A picture graph can look easy for exactly the reason it becomes difficult: the pictures are familiar. A child sees five symbols and answers “five” before checking what those symbols represent. The counting is correct. The interpretation is not.
When one symbol represents two books, five symbols represent ten books. The page contains five marks, but the data describes ten objects. That distinction is the whole lesson. A graph is not a collection of decorations. It is a representation with an agreed meaning.
This guide teaches children to read that agreement, use it consistently, and check their answers. It begins with equal groups, moves through comparison and missing information, and finishes with original practice questions and explained answers. All classroom data below is invented for teaching.
The essential idea: one symbol can stand for a group
Start with six counters. Arrange them as three pairs. Place a small card beside each pair. There are now six counters and three cards. Each card stands for two counters.
Ask two different questions: “How many cards are there?” and “How many counters do the cards represent?” The answers are three and six. Neither is wrong; they answer different questions.
Now remove the counters but keep the cards. The cards still represent six counters because their agreed value has not changed. That is how a scaled picture graph works. A symbol is a carrier of information, not necessarily a picture of one object.
The relationship is:
Number of symbols × value of each symbol = represented quantity
A child should be able to explain each part in ordinary language before treating this as a formula. “There are four symbols. Each means three pupils. Four groups of three pupils make twelve pupils.”
Read the title, categories and key before calculating
Three parts give the graph meaning. The title tells us what the data is about. The category labels tell us which group each row describes. The key tells us how much each symbol represents.
Consider this text version of a picture graph. Every square is a symbol; it is not a single book.
Books borrowed by three classes
Key: each ■ represents 2 books
Class A ■ ■ ■
Class B ■ ■ ■ ■ ■
Class C ■ ■ ■ ■
Class A borrowed 6 books, Class B borrowed 10 books, and Class C borrowed 8 books. The key applies to every row of this graph. A child who changes the key between rows changes the data rather than interpreting it.
Before asking for totals, ask the child to complete this sentence: “In this graph, one square represents ___.” Then ask what the answer unit will be. Here it is books, not classes, children or squares.
Worked example 1: read one category
How many books did Class B borrow?
There are five symbols in the Class B row. Each represents two books. Therefore 5 × 2 = 10. Class B borrowed 10 books.
A useful check is repeated addition: 2 + 2 + 2 + 2 + 2 = 10. Another check is to place two counters under each symbol. These are different representations of the same relationship.
An answer of “5” tells the adult something specific: the child may have counted symbols without converting them into books. Repeating the same question more loudly will not repair that distinction. Return briefly to the cards-and-counters model.
Worked example 2: compare two categories
How many more books did Class B borrow than Class A?
Class B represents 10 books and Class A represents 6 books. The difference is 10 − 6 = 4 books.
There is also a shorter valid method. Class B has two more symbols than Class A. Because the same key applies to both rows, the two extra symbols represent 2 × 2 = 4 books.
Discuss why the second method works. It does not work merely because one row looks longer. It works because matching symbols have equal values. The two matching groups can be compared symbol by symbol, leaving two extra groups of two books.
The answer is not “two more books”. Two is the difference in symbols. Four is the difference in books.
Worked example 3: find the total
How many books did the three classes borrow altogether?
One route is to read each row first: 6 + 10 + 8 = 24 books.
Another route is to count all the symbols: 3 + 5 + 4 = 12 symbols. Twelve groups of two represent 12 × 2 = 24 books. A child who is not ready to calculate that product directly can combine the known row totals instead.
Both methods preserve the same key. Neither involves adding the number of symbols to the number of books. An expression such as 3 + 10 + 4 mixes two kinds of quantities. The arithmetic might be performed correctly, but the calculation no longer describes the graph.
Encourage the learner to label intermediate quantities: “12 symbols” is a useful temporary answer; it is not yet the answer to a question about books.
Worked example 4: recover a missing row
A second graph records 30 books borrowed by three classes. Its key is again one symbol for two books. Class A borrowed 6 books and Class B borrowed 10 books. Class C’s row is missing. How many symbols should it contain?
