Written and drawn by attnom. Free to read, print and share.
M2.4.1 Chapter 1
Fair shares: equal parts
My brother cut the bread in two. Is his piece really a half?
Your guess
Pick one before you read on. It is fine to be wrong.
By the end you can
Tell if a shape is cut into equal parts
Share a group of things fairly between 2, 3 or 4 people
Explain why two pieces of different sizes are not halves
Two friends share one flat bread. They cut it into two pieces. One piece is big. One piece is small. Is that fair?
A share is fair when every piece is the same size. We call these pieces equal parts.
Draw it first
Cut a strip into two equal parts
1 Draw a long strip. The whole strip is one whole.
2 Fold it so the two ends meet. Draw a line on the fold.
3 Shade one part. It is one half of the strip.
Both bars have two parts. Only the top bar has two equal parts.
See it three ways
With things
Put 6 beans in a line. Give one to each of 2 people in turn. Stop when none are left.
As a picture
In symbols
6 ÷ 2 = 3. Each person gets 3.
Try it
Fold a fair share
You need: a sheet of paper, or an old page, a pencil.
Fold the paper so that its edges meet.
Press the fold flat, then open it.
Draw along the fold. Now fold it again and look.
What you will see: One half lies exactly on top of the other. Nothing sticks out. That is how you know the two parts are equal.
If you cannot try it, this is what happens.
A wrong idea many people have
Any two pieces are two halves.
Why it feels right: People say "cut it in half" for any cut into two pieces.
The test that breaks it: Cut a bread into one big piece and one small piece. Ask your friend to pick. Nobody picks the small one.
What is true: Halves are always equal. Two pieces of different sizes are just two pieces.
Key words
whole
all of one thing, before it is cut.
equal parts
parts that are all the same size.
half
one of two equal parts of a whole.
Back to your guess
The answer is B: Only if the two pieces are the same size.
Two pieces are halves only when they are the same size. A cut down the middle is one way, not the only way. The sizes are what count.
Check yourself
Try each one before you look. A wrong answer tells you where to go next.
Which bar is cut into halves?
See the answer
Bar B
B has two parts of the same size. A has a small part and a big part.
If you chose A: You may be counting pieces. Bar A has two pieces, but they are not the same size.
Share 12 stones fairly between 3 friends. How many stones does each friend get?
See the answer
4 stones each
12 stones in 3 equal groups is 4 in each group. 4 + 4 + 4 = 12.
Ravi has 10 beans. He gives 6 to his sister and keeps 4. Is that a fair share?
See the answer
No. A fair share is 5 each.
10 beans shared by 2 is 5 each. 6 and 4 are not the same.
If you chose A: Both having some is not enough. A fair share means the same number each.
Draw a square. Cut it into 4 equal parts. Then find a second way to do it.
See the answer
One way: a line across and a line down, through the middle. Another way: three lines from top to bottom. They make 4 strips of the same width.
Both ways give 4 parts that are all the same size. There are many more ways.
Tariq wants 3 equal parts of a bread. Which bar shows 3 equal parts?
See the answer
Bar B
Bar B has 3 parts of the same size. In A and C some parts are bigger than others.
If you chose A: Bar A has 3 parts, but all three are different sizes. Counting the parts is not enough.
If you chose C: Two parts match. But the first part is twice as long as one of them. All 3 must match.
Amina shares 12 beans fairly between 4 friends. How many beans does each friend get?
See the answer
3 beans each
12 beans in 4 equal groups is 3 in each group. 3 + 3 + 3 + 3 = 12.
If you chose A: There are 4 friends, but that is not the share. Make 4 equal groups of the 12 beans. Each group has 3.
If you chose B: You took 4 away from 12. Sharing between 4 friends means making 4 equal groups.
Sipho has 15 beans for 3 children. Which plan is a fair share?
See the answer
Give 5 beans to each child.
15 beans in 3 equal groups is 5 in each group. 5 + 5 + 5 = 15.
If you chose B: Every child gets some, and 7 + 5 + 3 = 15. But a fair share gives every child the same number.
If you chose C: 3 + 3 + 3 = 9, so 6 beans are left over. A fair share of 15 beans uses all of them.
Leila cuts a bread into pieces to share. How can she check that the pieces are equal?
See the answer
Lay one piece on top of another. They match exactly.
Pieces that match exactly are the same size.
If you chose A: Counting tells how many pieces there are, not how big.
If you chose B: Straight cuts can still leave one big piece and one small piece.
Mei cuts a bread into two parts, as in the picture. She says each part is a half. What do you say?
See the answer
No. The two parts are not the same size.
One part is big and one part is small. Halves are two parts of the same size.
If you chose A: Two parts are not enough. Halves must be the same size.
If you chose C: Two parts of any size can make a whole bread. Halves must also be the same size.
Keep it
Copy this into your exercise book.
Equal parts are the same size. A half is one of two equal parts. Two pieces are only halves if they are equal.
Draw: A strip cut into two equal parts. Shade one part and write "one half".
M2.4.2 Chapter 2
One part: halves, quarters, thirds, fifths
Eight is more than four. So is 1/8 more than 1/4?
Your guess
Pick one before you read on. It is fine to be wrong.
By the end you can
Name one equal part as a half, quarter, third or fifth
Say what the bottom number of a fraction tells you
Tell which of two parts is smaller, such as 1/8 and 1/4
Last time you cut a whole into equal parts. Now we give one of those parts a name.
Draw it first
Name one part of a strip
1 Draw a long strip. The whole strip is one whole.
2 Cut it into 4 equal parts. Fold it twice to be sure.
3 Shade one part. It is one quarter of the strip, and we write 1/4.
The 4 on the bottom is the bottom number. It tells how many equal parts the whole is cut into.
The 1 on top says we take one part. 1/4 is a fraction. A fraction is a number for part of a whole.
Some people say fourth, not quarter. Both names mean one of 4 equal parts.
