Class 5 Mathematics Chapter 13: Animal Jumps

Kabir's flower bed

Kabir has twelve marigold plants in a tray. His grandmother wants them set out in the bed in equal rows. No plant may be left out. Every row must hold the same number. Kabir kneels down and starts moving them about.

The first bed

His first try is one long row of twelve. It works, but it looks like a fence. His grandmother laughs at it. He counts it out anyway and writes one times twelve is twelve.

Two more beds

Next he makes two rows of six. Then he makes three rows of four. Both beds look neat and both use every plant. Nothing is left in the tray. He writes two times six and three times four beside each bed.

Rows and columns

A neat block of rows and columns is called an array. Every array Kabir builds gives him a pair of numbers. Those numbers multiply to make twelve. Turning a bed sideways gives no new pair, so he counts each bed once.

The word factor

The numbers in those pairs are the factors of twelve. They are one, two, three, four, six and twelve. Each one divides twelve exactly, leaving nothing over. A factor is never bigger than the number it belongs to.

The word multiple

Look at it the other way round. Two threes make six and six twos make twelve. So twelve is a multiple of one, two, three, four, six and twelve. A multiple is never smaller than the number it comes from.

Always two friends

Every number has at least two factors. One is a factor of everything, because a row of ones always fits. The number itself is a factor too, because one long row of it always works.

The stubborn thirteen

Kabir tries thirteen plants the next day. Two rows leave one over. Three rows leave one over. Only one long row of thirteen works. Thirteen has just two factors.

Numbers like that

A number with only two factors is called a prime number. Thirteen is prime. So are seventeen and twenty-three. Twelve is not prime, because it has six factors. Kabir checks fifteen and finds three rows of five, so fifteen is not prime either.

Remember

• A factor divides a number exactly.
• Every array shows a pair of factors.
• A number is a multiple of each of its factors.
• A prime number has only two factors.

Scenario:

A class arranges 12 pebbles into neat blocks. They find 1 by 12, 2 by 6 and 3 by 4. Imran says there must be more. Their teacher asks him to try 5. Twelve pebbles cannot make equal rows of 5; one row comes out short. The class lists what worked: 1, 2, 3, 4, 6 and 12. Those are the factors of 12. Imran sees why 5 failed. A factor has to divide the number exactly with nothing left over, and every neat block shows one pair of factors sitting together.

Analysis:

Five is not excluded by chance. A factor must leave nothing over, and the leftover pebble is the proof that five is not one.

Activity:

Arrange 18 objects into every neat block you can. List the factors of 18 from what you find.

Title:

Pebble arrays on the floor

What You Notice:

Twenty-four gives many arrays, so it has many factors. Twenty-three gives only one long row, because it is prime. The pebbles show the difference before the words do.

The number one is a factor of every whole number there is.

Two is the only even number that is prime.

Factor or multiple — Factors are small, multiples are tall. — A factor is never bigger than the number, while a multiple is never smaller than it.

Chalk squares in the lane

Nita draws a long line of chalk squares in the lane. She numbers them carefully from zero upwards. Her cousin Arif joins her after school. They decide to hop along the line instead of walking down it.

Two different hops

Nita can hop four squares at a time. Arif has longer legs and hops six at a time. Both of them always start at the square marked zero. Starting at zero matters, because the first hop must land on the number itself.

Where Nita lands

Nita lands on four, then eight, then twelve. She goes on to sixteen, twenty and twenty-four. Those are the multiples of four. She chalks a small orange dot on each one as she goes.

Where Arif lands

Arif lands on six, then twelve, then eighteen, then twenty-four. Those are the multiples of six. His footprints are further apart, because his hop is longer. He marks his squares in green.

The shared squares

Some chalk squares carry both sets of footprints. Twelve is one. Twenty-four is another. Thirty-six would be the next, if the lane were long enough. The shared squares come at a steady gap of twelve.

A name for them

A square that both hoppers reach is called a common multiple of four and six. Twelve is the first one. The meetings then repeat forever, as long as the children keep hopping.

The quick way

You need not draw the whole lane. Write the two times tables side by side. Then look down them for a number in both lists. The smallest shared number is the first meeting place.

A sure meeting place

Nita spots something useful. Four sixes make twenty-four, and that is a meeting square. Multiplying two numbers always gives one of their common multiples. It may not be the first one, but it is always shared.

