Statistics
By the end of this chapter you will be able to
- Count the objects in each group and say how many there are
- Read a tally chart, with groups of five marks
- Read a pictogram, where each picture stands for one thing
- Read a block diagram, where each block stands for one thing
- Read a simple table and find a number in it
- Find the most, the fewest and the total, and compare two counts
You should already know
- Counting objects and reading numbers to 100 (Chapter 1)
- Adding and subtracting to 20 and 100 (Chapter 2)
- Counting in 2s, 5s and 10s (Chapter 3)
- Shapes and their sides (Chapter 8)
Where this leads
- Drawing and interpreting bar charts and pictograms with a key (Year 3, Chapter 9)
- Organising and presenting data in tables and charts (later years)
National Curriculum for England, Key Stage 1 mathematics, Year 2. Content: Interpret and construct simple pictograms, tally charts, block diagrams and simple tables, Ask and answer simple questions by counting the number of objects in each category and sorting the categories by quantity, Ask and answer questions about totalling and comparing categorical data.
Exercise P Do you remember?
Non-calculatorMethod. Count the cubes once each, touching each one.
- Touch each one once. Say the number as you touch each one.1 2 3 4 5 6 7
- Read the last number. The last number is how many there are.7
- Write the answer. 7
Check. Count them again from the other end. You get 7 again. ✓
Watch out. Move each one to one side as you count it, so you do not count it twice.
Method. Find each count, then add them all together.
- Write down each count. Each count is one row.Ravi 6, Mia 3
- Add the counts together. Add them all, so none is missed.6 + 3 = 9
- Write the answer. 9
Check. Add them in the other order to check: 3 + 6 = 9. ✓
Watch out. Make sure every row is in your total, and no row is counted twice.
Method. Compare the counts. The biggest count is the most.
- Write down each count. Put each count next to its name.Sam 8, Ali 5
- Find the biggest count. 8 is the biggest count.biggest: 8
- Write the answer. The Sam row has 8.Sam
Check. Every other count is smaller than 8. ✓
Watch out. Check every count, not only the first two. The most is the count that is bigger than all the others.
Method. Count on in tens. Each jump is 10.
- Look at the numbers. Each number is 10 more than the one before.30, 40, 50
- Add 10 to the last number. The next number is 10 more than 50.50 + 10 = 60
- Write the answer. 60
Check. Count back in tens from 60: 60, 50, 40, 30. ✓
Watch out. Jump by 10 each time. Do not jump by 1.
Method. Count the sides of the shape.
- Count the straight sides. A square has four sides that are all the same length.4 sides
- Name the shape. A flat shape with 4 equal sides is a square.square
Check. Its four corners are square corners. ✓
Watch out. A rectangle has 4 sides too, but its long sides are not the same length as its short sides.
Sorting and counting into groups
We can sort things into groups. Each group holds one kind of thing.
To find out how many are in a group, we count it. Touch each one once as you count.
When we know every count, we can compare them. We can say which group has the most. We can also say which group has the fewest.
Key ideas
- A group holds one kind of thing.
- Count each object once, and only once.
- The group with the most objects has the most.
- Put the groups in order, from fewest to most, using the counts.
Method. Count the coats once each, touching each one.
- Touch each one once. Say the number as you touch each one.1 2 3 4 5 6
- Read the last number. The last number is how many there are.6
- Write the answer. 6
Check. Count them again from the other end. You get 6 again. ✓
Watch out. Move each one to one side as you count it, so you do not count it twice.
Method. Compare the counts. The smallest count is the fewest.
- Write down each count. Put each count next to its name.hats 3, coats 6, socks 4
- Find the smallest count. 3 is the smallest count.smallest: 3
- Write the answer. The hats row has 3.hats
Check. Every other count is bigger than 3. ✓
Watch out. Check every count, not only the first two. The fewest is the count that is smaller than all the others.
Method. Find each count, then put the rows in order of their counts, fewest first.
- Write down each count. Put each count next to its name.hats 3, coats 6, socks 4
- Put the counts in order, fewest first. Start from the smallest count.3 → 4 → 6
- Write the order. hats, socks, coats
Check. Check that each count is bigger than the one before it. ✓
Watch out. Compare the counts, not the names. The longest name does not mean the biggest count.
Watch out
Watch out: a group that looks big may have fewer objects. Count every one before you compare.
Exam tip
Grown-up tip. Put out about 20 buttons or coins in small piles. Ask: “How many in this pile?” Then ask: “Which pile has the most? Which has the fewest?”
Exercise 10A Sorting and counting into groups
Non-calculatorMethod. Count the balls once each, touching each one.
- Touch each one once. Say the number as you touch each one.1 2 3 4 5
- Read the last number. The last number is how many there are.5
- Write the answer. 5
Check. Count them again from the other end. You get 5 again. ✓
Watch out. Move each one to one side as you count it, so you do not count it twice.
Method. Count the teddy bears once each, touching each one.
- Touch each one once. Say the number as you touch each one.1 2
- Read the last number. The last number is how many there are.2
- Write the answer. 2
Check. Count them again from the other end. You get 2 again. ✓
Watch out. Move each one to one side as you count it, so you do not count it twice.
Method. Count the kites once each, touching each one.
- Touch each one once. Say the number as you touch each one.1 2 3 4 5 6 7
- Read the last number. The last number is how many there are.7
- Write the answer. 7
Check. Count them again from the other end. You get 7 again. ✓
Watch out. Move each one to one side as you count it, so you do not count it twice.
Method. Count the cars once each, touching each one.
- Touch each one once. Say the number as you touch each one.1 2 3 4
- Read the last number. The last number is how many there are.4
- Write the answer. 4
Check. Count them again from the other end. You get 4 again. ✓
Watch out. Move each one to one side as you count it, so you do not count it twice.
Method. Compare the counts. The biggest count is the most.
- Write down each count. Put each count next to its name.balls 5, teddy bears 2, kites 7, cars 4
- Find the biggest count. 7 is the biggest count.biggest: 7
- Write the answer. The kites row has 7.kites
Check. Every other count is smaller than 7. ✓
Watch out. Check every count, not only the first two. The most is the count that is bigger than all the others.