Cut a whole into 2 equal parts. One part is one half, 1/2. Cut it into 8 equal parts. One part is one eighth, 1/8.
One part of each circle is shaded.
Thirds and fifths are clearer on a bar. A circle is hard to cut into 3 or 5 equal parts. Cut a bar into 3 equal parts. Each part is a third, 1/3.
Cut a bar into 5 equal parts. Each part is a fifth, 1/5.
The two bars are the same length.
Now look at the bottom numbers. 8 is more than 4. But one eighth is a smaller part than one quarter.
Why? The more parts you cut, the smaller each part is.
Going down, the bottom number grows and the shaded part shrinks.
So 1/8 is smaller than 1/4. And 1/4 is smaller than 1/2.
See it three ways
With things
Fold a paper into 4 equal parts. Open it. Colour one part.
As a picture
In symbols
1 part of 4 equal parts is 1/4.
Try it
Fold and look at one part
You need: a sheet of paper, or an old page, a pencil.
Fold the paper in half. Open it. You have 2 equal parts. Look at one part.
Fold it in half, then in half again. Open it. You have 4 equal parts.
Fold it in half 3 times. Open it. You have 8 equal parts.
Colour one part each time. Compare how big each part is.
What you will see: The more times you fold, the more parts you get. Each part is smaller. One eighth is the smallest part.
If you cannot try it, this is what happens.
A wrong idea many people have
The bigger the bottom number, the bigger the part.
Why it feels right: Bigger numbers usually mean more. 8 is more than 4.
The test that breaks it: Take two breads of the same size. Cut one into 4 equal parts. Cut the other into 8. Pick one part from each. Which part is bigger?
What is true: The bottom number counts the parts. More parts means smaller parts. So 1/8 is smaller than 1/4.
Key words
bottom number
the number under the line. It tells how many equal parts the whole is cut into.
quarter
one of 4 equal parts of a whole. It is written 1/4. Also called a fourth.
third
one of 3 equal parts of a whole. It is written 1/3.
fifth
one of 5 equal parts of a whole. It is written 1/5.
Back to your guess
The answer is B: No. One eighth is a smaller part than one quarter.
The bottom number counts the equal parts. More parts means each part is smaller. So 1/8 is smaller than 1/4.
Check yourself
Try each one before you look. A wrong answer tells you where to go next.
Which bar has 1/4 shaded?
See the answer
Bar A
Bar A has 4 equal parts and one is shaded. That is 1/4.
If you chose B: Count the equal parts. Bar B has 3 parts, so one part is a third, 1/3.
A bread is cut into 5 equal parts. What is one part called?
See the answer
one fifth
The bottom number is the number of equal parts. 5 parts gives fifths.
If you chose B: A fourth is one of 4 equal parts. Count again. This bread has 5.
If you chose C: A third is one of 3 equal parts. This bread has 5.
Which is the smaller part, one quarter or one eighth?
See the answer
one eighth
Cutting a whole into 8 parts gives smaller parts than 4 parts.
If you chose A: 8 is more than 4, but that means more parts. More parts are smaller parts.
From an earlier chapter
Mei cuts a flat bread into 4 pieces. The pieces are not the same size. Is one piece a quarter?
See the answer
No. The parts must be equal.
Quarters, halves and thirds are always equal parts. Pieces of different sizes are not.
If you chose A: Counting pieces is not enough. A quarter must be one of 4 equal parts. Go to M2.4.1.
Draw a bar. Cut it into 3 equal parts. Shade one part. What is the shaded part called, and how do you write it?
See the answer
One third, written 1/3.
The bar has 3 equal parts, so the bottom number is 3. You took one part, so the top is 1.
Ravi cuts a bread into 5 equal parts and takes one. How does he write his part?
See the answer
1/5
The bread has 5 equal parts, so the bottom number is 5. Ravi takes 1 part, so the top is 1.
If you chose A: The bottom number counts all 5 equal parts. It does not count the parts left over.
If you chose B: The top number says how many parts Ravi takes, which is 1. The bottom counts all the equal parts.
Mei shades one part of the bar. How do you write the shaded part?
See the answer
1/4
The bar has 4 equal parts, so the bottom number is 4. One part is shaded, so the top is 1.
If you chose B: You counted the parts left over, which is 3. The bottom number counts all the equal parts.
If you chose C: You counted the lines, which is 5. Count the parts between the lines.
Three breads are the same size. Which part is smallest: 1/2, 1/4 or 1/8?
See the answer
1/8
The bread cut into 8 parts has the most parts. So each part is the smallest.
If you chose A: 2 is the smallest bottom number. But fewer parts means each part is bigger.
If you chose C: Each one is 1 part. But the bottom number says how many parts. More parts means smaller parts.
Which bar is cut into quarters?
See the answer
Bar C
Bar C has 4 parts of the same size. Bar B has 4 parts that are not equal.
If you chose A: Bar A has 3 equal parts. That makes thirds. Count the parts, not the lines.
If you chose B: Bar B has 4 parts, but they are not the same size. Quarters are equal parts. Go to M2.4.1.
Keep it
Copy this into your exercise book.
The bottom number says how many equal parts. One part of 4 is 1/4. More parts means smaller parts, so 1/8 is smaller than 1/4.
Draw: Two strips of the same length. Cut one into 4 equal parts and one into 8. Shade one part in each.
M2.4.3 Chapter 3
More than one part
If 1/4 is one part, what do I call three parts?
Your guess
Pick one before you read on. It is fine to be wrong.
By the end you can
Read and write a fraction such as 3/4
Say what the top and bottom numbers tell you
Write a whole as 4/4 or 5/5
A fraction names some equal parts of a whole. Last time you named one part. Now you can take more.
Cut a strip into 4 equal parts. Each part is 1/4. Count the parts as you take them.
