Class 6 Mathematics Chapter 1: Patterns in Mathematics

The Staircase of Bricks

Meghna lives in Nagpur. A mason is building steps at the back of her house. She watches him lay one brick for the lowest step, two for the next, three for the one above that. She counts four on the following step. Then she says the fifth will take five bricks, before he has laid a single one. She is right. The mason laughs and asks how she knew.

What Meghna Actually Did

Meghna did something ordinary and something powerful. She looked at a few cases. She found the rule sitting behind them. Then she used that rule to say what came next, before it existed. That is the shape of nearly all mathematical work. It begins with looking, not with a formula.

Two Jobs, Not One

Mathematics has two jobs. The first is to spot a pattern. The second is to explain why that pattern must hold. Most people stop once the first job is done. Mathematicians do not. A pattern you can only point at is a guess. A pattern you can explain is something you can build on.

When a Pattern Lets You Down

Suppose the mason had changed his plan at the sixth step. Meghna's rule would have collapsed. Nothing was protecting it, because she had never asked why the steps grew that way. Once she knows the reason, she is safe. One extra brick each time keeps every step the same height.

Patterns All Around

Look for patterns and you find them quickly. The moon changes shape in a fixed order. Rows of a rangoli repeat. Bus fares climb by the same amount at each stop. Cooking, shopping, cricket scores and tiled floors all carry patterns. Mathematics is the habit of noticing them on purpose.

An Art and a Science

Mathematicians call their subject both an art and a science. It is a science because every claim has to be tested and justified. It is an art because choosing what to look at takes imagination. Two people can explain the same pattern in two different ways, and both can be beautiful.

A Reason Can Travel

Here is why the second job matters so much. A reason is not stuck to the place where it was found. It can be picked up and carried somewhere else entirely. The pattern was noticed in one corner of the world. The explanation then works in corners nobody had looked at yet.

From the Night Sky to Rockets

People watched the sun, the moon, the planets and their satellites for centuries. Out of those patterns grew the theory of gravitation. That single explanation now lifts our own satellites into orbit. It steers spacecraft towards the Moon and towards Mars. Nobody staring at the sky long ago planned any of that.

Patterns Inside Us

Medicine tells a similar story. Scientists studied patterns in genomes, the coded instructions carried inside living cells. Reading those patterns has helped doctors recognise diseases and treat them. A pattern found in a quiet laboratory ends up helping a patient in a hospital ward.

The Plainest Numbers of All

The most basic patterns of mathematics live among the whole numbers: 0, 1, 2, 3, 4, and on for ever. Nothing could look emptier than that list. Yet these numbers hide surprises that took great mathematicians years to explain. You will meet several of those surprises in this very chapter.

Number Theory

The part of mathematics that hunts for patterns among whole numbers carries a name. It is called number theory. A number theorist may spend months on a question a child could ask in a minute. That is normal here. Easy questions about whole numbers are often the hardest ones to settle.

Ten Sequences Worth Knowing

A number sequence is simply a list with a rule behind it. Table 1 of the chapter sets out ten famous ones. All 1s. Counting numbers. Odd numbers. Even numbers. Triangular numbers. Squares. Cubes. Virahānka numbers. Powers of 2. Powers of 3. Every one of them comes back later in the year.

Reading a Rule off a List

Take the odd numbers: 1, 3, 5, 7, 9, 11, 13. Each term is two more than the one before it. Now take the squares: 1, 4, 9, 16, 25, 36, 49. Here each term is a counting number multiplied by itself. Say the rule aloud in your own words and the list will stay with you.

The Virahānka Numbers

One sequence on the list grows in an unusual way: 1, 2, 3, 5, 8, 13, 21. Add any two terms that sit side by side and you land on the next one. So 5 and 8 give 13. These numbers are named for Virahānka, an Indian scholar of long ago.

Doubling and Tripling

Two sequences on the list are built by multiplying again and again. Powers of 2 double each time: 1, 2, 4, 8, 16, 32, 64. Powers of 3 triple each time: 1, 3, 9, 27, 81, 243, 729. Doubling looks gentle at the start. It turns enormous very quickly.

