Class 5 Mathematics Chapter 9: Coconut Farm

Coconuts in neat rows

Susie and Sunitha run a coconut farm in Kerala. This morning they lay the coconuts out on a mat. There are 5 rows, and each row holds 7 coconuts. Counting one by one takes ages. Is there a faster way to know the total?

One picture, one times fact

The rows make a neat block. Five rows of seven give 5 x 7 = 35 coconuts. The same block can be read the other way too. Seven columns of five also give 35. One picture, one times fact.

Two sharing facts hide inside

Now read the block as sharing. Split 35 coconuts into 5 equal rows and each row gets 7. That is 35 divided by 5 equals 7. Split them into 7 equal columns and each gets 5. That is 35 divided by 7 equals 5.

Naming the three parts

The number being shared is the dividend. The number of equal groups is the divisor. What each group gets is the quotient. They fit together neatly. The dividend equals the divisor times the quotient.

Checking your answer

Sunitha always checks her work. She has found that 35 divided by 7 is 5. So she multiplies 7 by 5 and gets 35 back. Multiplying undoes dividing. If the check fails, the answer is wrong.

Bigger numbers, the same rule

Take 400 x 8 = 3,200. Two sharing facts fall straight out of it. First, 3,200 divided by 8 is 400. Second, 3,200 divided by 400 is 8. The rule never changes, however big the numbers grow.

Sharing into tens

Look at what dividing by 10 does. 40 divided by 10 is 4. Then 400 divided by 10 is 40. Then 4,000 divided by 10 is 400. Every digit simply slides one place to the right.

Sharing into hundreds

Dividing by 100 slides the digits two places. So 1,400 divided by 100 is 14. And 4,200 divided by 100 is 42. A zero at the end of the dividend is what makes this work so neatly.

When the dividend grows

Try 220 divided by 20, which is 11. Now make the dividend ten times bigger. 2,200 divided by 20 is 110. Ten times more to share means each group gets ten times more.

The family that answers ten

Some pairs always give 10. Look at 700 divided by 70. Look at 440 divided by 44. Look at 630 divided by 63. In each pair the dividend is exactly ten times the divisor.

A missing number puzzle

What goes in the gap here? A number divided by 18 gives 100. Use the rule. The dividend equals the divisor times the quotient. So 18 x 100 = 1,800 fills the gap.

Remember

• Every times fact gives you two division facts.
• Dividend equals divisor times quotient.
• Check a division by multiplying back.
• Dividing by 10 or 100 slides the digits to the right.

Scenario:

A class lays out 24 coconuts in 4 rows of 6. Their teacher asks what the block shows. Ravi says one times fact, 4 times 6 is 24. Meena spots two more. Sharing 24 coconuts into 4 rows gives 6 each, and sharing them into 6 columns gives 4 each. So the block holds one multiplication and two divisions. Ravi sees that he had read the block one way only. The class names the parts. The 24 is the dividend, the number of groups is the divisor, and what each group gets is the quotient.

Analysis:

One arrangement of coconuts and three facts in it. Reading the block by rows and then by columns is what produces the two divisions.

Activity:

Lay out 36 small objects in a block. Write the one times fact and the two division facts it shows.

Title:

Build fact families with beans

Safety:

Do not put the beans or seeds in your mouth.

What You Notice:

The same 24 beans make several different blocks. Each block gives one times fact and two division facts.

A single coconut palm can give more than 60 coconuts in one year.

The word quotient comes from a Latin word meaning how many times.

The division rule — Divisor times quotient brings the dividend back. — It works both ways. Use it to check an answer, and use it to fill a gap when the dividend is missing.

Six lorries at the gate

Susie has 582 coconuts to send out today. Six regular customers are waiting, and each one must get the same number. Sunitha reaches for a pencil. Susie says there are quicker paths through a division like this. Let us walk down a few of them.

Split into friendly parts

Start with an easier sum, 1,248 divided by 6. Break 1,248 into 1,200 and 48. Both parts are easy. 1,200 divided by 6 is 200, and 48 divided by 6 is 8. Add the two answers to get 208.

Choose parts your tables reach

The splitting only helps if both parts are friendly. For 545 divided by 5, split into 500 and 45. That gives 100 and 9, so the answer is 109. Splitting 545 into 300 and 245 would help nobody.

Go past, then come back

Some numbers sit just under a round one. Take 1,992 divided by 4. Notice that 1,992 is 2,000 take away 8. Now 2,000 divided by 4 is 500, and 8 divided by 4 is 2. So the answer is 498.

Halve, and halve again

Dividing by 4 is the same as halving twice. Try 128 divided by 4. Half of 128 is 64. Half of 64 is 32. So the answer is 32, without a single written sum.

Three halvings for eight

Dividing by 8 means halving three times. Try 168 divided by 8. Half of 168 is 84. Half of 84 is 42. Half of 42 is 21. So 168 divided by 8 is 21.

Guess before you divide

Back to 582 coconuts for 6 customers. Will each one get more than 100? Six hundreds would need 600 coconuts, and Susie has fewer. So each customer gets a little under 100. A guess like this catches silly answers early.

Susie's careful way

Susie takes away groups of 6 bit by bit. She hands out 20 to each customer four times over, using 480. That leaves 102. Ten more each uses 60, leaving 42. Seven more each clears it. So 20, 20, 20, 20, 10 and 7 make 97.

Sunitha's shorter way

Sunitha takes a bigger bite. She gives 90 to each customer at once, which uses 540. Only 42 coconuts are left. Seven more each finishes the job. She reaches 97 in two steps, not six.