Draw it first
Count quarters to make three quarters
1 Draw a strip. Cut it into 4 equal parts.
2 Shade one part. That is 1/4.
3 Shade a second part. Now 2 parts are shaded. That is 2/4.
4 Shade a third part. Now 3 parts are shaded. That is 3/4.
You counted 1, 2, 3 quarters. So the fraction is 3/4. We say three quarters.
The bottom number is 4. It tells how many equal parts make the whole. It is also called the denominator.
The top number is 3. It tells how many parts you take. It is called the numerator.
Fifths work the same way. Cut a bar into 5 equal parts. Count the fifths you take.
Each bar has 5 equal parts. The top number counts the shaded parts.
What if you take every part? You have the whole. All 4 quarters make 4/4. All 5 fifths make 5/5.
When the top number equals the bottom number, you have one whole.
See it three ways
With things
Fold a paper into 4 equal parts. Open it. Colour 3 of the parts.
As a picture
In symbols
3 parts of 4 equal parts is 3/4.
Worked example
A bar has 5 equal parts. Mei shades 2 parts. What fraction is shaded?
Count all the equal parts. There are 5. That is the bottom number.
Count the shaded parts. There are 2. That is the top number.
Write the top number over the bottom number.
Answer:2/5
Try it
Fold and colour three quarters
You need: a sheet of paper, or an old page, a pencil.
Fold the paper in half. Fold it in half again.
Open it. Count the equal parts. There are 4.
Colour 3 of the parts.
Count the coloured parts, then count all the parts.
What you will see: 3 coloured parts out of 4 equal parts is 3/4. The part left over is 1/4. Together they make 4/4, the whole.
If you cannot try it, this is what happens.
A wrong idea many people have
The bottom number says how many parts you take.
Why it feels right: In 3/4, the 4 is the bigger number. It looks like the count.
The test that breaks it: Fold a paper into 4 equal parts. Colour 3. Count the parts you coloured. Then count all the parts.
What is true: You coloured 3, and the paper has 4 parts. The top number counts the parts you take. The bottom number counts all the equal parts.
Key words
fraction
a number that names equal parts of a whole, like 3/4.
numerator
the top number. It tells how many parts you take.
denominator
the bottom number. It tells how many equal parts make the whole.
Back to your guess
The answer is B: It is 3/4, three of the quarter parts.
Count the quarter parts you take. Three quarters is 3/4. The top number counts the parts you take.
Check yourself
Try each one before you look. A wrong answer tells you where to go next.
What fraction of the bar is shaded?
See the answer
3/5
There are 5 equal parts, so the bottom number is 5. 3 are shaded, so the top number is 3.
If you chose B: You counted the parts that are not shaded. The top number counts the shaded parts.
If you chose C: You may have put shaded parts over parts not shaded. The bottom number counts all the equal parts. That is 5, not 2. Go to M2.4.2.
Which bar shows 2/3?
See the answer
Bar A
Bar A has 3 equal parts and 2 are shaded. That is 2/3.
If you chose B: Bar B has 5 equal parts, so it shows 2/5. The bottom number must be 3. Go to M2.4.2.
A bread is cut into 4 equal parts. Kofi eats all 4 parts. What fraction did he eat?
See the answer
4/4, the whole bread
The bread has 4 equal parts. He took 4 of them. So the top and bottom numbers are both 4.
If you chose B: That is one part. Kofi ate 4 parts.
If you chose C: The bottom number is the number of equal parts in the whole. That is 4. Go to M2.4.2.
From an earlier chapter
Which is the biggest part: one third, one quarter or one fifth?
See the answer
one third
Cutting into 3 parts gives bigger parts than 4 or 5.
If you chose B: Cutting into 3 parts gives bigger parts than 4. Go to M2.4.2.
If you chose C: More parts means smaller parts. Cutting into 5 gives the smallest part here. Go to M2.4.2.
Draw a bar in 5 equal parts. Shade 3 parts. Write the fraction, and name the top and bottom numbers.
See the answer
3/5. The top number is 3. The bottom number is 5.
The bar has 5 equal parts. You shaded 3 of them. The top number counts shaded parts.
Sipho cuts a bread into 8 equal parts and eats 3. What does the 8 in 3/8 tell you?
See the answer
How many equal parts the whole bread has.
The bread is cut into 8 equal parts. The bottom number counts all of them.
If you chose A: Sipho ate 3 parts, the top number. The bottom number counts all the equal parts.
If you chose C: The bottom number counts the equal parts. It does not say which part he ate.
Hana shades the whole bar. Which fraction names what she shaded?
See the answer
8/8
The bar has 8 equal parts and Hana shaded all 8. The top and bottom numbers are both 8.
If you chose A: That is one part. Hana shaded every part.
If you chose B: The bottom number counts the equal parts in the whole. That is 8, not 1.
What fraction of the flat bread is shaded?
See the answer
5/8
There are 8 equal slices, so the bottom number is 8. 5 are shaded, so the top number is 5.
If you chose A: You put shaded parts over parts not shaded. The bottom number counts all the equal parts, 8.
If you chose C: You counted the parts that are not shaded. The top number counts the shaded parts.
Which bar shows 3/4?
See the answer
Bar C
Bar C has 4 equal parts and 3 are shaded. That is 3/4.
If you chose A: You counted the parts that are not shaded. The top number counts the shaded parts.
If you chose B: Bar B has 3 parts shaded. But its 4 parts are not equal. A fraction needs equal parts. Go to M2.4.1.
Keep it
Copy this into your exercise book.
The bottom number counts all the equal parts. The top number counts the parts you take. 3/4 is three quarters. 4/4 is one whole.
Draw: A strip in 4 equal parts. Shade 3 parts and write 3/4 under it.
M2.4.4 Chapter 4
Fractions of a group
I have 12 beans. How many beans is 1/3 of them?
Your guess
Pick one before you read on. It is fine to be wrong.
