PATTERNS IN MATHEMATICS
1.1 What is Mathematics?
PATTERNS IN MATHEMATICS: Mathematics is, in large part, the search for patterns, and for the explanations as to why those patterns exist.
Such patterns indeed exist all around us—in nature, in our homes and schools, and in the motion of the sun, moon, and stars. They occur in everything that we do and see, from shopping and cooking, to throwing a ball and playing games, to understanding weather patterns and using technology.
The search for patterns and their explanations can be a fun and creative endeavour. It is for this reason that mathematicians think of mathematics both as an art and as a science. This year, we hope that you will get a chance to see the creativity and artistry involved in discovering and understanding mathematical patterns.
It is important to keep in mind that mathematics aims to not just find out what patterns exist, but also the explanations for why they exist. Such explanations can often then be used in applications well beyond the context in which they were discovered, which can then help to propel humanity forward.
Figure it Out (Page 2)
1. Can you think of other examples where mathematics helps us in our everyday lives?
Mathematics is fundamental to our daily lives in countless ways:
- Time Management: Reading clocks, scheduling appointments, and calculating durations.
- Budgeting and Finance: Managing money, calculating discounts, interest rates, and taxes.
- Cooking and Baking: Measuring ingredients, converting units, and adjusting recipe quantities.
- Health and Fitness: Calculating calorie intake, monitoring heart rate, and tracking workout progress.
- Travel and Navigation: Using maps, calculating distances, fuel consumption, and travel times.
2. How has mathematics helped propel humanity forward?
Mathematics is the backbone of almost all technological and scientific advancements:
- Science and Engineering: From developing the theory of gravitation to designing bridges and sending rockets to Mars, math provides the models and tools for understanding and shaping the physical world.
- Medicine: Understanding patterns in genomes helps in diagnosing and curing diseases, while statistical analysis is crucial for clinical trials.
- Technology: The algorithms that run computers, mobile phones, and the internet are purely mathematical constructs.
- Economy and Democracy: Statistics and mathematical models are used to run economies, conduct censuses, and ensure fair democratic processes.
1.2 Patterns in Numbers
Among the most basic patterns that occur in mathematics are patterns of numbers, particularly patterns of whole numbers: 0, 1, 2, 3, 4, …
The branch of Mathematics that studies patterns in whole numbers is called number theory.
Number sequences are the most basic and among the most fascinating types of patterns that mathematicians study. Table 1 shows some key number sequences that are studied in Mathematics.
Table 1: Examples of number sequences (Page 3)
| Sequence | Name |
|---|---|
| 1, 1, 1, 1, 1, 1, 1, … | (All 1’s) |
| 1, 2, 3, 4, 5, 6, 7, … | (Counting numbers) |
| 1, 3, 5, 7, 9, 11, 13, … | (Odd numbers) |
| 2, 4, 6, 8, 10, 12, 14, … | (Even numbers) |
| 1, 3, 6, 10, 15, 21, 28, … | (Triangular numbers) |
| 1, 4, 9, 16, 25, 36, 49, … | (Squares) |
| 1, 8, 27, 64, 125, 216, … | (Cubes) |
| 1, 2, 3, 5, 8, 13, 21, … | (Virahāṅka numbers) |
| 1, 2, 4, 8, 16, 32, 64, … | (Powers of 2) |
| 1, 3, 9, 27, 81, 243, 729, … | (Powers of 3) |
Figure it Out (Page 3)
2. Rewrite each sequence of Table 1 in your notebook, along with the next three numbers in each sequence! After each sequence, write in your own words what is the rule for forming the numbers in the sequence.
- All 1’s: 1, 1, 1, 1, 1, 1, 1, … Next three: 1, 1, 1. Rule: The number is always 1.
- Counting numbers: 1, 2, 3, 4, 5, 6, 7, … Next three: 8, 9, 10. Rule: Start with 1 and add 1 to get the next number.
- Odd numbers: 1, 3, 5, 7, 9, 11, 13, … Next three: 15, 17, 19. Rule: Start with 1 and add 2 to get the next number.
- Even numbers: 2, 4, 6, 8, 10, 12, 14, … Next three: 16, 18, 20. Rule: Start with 2 and add 2 to get the next number.
- Triangular numbers: 1, 3, 6, 10, 15, 21, 28, … Next three: 36, 45, 55. Rule: Start with 1, then add 2, then add 3, then add 4, and so on. (e.g., 28 + 8 = 36).