Bigger bites, fewer steps

Both girls get 97. Susie's way is safe and slow. Sunitha's way needs a better first guess, but it saves work. As your estimating improves, take bigger bites.

Packing the bags

Each customer's 97 coconuts go into bags of 25. Three bags hold 75. That leaves 22 coconuts over, so a fourth bag is needed. Four bags it is, even though the last one is not full.

Remember

• Split a number into parts your tables reach.
• Go past to a round number, then come back.
• Dividing by 4 is halving twice, and by 8 is halving three times.
• Estimate first, then take the biggest bite you dare.

Scenario:

A farmer shares 96 coconuts among 4 carts. A class finds 96 hard to divide in one step. Nita splits it into 80 and 16. Eighty shared by 4 is 20, and 16 shared by 4 is 4. Together that is 24 each. Imran tries a different split, 40 and 56, and also reaches 24. Their teacher asks whether any split works. The class tests one. Splitting into 90 and 6 gives an awkward remainder in each part, so the answer is harder to see. They write the rule down. Split into parts your times tables reach.

Analysis:

Any split is arithmetically allowed and only some are useful. The point of splitting is to land on numbers the tables already cover.

Activity:

Divide 84 by 4 using two different splits. Write which split was easier and why.

Title:

Race the two methods

What You Notice:

Splitting wins when the number breaks neatly. Big bites win when the divisor is awkward and the dividend is large.

Shopkeepers rarely use long division. They split numbers into friendly parts instead.

A coconut farm in Kerala may harvest over a thousand nuts in a single month.

Dividing by four and eight — Four is two halves, eight is three. — Halve twice and you have divided by 4. Halve a third time and you have divided by 8.

Partial quotients — Take the biggest bite you dare. — Each bite is a group of the divisor handed out. Bigger bites mean fewer steps, so estimate well first.

Sixty two candies, five children

Sunitha's mother comes home with 62 candies. Five children are waiting, and every one must get the same number. She does not guess. She lays the candies out in bundles of ten and starts with those.

Share the tens first

There are 6 bundles of ten and 2 loose candies. Share the 6 tens among 5 children. Each child takes 1 ten, and one bundle is left over. Open that bundle. Now there are 10 and 2, which is 12 loose candies.

Share the ones next

Share the 12 loose candies among the 5 children. Each child takes 2 more. That uses 10 candies, and 2 are left on the table. So each child gets 12 candies in all.

The leftover has a name

Those 2 candies are the remainder. A remainder is always smaller than the divisor. If it were 5 or more, every child could take one more. Here 2 is less than 5, so the sharing is finished.

The rule that ties it together

Multiply the divisor by the quotient and add the remainder. That brings the dividend back. Check this sharing: 5 times 12 is 60, and 2 more makes 62. The rule works for every division.

When the tens will not share

Try 75 shared among 8. There are 7 tens, and 7 tens cannot be split among 8 children. So open every bundle. That gives 75 ones. Each child takes 9, using 72, and 3 are left over.

A zero in the middle

Now share 324 among 3. The 3 hundreds give 1 hundred each. Next come 2 tens, and 2 tens cannot be split among 3. Write 0 in the tens place. Then open them to make 24 ones, and each child takes 8. The answer is 108.

Why that zero matters

Suppose the zero is dropped and 18 is written. Check it. 3 times 18 is only 54, nowhere near 324. The zero is holding the tens place open. Never leave a place empty.

Four digits, the same steps

Share 7,032 among 6. One thousand each uses 6,000. One hundred each uses 600. Seven tens each uses 420, and two ones each uses 12. The quotient is 1,172, with nothing left over.

A two-digit divisor

Share 902 among 16. Fifty each uses 800. Six more each uses 96. Together that is 896, and 6 are left. So the answer is 56 remainder 6. Check: 16 times 56 is 896, and 6 more gives 902.

Oil from the coconut farm

Susie gets 1 litre of oil from 8 coconuts. This year she used 4,376 coconuts. Share 4,376 by 8. Five hundred each uses 4,000, forty each uses 320 and seven each uses 56. She made 547 litres.

Remember

• Share the biggest place first, then regroup what is left.
• Write a 0 whenever a place cannot be shared.
• The remainder is always smaller than the divisor.
• Check with divisor times quotient, plus the remainder.

Scenario:

Ramkishun grows vegetables on a small farm in Bihar. Every Saturday he sends a cart of them to the weekly market. He keeps a record book so that nothing is lost or forgotten. This week he wrote that brinjals sold at 24 rupees a kilogram, and that he sent 65 kilograms of them. He also noted that his onions sold at 18 rupees a kilogram and brought in 2,160 rupees. Beside that he wrote 96 kilograms of carrots, which earned 2,880 rupees. His little grandson has rubbed out three of the numbers while playing. Ramkishun asks the class to work them out again.

Analysis:

Multiplying and dividing fill the gaps in a real record book, and show how a farmer tracks what each crop earns.

Activity:

Collect a shop bill at home and check whether every total on it is correct.

Title:

Bundles of ten

Safety:

Ask an adult before using matchsticks, and do not light them.

What You Notice:

The leftover is always smaller than the number of friends. Opening a bundle is exactly what regrouping means on paper.

About 8 coconuts are needed to press one litre of coconut oil.

Coconut husk is spun into coir, which is used in mattresses, ropes and garden pots.

Checking a division with a remainder — Divisor times quotient, add the leftover. — Multiply the two, add the remainder, and you should land back on the dividend. If you do not, look again at the working.

A place that will not share — No share here, write a zero. — Leaving the place empty moves every later digit one column left, and the answer becomes ten times too small.