By the end you can
Find a fraction of a group of things
Make equal groups, then take some of them
Explain why 2/3 of 12 is 8
So far the whole was one thing, like a bread. A whole can be a group of things too.
A group is a number of things together. 12 beans are a group. We can take a fraction of them.
Take 1/3 of 12 beans. The bottom number is 3. Make 3 equal groups. Equal groups have the same number of things.
Draw it first
Find one third of 12 beans
1 Start with 12 beans. That is the whole group.
2 Make 3 equal groups. Each group has 4 beans.
3 Take 1 group. 1/3 of 12 is 4.
The bottom number makes the groups. The top number says how many groups to take.
For 2/3 of 12, make 3 equal groups. Then take 2 groups.
Worked example
Find 2/3 of 12 beans.
The bottom number is 3. Make 3 equal groups.
12 beans in 3 equal groups is 4 in each group.
The top number is 2. Take 2 groups.
2 groups of 4 is 4 + 4 = 8.
Answer: 8 beans
The same 12 beans can make 4 equal groups. Then each group is smaller. It holds fewer beans.
More groups means fewer beans in each group.
See it three ways
With things
Put 8 stones in a pile. Share them into 4 equal groups. Take 3 groups.
As a picture
In symbols
8 ÷ 4 = 2. 2 + 2 + 2 = 6. So 3/4 of 8 is 6.
Try it
Share beans into groups
You need: 12 beans, or 12 small stones, a flat table.
Count out 12 beans.
Share them one at a time into 3 groups. Stop when none are left.
Count one group.
Now push 2 groups together. Count them.
What you will see: Each group has 4. One group is 1/3 of 12, which is 4. Two groups are 2/3 of 12, which is 8.
If you cannot try it, this is what happens.
A wrong idea many people have
1/3 of 12 is 3, because of the 3.
Why it feels right: The 3 is the only number you see in the fraction. It looks like the answer.
The test that breaks it: Count out 12 beans. Make 3 equal groups. Count the beans in one group.
What is true: The 3 says how many groups to make. Each group has 4 beans, so 1/3 of 12 is 4.
Key words
group
a number of things together.
equal groups
groups that all have the same number of things.
Back to your guess
The answer is B: 4 beans.
Make 3 equal groups of the 12 beans. Each group has 4 beans. So 1/3 of 12 is 4.
Check yourself
Try each one before you look. A wrong answer tells you where to go next.
Look at the picture. How many stones are in 1/5 of 10 stones?
See the answer
2 stones
10 stones in 5 equal groups is 2 in each group. 1/5 of 10 is one group.
If you chose B: The 5 tells how many groups to make. It does not tell how many stones.
If you chose C: You counted the stones not filled in. That is 4/5 of 10. 1/5 is 1 group.
What is 1/4 of 12 beans?
See the answer
3 beans
12 beans in 4 equal groups is 3 in each group. One group is 3 beans.
What is 2/3 of 9 stones?
See the answer
6 stones
9 stones in 3 equal groups is 3 in each group. Two groups is 3 + 3 = 6.
From an earlier chapter
A bar has 5 equal parts. 4 parts are shaded. What fraction is shaded?
See the answer
4/5
The bar has 5 equal parts. 4 are shaded. So the fraction is 4/5.
If you chose B: That is one part. Here 4 parts are shaded, so the top number is 4. Go to M2.4.3.
If you chose C: The bottom number is all the equal parts. That is 5. Go to M2.4.3.
Mei has 12 beans. She gives 3/4 of them to Kofi. How many beans does Kofi get? How many does Mei keep?
See the answer
Kofi gets 9 beans. Mei keeps 3 beans.
12 beans in 4 equal groups is 3 in each group. Kofi takes 3 groups, which is 9. Mei keeps 1 group, which is 3.
Amina finds 2/3 of 12 beans. Which working is right?
See the answer
Make 3 equal groups of 4. Take 2 groups. That makes 8 beans.
12 beans in 3 equal groups is 4 in each group. 2 groups is 4 + 4 = 8.
If you chose A: The bottom number, 3, makes the groups. The top number says how many to take.
If you chose C: That is 1/3 of 12. The top number is 2, so take 2 groups.
Hana has 10 beans, as in the picture. What is 3/5 of them?
See the answer
6 beans
10 beans in 5 equal groups is 2 in each group. 3 groups is 2 + 2 + 2 = 6.
If you chose B: That is one group, 1/5 of 10. The top number is 3, so take 3 groups.
If you chose C: You took 5 away from 10. The bottom number says how many equal groups to make.
Tariq finds 3/4 of 8 stones. How many equal groups does he make first?
See the answer
4 groups
The bottom number is 4. So Tariq makes 4 equal groups, with 2 stones in each.
If you chose B: The top number says how many groups to take. The bottom number says how many to make.
If you chose C: There are 2 stones in each group. But Tariq makes 4 groups.
Priya has 6 stones in equal groups, as in the picture. She takes 1 group. What fraction of the stones is that?
See the answer
1/3
The 6 stones are in 3 equal groups. Priya takes 1 group, so she takes 1/3.
If you chose A: You used the 2 stones in a group. The bottom number counts the groups, which is 3.
If you chose C: That is one stone out of 6. Priya took a whole group of stones.
Keep it
Copy this into your exercise book.
The bottom number says how many equal groups to make. The top number says how many groups to take. 2/3 of 12 is 8.
Draw: 12 dots in 3 groups of 4. Fill in 2 groups. Write 4 + 4 = 8.
M2.4.5 Chapter 5
The number line, past 1
I saw 5/4 on a page. Can a fraction be more than 1?
Your guess
Pick one before you read on. It is fine to be wrong.
By the end you can
Find a fraction on a number line
Tell if a fraction is before 1, at 1, or past 1
Count steps past 1 to place a fraction such as 5/4
A number line is a straight line with numbers on it. Each number has its own place.
Fractions are numbers too. They have places on the line, between the whole numbers.
