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Primary Maths · Year 4Chapter 11 · Statistics
Year 4 · Chapter 11

Year 4 Pictograms, Bar Charts and Time Graphs: worked examples and questions with answers

By the end of this chapter you will be able to

  • Collect data with a tally chart and write the totals in a frequency table
  • Read and draw pictograms, using a key where one symbol stands for 2, 5, 10 or more, and half symbols
  • Read bar charts (vertical and horizontal) with scales going up in 1s, 2s, 5s, 10s, 20s and 50s, including values between the marks
  • Draw and complete bar charts, and choose a sensible scale
  • Read tables and two-way tables, and use the totals to find missing numbers
  • Tell the difference between data you count and data you measure, and read and draw time graphs
  • Solve “how many more”, “how many altogether” and “what is the difference” problems, and explain whether a statement about a chart is true

You should already know

  • Counting in 2s, 5s and 10s (Years 1 to 3)
  • Adding and subtracting numbers up to 4 digits (Chapter 2)
  • Times tables up to 12 × 12 (Chapter 3)
  • Telling the time and the 24-hour clock (Chapter 8)
  • Reading a number line and scale (Chapters 1 and 7)
  • Carroll diagrams and Venn diagrams (Chapter 9)

Where this leads

  • Line graphs and tables, including timetables (Year 5)
  • Pie charts, line graphs and the mean (Year 6)
  • Averages and charts at GCSE

National Curriculum for England, Key Stage 2 mathematics, Year 4. Content: Interpret and present discrete and continuous data using appropriate graphical methods, including bar charts and time graphs, Solve comparison, sum and difference problems using information presented in bar charts, pictograms, tables and other graphs, Tally charts and frequency tables.

Printable worksheets. Every lesson in this chapter has a free worksheet and an answer sheet to download and print.

Exercise P  Do you remember?

Non-calculator
1
FCount in fives. What comes next? 35, 40, 45, …
2
FWork out 24 + 17.
3
FWork out 9 × 5.
4
FWork out 53 − 28.
5
FA number line goes up in 2s: 0, 2, 4, 6, 8. What number is halfway between 6 and 8?
11.1

Tally charts and frequency tables

Year 4
Tally charts and frequency tables, Interpret and present discrete data

Printable: Worksheet (PDF) · Answers (PDF) · Open this lesson on its own page

Video lesson: Tally charts and frequency tablesComing soon

Data is information. When we collect data we find things out. Priya wants to know which fruit is the most popular in her class. She asks every child and writes down their answer. But counting a long list is hard, so she makes a tally chart.

FruitTallyFrequency
Apple8
Banana13
Grapes6
Orange11
A tally chart (the Tally column) and a frequency table (the Frequency column) for 38 children.

Each time a child chooses a fruit, Priya draws one tally mark in that row. She draws four strokes and then the fifth stroke goes across the four. That makes a group of 5. Groups of five are quick to count: 5, 10, 15, 20 … and then you add the single strokes.

When the voting is finished, Priya counts the tally marks in each row and writes the number in the Frequency column. The frequency is how many there are of one kind. A chart with the Item, Tally and Frequency columns is called a frequency table. Always check the frequencies: they must add up to the number of children asked.

Key ideas

  • A tally mark is one stroke for each thing you count. The fifth stroke goes across the first four to make a group of 5.
  • To count a tally: count in fives for the groups, then count on the single strokes.
  • The frequency is the total number in a row. Write it as a number.
  • The frequencies add up to the total number of things counted.
Worked example 11.1.1
Amira counted birds in the park and made these tally marks. How many birds did she count?

Key words. A tally mark is a small stroke we draw for each thing we count. Every fifth stroke goes across the other four, so each group of 5 is easy to see. The frequency is the total number of tally marks in one row.

Method. Count the groups of five first (count in 5s), then add on the single strokes.

  1. Count the groups of five. Each group is four strokes with one across them. That group is worth 5.
    2 groups: count in fives: 5, 10
    2 × 5 = 10
  2. Count the single strokes at the end. These are not in a group of five yet.
    4 single strokes = 4
  3. Add the two amounts. The groups and the single strokes make the total.
    10 + 4 = 14

Check. Count the strokes one by one: you should reach 14. ✓

Watch out. A group with a line across is worth 5, not 1.

Worked example 11.1.2
Draw tally marks to show 18.

Key words. A tally mark is a small stroke we draw for each thing we count. Every fifth stroke goes across the other four, so each group of 5 is easy to see. The frequency is the total number of tally marks in one row.

Method. Draw one group of five for every five, then draw the strokes that are left.

  1. Split the number into groups of five. Tally marks are drawn in fives, so we find how many fives there are.
    18 ÷ 5 = 3 remainder 3
  2. Draw the groups of five. Draw four strokes, then a fifth stroke across the four.
    3 groups of five
  3. Draw the strokes that are left. These make an unfinished group. Do not draw a line across.
    3 single strokes
  4. Put it together and check. Count in fives, then count on.
    5, 10, 15, then 3 more: 18

Check. 15 + 3 = 18. ✓

Watch out. The fifth stroke goes across the other four. Do not draw five strokes in a row.

Worked example 11.1.3
The tally chart shows the games children chose at playtime. How many children chose football?
GameTally
Skipping
Football
Tag
Hopscotch

Key words. A tally mark is a small stroke we draw for each thing we count. Every fifth stroke goes across the other four, so each group of 5 is easy to see. The frequency is the total number of tally marks in one row.

Method. Look at the tally carefully and read off the number for Football.

  1. Count in fives. Each group of 4 strokes with a line across is 5 tally marks. Count the fives first, then the single strokes.
    Count: 5, 10, 15, 20 ...
  2. Count the tally marks for Football. Groups of five first, then the strokes left over.
    Football: 2 groups of 5 = 10, and 4 more = 10 + 4 = 14
  3. Write the answer.
    Football = 14 children
    GameTally
    Skipping
    Football
    Tag
    Hopscotch

Check. Look again at the picture: Football is 14. ✓

Watch out. Count each crossed group as 5, not as 1. Then add the single marks.

Worked example 11.1.4
Noah wrote down the colours of 20 cars that passed the school gate: red, blue, white, red, silver, white, blue, red, white, white, silver, red, blue, white, silver, white, red, blue, white, silver. Make a frequency table.

Key words. A tally mark is a small stroke we draw for each thing we count. Every fifth stroke goes across the other four, so each group of 5 is easy to see. The frequency is the total number of tally marks in one row.

Method. Go through the list once. Make a tally mark for each item, then count the marks in each row.

  1. Make a table with a row for each kind. We need a place to put the tally marks for each kind.
    Rows: red, blue, white, silver
  2. Go through the list one item at a time. Make one tally mark in the right row for each item. Cross off each item as you go, so you do not count it twice.
    red, blue, white, red, silver
    white, blue, red, white, white
    silver, red, blue, white, silver
    white, red, blue, white, silver
  3. Count the tally marks in each row. Count in fives, then add the single strokes. Write each total in the Frequency column.
    ColourTallyFrequency
    red5
    blue4
    white7
    silver4
  4. Check the total. Every item in the list is in one row, so the frequencies must add up to 20.
    5 + 4 + 7 + 4 = 20, and the list has 20 items.

Check. The frequencies add up to 20, the same as the number of items in the list. ✓

Watch out. Cross off each item as you tally it. Then you will not miss one or count one twice.

Worked example 11.1.5
How many children chose a game altogether in the tally chart above?
GameTally
Skipping
Football
Tag
Hopscotch

Key words. A tally mark is a small stroke we draw for each thing we count. Every fifth stroke goes across the other four, so each group of 5 is easy to see. The frequency is the total number of tally marks in one row.

Method. Read each number from the picture, then add them all.

  1. Count in fives. Each group of 4 strokes with a line across is 5 tally marks. Count the fives first, then the single strokes.
    Count: 5, 10, 15, 20 ...
  2. Read every number. We need these numbers for the calculation.
    Skipping = 9
    Football = 14
    Tag = 11
    Hopscotch = 6
    GameTally
    Skipping
    Football
    Tag
    Hopscotch
  3. Add the numbers together. Altogether means add everything. Add one number at a time.
    9 + 14 + 11 + 6
    9 + 14 = 23
    23 + 11 = 34
    34 + 6 = 40
  4. Write the answer.
    Total = 40 children

Check. Add in the opposite order: 6 + 11 + 14 + 9 = 40. ✓

Watch out. Do not leave out a row or a bar. Check you have used every one.

Worked example 11.1.6
How many more children chose football than hopscotch?
GameTally
Skipping
Football
Tag
Hopscotch

Key words. The difference between two numbers is how much bigger one is than the other. We find it by subtracting. “How many more” and “how many fewer” questions are difference questions. A tally mark is a small stroke we draw for each thing we count. Every fifth stroke goes across the other four, so each group of 5 is easy to see. The frequency is the total number of tally marks in one row.

Method. To find “how many more”, read both numbers and then subtract the smaller from the bigger.

  1. Count in fives. Each group of 4 strokes with a line across is 5 tally marks. Count the fives first, then the single strokes.
    Count: 5, 10, 15, 20 ...
  2. Read the numbers we need from the picture. We need these numbers for the calculation.
    Football = 14
    Hopscotch = 6
    GameTally
    Skipping
    Football
    Tag
    Hopscotch
  3. Find the difference. To find how many more (or fewer), subtract the smaller number from the bigger number.
    14 − 6 = 8
  4. Write the answer.
    8 more

Check. Add back: 6 + 8 = 14. ✓

Watch out. Subtract: do not add. The question asks for the difference.

Worked example 11.1.7
30 children voted for their favourite colour. Red got 8 votes, blue got 11 and yellow got 6. How many children voted for green?
811?630 children altogether

Key words. A table shows information in rows (going across) and columns (going down). A total is the answer when you add all the numbers together.

Method. The numbers in the table must add up to the total. Add the numbers we know, then subtract from the total.

  1. Add the numbers we know. The missing number is the part of the total that is left.
    8 + 11 + 6 = 25
  2. Subtract from the total. Total minus the parts we know leaves the missing part.
    30 − 25 = 5
  3. Write the answer.
    Green = 5
    811?630 children altogether

Check. Add them all: 8 + 11 + 6 + 5 = 30. ✓

Watch out. Use subtraction to find a missing part. Do not add the total.

Watch out

A group of five has a line across it, and it counts as 5, not 1. Also, do not stop counting early: add the single strokes at the end. Check that your frequencies add up to the total.

Exercise 11A  Tally charts and frequency tables

Non-calculator
1
FHow many tally marks are there? Write each number.
(a)
(b)
(c)
(d)
2
FDraw tally marks to show each number.
(a)6
(b)9
(c)15
(d)21
3
FChildren in Class 4 chose their favourite pet. The tally chart shows the results.
PetTally
Dog
Cat
Rabbit
Fish
(a)How many children chose cats?
(b)Which pet got the fewest votes?
(c)How many children voted altogether?
(d)How many more children chose dogs than fish?
4
FZainab made a tally chart of favourite colours. Write the frequency for each colour. Here is the chart with the Frequency column to fill in.
ColourTallyFrequency
Red  ?  
Blue  ?  
Green  ?  
Yellow  ?  
(a)Red
(b)Blue
(c)Green
(d)Yellow
5
FJack counted the shapes in a pattern: circle, square, triangle, circle, circle, square, triangle, circle, square, circle, triangle, square, circle, triangle, circle. Make a frequency table. Write the frequency of each shape.
6
FLeo rolled a dice 24 times. Here are his results: 3, 5, 1, 6, 2, 4, 6, 3, 3, 5, 2, 6, 1, 4, 5, 6, 3, 2, 6, 4, 1, 5, 6, 3.
(a)How many times did he roll a 6?
(b)Make a frequency table of his results. Write the frequency of each number from 1 to 6.
(c)Which number did he roll most often?
(d)How many times did he roll an even number (2, 4 or 6)?
7
PSMei asked children where they would like to go on a school trip. The table shows the votes.
TripFrequency
Zoo14
Beach9
Museum12
Farm5
(a)How many children voted for the museum?
(b)How many children voted altogether?
(c)How many more children voted for the zoo than for the farm?
(d)How many children voted for the beach or the farm?
8
PS50 children voted for their favourite snack. Fruit got 17 votes, crisps got 12 and cereal bars got 9. The rest voted for popcorn. How many voted for popcorn?
9
PSLiam asked 40 children how they come to school. Walk 16, bus 9, car 11. The rest cycle. How many cycle?
10
RNoah looks at this tally and says, “It shows 5, because there are 3 groups and 2 strokes, and 3 + 2 = 5.” What mistake has he made? What number does the tally show?
11
RElla says, “We draw the fifth tally mark across the four so that it is easy to count in fives.” Is she right? Explain.
12
RAisha made a tally chart of the vowels in a sentence. Red has 9 tally marks, blue has 4 more than green, and the total is 25. How many tally marks does blue have, and how many does green have?
11.2

Pictograms

Year 4
Interpret and present discrete data using pictograms, Solve comparison, sum and difference problems using pictograms

Printable: Worksheet (PDF) · Answers (PDF) · Open this lesson on its own page

Video lesson: PictogramsComing soon

A pictogram uses small pictures, called symbols, to show data. Each symbol is one thing that is counted. The picture below shows the pets that children in Class 4 have. Each heart stands for 1 pet.