Scenario:

A class looks at the floor tiles in the school corridor and tries to predict what comes next. Rohit counts the tiles in one band and gets six. He predicts six in the next band and is wrong, because it holds eight. Their teacher asks him to count three bands before predicting. He finds four, six and eight. Now the rule is visible and the next band is ten. The class checks it against the corridor and the prediction holds all the way down. Sana writes what changed. One observation cannot show a pattern, because a pattern is a relation between several, not a property of any one.

Analysis:

Rohit guessed from a single band. Three bands give a difference, and a difference is what can be extended.

Activity:

Find a repeating pattern of tiles or bricks near you. Count three parts of it, state the rule, and check your prediction.

Title:

A pattern hunt at home

What You Notice:

Most people fill the list of patterns easily and then stall on the reasons. That gap is exactly the gap between the first job of mathematics and the second.

Powers of 3 climb to 729 in seven terms, while the counting numbers have only reached 7.

The whole numbers open with 0, so the very first whole number is the one that counts nothing at all.

In this book the Virahānka numbers run 1, 2, 3, 5, 8, 13, 21, and each term after the first two is a sum of the two before it.

The two jobs of mathematics — SPOT, then SHOW — SPOT the pattern. SHOW why it must hold. Stop after spotting and you are only guessing.

Virahānka numbers — Two behind, one ahead — Add the two terms behind you to reach the one ahead. So 2 and 3 give 5, and 3 and 5 give 8.

Dots on the Doorstep

Tara is drawing a rangoli outside her door in Madurai. She starts with one dot of rice flour. Under it she sets a row of two, then a row of three, then a row of four. Her brother counts the dots at each stage and calls out the totals: 1, then 3, then 6, then 10. He wants to know the next total before she draws it. Tara says he can see the answer, not just calculate it.

A Picture Carries the Rule

Here is why her drawing helps. A rule written in words has to be remembered. A picture shows what the rule is doing. When you can see the next row being added, you can see why the total grows the way it does. Many number sequences can be drawn like this, and the drawing usually explains more than the list does.

Triangular Numbers

Place one dot. Add a row of two below it, then a row of three, then a row of four. Count after each row and you get 1, 3, 6, 10, 15, 21, 28. The picture is always a triangle, and that is where the name comes from. Each new row is one dot longer than the row above it.

Square Numbers

Now arrange the dots in a square grid instead. One dot, then two rows of two, then three rows of three. The counts run 1, 4, 9, 16, 25, 36, 49. Each term is a counting number multiplied by itself, since 5 x 5 = 25 and 7 x 7 = 49. Seeing the grid makes the word square obvious.

Cubes

Stack the dots into a solid box, equally wide, deep and tall. A box two dots each way holds 8. A box three dots each way holds 27, because 3 x 3 x 3 = 27. Carry on and the counts run 1, 8, 27, 64, 125, 216. These are the cubes, named for the solid shape that holds them.

One Number, Two Shapes

Look carefully at 36. It sits among the squares, since a six by six grid holds 36 dots. It also sits among the triangular numbers, since a triangle with eight rows holds 36 dots. The same number can therefore play two different parts. Which part it plays depends on the question you are asking.

Hexagonal Numbers

Try ringing a single dot with hexagons instead. The counts come out as 1, 7, 19, 37, and the next one is 61. There is a neat way to reach them. Multiply a triangular number by 6 and add the middle dot: 6 sixes is 36, and one more makes 37. Add these counts up and the cubes appear: 1, then 8, then 27, then 64.

Adding the Odd Numbers

Now put the pictures aside and add. One is 1. One and three make 4. Adding five gives 9, adding seven gives 16, adding nine gives 25, adding eleven gives 36. Every total is a square number. That is startling enough to be worth doubting. Does it really carry on for ever, or will it break somewhere further along?

The Grid Gives the Reason

Draw a six by six grid of dots. Ring the single dot in one corner. Around it, ring the bent strip of three that completes a two by two square. Around that, the bent strip of five. Then seven, then nine, then eleven. Every dot is used once and none is used twice. So the grid shows that 1 + 3 + 5 + 7 + 9 + 11 = 36.

Why It Cannot Stop

The important part is not the picture itself. It is that the same picture can be drawn for a square of any size at all. Each new bent strip is exactly two dots longer than the one inside it, which is what keeps the strips odd. So the pattern has no last case. It is explained rather than merely observed.