Draw it first
Find 5/4 on a number line
1 Draw a line. Mark 0, 1 and 2 with equal gaps.
2 Cut each whole into 4 equal steps. Each step is 1/4.
3 Count 5 steps from 0. You land on 5/4. It is past 1.
Four steps of 1/4 reach 1. So 4/4 is the same place as 1.
A is 1/2. It is before 1. B is 3/2. It is past 1.
Look at the top and bottom numbers. Is the top bigger than the bottom? Then the fraction is past 1.
See it three ways
With things
Take two strings, each 1 metre long. Cut each one into 4 equal pieces. Lay 5 pieces end to end.
As a picture
In symbols
5/4 = 4/4 + 1/4. That is 1 and one more quarter.
Try it
Walk past 1 with paper strips
You need: two strips of paper of the same length, a pencil.
Fold one strip in half. Fold it in half again. Open it.
Do the same with the other strip. Each strip has 4 equal parts.
Lay the strips end to end. Write 0 at the left end.
Count 4 parts from 0. Where the first strip ends, write 1.
Count one more part. Write 5/4 at the end of that part.
What you will see: The fifth part is on the second strip. So 5/4 is past 1.
If you cannot try it, this is what happens.
A wrong idea many people have
A fraction is always less than 1.
Why it feels right: Most fractions we cut are pieces of one whole.
The test that breaks it: Cut two breads of the same size, each into 4 equal pieces. Take 5 pieces. You hold more than one whole bread.
What is true: A fraction can be less than 1. It can equal 1. It can be more than 1. If the top is bigger than the bottom, it is past 1.
Key words
number line
a line where every number has its own place.
steps
the equal parts of a whole, on a number line.
Back to your guess
The answer is B: Yes. Count more parts than one whole has. You go past 1.
A fraction is a number. It has a place on the number line. Five quarters sits after 1.
Check yourself
Try each one before you look. A wrong answer tells you where to go next.
Count the steps from 0. Which point is at 5/4?
See the answer
Point B
B is 5 steps from 0, and each step is 1/4. So B is 5/4.
If you chose A: A is 3 steps from 0. That is 3/4, before 1.
If you chose C: C is 7 steps from 0. That is 7/4, past 1 but too far.
Is 6/4 before 1, at 1, or past 1?
See the answer
Past 1
4 steps of 1/4 reach 1. 6/4 has 6 steps, so it is past 1.
If you chose A: You may think a fraction is always small. Here the top, 6, is bigger than the bottom, 4.
If you chose B: Only 4/4 is at 1. 6/4 is two more steps along.
Each whole is cut into 3 equal steps. Point P is the first tick after 1. What fraction is P?
See the answer
4/3
1 is 3 steps from 0. One more step makes 4 steps. Each step is 1/3, so P is 4/3.
From an earlier chapter
Amina eats 3 of the 4 equal parts of a bread. What fraction of the bread did she eat?
See the answer
3/4
The bread has 4 equal parts, so the bottom is 4. She ate 3, so the top is 3.
If you chose B: That is one part. She ate 3 parts. Go to M2.4.3.
If you chose C: The bottom number is how many equal parts the whole has. The whole has 4. Go to M2.4.3.
Draw a number line from 0 to 2. Cut each whole in half. Mark 3/2.
See the answer
Mark 0, 1 and 2. Mark the middle of each whole. Count 3 half steps from 0. You land on 3/2, halfway between 1 and 2.
Each step is 1/2. Two steps reach 1, and one more step makes 3/2.
What fraction is point Q?
See the answer
5/2
Each step is 1/2. Q is 5 steps from 0, so Q is 5/2.
If you chose A: The bottom number is how many steps make one whole. That is 2. The top number counts the steps from 0.
If you chose C: You counted the tick at 0 as a step. Count the steps after 0.
Which of these fractions is past 1?
See the answer
9/6
6 steps of 1/6 reach 1. 9/6 has 9 steps, so it is past 1.
If you chose A: 5 steps of 1/8 do not reach 1. You need 8 steps.
If you chose B: 6 steps of 1/6 reach 1 exactly. It is at 1, not past it.
Kofi counts steps of 1/3 from 0 and lands on 5/3. How many steps does he count?
See the answer
5 steps
The top number counts the steps from 0. 5/3 is 5 steps of 1/3.
If you chose B: There are 3 steps in each whole. Kofi went past 1, so he counted more than 3.
If you chose C: There are 2 steps past 1. Kofi counted every step from 0.
Priya says 7/4 is past 1. She says the top is bigger than the bottom. Is she right?
See the answer
Yes. The top is bigger than the bottom.
4 steps of 1/4 reach 1. 7/4 has 7 steps, so it is past 1.
If you chose A: A fraction can be more than 1. 7 steps of 1/4 go past the 4 steps that reach 1.
If you chose C: Only 4 quarters make one whole, 4/4. 7/4 is 3 more steps along, so it is past 1.
Keep it
Copy this into your exercise book.
A fraction is a number with a place on the number line. Cut each whole into equal steps. Count the steps from 0. Top bigger than bottom? You are past 1.
Draw: A line from 0 to 2. Cut each whole into 4 steps. Mark 5/4, just past 1.
M2.4.6 Chapter 6
Same amount, different names
Is 2/4 more than 1/2, since the numbers are bigger?
Your guess
Pick one before you read on. It is fine to be wrong.
By the end you can
Show that two fractions name the same amount
Find a fraction equal to 1/2, 1/3 or 3/4
Explain why 2/4 is not more than 1/2
Cut a bread into 2 equal halves. Then cut each half into 2 equal pieces. Now you have 4 pieces in all.
You still have the same bread. It has the same size. Only the cuts are new. One half is now 2 of the 4 pieces.
Fractions that name the same amount are equal fractions. We write 1/2 = 2/4.