Pets in Class 4DogsCatsFishRabbitsKey:= 1 pets
A pictogram. The key says 1 heart = 1 pet. Cats has 8 hearts, so 8 children have a cat.

When there is lots of data, we do not want to draw hundreds of pictures. So one symbol can stand for 2, 5 or 10. The key tells us how many each symbol is worth. Always read the key first!

Cakes sold at the school saleMonTueWedThuKey:= 5 cakes
Here 1 symbol = 5 cakes. Tuesday has 7 symbols, so 7 × 5 = 35 cakes.

Sometimes the data does not fit exactly. A half symbol is worth half of what a whole symbol is worth. If the key says 1 symbol = 10 people, then a half symbol is 5 people. To draw a pictogram, divide each number by the key to find how many symbols to draw.

Favourite colourRedBlueGreenKey:= 10 pupilsand= 5 pupils
Key: 1 square = 10 pupils and half a square = 5 pupils. Red has 2 squares and a half: 20 + 5 = 25.

Key ideas

  • The key says how many one symbol stands for. Read it first.
  • To read a row: count the symbols, then multiply by the key. A half symbol is worth half the key.
  • To draw a row: divide the number by the key. Use a half symbol for a remainder that is half the key.
  • Draw all the symbols the same size, in a straight line with equal gaps.
Worked example 11.2.1
The pictogram shows the pets in Class 4. How many children have a cat?
Pets in Class 4DogsCatsFishRabbitsKey:= 1 pets

Key words. A pictogram uses small pictures (symbols) to show data. The key tells you how many each symbol stands for. A half symbol stands for half of that number.

Method. Look at the picto carefully and read off the number for Cats.

  1. Read the key. The key tells us what one symbol is worth. Here a symbol is not worth 1.
    1 symbol = 1 pets
  2. Count the symbols in the row for Cats. Multiply the number of symbols by what one symbol is worth.
    Cats: 8 symbols
    8 × 1 = 8
  3. Write the answer.
    Cats = 8 pets
    Pets in Class 4DogsCatsFishRabbitsKey:= 1 pets

Check. Count back: 8 ÷ 1 = 8 symbols. ✓

Watch out. Do not forget the key. A symbol is not always worth 1. A half symbol is worth half.

Worked example 11.2.2
The pictogram shows how many stickers Class 4 won each day. How many stickers did they win altogether?
Stickers won by Class 4MonTueWedThuFriKey:= 2 stickers

Key words. A pictogram uses small pictures (symbols) to show data. The key tells you how many each symbol stands for. A half symbol stands for half of that number.

Method. Read each number from the picture, then add them all.

  1. Read the key. The key tells us what one symbol is worth. Here a symbol is not worth 1.
    1 symbol = 2 stickers
  2. Read every number. We need these numbers for the calculation.
    Mon = 6
    Tue = 10
    Wed = 4
    Thu = 8
    Fri = 12
    Stickers won by Class 4MonTueWedThuFriKey:= 2 stickers
  3. Add the numbers together. Altogether means add everything. Add one number at a time.
    6 + 10 + 4 + 8 + 12
    6 + 10 = 16
    16 + 4 = 20
    20 + 8 = 28
    28 + 12 = 40
  4. Write the answer.
    Total = 40 stickers

Check. Add in the opposite order: 12 + 8 + 4 + 10 + 6 = 40. ✓

Watch out. Do not leave out a row or a bar. Check you have used every one.

Worked example 11.2.3
How many more goals did the team score in Match 3 than in Match 2?
Goals scored by the school teamMatch 1Match 2Match 3Match 4Key:= 5 goals

Key words. The difference between two numbers is how much bigger one is than the other. We find it by subtracting. “How many more” and “how many fewer” questions are difference questions. A pictogram uses small pictures (symbols) to show data. The key tells you how many each symbol stands for. A half symbol stands for half of that number.

Method. To find “how many more”, read both numbers and then subtract the smaller from the bigger.

  1. Read the key. The key tells us what one symbol is worth. Here a symbol is not worth 1.
    1 symbol = 5 goals
  2. Read the numbers we need from the picture. We need these numbers for the calculation.
    Match 3 = 25
    Match 2 = 10
    Goals scored by the school teamMatch 1Match 2Match 3Match 4Key:= 5 goals
  3. Find the difference. To find how many more (or fewer), subtract the smaller number from the bigger number.
    25 − 10 = 15
  4. Write the answer.
    15 more

Check. Add back: 10 + 15 = 25. ✓

Watch out. Subtract: do not add. The question asks for the difference.

Worked example 11.2.4
The key says 1 symbol = 10 cars. How many cars were in the car park on Monday?
Cars in the school car parkMonTueWedThuFriKey:= 10 carsand= 5 cars

Key words. A pictogram uses small pictures (symbols) to show data. The key tells you how many each symbol stands for. A half symbol stands for half of that number.

Method. Look at the picto carefully and read off the number for Mon.

  1. Read the key. The key tells us what one symbol is worth. Here a symbol is not worth 1.
    1 symbol = 10 cars
    Half a symbol = 10 ÷ 2 = 5 cars
  2. Count the symbols in the row for Mon. Multiply the number of symbols by what one symbol is worth.
    Mon: 4 whole symbols and 1 half symbol
    4 × 10 = 40
    Half symbol = 5
    40 + 5 = 45
  3. Write the answer.
    Mon = 45 cars
    Cars in the school car parkMonTueWedThuFriKey:= 10 carsand= 5 cars

Check. Count back: 45 ÷ 10 = 4 and a half symbols. ✓

Watch out. Do not forget the key. A symbol is not always worth 1. A half symbol is worth half.

Worked example 11.2.5
Priya borrowed 20 maths books. The pictogram shows 4 symbols in the Maths row, but the key is hidden. How much is one symbol worth?
Books borrowedMathsReadingKey:= ?

Key words. A pictogram uses small pictures (symbols) to show data. The key tells you how many each symbol stands for. A half symbol stands for half of that number.

Method. The key is how much one symbol is worth. Divide the number by the number of symbols.

  1. Count the symbols in the row. We need to know how many symbols make the number.
    Books: 4 symbols
  2. Divide the number by the symbols. If 4 symbols are worth 20, one symbol is worth 20 ÷ 4.
    20 ÷ 4 = 5
  3. Write the answer.
    1 symbol = 5

Check. 4 × 5 = 20. ✓

Watch out. Divide: do not add or subtract.

Worked example 11.2.6
The table shows how many pupils walked to school. Draw the symbols for Tuesday and Thursday on the pictogram. The key is 1 symbol = 10 pupils.
DayMonTueWedThuFri
Pupils4025301535
Pupils who walked to schoolMonTueWedThuFriKey:= 10 pupilsand= 5 pupils

Key words. A pictogram uses small pictures (symbols) to show data. The key tells you how many each symbol stands for. A half symbol stands for half of that number.

Method. Divide each number by the key to find how many symbols to draw.

  1. Read the key. The key tells us what one symbol is worth. Here a symbol is not worth 1.
    1 symbol = 10 pupils
    Half a symbol = 10 ÷ 2 = 5 pupils
  2. Work out how many symbols each row needs. Divide each number by the key. A half symbol is used for half a key.
    Tue: 25 ÷ 10 = 2 remainder 5, so 2 symbols and a half symbol
    Thu: 15 ÷ 10 = 1 remainder 5, so 1 symbol and a half symbol
  3. Draw the symbols. Draw them neatly in a line, with the same size and the same gaps.
    Pupils who walked to schoolMonTueWedThuFriKey:= 10 pupilsand= 5 pupils

Check. Count the symbols you drew. They should be worth 25, 15. ✓

Watch out. Use a half symbol if the number is half way between two whole symbols.

Worked example 11.2.7
Mia recorded the books she read: Mon 8, Tue 12, Wed 6, Thu 10. Draw a pictogram. Use 1 symbol = 2 books.

Key words. A pictogram uses small pictures (symbols) to show data. The key tells you how many each symbol stands for. A half symbol stands for half of that number.

Method. Choose a key, divide every number by the key to find the symbols for each row, then draw the pictogram with its key.

  1. Choose a key. The key should divide into every number. The biggest number is 12. With a key of 2 we need only 6 symbols for the longest row.
    1 symbol = 2 books
  2. Work out the symbols for each row. Divide every number by the key.
    Mon: 8 ÷ 2 = 4
    Tue: 12 ÷ 2 = 6
    Wed: 6 ÷ 2 = 3
    Thu: 10 ÷ 2 = 5
  3. Draw the pictogram and the key. Draw the symbols for each row. Do not forget to write the key.
    Books readMonTueWedThuKey:= 2 books

Check. Count the symbols in one row and multiply by 2 to check. ✓

Watch out. Always draw the key. Without it nobody knows what a symbol is worth.

Watch out

Do not read a symbol as 1 when the key says it is worth more. Always read the key first. A half symbol is half of the key, not half of 1.

Exercise 11B  Pictograms

Non-calculator
1
FThe key says 1 face = 1 child. Read the pictogram.
Favourite clubsFootballSwimmingDancingJudoKey:= 1 children
(a)How many children chose football?
(b)How many children chose dancing?
(c)How many children chose judo?
(d)How many children chose football or judo?
2
FThe key says 1 star = 2 kg of fruit and half a star = 1 kg.
Fruit sold at the marketApplesPearsPlumsCherriesKey:= 2 kgand= 1 kg
(a)How many kilograms of apples were sold?
(b)How many kilograms of pears were sold?
(c)How many kilograms of plums were sold?
(d)How many kilograms of cherries were sold?
3
FEach book symbol stands for 5 books.
Books read this termClass 3Class 4Class 5Class 6Key:= 5 books
(a)How many books did Class 4 read?
(b)Which class read the fewest books?
(c)How many more books did Class 6 read than Class 5?
(d)How many books did the four classes read altogether?
4
FThe key says 1 triangle = 10 people and half a triangle = 5 people.
How people travel to the town centreBusCarWalkCycleTrainKey:= 10 peopleand= 5 people
(a)How many people travel by car?
(b)How many people travel by cycle?
(c)Which way of travelling is the most popular?
(d)How many more people go by car than by bus?
5
FPut the days in order, from the most cars to the fewest cars.
Cars in the school car parkMonTueWedThuFriKey:= 10 carsand= 5 cars
6
FHow many symbols are needed to show each number? One symbol = the key given. Write the number of symbols.
(a)12 when 1 symbol = 2
(b)15 when 1 symbol = 5
(c)40 when 1 symbol = 10
(d)45 when 1 symbol = 10
7
FThe table shows the sandwiches sold each day. Part of the pictogram is drawn. Work out the symbols needed for Wednesday and Friday. Key: 1 circle = 5 sandwiches.
DayMonTueWedThuFri
Sandwiches2030152535
Sandwiches soldMonTueWedThuFriKey:= 5 sandwiches
(a)How many symbols for Wednesday (15 sandwiches)?
(b)How many symbols for Friday (35 sandwiches)?
8
PSThe key is hidden in each pictogram. Work out how much one symbol is worth.
PointsBlue teamRed teamKey:= ?
TeaMonTueKey:= ?
TravelWalkBusCarKey:= ?
(a)The Blue team row has 4 stars. The Blue team scored 12 points.
(b)The Tuesday row has 6 hearts. 24 cups of tea were drunk on Tuesday.
(c)The Bus row has 5 triangles. 50 pupils came by bus.
9
FDraw a pictogram for the data in the table. Use 1 heart = 2 pets. Write how many hearts are needed in each row.
PetCatsDogsBirdsFish
Number812610
10
FDraw a pictogram for the plants sold. Use 1 star = 5 plants.
PlantRoseTulipDaisyLily
Sold35204015
11
RZainab draws a pictogram of the stickers some children have. Her key says 1 symbol = 4 stickers. Leo draws the same data with the key 1 symbol = 2 stickers. How many symbols does each child need in Leo’s pictogram?
(a)Amira has 12 stickers
(b)Noah has 8 stickers
(c)Priya has 20 stickers
12
RTom looks at the pictogram. He says, “Tuesday has 8 stars, because I counted 8 symbols.” Is he right? Explain.
Gold stars earnedMonTueWedKey:= 2 stars
13
PSThe pictogram shows the cakes sold at the school sale. On Saturday the school sold 30 more cakes than on Wednesday. How many cakes were sold on Saturday?
Cakes soldMonTueWedThuFriKey:= 5 cakes
14
RA pictogram uses 1 symbol = 10 pupils. Red has 2 whole symbols and a half. Blue has 3 whole symbols. Green has the rest of 85 pupils altogether. How many symbols, and half symbols, does Green have?
11.3

Reading bar charts

Year 4
Interpret discrete data using bar charts, Solve comparison, sum and difference problems using bar charts

Printable: Worksheet (PDF) · Answers (PDF) · Open this lesson on its own page

Video lesson: Reading bar chartsComing soon

A bar chart shows data using bars. The longer the bar, the bigger the number. Every bar chart has a title, labels for each bar, and a scale: the row of numbers on one side. The bars can stand up (vertical) or lie across (horizontal).