First find the books already accounted for: 6 + 10 = 16 books. Then find Class C’s quantity: 30 − 16 = 14 books. Finally convert the quantity into symbols: 14 ÷ 2 = 7 symbols.
This problem runs the representation in reverse. Reading a graph converts symbols into a quantity. Completing a graph converts a quantity into symbols.
Check by rebuilding the total: 6 + 10 + 14 = 30. Then check the representation: seven symbols, each worth two books, do represent fourteen books.
Notice that the final answer is symbols because the question asks what to draw. The correct unit depends on the question, not merely on the graph’s title.
Worked example 5: discover the key from known information
Four symbols represent twenty stickers. What does each symbol represent?
The twenty stickers are shared equally among four symbols. Therefore 20 ÷ 4 = 5 stickers per symbol.
Now suppose another row contains three symbols. It represents 3 × 5 = 15 stickers.
The key must be established before the second row is interpreted. A child who guesses “one symbol means two” because that was the previous graph’s key has transferred the appearance of the earlier task, not its reasoning.
A reliable explanation is: “I know four equal groups make twenty. One group is five. Every symbol in this graph therefore stands for five stickers.”
Worked example 6: compare two graphs with different keys
Graph A has six symbols and a key of two shells per symbol. Graph B has four symbols and a key of three shells per symbol. Which represents more shells?
Graph A represents 6 × 2 = 12 shells. Graph B represents 4 × 3 = 12 shells. They represent the same number of shells.
Counting symbols alone would suggest that Graph A represents more. The different keys make that comparison invalid. Convert both representations into the common quantity, shells, before comparing.
This is a useful transfer question because it checks whether the child understands the scale rather than merely following a repeated layout. The visual surface changes; the quantity relationship remains available.
What a graph does not tell us
Suppose a graph records books borrowed by three classes. It does not automatically tell us how many pupils borrowed them. One pupil might have borrowed more than one book. It does not tell us which book was most popular unless titles are separately recorded. It does not tell us why one class borrowed more.
Ask the child to separate “shown by the graph” from “possible but not shown”. This is not a trick. It is part of reading information accurately.
For example, “Class B borrowed four more books than Class A” follows from our first graph. “Class B likes reading more” does not follow from those numbers alone. The classes may have different numbers of pupils, different borrowing opportunities, or other differences that the graph does not describe.
A complete mathematical answer sometimes includes “There is not enough information.” That is better than inventing an explanation.
Four common errors and the repair each needs
Counting pictures as objects. Ask for the number of symbols and the represented quantity separately. Rebuild each symbol as an equal group of counters. The goal is to recover the meaning of the key.
Remembering the previous graph’s key. Place two graphs side by side with different keys. Ask the child to state each key before calculating. Do not let visual familiarity decide the value.
Stopping at the difference in symbols. Compare two rows, identify the extra symbols, and then convert those symbols using the key. Ask, “Two more what?” before accepting a bare number.
Losing the answer unit. Ask the child to finish “The question wants the number of ___.” Reading a category asks for represented objects; completing a row may ask for symbols. This distinction should be visible in the final answer.
An error is evidence about a particular task, not a diagnosis of a child’s ability. Check the interpretation with a second example before deciding what needs practice.
A short diagnostic before another worksheet
Use a graph with a key of three counters per symbol. Show a row with four symbols. Ask the child to identify the key, count the symbols, state the represented quantity, explain the calculation, and draw a row representing six counters.
The expected answers are three counters per symbol, four symbols, twelve counters, four equal groups of three, and two symbols.
If counting is secure but interpretation fails, practise the key. If interpretation is secure but the multiplication is slow, let the child use equal groups while building the relevant number fact. If reading works but drawing fails, practise the reverse direction. These are different teaching jobs.
Do not turn this small check into a timed test. Its purpose is to make the child’s reasoning observable.