Method. Compare the counts. The smallest count is the fewest.
- Write down each count. Put each count next to its name.balls 5, teddy bears 2, kites 7, cars 4
- Find the smallest count. 2 is the smallest count.smallest: 2
- Write the answer. The teddy bears row has 2.teddy bears
Check. Every other count is bigger than 2. ✓
Watch out. Check every count, not only the first two. The fewest is the count that is smaller than all the others.
Method. Find each count. Then take the smaller count away from the bigger one.
- Count the kites. There are 7.kites: 7
- Count the cars. There are 4.cars: 4
- Take the smaller count away from the bigger one. The difference is how many more or fewer.7 - 4 = 3
- Write the answer. 3
Check. Add the difference back on: 4 + 3 = 7. ✓
Watch out. Take away the smaller number from the bigger one. Do not add them.
Method. Find each count. Then take the smaller count away from the bigger one.
- Count the balls. There are 5.balls: 5
- Count the teddy bears. There are 2.teddy bears: 2
- Take the smaller count away from the bigger one. The difference is how many more or fewer.5 - 2 = 3
- Write the answer. 3
Check. Add the difference back on: 2 + 3 = 5. ✓
Watch out. Take away the smaller number from the bigger one. Do not add them.
Method. Find each count, then put the rows in order of their counts, fewest first.
- Write down each count. Put each count next to its name.balls 5, teddy bears 2, kites 7, cars 4
- Put the counts in order, fewest first. Start from the smallest count.2 → 4 → 5 → 7
- Write the order. teddy bears, cars, balls, kites
Check. Check that each count is bigger than the one before it. ✓
Watch out. Compare the counts, not the names. The longest name does not mean the biggest count.
Method. Find each count, then add them all together.
- Write down each count. Each count is one row.teddy bears 2, kites 7
- Add the counts together. Add them all, so none is missed.2 + 7 = 9
- Write the answer. 9
Check. Add them in the other order to check: 7 + 2 = 9. ✓
Watch out. Make sure every row is in your total, and no row is counted twice.
Method. Take the 2 away from the kites, then compare with the cars.
- Count the kites. There are 7.kites: 7
- Take 2 away. 7 - 2 = 5.7 - 2 = 5
- Compare with the cars. There are 4 cars.5 and 4 are different
- Decide. Is Ali right?No
Check. Count the kites again after taking 2 away, to check: 5. ✓
Watch out. Take the number away first. Then compare the two numbers.
Tally charts
A tally chart counts with marks. Each mark stands for one thing.
We make a group of five marks. Four marks go up, and one mark goes across them.
To count a tally, count the full groups of five first. Then count the marks left over.
Key ideas
- Each tally mark stands for one child.
- Five marks make one full group of five.
- Count the full groups of five first, then the marks left over.
- The row with the most marks is the most common way to travel.
Method. Count the bike marks in groups of five, then count the marks left over.
- Count the full groups of five. Each group has four upright marks and one line across them.2 groups of five = 10 marks
- Count the marks left over. These are not in a full group.1 left over
- Add them together. 10 and 1 make 11.10 + 1 = 11
- Write the answer. 11
Check. Count every mark one at a time, and you also get 11. ✓
Watch out. Count the groups of five first. Then count the single marks that are left over.
Method. Compare the counts. The smallest count is the fewest.
- Write down each count. Put each count next to its name.bikes 11, walk 8, bus 6
- Find the smallest count. 6 is the smallest count.smallest: 6
- Write the answer. The bus row has 6.bus
Check. Every other count is bigger than 6. ✓
Watch out. Check every count, not only the first two. The fewest is the count that is smaller than all the others.
Method. Find each count. Then take the smaller count away from the bigger one.
- Count the bikes. There are 11.bikes: 11
- Count the walk. There are 8.walk: 8
- Take the smaller count away from the bigger one. The difference is how many more or fewer.11 - 8 = 3
- Write the answer. 3
Check. Add the difference back on: 8 + 3 = 11. ✓
Watch out. Take away the smaller number from the bigger one. Do not add them.
Watch out
Watch out: count the marks in each group of five, one at a time. A full group always has five marks.
Exam tip
Grown-up tip. Make tally marks on paper for the number of socks in a pair of drawers. Ask: “How many full groups of five? How many left over?” Then count all the marks together.
Exercise 10B Tally charts
Non-calculatorMethod. Count the walk marks in groups of five, then count the marks left over.
- Count the full groups of five. Each group has four upright marks and one line across them.1 group of five = 5 marks
- Count the marks left over. These are not in a full group.4 left over
- Add them together. 5 and 4 make 9.5 + 4 = 9
- Write the answer. 9
Check. Count every mark one at a time, and you also get 9. ✓
Watch out. Count the groups of five first. Then count the single marks that are left over.
Method. Count the bike marks in groups of five, then count the marks left over.
- Count the full groups of five. Each group has four upright marks and one line across them.no full group = 0 marks
- Count the marks left over. These are not in a full group.4 left over
- Add them together. 0 and 4 make 4.0 + 4 = 4
- Write the answer. 4
Check. Count every mark one at a time, and you also get 4. ✓
Watch out. Count the groups of five first. Then count the single marks that are left over.
Method. Count the car marks in groups of five, then count the marks left over.
- Count the full groups of five. Each group has four upright marks and one line across them.2 groups of five = 10 marks
- Count the marks left over. These are not in a full group.3 left over
- Add them together. 10 and 3 make 13.10 + 3 = 13
- Write the answer. 13
Check. Count every mark one at a time, and you also get 13. ✓
Watch out. Count the groups of five first. Then count the single marks that are left over.
Method. Count the bus marks in groups of five, then count the marks left over.
- Count the full groups of five. Each group has four upright marks and one line across them.1 group of five = 5 marks
- Count the marks left over. These are not in a full group.2 left over
- Add them together. 5 and 2 make 7.5 + 2 = 7
- Write the answer. 7
Check. Count every mark one at a time, and you also get 7. ✓
Watch out. Count the groups of five first. Then count the single marks that are left over.