Ten Odd Numbers, and a Hundred

Once you trust the picture, the adding stops. The first ten odd numbers fill a ten by ten grid, and 10 x 10 = 100. The first hundred odd numbers fill a hundred by hundred grid. Multiply a hundred by itself and the answer is 10,000. No child should ever add that list by hand.

Climbing Up and Down

Here is a second route to the same squares. Climb the counting numbers to a chosen term and then walk back down. One is 1. Then 1 + 2 + 1 = 4. Then 1 + 2 + 3 + 2 + 1 = 9. Then 1 + 2 + 3 + 4 + 3 + 2 + 1 = 16. Squares again. Climbing to a hundred and back down gives 10,000.

The Plainest Sequence of All

Take the sequence of all 1s and add it up as you go. The totals are 1, 2, 3, 4, 5, which are the counting numbers. Now add that same sequence up and then down. The totals become 1, 3, 5, 7, 9, which are the odd numbers. The dullest list in Table 1 produces two of the others.

Counting Numbers Added Up

Add the counting numbers as you go and the totals run 1, 3, 6, 10, 15. Those are the triangular numbers, and Tara's rangoli explains why. Each fresh counting number is simply the new row of dots laid under the triangle. The sum after each step is the whole triangle so far.

Triangles in Pairs, and Powers of 2

Two last surprises. Add triangular numbers that sit side by side: 1 + 3 = 4, 3 + 6 = 9, 6 + 10 = 16, 10 + 15 = 25. Squares once more, because two triangles slot together into a square grid. Now add powers of 2 as you go, giving 1, 3, 7, 15, 31. Add one to each total and the powers of 2 return.

Scenario:

A class builds the counting numbers as rows of dots: one dot, then two, then three. Meera adds the rows up as she goes and writes 1, 3, 6, 10. She says the totals look random. Their teacher asks the class to arrange each total as a shape rather than a list. The children find that each one makes a triangle of dots. Meera lays the fourth triangle beside a copy of itself and gets a neat rectangle, four by five. She sees why the totals grow the way they do. The picture explains the sequence, and the list alone had hidden the reason.

Analysis:

The same numbers as a list look arbitrary and as triangles look inevitable. Two copies forming a rectangle is what explains the growth.

Activity:

Draw the first five triangular numbers as dots. Put two copies of one together and write the rectangle you get.

Title:

Build the square from bent strips

What You Notice:

The five colours use 1, 3, 5, 7 and 9 squares, and the grid holds 25 in all. Draw the same thing on a six by six grid and a sixth strip of eleven appears.

The first hundred odd numbers add to 10,000, and nobody needs to do the addition to know it.

Adding hexagonal numbers gives the cubes: 1, then 8, then 27, then 64.

Adding the sequence of all 1s up and down gives the odd numbers, while adding it only upwards gives the counting numbers.

Odd numbers add to squares — Bent corners fill a square — Each odd number is one bent corner of dots. Lay the corners inside each other and a square grid is filled.

Hexagonal numbers — Six times a triangle, plus the middle — Take a triangular number, six of them ring the centre, and the single middle dot is the plus one.

The Tiles Outside the Temple

Joseph is waiting for his family outside a temple in Thrissur. The paving in front of him is laid in bands. The first band is set with three-sided tiles, the next with four-sided ones, then five, then six. He starts counting corners instead of sides and gets the very same numbers. A pattern of shapes has quietly handed him a pattern of numbers.

Geometry

Shapes carry patterns as faithfully as numbers do. The branch of mathematics that studies them is called geometry. Its shapes may be flat, like a tile, or solid, like a brick. Mathematicians also work with shapes in more dimensions than three, which nobody can draw but everybody can reason about.

What a Shape Sequence Is

A shape sequence is a run of shapes built by a rule. You can look at the third shape and say what the fourth must be. Table 3 of the chapter sets out five of them: regular polygons, complete graphs, stacked squares, stacked triangles and the Koch snowflake. Each one is worth meeting on its own.

Regular Polygons

The first sequence simply adds a side each time. Its members have 3, 4, 5, 6, 7, 8, 9 and 10 sides. Their names in that order are triangle, quadrilateral, pentagon, hexagon, heptagon, octagon, nonagon and decagon. A quadrilateral with four equal sides and equal angles is the square you already know.