When one hop fits the other

Arif tries hops of three against hops of six. Every six is already a three, so they meet at six, twelve and eighteen. The bigger number is the first meeting. That happens whenever one number is a multiple of the other.

Remember

• Jump from zero to find multiples.
• A common multiple is a square both hoppers reach.
• Four and six first meet at twelve.
• The meetings then repeat at a steady gap.
• The product of two numbers is always a common multiple.

Scenario:

Two frogs start at zero on a number line. One jumps in fours and the other in sixes. A class is asked where they first land together. Ravi says at 10, adding the two jump sizes. Their teacher asks the class to mark the landings. The first frog hits 4, 8, 12, 16. The second hits 6, 12, 18. They first meet at 12. Ravi sees that adding was the wrong move. The meeting point has to appear on both lists, and 10 appears on neither. The class names it a common multiple of 4 and 6.

Analysis:

Adding the jump sizes gave 10, which neither frog ever lands on. A meeting point has to be in both sequences.

Activity:

Draw a number line to 30. Mark jumps of 3 and jumps of 5 and circle where they meet.

Title:

Chalk the hopping lane

What You Notice:

The circled squares come at a steady gap all along the lane. That gap is the first common multiple, and every later meeting is one more gap further on.

Two traffic lights that change every four and every six seconds turn together every twelve seconds.

Common multiples never stop. After the first one they go on forever.

Finding a common multiple — Two tables, one shared number. — Write out both times tables and look for a number that appears in each of them.

A long chalk trail

Nita and Arif draw a longer trail this time. It runs from zero right up to thirty-six. Their friends hide little paper flags on some of the squares. The hopper who reaches a flag may keep it.

Reaching a flag

Nita hops in fours again. She reaches twelve, sixteen and twenty-four. She never reaches twenty-two, however hard she tries. A jump of four skips right over it, from twenty to twenty-four.

The rule behind it

A jump reaches a number only if the jump size divides that number exactly. Four divides twenty-four, so she lands there. Four does not divide twenty-two, so that flag stays where it is.

Two flags at once

Two flags are hidden, one on twenty-four and one on thirty-six. Which jump size reaches both of them? The children try one size after another and write down what happens.

Trying them out

Jumps of two reach both. So do jumps of three, four, six and twelve. Jumps of five reach neither flag. Jumps of eight reach only twenty-four, so they are no use here.

The word for it

A size that reaches both flags is a common factor of twenty-four and thirty-six. Those sizes are one, two, three, four, six and twelve. Twelve is the largest of them.

One always works

A jump of one reaches every square on the trail. So one is a common factor of any pair of numbers you choose. It is the size that never fails.

Neighbours share little

Arif tries the flags on fourteen and fifteen. Only a jump of one reaches both of them. Two numbers that follow each other share nothing else at all.

Looking at the tail

Some tests are quick. A number ending in an even digit takes jumps of two. A number ending in five or zero takes jumps of five. Only a number ending in zero takes jumps of ten. For other sizes you must divide.

Why it helps

Common factors are useful outside the game too. A shopkeeper with two kinds of goods uses them to pack equal boxes. The largest common factor makes the fullest boxes, with nothing left on the shelf.

Remember

• A jump reaches a number only if it divides it.
• A common factor divides both numbers exactly.
• One is a common factor of every pair.
• Look at the last digit for jumps of two, five and ten.

Scenario:

A school in Nagaland is packing kits for a science fair. The store holds 24 magnifying lenses and 36 small torches. The teacher wants identical kits. The lenses must be shared equally. The torches must be shared equally too. Nothing may be left in the store. The head girl suggests eight kits. Eight lenses do fit three times over. But her friend checks the torches. They would be left in an awkward heap. So the children try again. They write every size that divides twenty-four. Then they write every size that divides thirty-six. They ring the sizes in both lists. Last of all they pick the largest ring. That gives the fullest kits.

Analysis:

A packing problem shows why common factors matter, and links number work to school science.

Activity:

Count two kinds of things at home and find the largest number of equal shares they both allow.

Two numbers that follow each other, such as fourteen and fifteen, share only the factor one.

Every number that takes jumps of ten also takes jumps of two and of five.

Testing for two, five and ten — Check the tail, not the whole animal. — Only the last digit matters for jumps of two, five and ten.