- Squares: 1, 4, 9, 16, 25, 36, 49, … Next three: 64, 81, 100. Rule: Multiply the term’s position by itself (1×1, 2×2, 3×3, …, 8×8=64).
- Cubes: 1, 8, 27, 64, 125, 216, … Next three: 343, 512, 729. Rule: Cube the term’s position (1x1x1, 2x2x2, …, 7x7x7=343).
- Virahāṅka numbers (Fibonacci): 1, 2, 3, 5, 8, 13, 21, … Next three: 34, 55, 89. Rule: Add the two previous numbers to get the next one (e.g., 13 + 21 = 34).
- Powers of 2: 1, 2, 4, 8, 16, 32, 64, … Next three: 128, 256, 512. Rule: Start with 1 and multiply by 2 to get the next number.
- Powers of 3: 1, 3, 9, 27, 81, 243, 729, … Next three: 2187, 6561, 19683. Rule: Start with 1 and multiply by 3 to get the next number.
1.3 Visualising Number Sequences
Many number sequences can be visualised using pictures. Visualising mathematical objects through pictures or diagrams can be a very fruitful way to understand mathematical patterns and concepts.
Figure it Out (Page 5)
2. Why are 1, 3, 6, 10, 15, … called triangular numbers? Why are 1, 4, 9, 16, 25, … called square numbers? Why are 1, 8, 27, 64, 125, … called cubes?
They are named after the geometric shapes they can form when represented by dots or objects, as shown on page 4.
- Triangular numbers: The dots can be arranged to form an equilateral triangle.
- Square numbers: The dots can be arranged to form a square.
- Cube numbers: The objects can be arranged to form a cube.
3. You will have noticed that 36 is both a triangular number and a square number! Make pictures in your notebook illustrating this.
A number that is both triangular and square is known as a square triangular number. 36 is the 8th triangular number (1+2+3+4+5+6+7+8 = 36) and the 6th square number (6×6 = 36). As instructed in the book, you can illustrate this by drawing a 6×6 grid of dots and a separate triangle of dots with 8 rows.
4. What would you call the following sequence of numbers? 1, 7, 19, 37, … What is the next number in the sequence?
These are called centered hexagonal numbers because the dots can be arranged in the shape of a hexagon with a dot in the center.
The pattern is formed by adding multiples of 6:
1
1 + 6 = 7
7 + 12 = 19
19 + 18 = 37
The next number in the sequence is found by adding the next multiple of 6 (which is 24):
37 + 24 = 61.
1.4 Relations among Number Sequences
Sometimes, number sequences can be related to each other in surprising ways.
Example: What happens when we start adding up odd numbers?
1 = 1 (= 1x1)
1 + 3 = 4 (= 2x2)
1 + 3 + 5 = 9 (= 3x3)
1 + 3 + 5 + 7 = 16 (= 4x4)
1 + 3 + 5 + 7 + 9 = 25 (= 5x5)
This beautiful pattern shows that the sum of the first ‘n’ odd numbers is equal to n².
A picture can explain it (Page 7)
We can see why this works by partitioning a square grid of dots into L-shaped groups called gnomons. Each L-shape represents the next odd number.
The picture for a 6×6 square makes it evident that 1 + 3 + 5 + 7 + 9 + 11 = 36. Because a similar picture can be made for a square of any size, this explains why adding up odd numbers always gives square numbers.
Figure it Out (Page 7)
By drawing a similar picture, can you say what is the sum of the first 10 odd numbers?
Based on the pattern, the sum of the first 10 odd numbers is 10 x 10 = 100.
Now by imagining a similar picture… can you say what is the sum of the first 100 odd numbers?
The sum of the first 100 odd numbers is 100 x 100 = 10000.
Another example: Adding up and down (Page 7)
Look at this pattern:
1 = 1 (= 1²)
1 + 2 + 1 = 4 (= 2²)
1 + 2 + 3 + 2 + 1 = 9 (= 3²)
1 + 2 + 3 + 4 + 3 + 2 + 1 = 16 (= 4²)
1 + 2 + 3 + 4 + 5 + 4 + 3 + 2 + 1 = 25 (= 5²)
This gives another way of getting square numbers: by adding counting numbers up to a number ‘n’ and then back down to 1, the result is n².