Method. Share the 13 marks into groups of five. Count the full groups.
- Make groups of five. Each full group has five marks.13 marks = 2 groups of five and 3 left
- Count the full groups. Only full groups of five count here.2 full groups
- Write the answer. 2
Check. 2 groups of five make 10 marks, and 3 are left over. ✓
Watch out. A group that is not full is not counted as a group.
Method. Share the 13 marks into groups of five. Count the marks left over.
- Make groups of five. Each full group has five marks.13 marks = 2 groups of five and 3 left
- Count the marks left over. These are not in a full group.3 left over
- Write the answer. 3
Check. 10 marks in full groups and 3 left over make 13. ✓
Watch out. Count only the marks that are not in a full group of five.
Method. Compare the counts. The biggest count is the most.
- Write down each count. Put each count next to its name.walk 9, bike 4, car 13, bus 7
- Find the biggest count. 13 is the biggest count.biggest: 13
- Write the answer. The car row has 13.car
Check. Every other count is smaller than 13. ✓
Watch out. Check every count, not only the first two. The most is the count that is bigger than all the others.
Method. Compare the counts. The smallest count is the fewest.
- Write down each count. Put each count next to its name.walk 9, bike 4, car 13, bus 7
- Find the smallest count. 4 is the smallest count.smallest: 4
- Write the answer. The bike row has 4.bike
Check. Every other count is bigger than 4. ✓
Watch out. Check every count, not only the first two. The fewest is the count that is smaller than all the others.
Method. Find each count. Then take the smaller count away from the bigger one.
- Count the car. There are 13.car: 13
- Count the bike. There are 4.bike: 4
- Take the smaller count away from the bigger one. The difference is how many more or fewer.13 - 4 = 9
- Write the answer. 9
Check. Add the difference back on: 4 + 9 = 13. ✓
Watch out. Take away the smaller number from the bigger one. Do not add them.
Method. Add the counts for each pair of rows. Find the pair that makes the total.
- List the pairs and their totals. Add the counts for each pair.walk 9 + bike 4 = 13; walk 9 + car 13 = 22; walk 9 + bus 7 = 16; bike 4 + car 13 = 17; bike 4 + bus 7 = 11; car 13 + bus 7 = 20
- Find the pair that makes 16. Only walk and bus make 16.16 = 16
- Write the answer. walk and bus
Check. walk and bus together make 16, and no other pair makes 16. ✓
Watch out. Check every pair. Two rows that are both bigger than the target cannot make it.
Method. Count the children who walked, then compare the number with what is said.
- Count the children who walked. There are 9.9
- Compare with what is said. Ben says 10.9 is not the same as 10
- Decide. Is Ben right?No
Check. Count them again to check: 9. ✓
Watch out. Count every one, then compare. Do not guess from the picture.
Pictograms
A pictogram uses pictures to show counts. Each picture stands for one thing.
Each row has a name above it. Count the pictures in the row to find the count.
A row with more pictures has the bigger count.
Key ideas
- Each picture stands for one thing.
- Read the name above each row.
- Count the pictures in one row, then go to the next row.
- A row with more pictures has a bigger count.
Method. Count the ducks once each, touching each one.
- Touch each one once. Say the number as you touch each one.1 2 3 4
- Read the last number. The last number is how many there are.4
- Write the answer. 4
Check. Count them again from the other end. You get 4 again. ✓
Watch out. Move each one to one side as you count it, so you do not count it twice.
Method. Find each count. Then take the smaller count away from the bigger one.
- Count the hens. There are 7.hens: 7
- Count the cows. There are 2.cows: 2
- Take the smaller count away from the bigger one. The difference is how many more or fewer.7 - 2 = 5
- Write the answer. 5
Check. Add the difference back on: 2 + 5 = 7. ✓
Watch out. Take away the smaller number from the bigger one. Do not add them.
Method. Find each count, then add them all together.
- Write down each count. Each count is one row.ducks 4, hens 7, cows 2
- Add the counts together. Add them all, so none is missed.4 + 7 + 2 = 13
- Write the answer. 13
Check. Add them in the other order to check: 2 + 7 + 4 = 13. ✓
Watch out. Make sure every row is in your total, and no row is counted twice.
Watch out
Watch out: each picture stands for one pet here. Count the pictures, not the rows.
Exam tip
Grown-up tip. Draw a pictogram of the pets, toys or fruit at home. Use one drawing for each one. Ask: “Which row has the most? How many pictures in that row?”
Exercise 10C Pictograms
Non-calculatorMethod. Count the cats once each, touching each one.
- Touch each one once. Say the number as you touch each one.1 2 3 4 5
- Read the last number. The last number is how many there are.5
- Write the answer. 5
Check. Count them again from the other end. You get 5 again. ✓
Watch out. Move each one to one side as you count it, so you do not count it twice.
Method. Count the dogs once each, touching each one.
- Touch each one once. Say the number as you touch each one.1 2 3
- Read the last number. The last number is how many there are.3
- Write the answer. 3
Check. Count them again from the other end. You get 3 again. ✓
Watch out. Move each one to one side as you count it, so you do not count it twice.
Method. Count the rabbits once each, touching each one.
- Touch each one once. Say the number as you touch each one.1 2 3 4 5 6
- Read the last number. The last number is how many there are.6
- Write the answer. 6
Check. Count them again from the other end. You get 6 again. ✓
Watch out. Move each one to one side as you count it, so you do not count it twice.
Method. Count the fish once each, touching each one.
- Touch each one once. Say the number as you touch each one.1 2
- Read the last number. The last number is how many there are.2
- Write the answer. 2
Check. Count them again from the other end. You get 2 again. ✓
Watch out. Move each one to one side as you count it, so you do not count it twice.
Method. Compare the counts. The biggest count is the most.
