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Patterns in Mathematics — Class 6 Mathematics

Ganita Prakash · Chapter 1Rising

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Chapter summary

This opening chapter presents mathematics as the search for patterns and the explanations of why those patterns exist. It introduces number sequences as the most basic patterns of whole numbers studied in number theory. Key sequences include counting numbers, odd numbers, even numbers, triangular numbers , square numbers , cubes , Virah\u0101nka (Fibonacci-like) numbers, and powers of 2 and 3. Many sequences can be visualised with dot pictures, and the chapter shows beautiful relations: the sum of the first odd numbers equals (so ), adding counting numbers up and down also gives square numbers, and adding consecutive triangular numbers gives squares. It then turns to patterns of shapes \u2014 regular polygons, complete graphs, stacked triangles and squares, and the Koch snowflake \u2014 studied in geometry, and links these shape sequences back to number sequences (sides of regular polygons give counting numbers from 3, lines in complete graphs give triangular numbers).

What is Mathematics?Patterns in NumbersVisualising Number SequencesRelations among Number SequencesPatterns in ShapesRelation to Number Sequences

Key ideas & terms

Number sequence: An ordered list of numbers formed by a fixed rule, such as odd numbers or square numbers.
Triangular numbers: The sequence formed by dots that can be arranged in a triangle.
Square numbers: The sequence ; the number of dots arranged in a square.
Cubes: The sequence , i.e. dots arranged in a cube.
Regular polygon: A closed shape with all sides equal and all angles equal, such as an equilateral triangle or square.
Koch snowflake: A shape sequence with line segments (3 times powers of 4), formed by repeatedly replacing each segment with a bump.

Extra questions & answers

What is mathematics described as in this chapter?

It is the search for patterns and for the explanations of why those patterns exist; mathematicians view it as both an art and a science.

What branch of mathematics studies patterns in whole numbers?

Number theory studies patterns in whole numbers .

Why are called triangular numbers?

Because that many dots can be arranged exactly in the shape of a triangle.

What is the sum of the first 10 odd numbers?

It is , because adding the first odd numbers always gives .

What special number is both a triangular number and a square number?

36, since 36 dots can be arranged both in a triangle and in a square.

Which sequence do you get by adding counting numbers up and down ()?

The square numbers .

What do you get when you add consecutive triangular numbers, e.g. ?

The square numbers .

What number sequence is given by the number of sides of regular polygons?

The counting numbers starting at 3: (triangle, quadrilateral, pentagon, \dots).

Which number sequence comes from the number of lines in complete graphs ?

The triangular numbers .

What is the corresponding number sequence for line segments in the Koch snowflake?

, which is 3 times the powers of 4.

Figure it Out

1. Can you think of other examples where mathematics helps us in our everyday lives?

Yes. We use mathematics when we buy fruits and vegetables and calculate the total cost and change, when we measure ingredients for cooking, when we check the time on a clock, when we measure the length, breadth and area of our house or classroom, when we calculate distance, speed and time while travelling, and when we make patterns and designs in buildings, clothes and rangoli.

2. How has mathematics helped propel humanity forward? (You might think of examples involving: carrying out scientific experiments; running our economy and democracy; building bridges, houses or other complex structures; making TVs, mobile phones, computers, bicycles, trains, cars, planes, calendars, clocks, etc.)

Mathematics helps us find the reason and rule behind a pattern and then use that rule in many other places. For example, studying the pattern of motion of stars and planets gave us the theory of gravitation which lets us launch satellites and send rockets to the Moon and Mars. Studying patterns in numbers helps us count votes in democracy, manage money in banks and markets, design strong bridges, houses and machines, and make calendars, clocks, computers, mobile phones, TVs, bicycles, cars and planes. Understanding patterns in our bodies helps doctors diagnose and cure diseases.

Figure it Out

1. Can you recognise the pattern in each of the sequences in Table 1?

Yes, each sequence has a clear rule: All 's is always ; Counting numbers increase by each time; Odd numbers increase by starting from ; Even numbers increase by starting from ; Triangular numbers are where we add each time; Squares are ; Cubes are ; Virahānka numbers are formed by adding the previous two numbers; Powers of are doubling each time and Powers of are tripling each time.

