 1, 2, 4, 8, 16 and 1, 3, 9, 27, 81 by two terms each.
- 1.For the first sequence, multiply 16 by 2 to get 32, then multiply 32 by 2 to get 64.
- 2.For the second, multiply 81 by 3 to get 243, then multiply 243 by 3 to get 729.
- 3.Both use a fixed multiplier, but different multipliers produce different growth.
A few starting terms alone may fit more than one possible rule. Use the named rule in a task, or state your rule clearly and check every shown term. Also, “multiply by 2” and “add 2” agree from 2 to 4 but give different next terms: 8 and 6.
Quiz
Which statement describes the main mathematical habit in this chapter?
Which number belongs to the whole numbers but is not in the counting sequence shown here?
What comes after 1, 3, 6, 10, 15 under the triangular-number rule?
In the Virahānka sequence, what follows 8, 13, 21?
Which rule generates 1, 3, 9, 27, 81?
Why do the odd and even sequences differ despite both increasing by 2?
Practice Problems
- Write the next three terms of each of the ten sequences in the table, and explain each rule in your own words.
- Give two everyday examples of mathematics. Explain what numbers or shapes are involved and what decision they help you make.
- Write the odd and even numbers from 1 to 20. Use pairs of dots to explain the distinction.
- A student says 2, 4, 8, 16 grows by adding 2. Identify the first change that disproves this rule and give a rule that works.
- Find the missing terms in 1, 2, 3, 5, __, 13, __, 34. Explain which pair you used each time.
- Compare 1, 4, 9, 16 with 1, 8, 27, 64. Write each term as a product and explain how the products differ.
- Choose a familiar repeating pattern, such as weekdays or a tile arrangement. Describe the repetition and discuss why the pattern is useful.
Key Takeaways
• Mathematics looks for patterns and explanations that can support practical applications. • A sequence is an ordered list; each entry is a term. • Whole numbers include 0; the counting sequence used here begins with 1. • Check the starting value as well as the addition, multiplication, or pair-sum rule. • A useful rule accounts for all the displayed terms and helps produce new ones.
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Next · Lesson 2
Seeing Number Sequences