Class 6 Mathematics Chapter 7: Fractions

Mahi cuts the barfi

Mahi has one long slab of coconut barfi and four cousins waiting. She wants every cousin to get the same amount. Her first cut lands off centre, and one piece is clearly fatter than the rest. Her cousin Tejas laughs and says that this is not sharing, it is luck. Mahi measures the slab instead, marks three cuts at equal gaps, and tries again. Now nobody can complain, because every piece matches every other piece.

What a fair cut means

Four pieces only become quarters when they are the same size. The number of pieces is not enough on its own. A slab broken into four ragged lumps still has four pieces, and not one of them is a quarter. So the first question to ask about any picture of a fraction is whether the parts are genuinely equal.

Naming one piece

When a whole is cut into equal parts, one part is called a fractional unit. Cut into two and each part is 1/2. Cut into four and each part is 1/4. Cut into ten and each part is 1/10. These pieces always carry a 1 on top, because you are holding one piece of that size. Another name for them is unit fractions.

Counting the pieces

One piece is rarely the whole story. Mahi keeps 1/4 for herself and puts three quarters on a plate for her cousins. That plate holds 3/4 of the slab, which is three pieces of size 1/4. A fraction is really a count: pick a piece size, then say how many.

Reading 3/5 aloud

The usual reading of 3/5 is three fifths. There is a second reading that helps more while you are learning: three pieces of 1/5. It names the piece size out loud, so you can picture the strip before you do anything else. Try it with 7/8 and with 2/3.

The two numbers have names

The number above the line is the numerator. The number below the line is the denominator. In 5/6 the numerator is 5 and the denominator is 6. The denominator is the naming number and the numerator is the counting number.

More sharers, smaller share

Sunita shares one dosa among 5 friends. Girish shares an identical dosa among 3 friends. Girish's friends get more, because the same dosa is being split fewer ways. This catches many people out, since 5 is the bigger number. Ask who is sharing, not which number looks larger, and 1/5 lands below 1/3 where it belongs.

Sharing more than one thing

Four friends share 3 ribbons of the same length. Cut every ribbon into 4 equal parts and hand one part of each ribbon to each friend. Every friend walks away with three pieces of size 1/4, which is 3/4 of a ribbon. Sharing 3 things among 4 people gives each person 3/4.

Folding a paper strip

Take a paper strip and call its length one unit. Fold it in half and open it out, and the crease has made two equal parts of 1/2 each. Fold it again and you get four parts of 1/4. Fold once more for eight parts of 1/8. Each fold doubles the number of pieces and halves their size.

Building up from one piece

With the eighths strip in front of you, count along it. Two pieces make 2/8. Four pieces make 4/8, which reaches the halfway crease. Eight pieces make 8/8, and that is the whole strip back again. Whenever the count on top matches the name below, you are holding one whole.

Fractions on the number line

A number line can hold fractions as easily as it holds 1, 2 and 3. The distance from 0 to 1 is one unit long. Cut that distance into 5 equal steps and each step measures 1/5. The marks between 0 and 1 are then 1/5, 2/5, 3/5 and 4/5, in that order.

Marking 3/5

To mark 3/5, start at 0 and count three steps of 1/5. Draw a line from 0 to that mark and you have drawn a length of 3/5 of a unit. Nandini marks 4/5 the same way, one step further along, and sees at once that 4/5 is the longer length.

Between any two marks

Suppose the gap from 0 to 1 is cut into 10 steps instead of 5. Now 3/5 has a neighbour at 6/10 sitting on exactly the same spot, and new marks appear at 1/10, 3/10 and 7/10. Cutting into more steps never uses up the room. There is always another fraction to find between two you have already marked.

Old words, new symbols

The symbols are newer than the ideas. Indian languages have named these amounts for a very long time, and many of those words are still in daily use at the vegetable market. Ask at home for the local words for half, quarter and one and a half, and you will collect three fraction names before you reach the door.