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.

The next three numbers are: All 's: - rule: every term is . Counting numbers: - rule: start at and add each time. Odd numbers: - rule: start at and add each time. Even numbers: - rule: start at and add each time. Triangular numbers: - rule: add then then then and so on. Squares: - rule: for . Cubes: - rule: for . Virahānka numbers: - rule: each term is sum of previous two terms, . Powers of : - rule: multiply by each time, . Powers of : - rule: multiply by each time, .

Figure it Out

1. Copy the pictorial representations of the number sequences in Table 2 in your notebook, and draw the next picture for each sequence!

The next pictures are: All 's: one more single green dot labelled ; Counting numbers: six red dots in a row labelled ; Odd numbers: eleven blue dots labelled ; Even numbers: twelve green dots labelled ; Triangular numbers: a triangle with dots in the bottom row totalling dots; Squares: a grid of blue dots totalling dots; Cubes: a cube totalling small cubes.

2. Why are 1, 3, 6, 10, 15, ... called triangular numbers? Why are 1, 4, 9, 16, 25, ... called square numbers or squares? Why are 1, 8, 27, 64, 125, ... called cubes?

Because their dot pictures form those shapes. can be arranged as triangles with dots in each row, so they are triangular numbers. can be arranged as square grids so they are square numbers, . can be arranged as cubes so they are cube numbers, .

3. You will have noticed that 36 is both a triangular number and a square number! That is, 36 dots can be arranged perfectly both in a triangle and in a square. Make pictures in your notebook illustrating this! This shows that the same number can be represented differently, and play different roles, depending on the context. Try representing some other numbers pictorially in different ways!

so dots make a perfect square. Also so the same dots can be arranged in a triangle with dots in the bottom row. This shows a number can be both square and triangular. Other examples: is both triangular and square and cube; is square but can also be shown as ; is both square and cube . You can draw these in your notebook.

4. What would you call the following sequence of numbers? That's right, they are called hexagonal numbers! Draw these in your notebook. What is the next number in the sequence?

The pictures show arranged as hexagons. They are called hexagonal numbers. The pattern is to add each time. The next hexagonal number is . So the sequence is .

5. Can you think of pictorial ways to visualise the sequence of Powers of 2? Powers of 3?

Yes. Powers of : can be shown by doubling: a point , a line segment with points, a square with corners, a cube with corners, a -dimensional cube with corners, and so on. Powers of : can be shown as point, points on a line, points in a square, small cubes in a big cube, and each next step multiplies by .

Figure it Out

1. Can you find a similar pictorial explanation for why adding counting numbers up and down, i.e., 1, 1 + 2 + 1, 1 + 2 + 3 + 2 + 1, ..., gives square numbers?

Yes. Arrange dots in a square and count diagonally. For example, a square has dots. The diagonals contain dots, so . Similarly a square splits into giving . In general the square splits into .

2. By imagining a large version of your picture, or drawing it partially, as needed, can you see what will be the value of 1 + 2 + 3 + ... + 99 + 100 + 99 + ... + 3 + 2 + 1?

It is the square. So

3. Which sequence do you get when you start to add the All 1's sequence up? What sequence do you get when you add the All 1's sequence up and down?

Adding All 's up: , , , gives the counting numbers . Adding All 's up and down: , , , gives the odd numbers .

4. Which sequence do you get when you start to add the counting numbers up? Can you give a smaller pictorial explanation?

Adding counting numbers up: , , , , gives the triangular numbers . Pictorially, stacking rows of dots forms a triangle.

5. What happens when you add up pairs of consecutive triangular numbers? That is, take 1 + 3, 3 + 6, 6 + 10, 10 + 15, ... Which sequence do you get? Why? Can you explain it with a picture?

We get square numbers: Pictorially, two consecutive triangles fit together to make a square. For example a triangle with dots and the next triangle with dots together fill a square.

6. What happens when you start to add up powers of 2 starting with 1, i.e., take 1, 1 + 2, 1 + 2 + 4, 1 + 2 + 4 + 8, ... ? Now add 1 to each of these numbers—what numbers do you get? Why does this happen?