- Write down each count. Put each count next to its name.cats 5, dogs 3, rabbits 6, fish 2
- Find the biggest count. 6 is the biggest count.biggest: 6
- Write the answer. The rabbits row has 6.rabbits
Check. Every other count is smaller than 6. ✓
Watch out. Check every count, not only the first two. The most is the count that is bigger than all the others.
Method. Compare the counts. The smallest count is the fewest.
- Write down each count. Put each count next to its name.cats 5, dogs 3, rabbits 6, fish 2
- Find the smallest count. 2 is the smallest count.smallest: 2
- Write the answer. The fish row has 2.fish
Check. Every other count is bigger than 2. ✓
Watch out. Check every count, not only the first two. The fewest is the count that is smaller than all the others.
Method. Find each count. Then take the smaller count away from the bigger one.
- Count the rabbits. There are 6.rabbits: 6
- Count the dogs. There are 3.dogs: 3
- Take the smaller count away from the bigger one. The difference is how many more or fewer.6 - 3 = 3
- Write the answer. 3
Check. Add the difference back on: 3 + 3 = 6. ✓
Watch out. Take away the smaller number from the bigger one. Do not add them.
Method. Find each count, then add them all together.
- Write down each count. Each count is one row.cats 5, dogs 3, rabbits 6, fish 2
- Add the counts together. Add them all, so none is missed.5 + 3 + 6 + 2 = 16
- Write the answer. 16
Check. Add them in the other order to check: 2 + 6 + 3 + 5 = 16. ✓
Watch out. Make sure every row is in your total, and no row is counted twice.
Method. Work out the count you need, then find the row with that count.
- Work out the count you need. The fish has 2. Add 3: 5.5
- Find the row with 5. The cats row has 5.cats: 5
- Write the answer. cats
Check. Count the cats to check: 5. ✓
Watch out. Work out the number first. Then look for it in the rows.
Method. Find each count, then put the rows in order of their counts, fewest first.
- Write down each count. Put each count next to its name.cats 5, dogs 3, rabbits 6, fish 2
- Put the counts in order, fewest first. Start from the smallest count.2 → 3 → 5 → 6
- Write the order. fish, dogs, cats, rabbits
Check. Check that each count is bigger than the one before it. ✓
Watch out. Compare the counts, not the names. The longest name does not mean the biggest count.
Method. Count both rows, then check the claim.
- Count the dogs. There are 3.dogs: 3
- Count the cats. There are 5.cats: 5
- Check the claim. The claim says there are the same number of dogs as cats. The counts are 3 and 5.No
Check. 3 and 5 are different, so the claim is not right. ✓
Watch out. Compare the two counts, not how big the pictures look.
Block diagrams
A block diagram uses blocks to show counts. Each block stands for one thing.
Each colour has its own column. Count the blocks in a column to find its count.
The tallest column has the most blocks. The shortest column has the fewest.
Key ideas
- Each block stands for one thing.
- Count the blocks in one column.
- The tallest column has the most blocks.
- Add two counts together to find how many altogether.
Method. Count the pink blocks once each, touching each one.
- Touch each one once. Say the number as you touch each one.1 2 3 4 5 6
- Read the last number. The last number is how many there are.6
- Write the answer. 6
Check. Count them again from the other end. You get 6 again. ✓
Watch out. Move each one to one side as you count it, so you do not count it twice.
Method. Find each count, then add them all together.
- Write down each count. Each count is one row.pink 6, green 2, orange 5
- Add the counts together. Add them all, so none is missed.6 + 2 + 5 = 13
- Write the answer. 13
Check. Add them in the other order to check: 5 + 2 + 6 = 13. ✓
Watch out. Make sure every row is in your total, and no row is counted twice.
Method. Compare the counts. The smallest count is the fewest.
- Write down each count. Put each count next to its name.pink 6, green 2, orange 5
- Find the smallest count. 2 is the smallest count.smallest: 2
- Write the answer. The green row has 2.green
Check. Every other count is bigger than 2. ✓
Watch out. Check every count, not only the first two. The fewest is the count that is smaller than all the others.
Watch out
Watch out: read the label under each column before you count it. The label tells you the colour.
Exam tip
Grown-up tip. Stack coloured bricks, sweets or cubes in columns, one column for each colour. Ask: “How many in the red column? Which column is the tallest?”
Exercise 10D Block diagrams
Non-calculatorMethod. Count the red blocks once each, touching each one.
- Touch each one once. Say the number as you touch each one.1 2 3 4
- Read the last number. The last number is how many there are.4
- Write the answer. 4
Check. Count them again from the other end. You get 4 again. ✓
Watch out. Move each one to one side as you count it, so you do not count it twice.
Method. Count the blue blocks once each, touching each one.
- Touch each one once. Say the number as you touch each one.1 2 3 4 5 6 7
- Read the last number. The last number is how many there are.7
- Write the answer. 7
Check. Count them again from the other end. You get 7 again. ✓
Watch out. Move each one to one side as you count it, so you do not count it twice.
Method. Count the yellow blocks once each, touching each one.
- Touch each one once. Say the number as you touch each one.1 2 3
- Read the last number. The last number is how many there are.3
- Write the answer. 3
Check. Count them again from the other end. You get 3 again. ✓
Watch out. Move each one to one side as you count it, so you do not count it twice.
Method. Count the green blocks once each, touching each one.
- Touch each one once. Say the number as you touch each one.1 2 3 4 5
- Read the last number. The last number is how many there are.5
- Write the answer. 5
Check. Count them again from the other end. You get 5 again. ✓
Watch out. Move each one to one side as you count it, so you do not count it twice.
Method. Compare the counts. The biggest count is the most.
- Write down each count. Put each count next to its name.red 4, blue 7, yellow 3, green 5
- Find the biggest count. 7 is the biggest count.biggest: 7
- Write the answer. The blue row has 7.blue
Check. Every other count is smaller than 7. ✓
Watch out. Check every count, not only the first two. The most is the count that is bigger than all the others.
Method. Compare the counts. The smallest count is the fewest.