Carry this away

Check the parts are equal. Read the lower number as the piece size and the upper number as the count. Remember that more sharers means less each. With those three habits, every fraction you meet has a picture behind it.

Scenario:

A class measures a pencil with a scale marked only in whole centimetres. It ends between 13 and 14. Meena writes 13 and Rohit writes 14. Their teacher asks which is right. Neither, exactly. The class folds a paper strip into halves and then quarters and lays it against the gap. The pencil reaches about three quarters past 13. They write 13 and three quarters. Meena sees why whole numbers were not enough. The thing being measured does not stop at a mark, so the unit has to be cut smaller until it does, and that is what a fraction is for.

Analysis:

Two whole-number answers and both wrong. A fraction appears the moment the thing measured does not end on a mark.

Activity:

Measure three objects with a scale showing only whole centimetres. Use a folded strip to write the leftover as a fraction.

Title:

Fold a fraction strip and mark your own number line

What You Notice:

Each fold doubles the number of pieces and halves their size. The 4/8 crease on the first strip lands exactly on the 1/2 crease, and no mark on the fifths strip lines up with the quarters.

Indian languages had fraction words long before fraction symbols arrived. Hindi calls a quarter paav and three quarters teen paav, and Tamil says mukkaal for the same amount.

The Rig Veda already uses a name for three quarters, tri-pada, which shows how old these ideas are in India.

Shopkeepers still weigh in paav and aadha kilo, so a child who knows quarters and halves can already read a weighing scale.

Which number does which job — Below names the piece, above counts the piece — The number below the line tells you how big one piece is. The number above it tells you how many pieces have been collected.

Sharing among more people — More plates, less on each — The food does not grow when guests arrive. Cut the same sweet for more plates and every plate holds less than before.

Aslam counts his idlis

Aslam is packing idlis into boxes that hold 5 each. He has counted out 17 idlis, one fifth of a box at a time. Three boxes fill up neatly, and two idlis sit on the plate with nowhere to go. He writes the count two ways: 17 pieces of 1/5, which is 17/5, or three full boxes and 2/5 of a fourth. Both lines describe the same plate of idlis.

When a fraction passes one

Up to now the fractions have all stayed below one whole. They do not have to. Collect 7 pieces of 1/4 and you have more than the 4 pieces one whole needs. The clue sits in the two numbers. A top number smaller than the bottom means less than 1, and a top number larger than the bottom means more than 1.

Testing the clue

Look at 5/8, 9/8 and 8/8. In 5/8 the count is below the eight pieces a whole takes, so it is under 1. In 8/8 the count matches exactly, so it is 1. In 9/8 the count runs one piece past, so it is above 1. The test takes a second and never needs a drawing.

Finding the wholes inside

How many wholes hide in 17/5? Every 5 pieces make one whole, so take them away in fives. Five pieces gone leaves 12, another five leaves 7, another five leaves 2. Three wholes came out and 2/5 stayed behind. So 17/5 is three wholes and two fifths.

Writing it as a mixed fraction

That amount is written 3 2/5 and read as three and two fifths. The 3 is the whole part and the 2/5 is the fractional part. The fractional part is always smaller than one whole, because any further whole would have been pulled out already.

One more of the same

Try 23/6. Sixes come out three times, using 18 pieces, and 5 pieces remain. So 23/6 is 3 5/6. Check it the other way round: 3 wholes hold 18 sixths, and 18 pieces plus 5 pieces is 23 pieces.

Going back the other way

A mixed fraction converts back just as easily. Take 2 3/7. Each whole holds 7 pieces of 1/7, so 2 wholes hold 14 of them. Add the 3 loose pieces and you have 17 pieces, giving 17/7. In the same way 4 1/8 becomes 33/8, because 4 wholes hold 32 eighths.

Two strips, one length

Now fold a strip in half and shade one part. Fold a second strip of the same length into quarters and shade two parts. Lay one above the other. The shading stops at exactly the same place. The pieces are different sizes and the counts are different, yet the amount shaded is the same.