012345678DogCatRabbitFishHamsterFavourite petsNumber of pupilsPet
A bar chart. The scale goes up in 1s. The bar for cats reaches 6, so 6 pupils chose cats.

To read a bar: put your finger on the top of the bar, then slide it across to the scale. The scale does not always go up in 1s. It can go up in 2s, 5s, 10s, 20s, 50s and more. Before you read a bar, look at the numbers on the scale and work out what one gap is worth.

0246810RaviMeiTomEllaGoals scoredNumber of goalsPlayer
051015202530MonTueWedThuFriVisitors to the museumNumber of visitorsDay
Left: the scale goes up in 2s. Right: the scale goes up in 5s.

Sometimes a bar finishes between two numbers on the scale. Then you need to find out what each small gap is worth. If the numbers go up in 10s and there is a gridline halfway between, each small gap is 5. A bar that ends on the halfway line between 30 and 40 is 35.

01020304050MonTueWedThuFriBooks readNumber of booksDay
020406080FootballSwimmingChessArtChildren at clubsNumber of children
Left: labels go up in 10s with a gridline halfway, so each gap is 5. Right: a horizontal bar chart: labels go up in 20s, each gap is 10.

Key ideas

  • Read the title and the labels first. Then find the scale.
  • Work out what one gap is worth: the step between two numbers ÷ the number of gaps.
  • Look at the top of a vertical bar (or the end of a horizontal bar) and read across (or down) to the scale.
  • If a bar ends between two numbers, count on in small gaps.
  • “How many more” means subtract. “Altogether” means add.
Worked example 11.3.1
Use the bar chart. How many pupils chose cats?
012345678DogCatRabbitFishHamsterFavourite petsNumber of pupilsPet

Key words. A bar chart uses bars to show data. The height (or length) of a bar shows how many. The scale is the row of numbers on the side. It tells you how much each gridline is worth.

Method. Look at the bar carefully and read off the number for Cat.

  1. Find the scale. The numbers up the side are the scale. We must know what one gridline is worth.
    The numbers go up in 1s: 0, 1, 2, 3...
  2. Read the bar for Cat. Find the top of the bar, then look across to the scale (follow the orange dotted line).
    The top of the bar for Cat is exactly on the line for 6.
  3. Write the answer.
    Cat = 6 pupils
    012345678DogCatRabbitFishHamsterFavourite petsNumber of pupilsPet

Check. Look again at the picture: Cat is 6. ✓

Watch out. Work out what each gridline is worth before you read a bar. The scale does not always go up in 1s.

Worked example 11.3.2
The chart shows the goals scored by four players. How many more goals did Mei score than Tom?
0246810RaviMeiTomEllaGoals scoredNumber of goalsPlayer

Key words. The difference between two numbers is how much bigger one is than the other. We find it by subtracting. “How many more” and “how many fewer” questions are difference questions. A bar chart uses bars to show data. The height (or length) of a bar shows how many. The scale is the row of numbers on the side. It tells you how much each gridline is worth.

Method. To find “how many more”, read both numbers and then subtract the smaller from the bigger.

  1. Find the scale. The numbers up the side are the scale. We must know what one gridline is worth.
    The numbers go up in 2s: 0, 2, 4, 6...
  2. Read the numbers we need from the picture. We need these numbers for the calculation.
    Mei = 10
    Tom = 4
    0246810RaviMeiTomEllaGoals scoredNumber of goalsPlayer
  3. Find the difference. To find how many more (or fewer), subtract the smaller number from the bigger number.
    10 − 4 = 6
  4. Write the answer.
    6 more

Check. Add back: 4 + 6 = 10. ✓

Watch out. Subtract: do not add. The question asks for the difference.

Worked example 11.3.3
The scale goes up in 2s, and there is a gridline between the numbers. How many books were in the reading corner on Monday?
024681012MonTueWedThuFriBooks in the reading cornerNumber of booksDay

Key words. A bar chart uses bars to show data. The height (or length) of a bar shows how many. The scale is the row of numbers on the side. It tells you how much each gridline is worth.

Method. Look at the bar carefully and read off the number for Mon.

  1. Find the scale. The numbers up the side are the scale. Not every gridline has a number, so we must work out what each small gap is worth.
    The numbers go up in 2s: 0, 2, 4...
    Between two numbers there are 2 gaps.
    2 ÷ 2 = 1, so each small gap is worth 1.
  2. Read the bar for Mon. Find the top of the bar, then look across to the scale (follow the orange dotted line).
    The top of the bar for Mon is between 6 and 8.
    It is 1 small gap after 6: 6 + 1 = 7
  3. Write the answer.
    Mon = 7 books
    024681012MonTueWedThuFriBooks in the reading cornerNumber of booksDay

Check. Look again at the picture: Mon is 7. ✓

Watch out. Work out what each gridline is worth before you read a bar. The scale does not always go up in 1s.

Worked example 11.3.4
How many visitors came to the museum in the whole week?
051015202530MonTueWedThuFriVisitors to the museumNumber of visitorsDay

Key words. A bar chart uses bars to show data. The height (or length) of a bar shows how many. The scale is the row of numbers on the side. It tells you how much each gridline is worth.

Method. Read each number from the picture, then add them all.

  1. Find the scale. The numbers up the side are the scale. We must know what one gridline is worth.
    The numbers go up in 5s: 0, 5, 10, 15...
  2. Read every number. We need these numbers for the calculation.
    Mon = 15
    Tue = 25
    Wed = 10
    Thu = 30
    Fri = 20
    051015202530MonTueWedThuFriVisitors to the museumNumber of visitorsDay
  3. Add the numbers together. Altogether means add everything. Add one number at a time.
    15 + 25 + 10 + 30 + 20
    15 + 25 = 40
    40 + 10 = 50
    50 + 30 = 80
    80 + 20 = 100
  4. Write the answer.
    Total = 100 visitors

Check. Add in the opposite order: 20 + 30 + 10 + 25 + 15 = 100. ✓

Watch out. Do not leave out a row or a bar. Check you have used every one.

Worked example 11.3.5
The scale goes up in 10s. How many books were read on Monday?
01020304050MonTueWedThuFriBooks readNumber of booksDay

Key words. A bar chart uses bars to show data. The height (or length) of a bar shows how many. The scale is the row of numbers on the side. It tells you how much each gridline is worth.

Method. Look at the bar carefully and read off the number for Mon.

  1. Find the scale. The numbers up the side are the scale. Not every gridline has a number, so we must work out what each small gap is worth.
    The numbers go up in 10s: 0, 10, 20...
    Between two numbers there are 2 gaps.
    10 ÷ 2 = 5, so each small gap is worth 5.
  2. Read the bar for Mon. Find the top of the bar, then look across to the scale (follow the orange dotted line).
    The top of the bar for Mon is between 30 and 40.
    It is 1 small gap after 30: 30 + 5 = 35
  3. Write the answer.
    Mon = 35 books
    01020304050MonTueWedThuFriBooks readNumber of booksDay

Check. Look again at the picture: Mon is 35. ✓

Watch out. Work out what each gridline is worth before you read a bar. The scale does not always go up in 1s.

Worked example 11.3.6
This is a horizontal bar chart. How many children go to swimming club?
020406080FootballSwimmingChessArtChildren at clubsNumber of children

Key words. A bar chart uses bars to show data. The height (or length) of a bar shows how many. The scale is the row of numbers on the side. It tells you how much each gridline is worth.

Method. Look at the bar carefully and read off the number for Swimming.

  1. Find the scale. The numbers along the bottom are the scale. Not every gridline has a number, so we must work out what each small gap is worth.
    The numbers go up in 20s: 0, 20, 40...
    Between two numbers there are 2 gaps.
    20 ÷ 2 = 10, so each small gap is worth 10.
  2. Read the bar for Swimming. Find the end of the bar, then look down to the scale (follow the orange dotted line).
    The end of the bar for Swimming is between 40 and 60.
    It is 1 small gap after 40: 40 + 10 = 50
  3. Write the answer.
    Swimming = 50 children
    020406080FootballSwimmingChessArtChildren at clubsNumber of children

Check. Look again at the picture: Swimming is 50. ✓

Watch out. Work out what each gridline is worth before you read a bar. The scale does not always go up in 1s.

Worked example 11.3.7
The scale goes up in 50s and each small gap is 10. How many people visited the zoo on Monday?
050100150200250300350SatSunMonTueVisitors to the zooNumber of visitorsDay

Key words. A bar chart uses bars to show data. The height (or length) of a bar shows how many. The scale is the row of numbers on the side. It tells you how much each gridline is worth.

Method. Look at the bar carefully and read off the number for Mon.

  1. Find the scale. The numbers up the side are the scale. Not every gridline has a number, so we must work out what each small gap is worth.
    The numbers go up in 50s: 0, 50, 100...
    Between two numbers there are 5 gaps.
    50 ÷ 5 = 10, so each small gap is worth 10.
  2. Read the bar for Mon. Find the top of the bar, then look across to the scale (follow the orange dotted line).
    The top of the bar for Mon is between 100 and 150.
    It is 3 small gaps after 100: 100 + 3 × 10 = 130
  3. Write the answer.
    Mon = 130 visitors
    050100150200250300350SatSunMonTueVisitors to the zooNumber of visitorsDay

Check. Look again at the picture: Mon is 130. ✓

Watch out. Work out what each gridline is worth before you read a bar. The scale does not always go up in 1s.

Worked example 11.3.8
How many more people visited the zoo on Saturday than on Tuesday?
050100150200250300350SatSunMonTueVisitors to the zooNumber of visitorsDay

Key words. The difference between two numbers is how much bigger one is than the other. We find it by subtracting. “How many more” and “how many fewer” questions are difference questions. A bar chart uses bars to show data. The height (or length) of a bar shows how many. The scale is the row of numbers on the side. It tells you how much each gridline is worth.

Method. To find “how many more”, read both numbers and then subtract the smaller from the bigger.

  1. Find the scale. The numbers up the side are the scale. Not every gridline has a number, so we must work out what each small gap is worth.
    The numbers go up in 50s: 0, 50, 100...
    Between two numbers there are 5 gaps.
    50 ÷ 5 = 10, so each small gap is worth 10.
  2. Read the numbers we need from the picture. We need these numbers for the calculation.
    Sat = 350
    Tue = 170
    050100150200250300350SatSunMonTueVisitors to the zooNumber of visitorsDay
  3. Find the difference. To find how many more (or fewer), subtract the smaller number from the bigger number.
    350 − 170 = 180
  4. Write the answer.
    180 more

Check. Add back: 170 + 180 = 350. ✓

Watch out. Subtract: do not add. The question asks for the difference.

Watch out

Do not just count the gridlines. Look at the numbers on the scale. If the scale goes up in 10s, one gridline is not 1. Check every time what each gap is worth, and be careful when a bar ends between two numbers.