- Write down each count. Put each count next to its name.red 4, blue 7, yellow 3, green 5
- Find the smallest count. 3 is the smallest count.smallest: 3
- Write the answer. The yellow row has 3.yellow
Check. Every other count is bigger than 3. ✓
Watch out. Check every count, not only the first two. The fewest is the count that is smaller than all the others.
Method. Find each count, then add them all together.
- Write down each count. Each count is one row.red 4, blue 7, yellow 3, green 5
- Add the counts together. Add them all, so none is missed.4 + 7 + 3 + 5 = 19
- Write the answer. 19
Check. Add them in the other order to check: 5 + 3 + 7 + 4 = 19. ✓
Watch out. Make sure every row is in your total, and no row is counted twice.
Method. Find each count. Then take the smaller count away from the bigger one.
- Count the blue. There are 7.blue: 7
- Count the red. There are 4.red: 4
- Take the smaller count away from the bigger one. The difference is how many more or fewer.7 - 4 = 3
- Write the answer. 3
Check. Add the difference back on: 4 + 3 = 7. ✓
Watch out. Take away the smaller number from the bigger one. Do not add them.
Method. Find each count, then add them all together.
- Write down each count. Each count is one row.blue 7, yellow 3
- Add the counts together. Add them all, so none is missed.7 + 3 = 10
- Write the answer. 10
Check. Add them in the other order to check: 3 + 7 = 10. ✓
Watch out. Make sure every row is in your total, and no row is counted twice.
Method. Find each count, then put the rows in order of their counts, most first.
- Write down each count. Put each count next to its name.red 4, blue 7, yellow 3, green 5
- Put the counts in order, most first. Start from the biggest count.7 → 5 → 4 → 3
- Write the order. blue, green, red, yellow
Check. Check that each count is smaller than the one before it. ✓
Watch out. Compare the counts, not the names. The longest name does not mean the biggest count.
Method. Take the 2 away from the blue, then compare with the green.
- Count the blue. There are 7.blue: 7
- Take 2 away. 7 - 2 = 5.7 - 2 = 5
- Compare with the green. There are 5 green.5 and 5 are the same
- Decide. Is Ali right?Yes
Check. Count the blue again after taking 2 away, to check: 5. ✓
Watch out. Take the number away first. Then compare the two numbers.
Simple tables
A table puts counts in rows and columns. It is easy to read.
Find the name in the first column. Then read the number in the same row.
A total is all the numbers added together.
Key ideas
- Read along a row to find the number.
- The total is all the counts added together.
- Use the total to find a number that is hidden.
- The biggest number is the most.
Method. Find the row for reading. Then read the number in that row.
- Find the row for reading. The row for reading is in the table.reading
- Read the number in the right-hand column. That number is how many chose it.reading: 9
- Write the answer. 9
Check. Read the same row again. The number is still 9. ✓
Watch out. Read along the row to its number. Do not read the number from the row above or below.
Method. Find each count, then add them all together.
- Write down each count. Each count is one row.drawing 4, reading 9, baking 6
- Add the counts together. Add them all, so none is missed.4 + 9 + 6 = 19
- Write the answer. 19
Check. Add them in the other order to check: 6 + 9 + 4 = 19. ✓
Watch out. Make sure every row is in your total, and no row is counted twice.
Method. Compare the counts. The biggest count is the most.
- Write down each count. Put each count next to its name.drawing 4, reading 9, baking 6
- Find the biggest count. 9 is the biggest count.biggest: 9
- Write the answer. The reading row has 9.reading
Check. Every other count is smaller than 9. ✓
Watch out. Check every count, not only the first two. The most is the count that is bigger than all the others.
Watch out
Watch out: read the number in the same row as the name. Do not read the row above or below.
Exam tip
Grown-up tip. Make a small table of toys or snacks on paper, with a count in each row. Ask: “How many in this row? Which row is the biggest? What is the total?”
Exercise 10E Simple tables
Non-calculatorMethod. Find the row for cycling. Then read the number in that row.
- Find the row for cycling. The row for cycling is in the table.cycling
- Read the number in the right-hand column. That number is how many chose it.cycling: 9
- Write the answer. 9
Check. Read the same row again. The number is still 9. ✓
Watch out. Read along the row to its number. Do not read the number from the row above or below.
Method. Find the row for kites. Then read the number in that row.
- Find the row for kites. The row for kites is in the table.kites
- Read the number in the right-hand column. That number is how many chose it.kites: 4
- Write the answer. 4
Check. Read the same row again. The number is still 4. ✓
Watch out. Read along the row to its number. Do not read the number from the row above or below.
Method. Find the row for ball games. Then read the number in that row.
- Find the row for ball games. The row for ball games is in the table.ball games
- Read the number in the right-hand column. That number is how many chose it.ball games: 6
- Write the answer. 6
Check. Read the same row again. The number is still 6. ✓
Watch out. Read along the row to its number. Do not read the number from the row above or below.
Method. Find the row for running. Then read the number in that row.
- Find the row for running. The row for running is in the table.running
- Read the number in the right-hand column. That number is how many chose it.running: 7
- Write the answer. 7
Check. Read the same row again. The number is still 7. ✓
Watch out. Read along the row to its number. Do not read the number from the row above or below.
Method. Compare the counts. The biggest count is the most.
- Write down each count. Put each count next to its name.ball games 6, cycling 9, kites 4, running 7
- Find the biggest count. 9 is the biggest count.biggest: 9
- Write the answer. The cycling row has 9.cycling
Check. Every other count is smaller than 9. ✓
Watch out. Check every count, not only the first two. The most is the count that is bigger than all the others.
Method. Compare the counts. The smallest count is the fewest.
- Write down each count. Put each count next to its name.ball games 6, cycling 9, kites 4, running 7
- Find the smallest count. 4 is the smallest count.smallest: 4
- Write the answer. The kites row has 4.kites
Check. Every other count is bigger than 4. ✓
Watch out. Check every count, not only the first two. The fewest is the count that is smaller than all the others.
Method. Find each count, then add them all together.