Naming what you have seen

Fractions that measure the same amount are called equivalent fractions. From that strip, 1/2 and 2/4 are equivalent. Fold once more for eighths, shade four of them, and 4/8 joins the family. So 1/2, 2/4 and 4/8 all name one and the same length.

A wall of strips

Stack strips cut into halves, thirds, quarters, fifths and sixths, all the same length, and you have a fraction wall. Reading down a column shows which fractions agree. The wall shows at once that 1/3 and 2/6 match, and that 1/3 never matches any number of quarters.

Equal shares say the same thing

Equivalence also turns up in sharing. Share 2 cakes among 3 children and each child gets 2/3 of a cake. Share 4 cakes among 6 children and cut each cake into 6 parts, and each child again collects the same amount. So 2/3 and 4/6 are equivalent, which is why 2/3 is called the simplest form of 4/6.

Making an equivalent fraction to order

You do not need a strip every time. Take 3/5 and multiply the top and the bottom by 2, giving 6/10, since 3 x 2 = 6 and 5 x 2 = 10. Multiply both by 4 instead and 3/5 becomes 12/20. Each new fraction has more pieces, and each piece is smaller, so the amount does not move.

A family with no end

That means every fraction has an endless family of partners. Starting from 2/3 you can write 4/6, then 6/9, then 8/12, and carry on as long as you like. All of them mark the same point on a number line, because you are only ever changing how the whole is cut.

Using equivalence to compare

Mahi asks who gets more: 1 chikki split between 2 children, or 5 chikkis split among 8 children. The pieces are different sizes, so the counts cannot be compared yet. Rewrite 1/2 in eighths and it becomes 4/8. Now both amounts are counted in eighths, and 5/8 beats 4/8, so the group of 8 does better.

Carry this away

Read the two numbers to see whether a fraction has passed 1. Pull out the wholes to write a mixed fraction, and put them back to undo it. To build an equivalent fraction, multiply the top and the bottom by the same number, never one of them alone.

Scenario:

A recipe needs three halves of a cup of flour. A class is asked to measure it with a one-cup measure. Arjun fills the cup three times and gets three cups. Their teacher asks him to read the fraction again. Three halves means three pieces, each half a cup. The class fills half a cup three times and finds it comes to one and a half cups. Arjun sees where he went wrong. He read the three and ignored what it was counting. The class writes both forms side by side and checks they pour the same amount.

Analysis:

Three halves and three cups differ by double. The numerator counts pieces and the denominator says how big a piece is, so neither can be read alone.

Activity:

Measure five quarters of a cup of water two ways: as five quarter-cups, and as one and a quarter cups. Check they match.

Title:

Build a fraction wall and hunt for partners

What You Notice:

The half is matched by 2/4 and by 3/6, and the third is matched by 2/6. Nothing in the quarters row lines up with anything in the thirds row, which shows that not every pair of fractions has a partner.

A fraction wall is just a stack of strips, each cut into a different number of equal parts. Lay it flat and every equivalent pair lines up down a single column.

Shops write mixed amounts every day, such as one and a half kilos of tomatoes, without anyone calling it mathematics.

Any fraction has endlessly many equivalent partners, because you may multiply the top and bottom by 2, by 3, by 4 and onwards without stopping.

Spotting a fraction above one — Top heavy, over one — If the number on top outweighs the number below, you have collected more pieces than a single whole needs.

Turning a mixed fraction back — Wholes into pieces, then add the loose ones — Each whole holds as many pieces as the lower number says. Count those pieces, add the loose ones on top, and keep the same lower number.

Devraj doubts the rule

Devraj has been told to multiply the top and the bottom of a fraction by the same number. He does it, gets 4/6 from 2/3, and asks the question everyone should ask. How can both numbers change while the amount stays the same? His teacher hands him two strips of equal length and lets him find out for himself.

Two strips, one shading

Devraj cuts the first strip into 3 equal parts and shades 2 of them. He cuts the second strip into 6 equal parts and shades 4. Laying one on the other, the two shaded regions finish at the same point. The pieces are different and the counts are different, and the shaded amount is identical.