Exercise 11C  Reading bar charts

Non-calculator
1
FRead the bar chart. The scale goes up in 1s.
01234567891011FootballSwimmingDanceJudoChessFavourite clubsNumber of childrenClub
(a)How many children chose football?
(b)How many children chose dance?
(c)How many children chose judo?
(d)Which club is the least popular?
2
FThe scale goes up in 2s.
02468101214RaviMeiTomEllaKofiGoals scored this seasonNumber of goalsPlayer
(a)How many goals did Ravi score?
(b)How many goals did Ella score?
(c)Which player scored 10 goals?
(d)How many goals did the five players score altogether?
3
FThe numbers go up in 2s, but each gap is worth 1.
02468101214MonTueWedThuFriCups of tea sold at the cafeNumber of cupsDay
(a)How many cups of tea were sold on Mon?
(b)How many cups of tea were sold on Tue?
(c)How many cups of tea were sold on Wed?
(d)How many more cups were sold on Tuesday than on Friday?
4
FThe scale goes up in 5s.
0510152025303540SatSunMonTueWedCars washed at the fundraiserNumber of carsDay
(a)How many cars were washed on Saturday?
(b)How many cars were washed on Tuesday?
(c)How many cars were washed on Sunday and Monday together?
(d)How many cars were washed in the five days?
5
FThe numbers go up in 10s. There is a gridline halfway between the numbers.
0102030405060708090MonTueWedThuFriTickets sold for the school playNumber of ticketsDay
(a)How many tickets were sold on Mon?
(b)How many tickets were sold on Wed?
(c)How many tickets were sold on Fri?
(d)On which day were 30 tickets sold?
6
FThis is a horizontal bar chart. The scale goes up in 20s. Each small gap is worth 10.
020406080100CheeseHamEggTunaSaladSandwiches soldNumber of sandwiches
(a)How many cheese sandwiches were sold?
(b)How many egg sandwiches were sold?
(c)Which sandwich was the most popular?
(d)How many more tuna sandwiches than egg sandwiches were sold?
7
FThe scale goes up in 50s. Each small gap is worth 10.
050100150200250300350MonTueWedThuFriBooks borrowed from the libraryNumber of booksDay
(a)How many books were borrowed on Mon?
(b)How many books were borrowed on Tue?
(c)How many books were borrowed on Thu?
(d)On which day were 310 books borrowed?
8
FThe numbers on a scale go up in the step shown. Work out what one small gap is worth.
(a)The numbers go up in 2s. There are 2 gaps between each pair of numbers.
(b)The numbers go up in 10s. There are 5 gaps between each pair of numbers.
(c)The numbers go up in 20s. There are 4 gaps between each pair of numbers.
(d)The numbers go up in 50s. There are 5 gaps between each pair of numbers.
(e)The numbers go up in 10s. There are 2 gaps between each pair of numbers.
(f)The numbers go up in 5s. There are 5 gaps between each pair of numbers.
9
PSLook at the chart of children at the after-school club.
0102030405060MonTueWedThuFriChildren at the after-school clubNumber of childrenDay
(a)On which day were the most children at the club?
(b)How many children were at the club on Monday?
(c)What is the difference between the biggest number and the smallest number of children?
(d)How many more children were there on Wednesday than on Tuesday?
10
PSThe bar chart shows the stickers earned by two classes. The blue bars are Class 4A and the orange bars are Class 4B.
02468101214MonTueWedThuStickers earnedNumber of stickersClass 4AClass 4B
(a)How many stickers did Class 4A earn on Tuesday?
(b)How many stickers did Class 4B earn on Tuesday?
(c)How many stickers did Class 4A earn in the four days?
(d)How many stickers did Class 4B earn in the four days?
11
RWrite true or false for each statement about the chess club chart.
0102030405060MonTueWedThuFriPupils at the chess clubNumber of pupilsDay
(a)More pupils came on Wednesday than on Monday.
(b)Thursday had twice as many pupils as Monday.
(c)Friday had 25 pupils.
(d)Tuesday had fewer pupils than Friday.
12
RMr Khan makes a bar chart of the days it rained and the days it was dry. He says, “The bar chart has bars that go up to 10, so each gridline must be worth 1.” Why might he be wrong? What should he check?
13
RPart of the bar chart is covered by an ink blot. The bar for Thursday cannot be seen. In total, 125 visitors came in the week. How many came on Thursday?
05101520253035MonTueWedThuFriVisitors to the museumNumber of visitorsDay
11.4

Drawing bar charts and choosing a scale

Year 4
Present discrete data using bar charts, Choose a suitable scale

Printable: Worksheet (PDF) · Answers (PDF) · Open this lesson on its own page

Video lesson: Drawing bar charts and choosing a scaleComing soon

Now it is your turn to draw a bar chart. The data below shows the number of children at four clubs. Follow these steps.

  1. Give the chart a title that says what it shows. Label the bottom (the clubs) and the side (the number of children).
  2. Choose a scale. Look at the biggest number. The scale must go a little higher than this. Pick easy steps, such as 2, 5, 10 or 20.
  3. Draw the bars. Start at 0. Make every bar the same width, with gaps between them. Each bar goes up to the right number on the scale.
01020304050607080FootballDanceChessArtChildren at clubsNumber of childrenClub
01020304050607080FootballDanceChessArtChildren at clubsNumber of childrenClub
Left: the labelled axes with the scale (0 to 80 in 10s, with a gridline halfway). Right: the finished chart.

How to choose a scale. The biggest number here is 75. If we went up in 1s we would need 75 gridlines, which is far too many. Going up in 10s needs only 8 labels. But 75 is not a multiple of 10, so we put a gridline halfway between the numbers. Each small gap is then 5, and every number (40, 75, 20, 60) lands exactly on a gridline. For small numbers, up to about 10, a scale in 1s is fine. For numbers in the hundreds, steps of 50 or 100 work well.

Key ideas

  • Every chart needs a title and labels on both axes.
  • The scale starts at 0 and goes a little higher than the biggest number.
  • Choose easy steps (1, 2, 5, 10, 20, 50). Add gridlines halfway if some numbers fall between the labels.
  • All the bars start on the 0 line and are the same width.
  • Always check your chart by reading each bar back and comparing with the table.
Worked example 11.4.1
Mia has this data: 8, 5, 11, 6. What scale could she use for a bar chart?

Key words. A bar chart uses bars to show data. The height (or length) of a bar shows how many. The scale is the row of numbers on the side. It tells you how much each gridline is worth.

Method. Look at the biggest number. Choose easy steps so the scale is not too long, and make sure every number can be drawn exactly.

  1. Find the biggest number. The scale must reach at least this far.
    Numbers: 8, 5, 11, 6
    Biggest = 11
  2. Choose a step for the labels. The scale should have no more than about 8 numbers, and easy steps like 1, 2, 5, 10 or 20 are best.
    Going up in 2s to 12: 6 steps, which is a good size.
  3. Check every number can be shown. Some numbers are not multiples of 2, so they would be between the labels. Put 1 gridline between the numbers so that each gap is 1.
    2 ÷ 2 = 1
    8 = 8 × 1, 5 = 5 × 1, 11 = 11 × 1, 6 = 6 × 1
  4. Write the scale.
    The scale goes from 0 to 12 and the labels go up in 2s, with 1 gridline between the numbers (each gap = 1).

Check. The biggest bar will reach 11, which is below the top of the scale, 12. ✓

Watch out. A scale that is too short cuts off the tall bars. A scale that goes up in 1s for big numbers is too long.

Worked example 11.4.2
Priya has this data: 40, 75, 20, 60. What scale could she use for a bar chart?

Key words. A bar chart uses bars to show data. The height (or length) of a bar shows how many. The scale is the row of numbers on the side. It tells you how much each gridline is worth.

Method. Look at the biggest number. Choose easy steps so the scale is not too long, and make sure every number can be drawn exactly.

  1. Find the biggest number. The scale must reach at least this far.
    Numbers: 40, 75, 20, 60
    Biggest = 75
  2. Choose a step for the labels. The scale should have no more than about 8 numbers, and easy steps like 1, 2, 5, 10 or 20 are best.
    Going up in 10s to 80: 8 steps, which is a good size.
  3. Check every number can be shown. Some numbers are not multiples of 10, so they would be between the labels. Put 1 gridline between the numbers so that each gap is 5.
    10 ÷ 2 = 5
    40 = 8 × 5, 75 = 15 × 5, 20 = 4 × 5, 60 = 12 × 5
  4. Write the scale.
    The scale goes from 0 to 80 and the labels go up in 10s, with 1 gridline between the numbers (each gap = 5).

Check. The biggest bar will reach 75, which is below the top of the scale, 80. ✓

Watch out. A scale that is too short cuts off the tall bars. A scale that goes up in 1s for big numbers is too long.

Worked example 11.4.3
A shop sold these numbers of ice creams in four days: 120, 80, 150, 90. What scale could be used?

Key words. A bar chart uses bars to show data. The height (or length) of a bar shows how many. The scale is the row of numbers on the side. It tells you how much each gridline is worth.

Method. Look at the biggest number. Choose easy steps so the scale is not too long, and make sure every number can be drawn exactly.

  1. Find the biggest number. The scale must reach at least this far.
    Numbers: 120, 80, 150, 90
    Biggest = 150
  2. Choose a step for the labels. The scale should have no more than about 8 numbers, and easy steps like 1, 2, 5, 10 or 20 are best.
    Going up in 20s to 160: 8 steps, which is a good size.
  3. Check every number can be shown. Some numbers are not multiples of 20, so they would be between the labels. Put 1 gridline between the numbers so that each gap is 10.
    20 ÷ 2 = 10
    120 = 12 × 10, 80 = 8 × 10, 150 = 15 × 10, 90 = 9 × 10
  4. Write the scale.
    The scale goes from 0 to 160 and the labels go up in 20s, with 1 gridline between the numbers (each gap = 10).

Check. The biggest bar will reach 150, which is below the top of the scale, 160. ✓

Watch out. A scale that is too short cuts off the tall bars. A scale that goes up in 1s for big numbers is too long.

Worked example 11.4.4
Draw a bar chart for this table. Use a scale that goes up in 2s.
PetDogCatFishBird
Pupils812610
024681012DogCatFishBirdFavourite petsNumber of pupilsPet

Key words. A bar chart uses bars to show data. The height (or length) of a bar shows how many. The scale is the row of numbers on the side. It tells you how much each gridline is worth.

Method. Find each bar’s height on the scale, then draw bars of the same width up to that height.

  1. Check the scale. We need to know what each gridline is worth before we draw.
    The numbers go up in 2s: 0, 2, 4, 6...
  2. Find where each bar stops. Use the scale to find the height of each bar.
    Dog: 8 is on the line for 8
    Cat: 12 is on the line for 12
    Fish: 6 is on the line for 6
    Bird: 10 is on the line for 10
  3. Draw the bars. Start at the bottom. Draw each bar up to its height. Make every bar the same width, with a gap between the bars.
    024681012DogCatFishBirdFavourite petsNumber of pupilsPet

Check. Read each bar back from your chart: they should match the table. ✓

Watch out. Draw every bar from the bottom axis (the 0 line), with the same width.

Worked example 11.4.5
Complete the bar chart. The table says Wednesday is 45 and Friday is 50.
DayMonTueWedThuFri
Books3520453050
01020304050MonTueWedThuFriBooks readNumber of booksDay

Key words. A bar chart uses bars to show data. The height (or length) of a bar shows how many. The scale is the row of numbers on the side. It tells you how much each gridline is worth.

Method. Find each bar’s height on the scale, then draw bars of the same width up to that height.

  1. Check the scale. We need to know what each gridline is worth before we draw.
    The numbers go up in 10s: 0, 10, 20...
    Between two numbers there are 2 gaps.
    10 ÷ 2 = 5, so each small gap is worth 5.
  2. Find where each bar stops. Use the scale to find the height of each bar.
    Wed: 45 is 1 gap above 40
    Fri: 50 is on the line for 50
  3. Draw the bars. Start at the bottom. Draw each bar up to its height. Make every bar the same width, with a gap between the bars.
    01020304050MonTueWedThuFriBooks readNumber of booksDay

Check. Read each bar back from your chart: they should match the table. ✓

Watch out. Draw every bar from the bottom axis (the 0 line), with the same width.

Worked example 11.4.6
Noah drew a bar chart from this table. One bar is wrong. Which one?
DayMonTueWedThuFri
Visitors2540301535
0510152025303540MonTueWedThuFriVisitors to the parkNumber of visitorsDay

Key words. A bar chart uses bars to show data. The height (or length) of a bar shows how many. The scale is the row of numbers on the side. It tells you how much each gridline is worth.

Method. Read every bar and compare it with the table. Find the one that is different.

  1. Find the scale. The numbers up the side are the scale. We must know what one gridline is worth.
    The numbers go up in 5s: 0, 5, 10, 15...
  2. Read each bar. Write down the number each bar shows.
    Mon = 25, Tue = 40, Wed = 20, Thu = 15, Fri = 35
  3. Compare with the table. The numbers should be exactly the same.
    Mon: chart 25, table 25 ✓
    Tue: chart 40, table 40 ✓
    Wed: chart 20, table 30 × (different)
    Thu: chart 15, table 15 ✓
    Fri: chart 35, table 35 ✓
  4. Write the answer. The bar that does not match the table is wrong.
    The bar for Wed is wrong. It shows 20 but it should be 30.