- Write down each count. Each count is one row.ball games 6, cycling 9, kites 4, running 7
- Add the counts together. Add them all, so none is missed.6 + 9 + 4 + 7 = 26
- Write the answer. 26
Check. Add them in the other order to check: 7 + 4 + 9 + 6 = 26. ✓
Watch out. Make sure every row is in your total, and no row is counted twice.
Method. Find each count. Then take the smaller count away from the bigger one.
- Count the cycling. There are 9.cycling: 9
- Count the ball games. There are 6.ball games: 6
- Take the smaller count away from the bigger one. The difference is how many more or fewer.9 - 6 = 3
- Write the answer. 3
Check. Add the difference back on: 6 + 3 = 9. ✓
Watch out. Take away the smaller number from the bigger one. Do not add them.
Method. Add the counts you can see. Then take the sum away from the total.
- Add the counts you can see. These counts are shown in the table.6 + 4 + 7 = 17
- Take the sum away from the total of 26. The hidden count for cycling is what is left.26 - 17 = 9
- Write the answer. 9
Check. Add the hidden count back on: 17 + 9 = 26. ✓
Watch out. Use the total to find the hidden number. Do not guess it from the picture.
Method. Find each count, then put the rows in order of their counts, most first.
- Write down each count. Put each count next to its name.ball games 6, cycling 9, kites 4, running 7
- Put the counts in order, most first. Start from the biggest count.9 → 7 → 6 → 4
- Write the order. cycling, running, ball games, kites
Check. Check that each count is smaller than the one before it. ✓
Watch out. Compare the counts, not the names. The longest name does not mean the biggest count.
Method. Add up each side of the claim, then compare the two totals.
- Add the counts for running. They make 7 altogether.7 = 7
- Add the counts for ball games and kites. They make 10 altogether.6 + 4 = 10
- Compare the two totals. 7 is fewer than 10.No
Check. The claim compares running with ball games and kites. It is not right. ✓
Watch out. Add the counts on each side first. Then compare the totals, not the single rows.
Totals and comparisons
We can total the counts to find how many there are altogether.
We can compare two counts. We say how many more or how many fewer one has.
We can also put the counts in order, from fewest to most.
Always check your answer by counting again.
Key ideas
- To find how many altogether, add the counts together.
- To find how many more or fewer, take the smaller count from the bigger one.
- To sort the groups, compare their counts from the smallest to the biggest.
- Check each claim with the counts before you say Yes or No.
Method. Find each count, then add them all together.
- Write down each count. Each count is one row.ants 6, bees 3, frogs 5
- Add the counts together. Add them all, so none is missed.6 + 3 + 5 = 14
- Write the answer. 14
Check. Add them in the other order to check: 5 + 3 + 6 = 14. ✓
Watch out. Make sure every row is in your total, and no row is counted twice.
Method. Find each count. Then take the smaller count away from the bigger one.
- Count the ants. There are 6.ants: 6
- Count the bees. There are 3.bees: 3
- Take the smaller count away from the bigger one. The difference is how many more or fewer.6 - 3 = 3
- Write the answer. 3
Check. Add the difference back on: 3 + 3 = 6. ✓
Watch out. Take away the smaller number from the bigger one. Do not add them.
Method. Add the counts for each pair of rows. Find the pair that makes the total.
- List the pairs and their totals. Add the counts for each pair.ants 6 + bees 3 = 9; ants 6 + frogs 5 = 11; bees 3 + frogs 5 = 8
- Find the pair that makes 9. Only ants and bees make 9.9 = 9
- Write the answer. ants and bees
Check. ants and bees together make 9, and no other pair makes 9. ✓
Watch out. Check every pair. Two rows that are both bigger than the target cannot make it.
Watch out
Watch out: more or fewer? Take the smaller number from the bigger one, then read the question again.
Exam tip
Grown-up tip. Count two groups of beads or buttons and write both counts. Ask: “How many altogether? How many more is one than the other?”
Exercise 10F Totals and comparisons
Non-calculatorMethod. Count the shells once each, touching each one.
- Touch each one once. Say the number as you touch each one.1 2 3 4 5 6 7 8
- Read the last number. The last number is how many there are.8
- Write the answer. 8
Check. Count them again from the other end. You get 8 again. ✓
Watch out. Move each one to one side as you count it, so you do not count it twice.
Method. Count the crabs once each, touching each one.
- Touch each one once. Say the number as you touch each one.1 2 3
- Read the last number. The last number is how many there are.3
- Write the answer. 3
Check. Count them again from the other end. You get 3 again. ✓
Watch out. Move each one to one side as you count it, so you do not count it twice.
Method. Count the fish once each, touching each one.
- Touch each one once. Say the number as you touch each one.1 2 3 4 5
- Read the last number. The last number is how many there are.5
- Write the answer. 5
Check. Count them again from the other end. You get 5 again. ✓
Watch out. Move each one to one side as you count it, so you do not count it twice.
Method. Count the octopuses once each, touching each one.
- Touch each one once. Say the number as you touch each one.1 2
- Read the last number. The last number is how many there are.2
- Write the answer. 2
Check. Count them again from the other end. You get 2 again. ✓
Watch out. Move each one to one side as you count it, so you do not count it twice.
Method. Find each count, then add them all together.
- Write down each count. Each count is one row.shells 8, crabs 3, fish 5, octopuses 2
- Add the counts together. Add them all, so none is missed.8 + 3 + 5 + 2 = 18
- Write the answer. 18
Check. Add them in the other order to check: 2 + 5 + 3 + 8 = 18. ✓
Watch out. Make sure every row is in your total, and no row is counted twice.
Method. Find each count. Then take the smaller count away from the bigger one.
- Count the shells. There are 8.shells: 8
- Count the crabs. There are 3.crabs: 3
- Take the smaller count away from the bigger one. The difference is how many more or fewer.8 - 3 = 5
- Write the answer. 5
Check. Add the difference back on: 3 + 5 = 8. ✓
Watch out. Take away the smaller number from the bigger one. Do not add them.
Method. Find each count. Then take the smaller count away from the bigger one.
