Mathematics is not only about calculating an answer. It is also the search for patterns and explanations of why those patterns occur. Patterns appear in nature, at home and school, in the motion of the Sun, Moon and stars, and in activities such as shopping, cooking, games and technology. Looking for a rule helps us describe what we notice and predict what may happen next.
The textbook explains that mathematics can be viewed as both an art and a science: discovering a pattern calls for creativity, while explaining why it works calls for careful reasoning. An explanation may be useful far beyond the situation where the pattern was first noticed. For example, studying the motion of stars, planets and satellites contributed to the theory of gravitation and to sending satellites and rockets into space. Studying patterns in genomes has also helped with diagnosing and treating diseases.
When you investigate a pattern, ask three questions: What changes? What stays the same? What rule explains the change? A good mathematical explanation should do more than guess the next item; it should make the rule clear.
One basic type of pattern is a number sequence: a list of numbers arranged according to a rule. Whole-number patterns are studied in number theory. The chapter introduces several useful sequences. Counting numbers increase by one at a time; odd numbers increase by two starting from 1; even numbers increase by two starting from 2. Powers of 2 double at each step, while powers of 3 triple at each step.
Triangular numbers can be represented by dots arranged in triangular shapes: 1, 3, 6, 10, 15, 21, … . Each new term adds the next counting number. Square numbers are 1, 4, 9, 16, 25, 36, … and can be arranged in square grids. Cubes begin 1, 8, 27, 64, 125, … . The chapter also presents Virahānka numbers, beginning 1, 2, 3, 5, 8, 13, 21, …, in which each term after the first two is formed by adding the two previous terms.
To continue a sequence, first identify the rule rather than merely looking at the last number. For example, 2, 4, 6, 8, … adds 2 each time, so the next terms are 10, 12 and 14. For 1, 2, 4, 8, … each term doubles, so the next terms are 16, 32 and 64.
Pictures can make number patterns easier to see. One dot represents 1; two dots represent 2; square grids show square numbers; and dot arrangements in triangles show triangular numbers. Visualising is useful because it connects an abstract number with a shape that can be counted.
The number 36 is a helpful example from the chapter: it is both a square number and a triangular number. The same quantity can be arranged in more than one way. The chapter also introduces hexagonal numbers such as 1, 7, 19, 37, … through a pattern of dots arranged in hexagon-like layers. A picture can suggest a rule, but the rule should still be described clearly.
When drawing a sequence, label the first few terms and show how each new picture is formed. If a shape grows by adding a new row or layer, state what is added each time. This helps someone else reproduce the pattern and check the prediction.
Different number sequences can be linked. The chapter shows that adding consecutive odd numbers produces square numbers: 1 = 1, 1 + 3 = 4, 1 + 3 + 5 = 9, and 1 + 3 + 5 + 7 = 16. Continuing the pattern gives 1 + 3 + 5 + 7 + 9 = 25 and 1 + 3 + 5 + 7 + 9 + 11 = 36. A square picture explains why: each new odd number can be seen as an L-shaped layer added to the previous square to make a larger square.
Therefore, the sum of the first 10 odd numbers is 100, and the sum of the first 100 odd numbers is 10,000. These results follow the same growing-square pattern shown in the book.
Another pattern adds counting numbers upward and then downward: 1 = 1; 1 + 2 + 1 = 4; 1 + 2 + 3 + 2 + 1 = 9; and 1 + 2 + 3 + 4 + 3 + 2 + 1 = 16. These totals are also square numbers. The chapter invites students to find and explain further relationships, such as sums of consecutive triangular numbers or sums of powers of 2. Pictures are not decoration here; they help explain why a relationship works.
Patterns can be made from shapes as well as numbers. Geometry is the branch of mathematics that studies patterns in shapes. Shapes may be one-dimensional, two-dimensional or three-dimensional. The chapter displays sequences of regular polygons, complete graphs, stacked squares, stacked triangles and Koch snowflake iterations.
A regular polygon has sides of equal length and equal angles. The sequence of regular polygons starts with a triangle (3 sides), quadrilateral (4), pentagon (5), hexagon (6), heptagon (7), octagon (8), nonagon (9) and decagon (10). The number of sides follows the counting numbers starting at 3. Each polygon also has the same number of corners as sides.
Counting the pieces in a shape sequence can produce a number sequence. For example, the Koch snowflake line-segment counts shown in the chapter begin 3, 12, 48, …; each iteration replaces every segment by four smaller segments, so the total is multiplied by 4. Looking from shapes to numbers and back again is a way to discover, test and explain mathematical relationships.
| Pattern type | Beginning terms | Rule or visual idea |
|---|---|---|
| Counting numbers | 1, 2, 3, 4, 5, … | Add 1 each time. |
| Odd numbers | 1, 3, 5, 7, 9, … | Add 2 each time, beginning at 1. |
| Even numbers | 2, 4, 6, 8, 10, … | Add 2 each time, beginning at 2. |
| Triangular numbers | 1, 3, 6, 10, 15, … | Add successive counting numbers. |
| Square numbers | 1, 4, 9, 16, 25, … | Can be arranged as square grids. |
| Powers of 2 | 1, 2, 4, 8, 16, … | Double the previous term. |
| Powers of 3 | 1, 3, 9, 27, 81, … | Triple the previous term. |