Check. All the other bars match the table. ✓

Watch out. Check every bar, one at a time.

Worked example 11.4.7
Draw the bars for Football (40) and Dance (75) on this horizontal chart. The labels go up in 20s and each gap is 5.
020406080FootballDanceChessArtChildren at clubsNumber of children

Key words. A bar chart uses bars to show data. The height (or length) of a bar shows how many. The scale is the row of numbers on the side. It tells you how much each gridline is worth.

Method. Find each bar’s height on the scale, then draw bars of the same width up to that height.

  1. Check the scale. We need to know what each gridline is worth before we draw.
    The numbers go up in 20s: 0, 20, 40...
    Between two numbers there are 4 gaps.
    20 ÷ 4 = 5, so each small gap is worth 5.
  2. Find where each bar stops. Use the scale to find the height of each bar.
    Football: 40 is on the line for 40
    Dance: 75 is 3 gaps above 60
  3. Draw the bars. Start at the bottom. Draw each bar up to its height. Make every bar the same width, with a gap between the bars.
    020406080FootballDanceChessArtChildren at clubsNumber of children

Check. Read each bar back from your chart: they should match the table. ✓

Watch out. Draw every bar from the bottom axis (the 0 line), with the same width.

Watch out

Do not start a bar anywhere but 0. Do not make the bars different widths. And do not change the scale part way up: the numbers on the scale must go up by the same amount each time.

Exercise 11D  Drawing bar charts and choosing a scale

Non-calculator
1
FChoose a good scale for each set of numbers. Write what the scale goes up in, and what each gap is worth.
(a)7, 4, 9, 6
(b)30, 55, 20, 45
(c)150, 100, 200, 250
(d)12, 6, 10, 8
2
FThe scale on a chart has gaps that are worth 5. How many gaps high is the bar for each number?
(a)10
(b)25
(c)40
(d)35
3
FDraw a bar chart for the pets table. Use the axes below. The scale goes up in 2s and each gap is worth 1.
PetDogCatRabbitFish
Pupils6947
0246810DogCatRabbitFishFavourite petsNumber of pupilsPet
4
FComplete the bar chart for the library. Draw the bars for Tuesday and Thursday.
DayMonTueWedThuFri
Visitors2035154025
0510152025303540MonTueWedThuFriVisitors to the libraryNumber of visitorsDay
5
FDraw a horizontal bar chart for the table. The labels go up in 20s and each gap is worth 10.
SandwichCheeseHamEggTuna
Sold40703060
020406080CheeseHamEggTunaSandwiches soldNumber of sandwiches
6
FComplete the table using the bar chart. Write the number of cakes sold each day.
0102030405060MonTueWedThuFriCakes soldNumber of cakesDay
(a)Mon
(b)Tue
(c)Wed
(d)Thu
(e)Fri
7
RMei drew this bar chart of pupils at lunch clubs. She has made a mistake with the scale. What is the mistake? Write the labels the scale should have.
01020405060ChessArtChoirCodingPupils at lunch clubsNumber of pupilsClub
8
RNoah drew a bar chart from the park table. Which bar is wrong, and what should it be?
DayMonTueWedThuFri
Visitors2540301535
0510152025303540MonTueWedThuFriVisitors to the parkNumber of visitorsDay
9
RAmira has data from 40 to 90. She says, “I will use a scale that goes up in 1s.” Is this a good idea? Explain, and suggest a better scale.
10
PSJack made a tally chart of the colours of cars. Count the tally marks, then draw a bar chart. Use a scale that goes up in 2s with each gap worth 1.
ColourTally
Red
Blue
Green
Yellow
0246810RedBlueGreenYellowColours of carsNumber of carsColour
11
RTom draws a bar chart where each gap is worth 20. Which of these numbers can he show exactly on a gridline?
(a)60
(b)130
(c)90
(d)140
12
RZainab says, “When I draw a bar chart I can make the bars as wide or as narrow as I like, as long as they are all different.” Do you agree? Explain.
11.5

Tables and two-way tables

Year 4
Solve comparison, sum and difference problems using information presented in tables

Printable: Worksheet (PDF) · Answers (PDF) · Open this lesson on its own page

Video lesson: Tables and two-way tablesComing soon

A table is a neat way to show information in rows and columns. A row goes across. A column goes down. This table shows what the school shop sold. To find the number of rubbers sold on Wednesday, find the rubbers row and the Wed column, and look where they meet.

Item soldMonTueWedThu
Pencils1215914
Rubbers68104
Rulers5739
Rubbers row, Wed column: 10 rubbers were sold on Wednesday.

A two-way table shows two things at once. This one shows whether a pupil is a boy or a girl, and how they travel to school. It also has totals. The last number in each row is the total for that row. The last number in each column is the total for that column. The number in the bottom corner is the grand total: the number of pupils altogether.

Travel to schoolWalkBusCarCycleTotal
Boys865423
Girls793524
Total15158947
A two-way table with totals. 8 boys walk. 23 boys altogether. 15 pupils walk. 47 pupils altogether.

Missing numbers. The numbers in a row add up to the total of the row. The numbers in a column add up to the total of the column. So if a number is missing, add the numbers you know and take the answer away from the total.

Key ideas

  • A row goes across. A column goes down. To read a number, find the right row and the right column.
  • A total is all the numbers in a row or column added together.
  • In a two-way table the bottom-right number is the grand total.
  • To find a missing number: total − the numbers you know.
Worked example 11.5.1
How many pencils did the school shop sell on Tuesday?
Item soldMonTueWedThu
Pencils1215914
Rubbers68104
Rulers5739

Key words. In a table, a row goes across and a column goes down. A two-way table shows two things at once, for example boys and girls, and how they travel. The total of a row or column is all its numbers added together.

Method. Find the right row and the right column. The answer is the number where they meet.

  1. Find the row. Look down the left-hand side for Pencils.
    Row: Pencils
  2. Find the column. Look along the top for Tue.
    Column: Tue
  3. Find where the row and column meet. The number in that box is the answer.
    Pencils and Tue: 15
    Item soldMonTueWedThu
    Pencils1215914
    Rubbers68104
    Rulers5739

Check. Look again: row Pencils, column Tue, number 15. ✓

Watch out. Make sure you use the right row and the right column.

Worked example 11.5.2
How many rubbers were sold from Monday to Thursday altogether?
Item soldMonTueWedThu
Pencils1215914
Rubbers68104
Rulers5739

Key words. In a table, a row goes across and a column goes down. A two-way table shows two things at once, for example boys and girls, and how they travel. The total of a row or column is all its numbers added together.

Method. Add all the numbers in the row for Rubbers.

  1. Find the row. Look for Rubbers on the left.
    Item soldMonTueWedThu
    Pencils1215914
    Rubbers68104
    Rulers5739
  2. Add the numbers in the row. Altogether means add.
    6 + 8 + 10 + 4 = 28
  3. Write the answer.
    28 items

Check. Add backwards: 4 + 10 + 8 + 6 = 28. ✓

Watch out. Add every number in the row. Do not miss one.

Worked example 11.5.3
How many items were sold on Thursday altogether?
Item soldMonTueWedThu
Pencils1215914
Rubbers68104
Rulers5739

Key words. In a table, a row goes across and a column goes down. A two-way table shows two things at once, for example boys and girls, and how they travel. The total of a row or column is all its numbers added together.

Method. Add all the numbers in the column for Thu.

  1. Find the column. Look for Thu along the top.
    Item soldMonTueWedThu
    Pencils1215914
    Rubbers68104
    Rulers5739
  2. Add the numbers in the column. Altogether means add.
    14 + 4 + 9 = 27
  3. Write the answer.
    27 items

Check. Add backwards: 9 + 4 + 14 = 27. ✓

Watch out. Add every number in the column.

Worked example 11.5.4
How many more pencils than rulers were sold on Monday?
Item soldMonTueWedThu
Pencils1215914
Rubbers68104
Rulers5739

Key words. The difference between two numbers is how much bigger one is than the other. We find it by subtracting. “How many more” and “how many fewer” questions are difference questions. In a table, a row goes across and a column goes down. A two-way table shows two things at once, for example boys and girls, and how they travel. The total of a row or column is all its numbers added together.

Method. Read both numbers from the table, then subtract the smaller from the bigger.

  1. Find the first number. Pencils and Mon.
    12
  2. Find the second number. Rulers and Mon.
    5
    Item soldMonTueWedThu
    Pencils1215914
    Rubbers68104
    Rulers5739
  3. Subtract. The difference is the bigger number minus the smaller number.
    12 − 5 = 7

Check. 5 + 7 = 12. ✓

Watch out. Subtract: do not add.

Worked example 11.5.5
The two-way table shows how pupils in Class 4 travel to school. How many boys cycle?
Travel to schoolWalkBusCarCycleTotal
Boys865423
Girls793524
Total15158947

Key words. In a table, a row goes across and a column goes down. A two-way table shows two things at once, for example boys and girls, and how they travel. The total of a row or column is all its numbers added together.

Method. Find the right row and the right column. The answer is the number where they meet.

  1. Find the row. Look down the left-hand side for Boys.
    Row: Boys
  2. Find the column. Look along the top for Cycle.
    Column: Cycle
  3. Find where the row and column meet. The number in that box is the answer.
    Boys and Cycle: 4
    Travel to schoolWalkBusCarCycleTotal
    Boys865423
    Girls793524
    Total15158947

Check. Look again: row Boys, column Cycle, number 4. ✓

Watch out. Make sure you use the right row and the right column.

Worked example 11.5.6
How many girls are there in Class 4? Use the table above.
Travel to schoolWalkBusCarCycle
Boys8654
Girls7935

Key words. In a table, a row goes across and a column goes down. A two-way table shows two things at once, for example boys and girls, and how they travel. The total of a row or column is all its numbers added together.

Method. Add all the numbers in the row for Girls.

  1. Find the row. Look for Girls on the left.
    Travel to schoolWalkBusCarCycle
    Boys8654
    Girls7935
  2. Add the numbers in the row. Altogether means add.
    7 + 9 + 3 + 5 = 24
  3. Write the answer.
    24 pupils

Check. Add backwards: 5 + 3 + 9 + 7 = 24. ✓

Watch out. Add every number in the row. Do not miss one.

Worked example 11.5.7
Some numbers have been rubbed out. The total number of girls is 24. How many girls come by bus?
Travel to schoolWalkBusCarCycleTotal
Boys865423
Girls7  ?  3524
Total15158947

Key words. In a table, a row goes across and a column goes down. A two-way table shows two things at once, for example boys and girls, and how they travel. The total of a row or column is all its numbers added together.

Method. The numbers in a row (or column) add up to its total. Add the numbers you know, then subtract from the total.

  1. Look at the row or column with the missing number. We use the row for Girls. It has a total.
    Travel to schoolWalkBusCarCycleTotal
    Boys865423
    Girls7  ?  3524
    Total15158947
  2. Add the numbers we know. The missing number is the part of the total that is left.
    7 + 3 + 5 = 15
  3. Subtract from the total. Total minus the known parts leaves the missing part.
    24 − 15 = 9

Check. 7 + 3 + 5 + 9 = 24. ✓

Watch out. Use subtraction to find the missing number.

Worked example 11.5.8
Complete the two-way table. Write the missing numbers.
ClubSwimmingChessChoirTotal
Year 39  ?  1227
Year 4118  ?  26
Total2014  ?  53

Key words. In a table, a row goes across and a column goes down. A two-way table shows two things at once, for example boys and girls, and how they travel. The total of a row or column is all its numbers added together.

Method. Use the totals. For a blank number: total minus the numbers you know. For a blank total: add the numbers.

  1. Look for a row or column with only one blank. We can find that blank using the total of that row or column.
    ClubSwimmingChessChoirTotal
    Year 39  ?  1227
    Year 4118  ?  26
    Total2014  ?  53
  2. Work out each blank. For a blank number, take the numbers you know away from the total. For a blank total, add the numbers.
    Year 3, Chess: 27 − 9 + 12 = 6
    Year 4, Choir: 26 − 11 + 8 = 7
    Total for Choir: 12 + 7 = 19
  3. Write the completed table. Check that every row and column adds up.
    ClubSwimmingChessChoirTotal
    Year 3961227
    Year 4118726
    Total20141953

Check. The rows add up to the grand total, 53, and so do the columns. ✓

Watch out. Check across and down: both must add up.