- Count the shells. There are 8.shells: 8
- Count the fish. There are 5.fish: 5
- Take the smaller count away from the bigger one. The difference is how many more or fewer.8 - 5 = 3
- Write the answer. 3
Check. Add the difference back on: 5 + 3 = 8. ✓
Watch out. Take away the smaller number from the bigger one. Do not add them.
Method. Compare the counts. The biggest count is the most.
- Write down each count. Put each count next to its name.shells 8, crabs 3, fish 5, octopuses 2
- Find the biggest count. 8 is the biggest count.biggest: 8
- Write the answer. The shells row has 8.shells
Check. Every other count is smaller than 8. ✓
Watch out. Check every count, not only the first two. The most is the count that is bigger than all the others.
Method. Find each count, then put the rows in order of their counts, fewest first.
- Write down each count. Put each count next to its name.shells 8, crabs 3, fish 5, octopuses 2
- Put the counts in order, fewest first. Start from the smallest count.2 → 3 → 5 → 8
- Write the order. octopuses, crabs, fish, shells
Check. Check that each count is bigger than the one before it. ✓
Watch out. Compare the counts, not the names. The longest name does not mean the biggest count.
Method. Add the counts for each pair of rows. Find the pair that makes the total.
- List the pairs and their totals. Add the counts for each pair.shells 8 + crabs 3 = 11; shells 8 + fish 5 = 13; shells 8 + octopuses 2 = 10; crabs 3 + fish 5 = 8; crabs 3 + octopuses 2 = 5; fish 5 + octopuses 2 = 7
- Find the pair that makes 10. Only shells and octopuses make 10.10 = 10
- Write the answer. shells and octopuses
Check. shells and octopuses together make 10, and no other pair makes 10. ✓
Watch out. Check every pair. Two rows that are both bigger than the target cannot make it.
Method. Add up each side of the claim, then compare the two totals.
- Add the counts for fish. They make 5 altogether.5 = 5
- Add the counts for crabs and octopuses. They make 5 altogether.3 + 2 = 5
- Compare the two totals. 5 is the same as 5.No
Check. The claim compares fish with crabs and octopuses. It is not right. ✓
Watch out. Add the counts on each side first. Then compare the totals, not the single rows.
Chapter summary
| Skill | Method |
|---|---|
| Groups | Count the objects in each group, once each. Compare the counts to find the most and the fewest. |
| Tally chart | Each mark is one thing. Count the full groups of five first, then the marks left over. |
| Pictogram | Each picture stands for one thing. Count the pictures in a row. |
| Block diagram | Each block stands for one thing. Count the blocks in a column. |
| Table | Find the name, then read the number in the same row. Use the total to find a hidden number. |
| Totals and comparisons | Add the counts for a total. Take the smaller count from the bigger one for how many more or fewer. |
| Sorting | Put the counts in order, from the smallest to the biggest, then write the names in that order. |
Can you…?
- I can count the objects in each group
- I can read a tally chart, with groups of five marks
- I can read a pictogram, where each picture stands for one thing
- I can read a block diagram, where each block stands for one thing
- I can read a simple table and find a number in it
- I can find the most, the fewest and the total
- I can compare two counts to say how many more or fewer
- I can sort the categories from the fewest to the most
Key terms
- category
- a group of things that are alike, such as all the fruit.
- tally chart
- a chart that counts with marks. Each mark is one thing.
- group of five
- five tally marks: four going up and one going across them.
- pictogram
- a chart that uses pictures to show counts. Each picture stands for one thing.
- block diagram
- a chart that uses blocks in columns. Each block stands for one thing.
- table
- counts written in rows and columns, so they are easy to read.
- total
- all the counts added together.
- most
- the biggest count.
- fewest
- the smallest count.
- sort
- put things in order, for example from the fewest to the most.
Exercise 10Z Check-up
Non-calculatorMethod. Count the pizzas once each, touching each one.
- Touch each one once. Say the number as you touch each one.1 2 3 4 5 6 7 8 9
- Read the last number. The last number is how many there are.9
- Write the answer. 9
Check. Count them again from the other end. You get 9 again. ✓
Watch out. Move each one to one side as you count it, so you do not count it twice.
Method. Count the sandwiches once each, touching each one.
- Touch each one once. Say the number as you touch each one.1 2 3 4 5 6
- Read the last number. The last number is how many there are.6
- Write the answer. 6
Check. Count them again from the other end. You get 6 again. ✓
Watch out. Move each one to one side as you count it, so you do not count it twice.
Method. Compare the counts. The smallest count is the fewest.
- Write down each count. Put each count next to its name.apples 4, sandwiches 6, pizzas 9, carrots 2
- Find the smallest count. 2 is the smallest count.smallest: 2
- Write the answer. The carrots row has 2.carrots
Check. Every other count is bigger than 2. ✓
Watch out. Check every count, not only the first two. The fewest is the count that is smaller than all the others.
Method. Find each count. Then take the smaller count away from the bigger one.
- Count the pizzas. There are 9.pizzas: 9
- Count the apples. There are 4.apples: 4
- Take the smaller count away from the bigger one. The difference is how many more or fewer.9 - 4 = 5
- Write the answer. 5
Check. Add the difference back on: 4 + 5 = 9. ✓
Watch out. Take away the smaller number from the bigger one. Do not add them.
Method. Find each count, then add them all together.
- Write down each count. Each count is one row.apples 4, sandwiches 6, pizzas 9, carrots 2
- Add the counts together. Add them all, so none is missed.4 + 6 + 9 + 2 = 21
- Write the answer. 21
Check. Add them in the other order to check: 2 + 9 + 6 + 4 = 21. ✓
Watch out. Make sure every row is in your total, and no row is counted twice.
Method. Find each count, then put the rows in order of their counts, fewest first.
- Write down each count. Put each count next to its name.apples 4, sandwiches 6, pizzas 9, carrots 2
- Put the counts in order, fewest first. Start from the smallest count.2 → 4 → 6 → 9
- Write the order. carrots, apples, sandwiches, pizzas
Check. Check that each count is bigger than the one before it. ✓
Watch out. Compare the counts, not the names. The longest name does not mean the biggest count.