Figure it Out (Pages 8-9)
5. What happens when you add up pairs of consecutive triangular numbers? (e.g., 1+3, 3+6, 6+10, …) Which sequence do you get? Why? Can you explain it with a picture?
When you add consecutive triangular numbers, you get square numbers.
- 1 + 3 = 4 (= 2²)
- 3 + 6 = 9 (= 3²)
- 6 + 10 = 16 (= 4²)
- 10 + 15 = 25 (= 5²)
Pictorial Explanation: You can explain this with a picture by drawing two consecutive triangles and fitting them together. For example, a triangle with 3 dots (the 2nd triangular number) and a triangle with 6 dots (the 3rd triangular number) can be interlocked to form a 3×3 square of 9 dots.
8. What happens when you start to add up hexagonal numbers (1, 1+7, 1+7+19, …)? Which sequence do you get? Can you explain it using a picture of a cube?
When you add consecutive (centered) hexagonal numbers, you get the sequence of cube numbers.
- 1 = 1 (= 1³)
- 1 + 7 = 8 (= 2³)
- 1 + 7 + 19 = 27 (= 3³)
- 1 + 7 + 19 + 37 = 64 (= 4³)
Pictorial Explanation: This can be explained using a picture of a cube, as prompted on Page 9. Imagine a 2x2x2 cube made of 8 smaller cubes. It has 1 cube at its core and is surrounded by a ‘shell’ of 7 cubes. The next cube, 3x3x3=27, is formed by adding a new shell of 19 cubes around the 2x2x2 cube. The number of cubes in each shell corresponds to a hexagonal number.
1.5 Patterns in Shapes
Other important and basic patterns that occur in Mathematics are patterns of shapes. The branch of Mathematics that studies patterns in shapes is called geometry.
Shape sequences are an important type of shape pattern. Table 3 shows a few key shape sequences.
1.6 Relation to Number Sequences
Often, a shape sequence is related to a number sequence in a surprising way. Such a relationship can be helpful in studying and understanding both the shape sequence and the related number sequence.
Example: The number of sides in the sequence of Regular Polygons is given by the counting numbers starting at 3 (3, 4, 5, 6, …). That is why these shapes are called, respectively, regular triangle, quadrilateral, pentagon, hexagon, and so on.
Figure it Out (Pages 11-12)
2. Count the number of lines in each shape in the sequence of Complete Graphs. Which number sequence do you get?
The number of lines in the complete graphs K2, K3, K4, K5, K6 (from Page 10) are:
- K2: 1
- K3: 3
- K4: 6
- K5: 10
- K6: 15
This is the sequence of triangular numbers: 1, 3, 6, 10, 15, …
4. How many little triangles are there in each shape of the sequence of Stacked Triangles? Which number sequence does this give?
Counting the small unit triangles in the “Stacked Triangles” sequence (from Page 10):
- Shape 1 (1 row): 1
- Shape 2 (2 rows): 4
- Shape 3 (3 rows): 9
- Shape 4 (4 rows): 16
This gives the sequence of square numbers: 1, 4, 9, 16, … This is because the number of triangles in each row follows the pattern of odd numbers (1, 3, 5, 7,…), and we already know that the sum of consecutive odd numbers results in a square number.
5. How many total line segments are there in each shape of the Koch Snowflake? What is the corresponding number sequence?
The Koch Snowflake (from Page 10) starts with an equilateral triangle (3 segments). In each step, every line segment is replaced by 4 smaller segments. So, the total number of segments gets multiplied by 4 each time.
- Shape 1: 3
- Shape 2: 3 x 4 = 12
- Shape 3: 12 x 4 = 48
- Shape 4: 48 x 4 = 192
The sequence is 3, 12, 48, 192, …, which can be described as 3 times the powers of 4 (3×4⁰, 3×4¹, 3×4², 3×4³, …).
Summary (Page 12)
- Mathematics can be seen as the search for patterns and the explanations for why they exist.
- Number sequences are one of the most basic patterns, including counting, odd, even, triangular, square, and cube numbers.
- Number sequences can be related in remarkable ways, such as the sum of odd numbers resulting in square numbers.
- Visualizing sequences with pictures is a powerful tool for understanding them.
- Shape sequences, like regular polygons and fractals like the Koch snowflake, are another fundamental type of pattern in mathematics.
- Shape sequences often have interesting relationships with number sequences (e.g., the number of sides in polygons or lines in complete graphs).