What actually happened

Look closely at the second strip. Every third has been cut down the middle into two sixths. So each shaded third turned into two shaded sixths, and 2 shaded pieces became 4. The pieces halved in size and doubled in number. Neither change works alone, and together they leave the shading exactly where it was.

The rule, with its reason

This is why the rule says both numbers, always the same one, and never just one of them. Multiplying below cuts the pieces smaller. Multiplying above takes more of them. Multiply both by 3 and you get 6/9 from 2/3, since 2 x 3 = 6 and 3 x 3 = 9. The shading still does not move.

Running it backwards

The rule works in reverse too. If small pieces can be joined into larger ones, divide both numbers instead. In 18/24, groups of 6 twenty-fourths join into single quarters. Divide 18 and 24 by 6 and the fraction becomes 3/4. The bar looks simpler and shades the same length as before.

Simplest form

A fraction is in its simplest form when the only number going into both of its parts is 1. In 3/4 there is nothing left to divide by, so 3/4 is the simplest form of 18/24. It is also called lowest terms, because the two numbers are now as small as they can be.

Finding the factor to use

The quickest route divides by the highest common factor in one step, since HCF(18, 24) = 6. Going in steps is just as safe. Divide 18/24 by 2 to get 9/12, then divide by 3 to get 3/4. Both routes land on the same answer, so use whichever you can see.

A little practice

Take 20/50. Both numbers are divisible by 10, so the fraction becomes 2/5. Take 27/45. Both are divisible by 9, giving 3/5. Take 35/56. Both are divisible by 7, giving 5/8. In each case the amount is untouched and only the description has changed.

Why comparing needs care

Now think about which is larger, 3/5 or 5/8. The counts are 3 and 5, but they count pieces of different sizes, so the comparison means nothing yet. It is like saying five spoons beat three ladles. The pieces have to be made the same size before the counts can be trusted.

Choosing a common piece

Any number that both 5 and 8 divide will work. The product 40 always works, since 5 x 8 = 40, and it is easy to find. Sometimes the smallest common multiple is smaller and tidier. For 5 and 8 there is nothing smaller than 40, so 40 it is.

Comparing 3/5 and 5/8

Rewrite both over 40. For the first, 3 x 8 = 24 and 5 x 8 = 40, giving 24/40. For the second, 5 x 5 = 25 and 8 x 5 = 40, giving 25/40. Both now count fortieths, and 25 pieces beat 24. So 5/8 is the larger fraction, by a single fortieth.

A tidier common piece

Compare 5/6 and 7/9. The product 54 would work, but 18 is a common multiple of 6 and 9 and is smaller. For the first, 5 x 3 = 15 and 6 x 3 = 18, giving 15/18. For the second, 7 x 2 = 14 and 9 x 2 = 18, giving 14/18. Since 15 pieces beat 14, the answer is 5/6.

Putting a list in order

The same method sorts a whole list. Take 2/3, 5/9 and 7/12. All three fit over 36, giving 24/36, 20/36 and 21/36. Reading the counts, the order from smallest upwards is 5/9, then 7/12, then 2/3. Nothing was guessed and nothing was measured.

Two useful shortcuts

Some pairs need no work at all. With the same piece size, as in 4/9 and 7/9, the larger count wins straight away. With the same count, as in 5/6 and 5/11, the larger lower number means thinner pieces, so 5/6 is the bigger amount. Use these when they fit and the long method when they do not.

Carry this away

The rule is not magic. Multiplying both numbers re-cuts the whole, and dividing both joins pieces back together. Reduce to simplest form to tidy up, and match the piece sizes before you trust a comparison.

Scenario:

Two children compare their share of a chapati. Nita has two quarters and Imran has one half. Nita says she has more because two is bigger than one. Their teacher asks the class to cut two paper circles and shade each share. The shaded parts match exactly. Nita sees that she had compared only the top numbers. The class writes down why that fails. Her pieces are smaller as well as more numerous. A fraction is a relation between its two numbers, so a bigger top number can sit on smaller pieces and come to the same amount.

