Class 5 Mathematics Chapter 2: Fractions

Gurpreet's fraction kit

Gurpreet has a new fraction kit. It has a long strip for one whole. It also has smaller pieces marked 1/2, 1/3, 1/4, 1/6 and 1/12. Gurpreet wants to build one whole with two kinds of pieces. Can he do it? Let us play along and find out.

Equal parts only

A fraction needs equal parts. Cut a roti into 4 pieces of the same size. Each piece is 1/4 of the roti. What if one piece is big and one is tiny? Then they are not fourths. They are just pieces. So always check that the parts match in size.

Filling one whole

How many 1/5 pieces fill a whole strip? Five of them. Two 1/2 pieces fill a whole. Three 1/3 pieces fill a whole. The bottom number tells you how many same-size pieces make one whole.

Two kinds of pieces

Gurpreet puts one 1/2 piece on the whole strip. Half the strip is still empty. He fills it with two 1/4 pieces. It fits! So two 1/4 pieces take the same space as one 1/2 piece. We write this as 1/2 = 2/4.

Breaking a piece

Break a 1/2 piece into 2 equal parts. Each part is 1/4. Now break a 1/2 piece into 3 equal parts. Each part is 1/6. So one half is also 3/6. One half has many names: 1/2 = 2/4 = 3/6 = 6/12.

Two names for one part

Take a strip of 6 equal boxes. Colour 2 boxes. You have coloured 2/6 of the strip. Now draw a thick line around every 2 boxes. The strip has 3 big parts, and 1 of them is coloured. That is 1/3. So 1/3 = 2/6.

A new word: equivalent

Some fractions show the same amount of the same whole. We call them equivalent fractions. Equivalent means "equal in value". 1/3 and 2/6 are equivalent. 1/2 and 2/4 are equivalent too.

Sameer's squares

Sameer colours the middle column of a square. That is 1/3 of it. He draws one line across. Now there are 6 boxes, and 2 are coloured. That is 2/6. With two lines he gets 3/9. With three lines he gets 4/12. The coloured part never changes.

Spot the pattern

Look at 1/3, 2/6, 3/9 and 4/12. The top goes 1, 2, 3, 4. The bottom goes 3, 6, 9, 12. Top and bottom are multiplied by the same number. For 3/9, both 1 and 3 are multiplied by 3. So 1/3 = 5/15 as well.

Going back down

You can divide too. Take 4/12. Divide the top and the bottom by 4. You get 1/3. Now take 6/9. Divide both by 3. You get 2/3, not 1/3. So 6/9 and 1/3 are not equivalent.

Check before you say yes

Is 3/5 the same as 6/10? 3 × 2 = 6 and 5 × 2 = 10. Yes, they are equivalent. Is 2/3 the same as 3/4? Lay a 2/3 strip under a 3/4 strip. The 3/4 strip reaches further. So they are not equivalent.

Remember

• The parts of a fraction must be equal.
• Equivalent fractions show the same amount.
• Multiply or divide the top and bottom by the same number.
• Adding the same number to both does not work.

Scenario:

Two children fold chapatis. Nita folds hers into two equal parts and takes one. Imran folds his into four equal parts and takes two. Nita says she got more. Their teacher asks them to lay the pieces on top of each other. The shapes match exactly. So one half and two quarters are the same amount. Nita is surprised and then sees why. Her piece is bigger and she has only one. Imran's pieces are smaller and he has two. The class writes both names for the same share and calls them equivalent.

Analysis:

Two names and one amount. Laying the pieces on top of each other is what shows the share has not changed.

Activity:

Fold two paper circles, one into halves and one into quarters. Shade one half and two quarters and compare them.

Title:

Make a paper fraction kit

Safety:

Only folding is needed. If you want to cut the strips, ask an adult to help with scissors.

What You Notice:

Two 1/4 parts end where one 1/2 part ends. Two 1/6 parts end where one 1/3 part ends. Same amount, new name.