Watch out

Make sure you use the right row and the right column. Put a finger on the row and a finger on the column, and bring them together. And do not mix up a total with a number inside the table.

Exercise 11E  Tables and two-way tables

Non-calculator
1
FThe table shows the items sold in the school shop.
Item soldMonTueWedThu
Pencils1215914
Rubbers68104
Rulers5739
(a)How many pencils were sold on Wed?
(b)How many rubbers were sold on Tue?
(c)How many rulers were sold on Thu?
(d)On which day were 14 pencils sold?
2
FUse the school shop table to find these totals.
Item soldMonTueWedThu
Pencils1215914
Rubbers68104
Rulers5739
(a)How many pencils were sold from Monday to Thursday?
(b)How many rulers were sold from Monday to Thursday?
(c)How many items were sold on Tuesday?
(d)How many items were sold on Wednesday?
3
FUse the school shop table.
Item soldMonTueWedThu
Pencils1215914
Rubbers68104
Rulers5739
(a)How many more pencils than rubbers were sold on Monday?
(b)How many more rulers were sold on Thursday than on Monday?
(c)What is the difference between the most pencils sold in a day and the fewest pencils sold in a day?
4
FLook at the two-way table about how Class 4 travel to school.
Travel to schoolWalkBusCarCycle
Boys8654
Girls7935
(a)How many boys walk?
(b)How many girls bus?
(c)How many girls cycle?
(d)How many pupils come by car altogether?
5
FUse the same table to answer these.
Travel to schoolWalkBusCarCycle
Boys8654
Girls7935
(a)How many boys are there in Class 4?
(b)How many girls are there in Class 4?
(c)How many pupils walk to school?
(d)How many more girls than boys come by bus?
6
FSome numbers in the lunch table have been rubbed out. Work out each missing number.
Lunch choicePizzaPastaSaladTotal
Class 314  ?  730
Class 41112  ?  28
Class 5  ?  101331
Total33312589
(a)Class 3, pasta
(b)Class 4, salad
(c)Class 5, pizza
7
FComplete the two-way table about pets. Fill in every blank.
PetCatsDogsFishTotal
Boys79  ?  20
Girls  ?  65  ?  
Total17  ?  941
8
PSTwelve children said whether they are a boy or a girl, and what pet they have. Each child is shown with their answers: Ali (girl, cats); Mei (girl, dogs); Tom (boy, dogs); Zara (girl, cats); Leo (boy, cats); Amy (girl, dogs); Ben (boy, dogs); Isla (girl, cats); Omar (boy, cats); Eve (girl, dogs); Sam (boy, dogs); Noor (girl, cats). Complete the two-way table by counting.
PetCatsDogs
Boys  ?    ?  
Girls  ?    ?  
(a)Boys with cats
(b)Boys with dogs
(c)Girls with cats
(d)Girls with dogs
9
RWrite true or false. Use the table of clubs.
ClubFootballNetballRugby
Year 314129
Year 411158
(a)More Year 4 children than Year 3 children play netball.
(b)Year 3 has 35 children who go to a club.
(c)Rugby is the least popular club for Year 3 and for Year 4.
10
RNoah adds the girls’ numbers in the travel table: 7, 9, 3 and 5. He gets 25. The table says there are 24 girls. Who is right? Check by adding.
11
RBoys and girls entered three events on sports day. The grand total is 42. The total for boys is 20. Girls run is 4 and girls swim is 8. How many girls entered the jump?
11.6

Time graphs and continuous data

Year 4
Interpret and present discrete and continuous data, Time graphs, Solve comparison, sum and difference problems using time graphs

Printable: Worksheet (PDF) · Answers (PDF) · Open this lesson on its own page

Video lesson: Time graphs and continuous dataComing soon

There are two kinds of data. Discrete data is data you count. The number of goals in a match is 0, 1, 2, 3 … You cannot score 2½ goals. We often show discrete data in a pictogram or a bar chart.

Continuous data is data you measure. A plant’s height, a temperature or the time for a race can be any value, with lots of values in between. Continuous data changes smoothly, so we can show it on a time graph (also called a line graph).

024681012141618208am9am10am11am12pm1pm2pmTemperature on a spring dayTemperature (°C)Time of day
A time graph. The time is along the bottom. The temperature is up the side. Each dot is a reading. The dots are joined with straight lines.

To read a time graph, find the time along the bottom. Go up to the dot, then across to the scale. At 11am the temperature is 18°C. To find when something happens, do it the other way round: find the number on the scale, go across to the line, then down to the time. At the same time as 12pm and 1pm the line is flat, so the temperature stayed the same.

The line shows the trend: what the numbers are doing over time. If the line goes up, the number is rising. If it goes down, the number is falling. If it is flat, the number stays the same. To draw a time graph, plot a dot for each reading, then join the dots in order with a ruler.

Key ideas

  • Discrete data is counted (whole numbers). Continuous data is measured (any value in between).
  • A time graph has time along the bottom and the measurement up the side.
  • To read a value: go up from the time to the line, then across to the scale. To find a time: go across from the value to the line, then down.
  • The line going up means rising. Going down means falling. Flat means no change.
  • To find a change, subtract: the bigger reading minus the smaller reading.
Worked example 11.6.1
Is the number of children in a class discrete or continuous data?

Key words. Discrete data is data you count. It jumps from one whole number to the next (you cannot have 2½ pupils). Continuous data is data you measure. It can take any value in between, like a height or a time. Continuous data changes smoothly.

Method. Think about how we get the numbers. We count discrete data. We measure continuous data.

  1. Ask: do we count it, or measure it? Counting gives whole numbers that jump. Measuring gives numbers that can be anywhere between.
    We count the children: 24, 25, 26 … whole numbers only.
  2. Decide. If we count it, it is discrete. If we measure it, it is continuous.
    It is discrete data.
  3. Choose a graph. Different data suits different graphs.
    Discrete data is often shown on a bar chart or pictogram.

Check. Can there be a value in between? No: you cannot have a part of a whole thing. ✓

Watch out. Money is counted in pence, so it is discrete. Time, height and temperature are measured, so they are continuous.

Worked example 11.6.2
Is the height of a plant discrete or continuous data?

Key words. Discrete data is data you count. It jumps from one whole number to the next (you cannot have 2½ pupils). Continuous data is data you measure. It can take any value in between, like a height or a time. Continuous data changes smoothly.

Method. Think about how we get the numbers. We count discrete data. We measure continuous data.

  1. Ask: do we count it, or measure it? Counting gives whole numbers that jump. Measuring gives numbers that can be anywhere between.
    We measure the height with a ruler. It could be 12 cm, 12.5 cm, 12.7 cm … any value in between.
  2. Decide. If we count it, it is discrete. If we measure it, it is continuous.
    It is continuous data.
  3. Choose a graph. Different data suits different graphs.
    Continuous data is often shown on a time graph (line graph).

Check. Can there be a value in between? Yes: for example, halfway between two values. ✓

Watch out. Money is counted in pence, so it is discrete. Time, height and temperature are measured, so they are continuous.

Worked example 11.6.3
The graph shows the temperature on a spring day. What was the temperature at 11am?
024681012141618208am9am10am11am12pm1pm2pmTemperature on a spring dayTemperature (°C)Time of day

Key words. A time graph (or line graph) shows how something changes over time. The points are joined by lines. The numbers along the bottom show the time. The numbers up the side show the measurement.

Method. Look at the graph carefully and read off the number for 11am.

  1. Find the scale. The scale up the side tells us what each gridline is worth.
    The numbers up the side go up in 2s: 0, 2, 4, 6...
  2. Find 11am along the bottom and read the point. Go up from the time to the point on the line, then look across to the scale (follow the orange dotted lines).
    At 11am, the point is exactly on the line for 18°C.
  3. Write the answer.
    11am = 18°C
    024681012141618208am9am10am11am12pm1pm2pmTemperature on a spring dayTemperature (°C)Time of day

Check. Look again at the picture: 11am is 18. ✓

Watch out. Read the scale up the side carefully. Check which time you are looking at along the bottom.

Worked example 11.6.4
At what time was the temperature 14°C?
024681012141618208am9am10am11am12pm1pm2pmTemperature on a spring dayTemperature (°C)Time of day

Key words. A time graph (or line graph) shows how something changes over time. The points are joined by lines. The numbers along the bottom show the time. The numbers up the side show the measurement.

Method. Start at 14 on the scale, go across to the line, then go down to find the time.

  1. Find the scale. The scale up the side tells us what each gridline is worth.
    The numbers up the side go up in 2s: 0, 2, 4, 6...
  2. Find 14 on the scale up the side. Then go across to the line.
    Look across from 14 until you reach the line.
  3. Go down to the time. Look straight down from the point to the numbers along the bottom.
    The point is above 10am.
    024681012141618208am9am10am11am12pm1pm2pmTemperature on a spring dayTemperature (°C)Time of day
  4. Write the answer.
    10am

Check. At 10am the temperature is 14. ✓

Watch out. Go across first, then down.

Worked example 11.6.5
How much warmer was it at 12pm than at 8am?
024681012141618208am9am10am11am12pm1pm2pmTemperature on a spring dayTemperature (°C)Time of day

Key words. The difference between two numbers is how much bigger one is than the other. We find it by subtracting. “How many more” and “how many fewer” questions are difference questions. A time graph (or line graph) shows how something changes over time. The points are joined by lines. The numbers along the bottom show the time. The numbers up the side show the measurement.

Method. To find “how many more”, read both numbers and then subtract the smaller from the bigger.

  1. Find the scale. The scale up the side tells us what each gridline is worth.
    The numbers up the side go up in 2s: 0, 2, 4, 6...
  2. Read the numbers we need from the picture. We need these numbers for the calculation.
    12pm = 20
    8am = 10
    024681012141618208am9am10am11am12pm1pm2pmTemperature on a spring dayTemperature (°C)Time of day
  3. Find the difference. To find how many more (or fewer), subtract the smaller number from the bigger number.
    20 − 10 = 10
  4. Write the answer.
    10 °C warmer

Check. Add back: 10 + 10 = 20. ✓

Watch out. Subtract: do not add. The question asks for the difference.

Worked example 11.6.6
Describe what happened to the temperature from 8am to 2pm.
024681012141618208am9am10am11am12pm1pm2pmTemperature on a spring dayTemperature (°C)Time of day

Key words. A time graph (or line graph) shows how something changes over time. The points are joined by lines. The numbers along the bottom show the time. The numbers up the side show the measurement.

Method. Look at each part of the line. Say if it goes up, goes down or stays the same.

  1. Find the scale. The scale up the side tells us what each gridline is worth.
    The numbers up the side go up in 2s: 0, 2, 4, 6...
  2. Read every point. We need each number.
    8am = 10, 9am = 12, 10am = 14, 11am = 18, 12pm = 20, 1pm = 20, 2pm = 16
    024681012141618208am9am10am11am12pm1pm2pmTemperature on a spring dayTemperature (°C)Time of day
  3. Compare each point with the next one. Up if the next number is bigger. Down if it is smaller. The same if the numbers match.
    8am to 9am: 10 to 12, it goes up
    9am to 10am: 12 to 14, it goes up
    10am to 11am: 14 to 18, it goes up
    11am to 12pm: 18 to 20, it goes up
    12pm to 1pm: 20 to 20, it stays the same
    1pm to 2pm: 20 to 16, it goes down
  4. Join the parts that are the same. Write short sentences that cover the whole graph.
    From 8am to 12pm it goes up.
    From 12pm to 1pm it stays the same.
    From 1pm to 2pm it goes down.

Check. The sentences cover the whole graph from 8am to 2pm. ✓

Watch out. A flat line means the number stays the same.

Worked example 11.6.7
The time graph shows the height of Aisha’s sunflower. How tall was it in Week 3?
0102030Week 0Week 1Week 2Week 3Week 4Week 5Height of a sunflowerHeight (cm)Time

Key words. A time graph (or line graph) shows how something changes over time. The points are joined by lines. The numbers along the bottom show the time. The numbers up the side show the measurement.

Method. Look at the graph carefully and read off the number for Week 3.

  1. Find the scale. The scale up the side tells us what each gridline is worth.
    The numbers up the side go up in 10s.
    Between two numbers there are 5 gaps.
    10 ÷ 5 = 2, so each small gap is worth 2.
  2. Find Week 3 along the bottom and read the point. Go up from the time to the point on the line, then look across to the scale (follow the orange dotted lines).
    At Week 3, the point is between 10 and 20.
    10 + 3 × 2 = 16 cm
  3. Write the answer.
    Week 3 = 16 cm
    0102030Week 0Week 1Week 2Week 3Week 4Week 5Height of a sunflowerHeight (cm)Time

Check. Look again at the picture: Week 3 is 16. ✓

Watch out. Read the scale up the side carefully. Check which time you are looking at along the bottom.