Analysis:

Two beats one and the shares are equal. Comparing numerators alone ignores that the pieces being counted are different sizes.

Activity:

Shade two paper circles to compare three sixths with one half. Write whether they match and why.

Title:

Two bars, one length: prove the rule yourself

Safety:

Ask an adult before using the scissors, and cut away from your fingers.

What You Notice:

Every coloured length measures 8 cm, whichever way the bar was cut. The pieces get thinner as the cuts increase, and the count rises by the same factor, so the coloured part never moves.

Any common multiple of the two lower numbers will do for comparing. The product of the two always works, and the smallest common multiple keeps the numbers tidier.

A fraction that nothing but 1 will divide through cannot be reduced any further, which is why its simplest form is the only one.

Reducing to simplest form is how recipes are written. Nobody asks for 8/16 of a cup when 1/2 says the same thing.

Why nothing changes — Twice as many, half the size — Doubling the lower number halves each piece and doubling the upper number takes twice as many. The two moves undo each other.

Before comparing — Match the pieces, then count them — Comparing counts of different-sized pieces proves nothing. Make the pieces the same size first, and the larger count then wins.

Mahi and the ribbon border

Mahi is sewing a border along a cloth. She has one piece of lace 3/10 of a metre long and another 2/5 of a metre long. To know whether the lace will reach, she has to add the two lengths. The counts are 3 and 2, and adding those alone tells her nothing, because tenths and fifths are different-sized pieces.

The easy case first

Start where the pieces already match. Take 3/11 and 5/11. Both count pieces of 1/11, so three pieces and five pieces make eight pieces. The answer is 8/11. Nothing had to be re-cut, so the piece size is unchanged and 11 stays below the line.

Why the lower number stays

Picture a strip cut into elevenths. Shading three and then shading five more gives eight shaded pieces out of the same eleven. The pieces never changed size during the shading, so the number naming them cannot change either. This is the whole reason the lower number is carried straight down.

The classic mistake

Many people add both numbers and write 3/11 + 5/11 as 8/22. Test that idea on something you already know. Half a roti and half a roti make a whole roti, but adding both numbers would give 2/4, which is only half of it. Adding the lower numbers cannot be right.

When the answer passes a whole

Try 5/9 + 7/9. The counts add to 12, so the answer is 12/9. That is more than one whole, since 9 pieces already make a whole. Written as a mixed fraction it is 1 3/9, and reducing the fractional part gives 1 1/3.

Fractions that do not match

Now back to Mahi's problem, and to sums such as 1/5 + 1/6. Fifths and sixths are different sizes, so the counts cannot be added as they stand. The fix is the equivalence rule from the last topic. Rewrite both fractions so that they count pieces of one shared size.

Choosing the shared piece

Any number that both lower numbers divide will do. The product always works, since 5 x 6 = 30, and sometimes a smaller common multiple is tidier. For fifths and sixths there is nothing below 30, so thirtieths it is.

Doing the sum

Rewrite each fraction over 30. For the first, 1 x 6 = 6 and 5 x 6 = 30, giving 6/30. For the second, 1 x 5 = 5 and 6 x 5 = 30, giving 5/30. Both now count thirtieths, so add the counts to reach 11/30. Nothing divides both 11 and 30, so the answer is already tidy.

A second example

Add 3/8 and 1/6. The smallest number both 8 and 6 divide is 24. For the first, 3 x 3 = 9 and 8 x 3 = 24, giving 9/24. For the second, 1 x 4 = 4 and 6 x 4 = 24, giving 4/24. Adding the counts gives 13/24.

Naming the method

This plan is very old. The book credits Brahmagupta, who worked in 628 CE, with writing down the general rule for the first time. Find a common lower number, add the counts, then simplify if it helps. That is still the method used today.

Taking fractions away

Subtraction runs on the same rails. Matching pieces first, then take one count from the other. For 11/12 - 5/12 the pieces already match, so 11 pieces minus 5 pieces leaves 6/12, which reduces to 1/2.