Method. Count the pears marks in groups of five, then count the marks left over.
- Count the full groups of five. Each group has four upright marks and one line across them.2 groups of five = 10 marks
- Count the marks left over. These are not in a full group.3 left over
- Add them together. 10 and 3 make 13.10 + 3 = 13
- Write the answer. 13
Check. Count every mark one at a time, and you also get 13. ✓
Watch out. Count the groups of five first. Then count the single marks that are left over.
Method. Add the counts for each pair of rows. Find the pair that makes the total.
- List the pairs and their totals. Add the counts for each pair.apples 4 + sandwiches 6 = 10; apples 4 + pizzas 9 = 13; apples 4 + carrots 2 = 6; sandwiches 6 + pizzas 9 = 15; sandwiches 6 + carrots 2 = 8; pizzas 9 + carrots 2 = 11
- Find the pair that makes 10. Only apples and sandwiches make 10.10 = 10
- Write the answer. apples and sandwiches
Check. apples and sandwiches together make 10, and no other pair makes 10. ✓
Watch out. Check every pair. Two rows that are both bigger than the target cannot make it.
Method. Work out the count you need, then find the row with that count.
- Work out the count you need. The pizzas have 9. Take away 3: 6.6
- Find the row with 6. The sandwiches row has 6.sandwiches: 6
- Write the answer. sandwiches
Check. Count the sandwiches to check: 6. ✓
Watch out. Work out the number first. Then look for it in the rows.
Method. Add up each side of the claim, then compare the two totals.
- Add the counts for sandwiches and carrots. They make 8 altogether.6 + 2 = 8
- Add the counts for pizzas. They make 9 altogether.9 = 9
- Compare the two totals. 8 is fewer than 9.No
Check. The claim compares sandwiches and carrots with pizzas. It is not right. ✓
Watch out. Add the counts on each side first. Then compare the totals, not the single rows.
Total 15 marks. Allow about 15 minutes.
Exercise 10X End-of-chapter quiz
Non-calculatorMethod. Count the stars once each, touching each one.
- Touch each one once. Say the number as you touch each one.1 2 3 4 5 6 7 8
- Read the last number. The last number is how many there are.8
- Write the answer. 8
Check. Count them again from the other end. You get 8 again. ✓
Watch out. Move each one to one side as you count it, so you do not count it twice.
Method. Count the frogs marks in groups of five, then count the marks left over.
- Count the full groups of five. Each group has four upright marks and one line across them.2 groups of five = 10 marks
- Count the marks left over. These are not in a full group.1 left over
- Add them together. 10 and 1 make 11.10 + 1 = 11
- Write the answer. 11
Check. Count every mark one at a time, and you also get 11. ✓
Watch out. Count the groups of five first. Then count the single marks that are left over.
Method. Compare the counts. The biggest count is the most.
- Write down each count. Put each count next to its name.pink 6, orange 3, purple 8
- Find the biggest count. 8 is the biggest count.biggest: 8
- Write the answer. The purple row has 8.purple
Check. Every other count is smaller than 8. ✓
Watch out. Check every count, not only the first two. The most is the count that is bigger than all the others.
Method. Find each count. Then take the smaller count away from the bigger one.
- Count the hats. There are 4.hats: 4
- Count the bags. There are 2.bags: 2
- Take the smaller count away from the bigger one. The difference is how many more or fewer.4 - 2 = 2
- Write the answer. 2
Check. Add the difference back on: 2 + 2 = 4. ✓
Watch out. Take away the smaller number from the bigger one. Do not add them.
Method. Find each count, then add them all together.
- Write down each count. Each count is one row.pears 3, lemons 7, grapes 5
- Add the counts together. Add them all, so none is missed.3 + 7 + 5 = 15
- Write the answer. 15
Check. Add them in the other order to check: 5 + 7 + 3 = 15. ✓
Watch out. Make sure every row is in your total, and no row is counted twice.
Method. Find the row for lemons. Then read the number in that row.
- Find the row for lemons. The row for lemons is in the table.lemons
- Read the number in the right-hand column. That number is how many chose it.lemons: 7
- Write the answer. 7
Check. Read the same row again. The number is still 7. ✓
Watch out. Read along the row to its number. Do not read the number from the row above or below.
Method. Find each count, then put the rows in order of their counts, fewest first.
- Write down each count. Put each count next to its name.cats 3, bees 6, ducks 1
- Put the counts in order, fewest first. Start from the smallest count.1 → 3 → 6
- Write the order. ducks, cats, bees
Check. Check that each count is bigger than the one before it. ✓
Watch out. Compare the counts, not the names. The longest name does not mean the biggest count.
Method. Add the counts for each pair of rows. Find the pair that makes the total.
- List the pairs and their totals. Add the counts for each pair.boats 2 + kites 5 = 7; boats 2 + bikes 7 = 9; kites 5 + bikes 7 = 12
- Find the pair that makes 9. Only boats and bikes make 9.9 = 9
- Write the answer. boats and bikes
Check. boats and bikes together make 9, and no other pair makes 9. ✓
Watch out. Check every pair. Two rows that are both bigger than the target cannot make it.
Method. Add the counts you can see. Then take the sum away from the total.
- Add the counts you can see. These counts are shown in the table.6 + 4 = 10
- Take the sum away from the total of 20. The hidden count for biscuits is what is left.20 - 10 = 10
- Write the answer. 10
Check. Add the hidden count back on: 10 + 10 = 20. ✓
Watch out. Use the total to find the hidden number. Do not guess it from the picture.
Method. Add up each side of the claim, then compare the two totals.
- Add the counts for lemons. They make 7 altogether.7 = 7
- Add the counts for pears and grapes. They make 8 altogether.3 + 5 = 8
- Compare the two totals. 7 is fewer than 8.No
Check. The claim compares lemons with pears and grapes. It is not right. ✓
Watch out. Add the counts on each side first. Then compare the totals, not the single rows.