Worked example 11.6.8
The table shows the height of a sunflower. Plot the points for Week 2 and Week 4 on the graph, then join the points.
Week012345
Height (cm)2610162228
0102030Week 0Week 1Week 2Week 3Week 4Week 5Height of a sunflowerHeight (cm)Time

Key words. A time graph (or line graph) shows how something changes over time. The points are joined by lines. The numbers along the bottom show the time. The numbers up the side show the measurement.

Method. Plot each point: find the time on the bottom, go up to the number, and put a dot. Then join the dots with a ruler.

  1. Find the scale. The scale up the side tells us what each gridline is worth.
    The numbers up the side go up in 10s.
    Between two numbers there are 5 gaps.
    10 ÷ 5 = 2, so each small gap is worth 2.
  2. Find each point. Find the time along the bottom. Go up to the number. Put a dot there.
    Week 2: go up to 10 (on the line for 10)
    Week 4: go up to 22 (1 gap above 20)
  3. Join the dots. Use a ruler to join each dot to the next one with a straight line.
    0102030Week 0Week 1Week 2Week 3Week 4Week 5Height of a sunflowerHeight (cm)Time

Check. Read each dot back from your graph. They should match the table. ✓

Watch out. Mark the dots exactly where the lines cross. Then join them in order.

Worked example 11.6.9
A reading was not taken at 10:30am. Estimate the temperature halfway between 10am and 11am.
024681012141618208am9am10am11am12pm1pm2pmTemperature on a spring dayTemperature (°C)Time of day

Key words. A time graph (or line graph) shows how something changes over time. The points are joined by lines. The numbers along the bottom show the time. The numbers up the side show the measurement.

Method. The line joining two points shows how the number changes. Halfway in time is about halfway in value.

  1. Read the two points. The time we want is halfway between them.
    10am = 14, 11am = 18
  2. Find the number halfway. Halfway between two numbers: add them, then halve. Or add half the difference.
    14 + 18 = 32, and 32 ÷ 2 = 16
    024681012141618208am9am10am11am12pm1pm2pmTemperature on a spring dayTemperature (°C)Time of day
  3. Write the answer. This is an estimate. We are using the straight line between the points.
    About 16

Check. 14 and 18 are each 2 away from 16. ✓

Watch out. This gives an estimate. We cannot be sure what happened between the readings.

Watch out

Do not mix up time and measurement. The time is along the bottom. The measurement is up the side. And remember: the line between two dots only shows the trend. We cannot be sure of the exact value there.

Exercise 11F  Time graphs and continuous data

Non-calculator
1
FIs each type of data discrete (counted) or continuous (measured)?
(a)The number of goals scored in a match
(b)The time it takes to run 100 metres
(c)The number of cars in a car park
(d)The temperature of a cup of tea
(e)The height of a tree
(f)The number of pages in a book
2
FRead the time graph of the temperature.
024681012141618208am9am10am11am12pm1pm2pmTemperature on a spring dayTemperature (°C)Time of day
(a)What was the temperature at 8am?
(b)What was the temperature at 9am?
(c)What was the temperature at 11am?
(d)What was the temperature at 2pm?
3
FUse the same graph. At what time was the temperature:
024681012141618208am9am10am11am12pm1pm2pmTemperature on a spring dayTemperature (°C)Time of day
(a)12°C?
(b)18°C?
(c)16°C?
4
FUse the temperature graph.
024681012141618208am9am10am11am12pm1pm2pmTemperature on a spring dayTemperature (°C)Time of day
(a)How much warmer was it at 11am than at 9am?
(b)How much cooler was it at 2pm than at 1pm?
(c)What is the difference between the highest and the lowest temperature?
5
FUse the temperature graph. Does the temperature go up, go down or stay the same between these times? By how much?
024681012141618208am9am10am11am12pm1pm2pmTemperature on a spring dayTemperature (°C)Time of day
(a)8am and 10am
(b)12pm and 1pm
(c)1pm and 2pm
6
FUse the temperature graph.
024681012141618208am9am10am11am12pm1pm2pmTemperature on a spring dayTemperature (°C)Time of day
(a)At what time did the temperature first reach its highest value?
(b)At what time was it the coolest?
(c)Between which two times was the temperature rise the biggest?
7
FAisha measures her sunflower each week.
0102030Week 0Week 1Week 2Week 3Week 4Week 5Height of a sunflowerHeight (cm)Time
(a)How tall was the sunflower in Week 1?
(b)How tall was the sunflower in Week 3?
(c)How tall was the sunflower in Week 5?
(d)How much did it grow between Week 2 and Week 5?
8
FThe table shows the height of a candle as it burns. Plot the points on the graph and join them with a ruler.
Hour012345
Height (cm)2218141062
04812162024Hour 0Hour 1Hour 2Hour 3Hour 4Hour 5Height of a burning candleHeight (cm)Time
9
FThe graph shows the water in a paddling pool. Plot the points for 11am and 1pm. They are 12 litres and 20 litres.
048121620249am10am11am12pm1pm2pmWater in a paddling poolWater (litres)Time
10
PSThe graph shows how far Ravi was from home on a walk.
024689:009:3010:0010:3011:0011:3012:00Ravi's walkDistance from home (km)Time
(a)How far from home was Ravi at 10:00?
(b)How far from home was he at 12:00?
(c)Ravi stopped for a rest. From what time to what time was the line flat?
(d)For how long did Ravi rest? Write your answer in minutes.
11
RWrite true or false for each statement about the temperature graph.
024681012141618208am9am10am11am12pm1pm2pmTemperature on a spring dayTemperature (°C)Time of day
(a)The temperature was 20°C at 12pm and at 1pm.
(b)The temperature went down between 11am and 12pm.
(c)The temperature at 9am was 2°C higher than at 8am.
12
RDescribe the whole temperature graph in two or three sentences. Say where it goes up, stays the same and goes down.
024681012141618208am9am10am11am12pm1pm2pmTemperature on a spring dayTemperature (°C)Time of day
13
RLeo says, “At 10:30am the temperature must have been exactly 16°C, because that is halfway between 14 and 18.” What is wrong with what Leo says? Estimate the temperature.
11.7

Solving problems and reasoning with charts

Year 4
Solve comparison, sum and difference problems using information presented in bar charts, pictograms, tables and other graphs

Printable: Worksheet (PDF) · Answers (PDF) · Open this lesson on its own page

Video lesson: Solving problems and reasoning with chartsComing soon

Now you can read tally charts, pictograms, bar charts, tables and time graphs. In this section you will use them to solve problems. Many questions ask you to compare things (“how many more”), to find a sum (“how many altogether”) or to find a difference (“what is the difference between the tallest and the shortest”).

The best way is to do it in steps. 1. Read the question: what is it asking? 2. Find the numbers you need on the chart (read the key or scale first!). 3. Decide: add, subtract or multiply? 4. Write the answer with its unit and check it makes sense.

010203040FootballSwimmingChessArtDanceChildren at clubsNumber of childrenClub
How many more children go to football and swimming together than to chess and art together? Football + swimming = 35 + 25 = 60. Chess + art = 15 + 20 = 35. 60 − 35 = 25.

Which chart? Use a tally chart to collect data as it comes in. Use a pictogram or a bar chart to compare things you have counted (discrete data), such as favourite fruit. Use a table when you need exact numbers. Use a time graph when something changes over time and you measure it (continuous data), such as temperature or height.

Key ideas

  • Read the question. Find what it asks and which numbers you need.
  • Read the key (pictogram) or the scale (bar chart, time graph) first.
  • “How many more” and “the difference” → subtract. “Altogether” and “in total” → add.
  • Tally and bar chart or pictogram: counted data. Time graph: measured data over time.
  • To decide if a statement is true: find the numbers and check them. Then give a reason.
Worked example 11.7.1
How many more children go to football and swimming together than to chess and art together?
010203040FootballSwimmingChessArtDanceChildren at clubsNumber of childrenClub

Key words. The difference between two numbers is how much bigger one is than the other. We find it by subtracting. “How many more” and “how many fewer” questions are difference questions. A bar chart uses bars to show data. The height (or length) of a bar shows how many. The scale is the row of numbers on the side. It tells you how much each gridline is worth.

Method. Work out each total first. Then subtract the smaller total from the bigger total.

  1. Find the scale. The numbers up the side are the scale. Not every gridline has a number, so we must work out what each small gap is worth.
    The numbers go up in 10s: 0, 10, 20...
    Between two numbers there are 2 gaps.
    10 ÷ 2 = 5, so each small gap is worth 5.
  2. Read the numbers we need from the picture. We need these numbers for the calculation.
    Football = 35
    Swimming = 25
    Chess = 15
    Art = 20
    010203040FootballSwimmingChessArtDanceChildren at clubsNumber of childrenClub
  3. Add the numbers for football and swimming. Together means add.
    35 + 25 = 60
  4. Add the numbers for chess and art. Together means add.
    15 + 20 = 35
  5. Subtract. How many more means the bigger total minus the smaller total.
    60 − 35 = 25

Check. 35 + 25 = 60. ✓

Watch out. There are three steps here: add, add, then subtract.

Worked example 11.7.2
Which day had twice as many books borrowed as Monday?
Books borrowedMonTueWedThuFriKey:= 5 books

Key words. A pictogram uses small pictures (symbols) to show data. The key tells you how many each symbol stands for. A half symbol stands for half of that number.

Method. Find the number for the first one, double it, then look for that number in the chart.

  1. Read the key. The key tells us what one symbol is worth. Here a symbol is not worth 1.
    1 symbol = 5 books
  2. Read the numbers we need from the picture. Multiply the number of symbols by what one symbol is worth.
    Mon = 20
    Books borrowedMonTueWedThuFriKey:= 5 books
  3. Double the number. Twice as many means two times as many.
    2 × 20 = 40
  4. Find the number 40 in the chart. Look for the row or bar with this number.
    Tue = 40
    Books borrowedMonTueWedThuFriKey:= 5 books
  5. Write the answer.
    Tue

Check. 40 ÷ 2 = 20. ✓

Watch out. Twice as many means × 2, not + 2.

Worked example 11.7.3
The pictogram shows the lunches sold on Friday. A wrap costs £3. How much money was taken for wraps?
Lunches sold on FridayPizzaWrapSoupKey:= 5 lunches

Key words. A table shows information in rows (going across) and columns (going down). A total is the answer when you add all the numbers together. A pictogram uses small pictures (symbols) to show data. The key tells you how many each symbol stands for. A half symbol stands for half of that number.

Method. Read the number from the chart. Then multiply by the price of one.

  1. Read the key. The key tells us what one symbol is worth. Here a symbol is not worth 1.
    1 symbol = 5 lunches
  2. Read the numbers we need from the picture. Multiply the number of symbols by what one symbol is worth.
    Wrap = 20
    Lunches sold on FridayPizzaWrapSoupKey:= 5 lunches
  3. Multiply by the price. Each one costs £3, so multiply the number sold by £3.
    20 × £3 = £60
  4. Write the answer.
    £60

Check. £60 ÷ 20 = £3. ✓

Watch out. Check you use the right price for the right item.

Worked example 11.7.4
Mia measures the temperature of a room every hour through the day. Which chart should she use: a bar chart, a pictogram or a time graph? Give a reason.

Key words. Discrete data is data you count. It jumps from one whole number to the next (you cannot have 2½ pupils). Continuous data is data you measure. It can take any value in between, like a height or a time. Continuous data changes smoothly.

Method. Think about what kind of data it is, and what we want to show.

  1. Is it counted or measured? Counted data is discrete. Measured data that changes over time is continuous.
    The temperature is measured, and it keeps changing. It is continuous data.
  2. What do we want to show? Compare amounts, or show how something changes?
    We want to show how the temperature changes through the day.
  3. Choose the chart.
    A time graph

Check. This chart fits the data. ✓

Watch out. A time graph is for data that changes over time. Bar charts and pictograms compare separate groups.

Worked example 11.7.5
Water runs into a paddling pool at the same speed all day. How many litres will there be at 3pm if the pattern carries on?
048121620249am10am11am12pm1pm2pmWater in a paddling poolWater (litres)Time

Key words. A time graph (or line graph) shows how something changes over time. The points are joined by lines. The numbers along the bottom show the time. The numbers up the side show the measurement.

Method. Find how much the graph goes up each time. Then carry on the pattern.