Subtracting when the pieces differ

Mahi also needs 7/10 of a metre and has used 2/5 of a metre. Tenths and fifths are different, but 5 divides 10, so use tenths. Since 2 x 2 = 4 and 5 x 2 = 10, the amount used is 4/10. Taking it away leaves 7/10 - 4/10, which is 3/10 of a metre.

One that needs more work

Try 5/6 - 3/8. Twenty-fourths suit both. For the first, 5 x 4 = 20 and 6 x 4 = 24, giving 20/24. For the second, 3 x 3 = 9 and 8 x 3 = 24, giving 9/24. Taking 9 pieces from 20 leaves 11/24.

Back to the lace

Mahi's two pieces are 3/10 and 2/5 of a metre. Rewriting the second as 4/10 and adding gives 7/10 of a metre in all. Her border needs a full metre, so she is 3/10 of a metre short and must buy more lace.

Carry this away

One question settles every fraction sum: do both fractions count the same size of piece? If they do, add or subtract the counts and keep the lower number. If they do not, rewrite both over a common lower number first, and then do exactly the same thing.

Scenario:

The Class 6 garden club at a school in Nashik has one long bed for the winter planting. The children agree to give 1/4 of the bed to methi and 1/3 of it to palak, because those two crops need the most watering. The rest will hold marigolds, which keep some insects away. Devraj measures the bed and marks it into twelve equal strips, so that both fractions can be laid out exactly. The club also keeps a watering record. On Monday the tank falls from 9/10 full to 1/4 full, and the children want to know how much water the garden drank. Their teacher asks them to write every answer as a fraction of the whole bed or the whole tank.

Analysis:

One shared piece size turns two awkward fractions into plain counting, and the same idea measures the water used from the tank.

Activity:

Mark a 12-strip bed on paper, colour the methi and palak shares, and check that the marigold strip really is 5/12.

The Class 6 book credits Brahmagupta, working in 628 CE, with the first general written account of how to add and subtract fractions this way.

His plan is the one still taught: bring the fractions to a common lower number, then add or take away the counts.

Carpenters and tailors add fractions all day, in sixteenths of an inch or quarters of a metre, using exactly this matching step.

The first move in any fraction sum — Match the pieces before you count them — Different-sized pieces cannot be counted together. Rewrite both fractions over one lower number, and the sum turns into simple counting.

What the lower number does during a sum — Counts move, piece size stays — Adding fractions collects more pieces of one size. The size itself is untouched, so the lower number is never added.

A word that means broken

When Nandini asks her grandmother for the old word for a fraction, she gets bhinna, a Sanskrit word meaning broken. It is a good name. A fraction really is a whole thing broken into equal pieces, with only some of them in your hand. Two other old names were bhaga and ansha, and each of those means a part or a piece.

Older than the symbols

The names came first. People shared food, cloth and land in equal parts long before anyone worked out a neat way to write the amounts down. That is why the words for half and quarter in Indian languages are far older than the two-number symbol every school now uses.

Numbers one above the other

The chapter traces our way of writing fractions back to India. In old Indian mathematical writing the two numbers were placed one directly above the other. The Bakshali manuscript, from around 300 CE, writes a half exactly like that, with the 1 sitting over the 2 and nothing drawn between them.

Centuries of use

That way of writing did not fade. The chapter names mathematicians who went on using and developing it. They are Aryabhata in 499 CE, Brahmagupta in 628 CE, Sridharacharya around 750 CE and Mahaviracharya around 850 CE. Each of them worked with fractions as a matter of course.

The line arrives

The bar we now draw between the two numbers was a later addition. The chapter credits the Moroccan mathematician Al-Hassar with introducing it in the 12th century. From there the notation spread into Europe and then around the world, which is why a child in Nashik and a child in Lisbon write 3/4 the same way.