The word "fraction" comes from a Latin word that means "to break".

Fold a paper strip in half, then in half again, and you get fourths. One more fold gives eighths.

Making equivalent fractions — What you do to the top, do to the bottom. — Multiply both numbers by the same number. The pieces get smaller, but you take more of them, so the amount stays the same.

Tamanna's two chocolates

Tamanna has two chocolate bars. One is small and one is big. She says, "1/3 of my big bar is more than 1/2 of my small bar." Her friend laughs. "But 1/2 is more than 1/3!" Who is right?

Tamanna is right

Her small bar has 4 squares. Half of it is 2 squares. Her big bar has 9 squares. One third of it is 3 squares. So 1/3 of the big bar is more chocolate. Her two wholes were not the same size.

Same whole first

So before we compare, we check the wholes. They must be the same size. Half of a small roti is less than half of a big roti. From now on, every comparison uses the same whole.

The bottom number: denominator

Look at 3/4. The bottom number is 4. It is called the denominator. It tells how many equal parts make the whole. A denominator of 4 means the whole is cut into 4 equal parts.

The top number: numerator

Now look at the top number of 3/4. It is 3. It is called the numerator. It tells how many of the equal parts we take. So 3/4 means 3 parts out of 4 equal parts.

Sevi and Shami share chikki

Sevi and Shami share a piece of chikki. Sevi eats 1/3 of it. Shami eats the rest, which is 2/3. Who ate more? Both ate thirds, so all the pieces are the same size. Shami ate 2 pieces and Sevi ate 1. Shami ate more.

Same denominator rule

When the denominators match, the pieces are the same size. So just count the pieces. More pieces means a bigger fraction. So 7/8 is more than 3/8, and 2/6 is more than 1/6.

The signs > and <

The sign > means "is more than". The sign < means "is less than". The open side always faces the bigger number. So we write 2/3 > 1/3. We also write 1/4 < 3/4.

Same numerator, new puzzle

Next evening, Sevi eats 4/6 of a paratha. Shami eats 4/5 of a paratha of the same size. Both ate 4 pieces. But are the pieces the same size? No. A paratha cut into 6 has smaller pieces than one cut into 5.

More sharing, smaller shares

Share one paratha among 5 people, and each gets 1/5. Share it among 6, and each gets 1/6. More sharing makes smaller shares. So 1/6 < 1/5. Four small pieces are less than four bigger pieces. So 4/6 < 4/5.

Same numerator rule

When the numerators match, look at the denominators. The smaller denominator gives bigger pieces. So it makes the bigger fraction. That is why 3/7 > 3/8, and 5/6 > 5/7.

Which rule do I use?

First find the number that is the same. Same bottom? Count the pieces. Same top? Look for the bigger pieces. Take 5/9 and 2/9. The bottom is the same, so 5/9 > 2/9. Take 2/9 and 2/5. The top is the same, so 2/5 > 2/9.

Scenario:

Ravi eats half of a small roti. Meena eats a quarter of a very big roti. Ravi says he ate more because a half beats a quarter. His teacher asks the class a question first. Were the two rotis the same size? They were not. The class draws both and shades the shares. Meena's quarter is the bigger piece. Ravi sees the catch. A fraction tells you how a whole was cut, not how big the whole was. The class writes the rule down. Fractions can only be compared when the wholes are the same size.

Analysis:

A half can be smaller than a quarter when the wholes differ. The fraction describes the cutting, not the size of what was cut.

Activity:

Draw a small circle and a large one. Shade half the small one and a quarter of the large one, then compare.

Title:

Fold and compare

Safety:

Only folding is needed, so no scissors are used.

What You Notice:

3/4 is much longer than 3/8. Both have 3 coloured parts, but fourths are bigger pieces than eighths.

Chikki is a sweet made from jaggery and peanuts. Lonavala in Maharashtra is famous for it.

The signs > and < were first printed in a maths book almost 400 years ago.