  1. Find the scale. The scale up the side tells us what each gridline is worth.
    The numbers up the side go up in 4s.
    Between two numbers there are 2 gaps.
    4 ÷ 2 = 2, so each small gap is worth 2.
  2. Read the points. We need the numbers on the graph.
    9am = 4, 10am = 8, 11am = 12, 12pm = 16, 1pm = 20, 2pm = 24
    048121620249am10am11am12pm1pm2pmWater in a paddling poolWater (litres)Time
  3. Look for the pattern. Find how much the number changes each time.
    9am to 10am: 8 − 4 = 4
    10am to 11am: 12 − 8 = 4
    11am to 12pm: 16 − 12 = 4
    It goes up by 4 every time.
  4. Carry on the pattern. If the pattern continues, add the same amount again.
    24 + 4 = 28

Check. 28 − 4 × 1 = 24. ✓

Watch out. This is a prediction. It only works if the pattern carries on.

Worked example 11.7.6
What is the difference between the height of the tallest child and the shortest child? Give your answer in centimetres.
020406080100120140160AmiraBenChloeDevEllaHeights of five childrenHeight (cm)Child

Key words. The difference between two numbers is how much bigger one is than the other. We find it by subtracting. “How many more” and “how many fewer” questions are difference questions. A bar chart uses bars to show data. The height (or length) of a bar shows how many. The scale is the row of numbers on the side. It tells you how much each gridline is worth.

Method. Find the biggest and the smallest numbers, then subtract.

  1. Find the scale. The numbers up the side are the scale. Not every gridline has a number, so we must work out what each small gap is worth.
    The numbers go up in 20s: 0, 20, 40...
    Between two numbers there are 2 gaps.
    20 ÷ 2 = 10, so each small gap is worth 10.
  2. Find the biggest and the smallest. We need these numbers for the calculation.
    Dev = 160
    Chloe = 120
    020406080100120140160AmiraBenChloeDevEllaHeights of five childrenHeight (cm)Child
  3. Subtract. The difference between two numbers is the bigger minus the smaller.
    160 − 120 = 40
  4. Write the answer.
    40

Check. 120 + 40 = 160. ✓

Watch out. Be sure you have found the biggest and the smallest of all the numbers.

Worked example 11.7.7
Zainab says, “Tuesday had twice as many books borrowed as Monday.” Is she right?
Books borrowedMonTueWedThuFriKey:= 5 books

Key words. A pictogram uses small pictures (symbols) to show data. The key tells you how many each symbol stands for. A half symbol stands for half of that number.

Method. Read both numbers. Double the smaller one and see if it matches the bigger one.

  1. Read the key. The key tells us what one symbol is worth. Here a symbol is not worth 1.
    1 symbol = 5 books
  2. Read the numbers we need from the picture. We need these numbers for the calculation.
    Tue = 40
    Mon = 20
    Books borrowedMonTueWedThuFriKey:= 5 books
  3. Double the smaller number. “Twice as many” means two times as many.
    2 × 20 = 40
  4. Compare. Is Tue equal to 40?
    40 = 40
  5. Decide.
    The statement is true.

Check. 40 ÷ 2 = 20. ✓

Watch out. Twice as many means × 2, not + 2.

Watch out

Do not stop after the first step! In a two-step problem, check you have answered the whole question. Also check the unit (books, children, centimetres, £). And read the key or scale before you read any numbers.

Exercise 11G  Problems and reasoning

Non-calculator
1
RWhich chart is best for each? Choose from: a tally chart, a bar chart or a pictogram, a time graph.
(a)The number of children who choose each flavour of ice cream
(b)How the height of a baby changes each month
(c)The votes for a class pet, as the votes are called out
(d)The temperature of a cup of tea as it cools
2
PSUse the clubs bar chart.
010203040FootballSwimmingChessArtDanceChildren at clubsNumber of childrenClub
(a)How many more children go to dance than to art?
(b)How many children go to chess, art or dance altogether?
(c)How many more children go to football and dance together than to swimming and art together?
3
PSThe pictogram shows the books borrowed each day.
Books borrowedMonTueWedThuFriKey:= 5 books
(a)Which day had twice as many books borrowed as Monday?
(b)How many books were borrowed from Monday to Friday?
(c)How many more books were borrowed on Tuesday than on Thursday?
4
PSThe pictogram shows the lunches sold on Friday. Pizza costs £2, wraps cost £3 and soup costs £1.
Lunches sold on FridayPizzaWrapSoupKey:= 5 lunches
(a)How much money was taken for pizza?
(b)How much money was taken for soup?
(c)How much money was taken for wraps?
5
PSThe paddling pool is filling up at the same speed all day.
048121620249am10am11am12pm1pm2pmWater in a paddling poolWater (litres)Time
(a)How many litres will there be at 3pm?
(b)How many litres will there be at 5pm?
6
PSThe bar chart shows the heights of five children.
020406080100120140160AmiraBenChloeDevEllaHeights of five childrenHeight (cm)Child
(a)What is the difference between the tallest and the shortest child?
(b)How tall is Chloe?
(c)How much taller is Ben than Amira?
7
RThree charts show how many pets some children have. Which two charts show the same data?
024681012DogCatFishPets: chart ANumber of pets
Pets: chart BDogCatFishKey:= 2 pets
Pets: chart CDogCatFishKey:= 4 petsand= 2 pets
8
RPriya says, “Tuesday has 9 stickers.” She counted the symbols. What mistake has she made? What is the correct number?
Stickers wonMonTueWedKey:= 2 stickers
9
RNoah looks at the bar chart and says, “Wednesday is 2, because the bar reaches the second gridline.” What mistake has he made? What does the bar really show?
010203040MonTueWedThuCakes soldNumber of cakesDay
10
RIs each statement true or false? Use the books pictogram. Give a reason.
Books borrowedMonTueWedThuFriKey:= 5 books
(a)Thursday had the fewest books borrowed.
(b)More books were borrowed on Wednesday than on Friday.
(c)Altogether, 125 books were borrowed.
11
PSThe two bar charts show the stickers earned by two classes. Both use the same scale.
02468101214MonTueWedThuClass 4A stickersNumber of stickers
02468101214MonTueWedThuClass 4B stickersNumber of stickers
(a)Which class earned more stickers in total?
(b)How many more stickers did Class 4A earn than Class 4B in total?
12
RA bar chart has four bars for Mon, Tue, Wed and Thu. In total, 60 pupils are shown. Tuesday has 10 pupils. Monday has twice as many as Tuesday. Wednesday has 10 more than Tuesday. How many pupils does Thursday have?
13
RElla says, “A time graph is a good way to show the number of children who like each fruit.” Do you agree? Explain.

Chapter summary

SkillMethod
Tally chartOne stroke for each thing. The fifth stroke goes across four. Count in 5s, then add the single strokes. The total is the frequency.
PictogramRead the key first. Count the symbols, multiply by the key. A half symbol is half the key. To draw: divide the number by the key.
Bar chartWork out what each gridline or gap is worth. Read across from the top of the bar. Between the marks, count on in small gaps.
Drawing a bar chartTitle, labels, a scale from 0 going up in easy steps, bars the same width starting at 0. Check against the table.
Tables and two-way tablesFind the right row and column. Totals: add the row or column. Missing number: total minus the numbers you know.
Time graphUp from the time, then across to the scale. To find a time: across from the number, then down. Flat means no change.
Discrete and continuousDiscrete: counted, whole numbers (bar chart, pictogram). Continuous: measured, any value (time graph).
How many more / altogether / differenceMore, fewer, difference: subtract. Altogether, in total: add. Do the steps in order and check the unit.

Can you…?

  • I can make a tally chart and a frequency table, and count tally marks in fives.
  • I can read and draw a pictogram with a key, and use a half symbol.
  • I can read a bar chart with a scale in 1s, 2s, 5s, 10s, 20s or 50s, including values between the marks.
  • I can draw and complete a bar chart and choose a sensible scale.
  • I can read a table and a two-way table, and find missing numbers using totals.
  • I can read a time graph, find when something happens, and describe whether the line goes up, goes down or stays the same.
  • I can say if data is discrete or continuous, and choose a suitable chart.
  • I can solve “how many more”, “altogether” and “difference” problems with charts, and spot mistakes in them.

Key terms

Data
Information that we collect, such as numbers or answers to a question.
Tally chart
A chart where we draw one stroke for each thing we count. Every fifth stroke goes across the other four.
Frequency
How many times something happens, or how many there are of one kind.
Frequency table
A table that lists each kind of thing and its frequency.
Pictogram
A chart that uses small pictures (symbols) to show data.
Key
The part of a pictogram that tells you how many one symbol is worth.
Bar chart
A chart that uses bars to show data. The longer the bar, the bigger the number.
Scale
The numbers on the side (or bottom) of a chart that tell you how much each gridline is worth.
Axis
A line of a chart with a scale or labels. A chart has two axes (more than one are called axes).
Two-way table
A table that shows two things at once, with totals for the rows and columns.
Discrete data
Data that we count. It is a whole number.
Continuous data
Data that we measure. It can have any value in between.
Time graph
A line graph that shows how something changes over time.
Trend
What the data is doing: going up, going down or staying the same.
Difference
How much bigger one number is than another. Subtract to find it.

Exercise 11Z  Check-up (non-calculator)

Non-calculator
1
How many tally marks are there?
2
The key says 1 square = 5 pizzas. How many pizzas were sold on Monday?
Pizzas soldMonTueWedThuKey:= 5 pizzas
3
A pictogram uses 1 symbol for 10 children. How many symbols are needed to show 30 children?
4
How many visitors came to the museum on Thursday?
010203040MonTueWedThuFriVisitors to the museumNumber of visitorsDay
5
How many more visitors came on Tuesday than on Wednesday?
010203040MonTueWedThuFriVisitors to the museumNumber of visitorsDay
6
How many girls chose milk?
DrinkTeaJuiceMilk
Boys698
Girls7510
7
The time graph shows the depth of snow. How deep was the snow on Day 3?
02468101214Day 1Day 2Day 3Day 4Day 5Snow depthDepth (cm)Time
8
On which day was the snow 10 cm deep?
02468101214Day 1Day 2Day 3Day 4Day 5Snow depthDepth (cm)Time
9
Is the weight of a baby discrete or continuous data?
10
Kofi measures his plant every week for ten weeks. Which chart should he use: a bar chart, a pictogram or a time graph? Explain.

Total 19 marks. Allow about 19 minutes.

Exercise 11X  SATs-style questions

Non-calculator
1
FThe bar chart shows the number of pupils at four sports clubs. How many pupils go to rugby club?[1 mark]
051015202530354045FootballNetballRugbyHockeyPupils at clubsNumber of pupilsClub
2
FIn the pictogram, a whole heart is 10 pets and half a heart is 5 pets. How many cats are there?[1 mark]
Pets in Year 4CatsDogsFishBirdsKey:= 10 petsand= 5 pets
3
FAli counted cars and made these tally marks. How many cars did he count?
[1 mark]
4
FHow many more pupils go to rugby than to hockey? Use the bar chart about clubs.[2 marks]
051015202530354045FootballNetballRugbyHockeyPupils at clubsNumber of pupilsClub
5
FThe table shows how many sandwiches were sold. Complete the pictogram for Tuesday and Wednesday. 1 square = 5 sandwiches. Write how many symbols you need for each day.[2 marks]
DayMonTueWedThu
Sandwiches30452035
Sandwiches soldMonTueWedThuKey:= 5 sandwiches
6
PSSome numbers in the table are missing. 30 Year 4 children chose a club. How many Year 4 children chose art?[2 marks]
ChessArtChoirTotal
Year 38121030
Year 49  ?  1430
7
PSThe graph shows the water in a paddling pool. How many more litres were in the pool at 1pm than at 10am?[2 marks]
048121620249am10am11am12pm1pm2pmWater in a paddling poolWater (litres)Time
8
RAmira says, “Tuesday has twice as many children as Monday.” Is she right? Show how you know. Use the pictogram.[2 marks]
Children at the clubMonTueWedKey:= 2 children
9
PSThe table shows the number of cakes sold each day. (a) Complete the bar chart: draw the bars for Tuesday and Thursday. (b) How many cakes were sold altogether?[3 marks]
DayMonTueWedThuFri
Cakes2035153025
05101520253035MonTueWedThuFriCakes soldNumber of cakesDay
10
PSThe graph shows the temperature in the classroom. (a) What was the temperature at 10am? (b) Between which times did the temperature stay the same? (c) How many degrees warmer was it at 12pm than at 8am? (d) Is temperature discrete or continuous data?[3 marks]
0246810121416188am9am10am11am12pmTemperatureTemperature (°C)Time
← Chapter 10: Position and direction

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