Draw it first
Cut each part in two
1 Draw a bar. Cut it in 2 equal parts. Shade 1 part. That is 1/2.
2 Draw a bar the same length. Cut it in 4. Shade 2 parts. That is 2/4.
3 Draw one more bar the same length. Cut it in 8. Shade 4 parts. That is 4/8.
Bars with the same length have the same whole. Look down the bars. Each shaded part ends in the same place.
1/3 = 2/6. Cut each third in 2, and 1 part becomes 2 parts.On equal lines, 1/2 and 2/4 are in the same place.
See it three ways
With things
Take two breads of the same size. Cut one in 2 and one in 4. Put 1 piece of the first by 2 pieces of the second.
As a picture
In symbols
1/2 = 2/4 = 4/8. Each time, the top and the bottom both double.
Worked example
Make a fraction equal to 2/3 that has 6 parts.
Cut each third in 2. The 3 parts become 6 parts.
Each shaded part is cut in 2 too. The 2 shaded parts become 4.
The bottom doubled, from 3 to 6. The top doubled, from 2 to 4.
Answer:2/3 = 4/6
Try it
Two strips, one amount
You need: two strips of paper of the same length, a pencil.
Fold the first strip in half. Open it. Shade one half.
Fold the second strip in half, then in half again. Open it.
Shade 2 of the 4 parts on the second strip.
Put the strips side by side, with the left ends together.
What you will see: Both shaded parts end in the same place. So 1/2 and 2/4 are the same amount.
If you cannot try it, this is what happens.
A wrong idea many people have
2/4 is more than 1/2, because 2 and 4 are bigger numbers.
Why it feels right: With whole numbers, a bigger number means more.
The test that breaks it: Make the two strips from the Try it. Lay them side by side. The shaded parts end in the same place. Neither sticks out.
What is true: Cutting parts smaller gives more parts. The amount does not change. 2/4 is the same amount as 1/2.
Key words
equal fractions
fractions that name the same amount.
same whole
bars or lines that have the same length.
Back to your guess
The answer is B: No. They are the same amount.
Two quarters is the same amount as one half. Bigger numbers do not mean more. The parts are smaller, and there are more of them.
Check yourself
Try each one before you look. A wrong answer tells you where to go next.
Which fraction is equal to 1/2?
See the answer
2/4
Cut each half in 2. The bottom doubles from 2 to 4. The top doubles from 1 to 2.
If you chose B: One part of 4 is less than a half. A half is 2 parts of 4.
If you chose C: Adding 1 to both numbers changes the amount. Double both numbers.
This bar has 3 equal parts, with 1 shaded. Cut each part in 2. How many parts are shaded now?
See the answer
2 of 6, which is 2/6
Each third becomes 2 parts, so the bar has 6 parts. The shaded third becomes 2 parts.
Is 3/6 equal to 1/2?
See the answer
Yes. Both name the same amount.
3 parts is half of 6 parts. So 3/6 is the same amount as 1/2.
If you chose B: Bigger numbers do not mean more. Fold a strip in 6 and shade 3. Compare it with a half.
From an earlier chapter
A bread is cut into 4 equal parts. Mei eats one part. What fraction of the bread is that?
See the answer
1/4
The bread has 4 equal parts, so the bottom is 4. She ate 1 part.
If you chose B: You may have counted the parts left. The bottom number is all the equal parts. The bread has 4. Go to M2.4.2.
If you chose C: The bottom number is how many equal parts. The top is how many you take. Go to M2.4.2.
Fill the gap. 3/4 = ?/8. What number goes in place of ?
See the answer
6
Cut each of the 4 parts in 2. Now there are 8 parts. Each shaded part becomes 2, so 3 parts become 6.
Look at the bars. Is 2/4 more than 1/2?
See the answer
No. They are equal. The shaded parts have the same length.
Both shaded parts end in the same place. So they are the same amount.
If you chose A: Look at the length, not the number of parts. The parts are smaller, so 2 of them match 1 half.
If you chose C: The parts are smaller, but there are more of them. Together they fill the same length.
Which fraction is equal to 1/3?
See the answer
2/6
Cut each third in 2. The bottom doubles from 3 to 6. The top doubles from 1 to 2.
If you chose A: That is one part of 6, a smaller amount. When the bottom doubles, the top must double too.
If you chose B: Adding 1 to both numbers changes the amount. 2/4 is the same as 1/2.
This bar has 4 equal parts, with 3 shaded. Cut each part in 2. What fraction is shaded now?
See the answer
6/8
The bar has 8 parts now. Each shaded part becomes 2, so 3 shaded parts become 6.
If you chose B: You doubled the bottom number but not the top. Each shaded part becomes 2, so 3 shaded parts become 6.
If you chose C: Adding 1 to both numbers changes the amount. When each part is cut in 2, both numbers double.
Which pair of fractions names the same amount?
See the answer
2/6 and 1/3
Cut each third in 2 and 1/3 becomes 2/6. The amount stays the same.
If you chose A: Both have 1 part, but 1/8 is a smaller part. They are not the same amount.
If you chose C: Adding 1 to both numbers changes the amount. 3/4 is more than 2/3.
Keep it
Copy this into your exercise book.
Different fractions can name the same amount. 1/2, 2/4 and 4/8 are equal. Cut every part in two. The top and the bottom both double. The amount stays the same.
Draw: Three bars of the same length, cut in 2, 4 and 8. Shade the same amount in each.
M2.4.7 Chapter 7
Which is bigger
Which is more, 1/4 or 1/8, since 8 is bigger than 4?
Your guess
Pick one before you read on. It is fine to be wrong.
By the end you can
Compare two fractions with the same bottom number
Compare two fractions with the same top number
Explain why 1/8 is less than 1/4
To compare two fractions is to find which one is more. Or to find that they are the same.
Always draw bars with the same whole. Then the bars have the same length, and the picture is fair.