What Regular Means

The word regular is doing real work in that name. It tells you two things at once. Every side is the same length as every other side. Every angle is equal to every other angle. So the sides all look alike and the corners all look alike. Drop either condition and the shape is a polygon but not a regular one.

Sides and Corners Agree

Count the sides of each polygon and you get 3, 4, 5, 6, 7, 8, 9, 10. Now count the corners instead. The answer is the same list. That is not a coincidence. Walk around the outline and every side you cross ends at exactly one corner, so the two counts must march together.

Complete Graphs

The second sequence starts with dots rather than sides. Mark some dots, then join every pair of them with a line. Leave no pair out. These are the complete graphs, named K2, K3, K4, K5 and K6 by how many dots they carry. The drawings quickly turn into a web of crossing lines.

Counting the Lines

Now count the lines in each one. K2 has a single line. K3 has 3 lines, K4 has 6, K5 has 10 and K6 has 15. That list should look familiar, because 1, 3, 6, 10, 15 are the triangular numbers. Each new dot has to be joined to every dot already there, which is why a fresh row appears each time.

Stacked Squares

The third sequence is a square built from smaller squares. The first is one square on its own. The second is two rows of two, holding 4. The third is three rows of three, holding 9. The fourth holds 16. So the counts run 1, 4, 9, 16, and these are the square numbers.

Stacked Triangles

Now stack little triangles into a big triangle instead. The counts come out as 1, 4, 9, 16 all over again, which is startling until you look at the rows. The top row holds 1 little triangle, the next holds 3, then 5, then 7. Those are the odd numbers, and odd numbers added from 1 give squares.

Two Shapes, One Sequence

So the two stacking sequences look nothing alike and count the same. That is worth pausing over. A number sequence does not belong to one picture. It can arrive from several directions, and each route explains something the others do not.

The Koch Snowflake

The last sequence is the strangest. Begin with a triangle, which has 3 line segments. Now take every segment and replace it with a speed bump made of four shorter segments. Do it again to every new segment, and again. The outline turns from a plain triangle into a crinkled star, and then into something like frost on a window.

Counting Snowflake Segments

Count the segments at each step. The triangle has 3. After one round of bumps each segment has become four, so there are 12. After the next round there are 48. The counts 3, 12, 48 are three times the powers of 4. This sequence does not appear anywhere in Table 1.

Shapes Hand Over Numbers

Every sequence in this topic gave up a number sequence when something was counted. Sides gave counting numbers from 3. Lines in complete graphs gave triangular numbers. Both stackings gave squares. The snowflake gave a sequence of its own. Counting is the bridge between the two kinds of pattern.

Why the Bridge Is Useful

That bridge carries traffic both ways, and this is the point of the whole chapter. A shape can explain a number pattern, as the split grid did for odd numbers. A number pattern can also predict what a shape will do next, before anybody has drawn it. Neither kind of pattern has to be studied alone.

Scenario:

Six schools send one captain each to a district science fair in Belagavi. Before the judging starts, the organiser asks every captain to shake hands once with every other captain. Ishita is keeping the register and decides to count the handshakes. She draws six dots in her notebook, one for each captain, and a line for every handshake made. Ten minutes later the page is a web of crossing lines and she has lost count. Her teacher tells her to rub it out and build the picture one captain at a time instead, writing down the new lines at each stage. Ishita tries it and finds a sequence she met in her maths chapter last week.

Analysis:

A social question turns into a complete graph. Building it dot by dot turns an uncountable tangle into a sequence with a reason behind it.

Activity:

Try it with your own family. Add one person at a time, count the new handshakes each person brings, and check that your running totals are triangular numbers.

Title:

Fold your own snowflake step

What You Notice:

The triangle starts with 3 segments and your first round gives 12. Each segment you treat turns into four, so a full second round would reach 48.

A complete graph on six dots already carries 15 lines, and every extra dot joins to all the earlier ones.

The Koch snowflake keeps its outline inside a small patch of paper, yet its edge gets longer at every single step.

Stacked squares and stacked triangles look nothing alike, and yet both give the counts 1, 4, 9, 16.

Naming regular polygons — Three to ten: tri, quad, pent, hex, hept, oct, non, dec — Say the eight short beginnings in order and the side count comes with them, starting at three.

Koch snowflake segments — One becomes four — Every segment is swapped for four shorter ones, so the count is multiplied by four at each step.