Rules, not just symbols

Symbols are only half the story. The chapter says that general fractions, where the top number need not be 1, were introduced in India along with the rules for adding, subtracting, multiplying and dividing them. It adds that the Sulba-sutras show such rules were already in use in Vedic times.

Brahmagupta writes it down

Brahmagupta, in 628 CE, set out those rules formally and in a general form. His instruction for addition is the one you used in the last topic. Bring the fractions to a shared lower number, then add the counts, and for subtraction take one count from the other. The wording is ancient and the method is the one in your notebook.

Fractions elsewhere

Other civilisations had fractions too. The chapter mentions ancient Egypt and Babylon, where writing leaned almost entirely on unit fractions, the ones with 1 on top. A scribe there could write 1/2 or 1/7 easily enough, and had to work harder for anything else.

Building with unit pieces

Their way round it was to add different unit fractions together. So 3/4 was written as 1/2 and 1/4, and 7/10 as 1/2 and 1/5. Check the second one over tenths: 1/2 is 5/10, 1/5 is 2/10, and the counts add to 7. Such sums are still called Egyptian fractions.

A harder example

Some amounts take three pieces. Take 11/12. Over twelfths, 1/2 is 6/12, 1/3 is 4/12 and 1/12 is itself. The counts 6, 4 and 1 add to 11, so 11/12 is 1/2 and 1/3 and 1/12. Finding such a split is a puzzle, and there is often more than one answer.

The puzzle of one

Here is the question the chapter ends on. Can different unit fractions add to exactly 1? Two of them never can. Half is the largest unit fraction there is, and any second one must be smaller, so two different pieces always fall short of a whole.

Three pieces, one answer

With three pieces it can be done, and the book says there is only one way to do it. The answer is 1/2 and 1/3 and 1/6. Check it over sixths: those are 3/6, 2/6 and 1/6, and the counts add to 6, which is one whole. No other set of three different unit fractions works.

Four pieces, six answers

Allow four different unit pieces and the puzzle opens up. The book reports that this version has six solutions. One of them is 1/2, 1/4, 1/6 and 1/12. Over twelfths those are 6/12, 3/12, 2/12 and 1/12, and the counts add to 12. Try finding another set yourself before looking anything up.

Why the history matters

These pages are not decoration. They show that the fraction in your notebook has been carried by many hands across many centuries. The method you use for a sum has a name and a date. They also show that mathematics grew in several places at once and travelled between them.

Carry this away

The names bhinna, bhaga and ansha mean broken and part. The stacked numbers came from India and the bar was added later in Morocco. Unit fractions carried whole systems of writing elsewhere, and they still make a fine puzzle.

Scenario:

A class learns that fractions were written down in India and Egypt thousands of years ago. Rohit asks why anybody needed them before machines. Their teacher asks the class what the oldest jobs involved. Dividing land between heirs, sharing a harvest, and measuring cloth. None of those comes out in whole numbers reliably. The class works out that fractions arrived with sharing rather than with mathematics for its own sake. Rohit changes his view. The need came first and the notation followed, which is why several places invented ways of writing them without hearing of each other.

Analysis:

Fractions were not invented for arithmetic lessons. Land, harvests and cloth all have to be divided, and none divides evenly on demand.

Activity:

Ask at home how cloth or land is measured locally. Write any fraction or part-unit that is used.

Title:

Hunt for your own Egyptian fractions

What You Notice:

The first bar shows 11 of 12 parts coloured, so those three unit pieces make 11/12. The second bar fills completely, which is one of the six four-piece answers to the puzzle.

The chapter reports that the Sulba-sutras show rules for working with fractions were already known in Vedic times.

Brahmagupta, writing in 628 CE, set out the rules for calculating with fractions in a formal and general way.

Writing an amount as a sum of different unit fractions can be done in more than one way for most fractions, which is what makes it a puzzle rather than a sum.

What bhinna means — Bhinna means broken — A fraction is a broken piece of a whole, and the Sanskrit name says exactly that.

The order the notation arrived in — Numbers stacked first, bar added later — The two numbers were written one above the other long before anyone drew a line between them.