Draw it first
Compare 1/4 and 1/8
1 Draw two bars of the same length. Each bar is one whole.
2 Cut one bar in 4 equal parts. Cut the other in 8 equal parts.
3 Shade one part in each bar. The 1/8 part is shorter than the 1/4 part.
So 1/8 is less than 1/4. More parts in the whole means smaller parts.
First rule. The same bottom number means parts of the same size. More parts is more.
The parts are the same size. 5 parts are more than 3 parts.On the number line, further right means more.
Second rule. When the top numbers are the same, look at the bottom numbers.
Both show 3 parts. The parts of 3/4 are bigger.
A bigger bottom number means smaller parts. So 3/8 is less than 3/4.
See it three ways
With things
Take two breads of the same size. Cut one in 4 and one in 8. Take 1 piece from each.
As a picture
In symbols
1/4 > 1/8. The open side of the sign faces the bigger one.
Try it
Fold two strips
You need: two strips of paper of the same length, a pencil.
Fold the first strip in half. Fold it in half again. Open it. It has 4 parts.
Fold the second strip in half 3 times. Open it. It has 8 parts.
Shade one part on each strip.
Put the strips side by side, with the left ends together.
What you will see: The shaded part of the second strip is half as long. So 1/8 is less than 1/4.
If you cannot try it, this is what happens.
A wrong idea many people have
1/8 is more than 1/4, because 8 is bigger than 4.
Why it feels right: With whole numbers, 8 is more than 4. The same rule seems to work here.
The test that breaks it: Share one bread between 4 people. Share another bread of the same size between 8 people. Ask who gets the bigger piece.
What is true: The bottom number is how many parts the whole is cut into. More parts means smaller parts. So 1/8 is less than 1/4.
Key words
compare
to find which is more, or that they are the same.
same whole
bars or lines that have the same length.
Back to your guess
The answer is B: 1/4, because its parts are bigger.
The bottom number tells how many parts the whole is cut into. More parts means smaller parts. So 1/4 is more than 1/8.
Check yourself
Try each one before you look. A wrong answer tells you where to go next.
Which is more, 3/8 or 5/8?
See the answer
5/8
Both are eighths, so the parts are the same size. 5 of them are more than 3 of them.
If you chose A: The parts are the same size. 5 parts is more than 3 parts. Go to M2.4.3.
Which is less, 1/3 or 1/6?
See the answer
1/6
Sixths are smaller parts than thirds. So 1/6 is less.
If you chose A: You may think 6 is bigger, so 1/6 is more. More parts means smaller parts. Go to M2.4.2.
From an earlier chapter
Which fraction is equal to 1/3?
See the answer
2/6
Cut each third in 2. The bottom doubles from 3 to 6. The top doubles from 1 to 2.
If you chose B: Cut each third in 2. The shaded third becomes 2 parts, not 1. Go to M2.4.6.
If you chose C: Changing only the top changes the amount. Double both numbers. Go to M2.4.6.
Put these in order, from least to most: 1/2, 1/8, 1/4.
See the answer
1/8, 1/4, 1/2
All three have the top 1. The bigger the bottom number, the smaller the part. So 1/8 is the least and 1/2 is the most.
Kofi eats 3/4 of a bread. Lucía eats 3/8 of a bread of the same size. Who ate more?
See the answer
Kofi
Both ate 3 parts. A quarter is bigger than an eighth, so Kofi ate more.
If you chose B: Each has 3 parts. But eighths are smaller than quarters. Go to M2.4.2.
If you chose C: Both have 3 parts, but the parts are not the same size. Go to M2.4.2.
Ravi has 5/8 of a bread. Amina has 3/8 of a bread of the same size. Who has more?
See the answer
Ravi
Both are eighths, so the parts are the same size. 5 of them are more than 3.
If you chose B: Bigger number, smaller part is a rule for the bottom number. Here both bottoms are 8, so the top decides. 5 parts are more than 3.
If you chose C: Both are eighths. But Ravi has 5 of them and Amina has 3.
Which is more, 3/5 or 3/10?
See the answer
3/5
Both have 3 parts. Fifths are bigger parts than tenths, so 3/5 is more.
If you chose A: Tenths are smaller than fifths. 3 small parts are less than 3 bigger parts.
If you chose B: Both have 3 parts, but the parts are not the same size. Fifths are bigger than tenths.
Mei says 1/8 is more than 1/4. Her reason is that 8 is more than 4. Is she right?
See the answer
No. More parts in the whole means smaller parts.
Eighths are smaller parts than quarters. So 1/8 is less than 1/4.
If you chose A: That works for whole numbers. In a fraction, the bottom number counts the parts. More parts are smaller.
If you chose C: The top is the same, but the bottom numbers are not. An eighth is a smaller part than a quarter.
Put 3/4, 3/5 and 3/8 in order, least to most. Which list is right?
See the answer
3/8, 3/5, 3/4
All three have 3 parts. The bigger the bottom number, the smaller the part. So 3/8 is the least.
If you chose B: That puts the smallest bottom number first. Eighths are the smallest parts, so 3/8 is the least.
If you chose C: Each has 3 parts, but the parts are not the same size. Quarters, fifths and eighths differ.
Keep it
Copy this into your exercise book.
Compare fractions on bars of the same whole. Same bottom number: more parts is more. Same top number: a bigger bottom number means smaller parts. So 1/8 is less than 1/4.
Draw: Two bars of the same length. Cut one in 4 and one in 8. Shade one part in each.
M2.4.8 Chapter 8
Adding and taking away parts
My friend says 1/4 + 1/4 is 2/8. Is that right?
Your guess
Pick one before you read on. It is fine to be wrong.
By the end you can
Add fractions that have the same bottom number
Take away fractions that have the same bottom number
Show adding and taking away as hops on a number line
Two quarters of a bread make half the bread. Let us see why.
In this chapter the bottom numbers are always the same. Parts with the same bottom number are like parts.