Same top number — More friends, smaller share. — Share one pizza among 6 friends and each slice is smaller than when 5 friends share it. So 1/6 is less than 1/5.

The signs > and < — The hungry crocodile eats the bigger number. — The open mouth of the sign always faces the bigger fraction, as in 2/3 > 1/3.

Raman's paratha puzzle

Raman's father makes soft parathas. He cuts each one into halves or into fourths. Every day he asks Raman and Radhika a question. "How many whole parathas did everyone eat?" Today, Maa took 5 half pieces. How much is that?

Counting halves

Two halves make 1 paratha. So 4 halves make 2 parathas. Maa has 1 more half. So 5 halves are 2 parathas and a half more. We write this as 5/2 = 2 and 1/2.

Halves on a number line

Draw a line and mark 0, 1, 2 and 3. Cut each gap into 2 equal parts. Each small hop is 1/2. Start at 0 and make 5 hops. You land between 2 and 3, at 5/2.

Radhika's share

Radhika took 6 half pieces. Two halves make 1, so 6 halves make 3 parathas. On the number line, 6 hops of 1/2 land right on 3. So 6/2 = 3.

Now in fourths

Another day, the parathas are cut into fourths. Dadaji takes 9 pieces of 1/4. Four fourths make 1 paratha, so 8 fourths make 2. One piece is left over. So 9/4 = 2 and 1/4.

Fractions more than 1

Look at 5/2 and 9/4. The top number is bigger than the bottom. So there is more than one whole. When the top and bottom are equal, like 4/4, it is exactly 1. When the top is smaller, it is less than 1.

Pizza night

Raman's family of 6 orders 2 pizzas. Each pizza is cut into 3 equal slices. That makes 6 slices, one for each person. Dadiji and Dadaji give their slices to Raman. Now Raman has 3 slices. That is 3/3, or 1 whole pizza.

Radhika gets more

Maa and Baba give their slices to Radhika. She has 3/3 too. Raman can eat only 2 slices, so he gives her 1 more. Now Radhika has 4/3. That is 1 whole pizza and 1/3 more. Raman keeps 2/3.

Compare with 1

Sevi ate 7/8 of a paratha. Shami ate 8/6 of a paratha. Who ate less? 7/8 is less than 1, because 7 is less than 8. 8/6 is more than 1. So Sevi ate less, and 7/8 < 8/6.

Compare with half

A fraction is exactly half when the top is half of the bottom. So 4/8 and 3/6 are both half. Now Sevi eats 5/8 and Shami eats 3/6. Half of 8 is 4, so 5/8 is more than half. 3/6 is exactly half. So 5/8 > 3/6.

The ant line

An ant is 1/4 cm long. Sixteen ants walk in a line. How long is the line? It is 16/4 cm. Four fourths make 1 cm, so 16 fourths make 4 cm.

Scenario:

Lakshmi aunty runs a small tiffin kitchen in Pune. She packs lunch for people who work in offices. She cuts every roti into halves, so the pieces stay soft and fit the box. On Monday, one box gets 5 half rotis. Another box gets 3 half rotis. In the evening, 4 half rotis come back uneaten. Lakshmi aunty does not throw them away. She turns them into a crunchy snack for her helpers. She also plans better. She writes down how much each person really eats. Then she cooks less and wastes less food. Her notebook is full of fractions!

Analysis:

Fractions help us count food in halves and wholes. Knowing how much we really eat helps us cook the right amount and waste less.

Activity:

At dinner, count the roti or chapati pieces your family eats. Write the total as a fraction and as whole rotis.

A ruler is a number line you can hold. The small marks cut each centimetre into 10 equal parts.

An hour has 60 minutes. Half an hour is 30 minutes, and a quarter of an hour is 15 minutes.

Is it more than 1? — Top taller than bottom? More than a whole! — If the numerator is bigger than the denominator, the fraction is more than 1, like 7/4.

Exactly half — Double the top. Is it the bottom? Then it is half. — 4 doubled is 8, so 4/8 is exactly half.