We get , , , , which is one less than a power of . Adding gives which are the Powers of : . This is because so adding makes . The picture on page shows doubling filling all but one corner.

7. What happens when you multiply the triangular numbers by 6 and add 1? Which sequence do you get? Can you explain it with a picture?

Take triangular numbers Then This is the hexagonal numbers (with as the first). Six copies of a triangle around a central dot make a hexagon, so forms a hexagonal dot pattern.

8. What happens when you start to add up hexagonal numbers, i.e., take 1, 1 + 7, 1 + 7 + 19, 1 + 7 + 19 + 37, ... ? Which sequence do you get? Can you explain it using a picture of a cube?

We get cube numbers: Hexagonal layers are like layers of a cube seen from a corner. Adding successive hexagonal shells builds a cube: the first hexagonal number is the central small cube, adding small cubes around it makes a cube, adding makes a cube, and so on.

9. Find your own patterns or relations in and among the sequences in Table 1. Can you explain why they happen with a picture or otherwise?

One example: an even number plus an odd number is odd, and sum of two odd numbers is even. Another example: difference between consecutive squares is an odd number, which is the -th odd number. You can see this in the L-shaped layer added around a square. You can try to find more: for instance Powers of are all even except , and the sum of first counting numbers is the -th triangular number.

Figure it Out

1. Can you recognise the pattern in each of the sequences in Table 3?

Yes. Regular Polygons: number of sides increases by each shape: . Complete Graphs : each new point is joined to all previous points. Stacked Squares: squares of size . Stacked Triangles: triangles built from small triangles with small units. Koch Snowflake: each straight segment is replaced by a bump, multiplying segments by .

2. Try and redraw each sequence in Table 3 in your notebook. Can you draw the next shape in each sequence? Why or why not? After each sequence, describe in your own words what is the rule or pattern for forming the shapes in the sequence.

Next shapes: Regular Polygons: next is a -sided regular hendecagon, then -sided dodecagon, and so on, you can draw them. Complete Graphs: next is with points all joined. Stacked Squares: next is small squares. Stacked Triangles: next is a triangle of side with small triangles. Koch Snowflake: next has each of the segments replaced by smaller segments, so it has segments and looks more crinkly. All can be drawn, but Koch Snowflake quickly becomes very detailed and tiny, so you may draw it approximately. The rule for each is as described above.

Figure it Out

1. Count the number of sides in each shape in the sequence of Regular Polygons. Which number sequence do you get? What about the number of corners in each shape in the sequence of Regular Polygons? Do you get the same number sequence? Can you explain why this happens?

Number of sides: triangle , quadrilateral , pentagon , hexagon , heptagon , octagon , nonagon , decagon , so we get which is counting numbers starting from . Number of corners (vertices) is also . We get the same sequence because every closed polygon has the same number of sides as vertices; each corner is where two sides meet.

2. Count the number of lines in each shape in the sequence of Complete Graphs. Which number sequence do you get? Can you explain why?

has line, has lines, has lines, has lines, has lines. So we get which is the triangular numbers. Reason: the number of connections among points is which is triangular.

3. How many little squares are there in each shape of the sequence of Stacked Squares? Which number sequence does this give? Can you explain why?

The shapes contain little squares. This is the square numbers . Because the -th shape is an square, it contains unit squares.

4. How many little triangles are there in each shape of the sequence of Stacked Triangles? Which number sequence does this give? Can you explain why? (Hint: In each shape in the sequence, how many triangles are there in each row?)

The counts are which are square numbers. The -th large triangle has rows with or counting up and down, and the total is , , etc. In terms of small triangles, row has small unit triangles, and adding them or counting up and down gives .

5. To get from one shape to the next shape in the Koch Snowflake sequence, one replaces each line segment '—' by a 'speed bump' . As one does this more and more times, the changes become tinier and tinier with very very small line segments. How many total line segments are there in each shape of the Koch Snowflake? What is the corresponding number sequence? (The answer is 3, 12, 48, ..., i.e., 3 times Powers of 4; this sequence is not shown in Table 1.)

First shape (triangle) has segments. Each segment becomes segments in the next step, so the counts are This is for , i.e. times the Powers of sequence .