You can count like parts as you count beans. 1 quarter and 2 quarters make 3 quarters.
Draw it first
Add 1/4 and 2/4 with hops
1 Draw a number line from 0 to 1. Cut it into quarters.
2 Start at 0. Make a hop of 1/4 to the right.
3 Hop 2/4 more. You land on 3/4. So 1/4 + 2/4 = 3/4.
The hops are all quarters. So the bottom number stays 4. Only the tops are added.
Worked example
Add 1/4 + 2/4.
The bottom numbers are both 4. The parts are the same size.
Add the top numbers. 1 + 2 = 3.
Keep the bottom number. It is still 4.
Answer:3/4
To take away, hop to the left. Start at 5/6 and take away 2/6.
Hop left by 2/6. You land on 3/6.
Worked example
Take away 5/6 − 2/6.
The bottom numbers are both 6.
Take away the top numbers. 5 − 2 = 3.
Keep the bottom number. It is still 6.
Answer:3/6. That is the same amount as 1/2.
Sometimes you hop past 1. Then the top is bigger than the bottom. For example, 3/4 + 2/4 = 5/4.
Now back to the question. Is 1/4 + 1/4 the same as 2/8? Look at the bars.
1/4 + 1/4 = 2/4. Two quarters cover half the bar. Two eighths cover only a quarter.
See it three ways
With things
Cut a bread in 4 equal pieces. Eat 1 piece. Then eat 2 more. You ate 3 pieces of 4.
As a picture
In symbols
1/4 + 2/4 = 3/4. Add the tops. Keep the bottom.
Try it
Add on a paper strip
You need: a strip of paper, a pencil.
Fold the strip in half. Fold it in half again. Open it. It has 4 equal parts.
Shade 1 part.
Shade 2 more parts, next to the first.
Count the shaded parts.
What you will see: You shaded 3 of the 4 parts. So 1/4 + 2/4 = 3/4. Every part is a quarter, so the bottom stays 4.
If you cannot try it, this is what happens.
A wrong idea many people have
1/4 + 1/4 = 2/8.
Why it feels right: You add the tops, 1 + 1. Then you add the bottoms, 4 + 4. Adding everything seems fair.
The test that breaks it: Cut a bread in 4 equal parts. Eat 1 part, then 1 more. You ate half the bread. Now cut a bread in 8. Two of those pieces are only a quarter.
What is true: The parts stay quarters, so the bottom stays 4. 1/4 + 1/4 = 2/4. That is 1/2.
Key words
like parts
parts that have the same bottom number.
hop
a jump along the number line.
Back to your guess
The answer is B: 2/4. Add the tops and keep the bottom.
Two quarters of a bread are half the bread. That is 2/4. And 2/8 is only a quarter. Add the tops and keep the bottom.
Check yourself
Try each one before you look. A wrong answer tells you where to go next.
Add. 2/7 + 3/7
See the answer
5/7
Add the tops. 2 + 3 = 5. Keep the bottom, 7.
If you chose B: You added the bottoms too. The parts stay sevenths, so the bottom stays 7.
If you chose C: You may have multiplied the tops. Add them. 2 + 3 = 5.
Take away. 5/6 − 2/6
See the answer
3/6
5 − 2 = 3. Keep the bottom, 6. The answer 3/6 is also one half.
Ravi hops along this line. He hops 3/4, then 2/4. Where does he land?
See the answer
5/4
3 + 2 = 5, and the bottom stays 4. So he lands on 5/4. That is past 1.
From an earlier chapter
What is 1/3 of 12 stones?
See the answer
4 stones
Share 12 stones into 3 equal groups. Each group has 4. 4 + 4 + 4 = 12.
A jug has 7/8 litre of water. Ana pours out 3/8 litre. How much is left?
See the answer
4/8 litre
7 − 3 = 4. The bottom stays 8. 4/8 is the same as one half litre.
Leila eats 1/6 of a bread. Then she eats 3/6 more. How much has she eaten in all?
See the answer
4/6
Add the tops. 1 + 3 = 4. Keep the bottom, 6.
If you chose A: You added the bottoms too. All the parts are sixths, so the bottom stays 6.
If you chose C: That is only the second part she ate. In all means add both parts. 1 + 3 = 4.
Yusuf has 5/8 of a bread. He gives 2/8 away. How much is left?
See the answer
3/8
Take away the tops. 5 − 2 = 3. Keep the bottom, 8.
If you chose A: You added. Giving away is taking away, so do 5 − 2.
If you chose B: That is the part Yusuf gave away. The question asks what is left.
Hana starts at 5/6 and hops left by 3/6. Where does she land?
See the answer
2/6
A hop to the left takes away. 5 − 3 = 2. The bottom stays 6.
If you chose B: You hopped to the right. A hop to the left takes away.
If you chose C: That is the size of the hop. Where she lands is 5 − 3 = 2 sixths.
Mei hops to the left on the line. Which number sentence shows her hop?
See the answer
3/4 − 2/4 = 1/4
Mei starts at 3/4 and hops left 2/4. 3 − 2 = 1. The bottom stays 4.
If you chose A: The hop goes to the left, so it takes away. A hop to the right adds.
If you chose C: You wrote the size of the hop. 3 − 2 = 1, so she lands on 1/4.
Keep it
Copy this into your exercise book.
Same bottom number: add or take away the tops. Keep the bottom. 1/4 + 2/4 = 3/4. Two quarters of a bread make half the bread. 1/4 + 1/4 is 2/4, never 2/8.
Draw: A number line in quarters. Hop 1/4, then hop 2/4, and land on 3/4.
Key words, A to Z
Every key word in this book, what it means, and the chapter that teaches it.
bottom number
the number under the line. It tells how many equal parts the whole is cut into. M2.4.2
compare
to find which is more, or that they are the same. M2.4.7
denominator
the bottom number. It tells how many equal parts make the whole. M2.4.3