Original practice: read, compare, reconstruct
Use the following graph for Questions 1–6.
Tokens earned by four teams
Key: each ■ represents 3 tokens
Red ■ ■ ■
Blue ■ ■ ■ ■ ■
Green ■ ■
Yellow ■ ■ ■ ■
- How many tokens did Green earn?
- How many tokens did Blue earn?
- How many more tokens did Yellow earn than Red?
- How many tokens did Red and Green earn altogether?
- How many more tokens did Blue and Yellow earn together than Red and Green together?
- A Purple team earned 12 tokens. How many symbols should its row contain?
For Questions 7–10, use the information in each question rather than the key above.
- A graph uses one symbol for five cards. How many cards do four symbols represent?
- A row of six symbols represents twenty-four buttons. What is the value of each symbol?
- Graph A has four symbols worth five marbles each. Graph B has five symbols worth four marbles each. Which represents more marbles?
- A graph has a total of thirty stickers. Two known rows represent ten and fifteen stickers. Each symbol represents five stickers. How many symbols belong in the remaining row?
Explained answers
1. Green: 6 tokens. Two symbols represent two groups of three: 2 × 3 = 6.
2. Blue: 15 tokens. Five groups of three make fifteen: 5 × 3 = 15.
3. Yellow earned 3 more tokens. Yellow represents 12 and Red represents 9. Their difference is 3. Alternatively, the one extra symbol represents three tokens.
4. Red and Green: 15 tokens. Red’s 9 plus Green’s 6 gives 15. Their five symbols together also represent five groups of three.
5. The difference is 12 tokens. Blue and Yellow together represent 15 + 12 = 27. Red and Green represent 9 + 6 = 15. Therefore 27 − 15 = 12.
6. Purple needs 4 symbols. Twelve tokens divided into groups of three require four groups: 12 ÷ 3 = 4.
7. Twenty cards. Four symbols worth five cards each represent 4 × 5 = 20 cards.
8. Four buttons per symbol. The twenty-four buttons are represented by six equal groups: 24 ÷ 6 = 4.
9. They represent equal quantities. Graph A represents 4 × 5 = 20 marbles. Graph B represents 5 × 4 = 20 marbles. Different symbol counts can represent the same quantity.
10. One symbol. The known rows account for 10 + 15 = 25 stickers. The remaining row represents 30 − 25 = 5 stickers. Since one symbol represents five stickers, the row needs one symbol.
Optional extension: part of a symbol
Some picture graphs explicitly use part of a symbol. Introduce this only after whole-symbol scales are secure, and only when the key or task makes the partial symbol’s value clear.
For example, when a full symbol represents two objects, half of that symbol can represent one object. Three full symbols and a half-symbol then represent 6 + 1 = 7 objects. This is an optional extension here, not a claim that every P2 graph must contain partial symbols.
Do not let children guess from a cropped or damaged picture. A mathematical representation needs an interpretable key. If the mark is ambiguous, first clarify the representation rather than rewarding a lucky numerical answer.
A practical teaching sequence
Begin with physical objects grouped beneath symbol cards. Remove the objects and ask the learner to recover the represented quantities. Introduce a written key and two or three labelled rows. Ask direct reading questions before moving to differences and totals.
Next reverse the direction: give a quantity and ask how many symbols to draw. Finally vary the key, layout and context so the child must read the current graph rather than remember the previous example.
An adult can keep the same question structure while changing only one demand. For example, keep the categories and symbol counts fixed but change the key from two to three. Ask what changes and what stays the same. The number of symbols stays fixed; the represented quantities change.
In a different variation, keep the quantities fixed but change the key. Twelve objects require six symbols at two per symbol, but four symbols at three per symbol. This makes the distinction between representation and quantity concrete.
A brief home routine
Choose a small collection such as buttons or building blocks. Make groups of two, three or five. Draw one symbol for each group and write the key. Ask one reading question, one comparison question and one reverse question.
Stop after the child has explained a useful idea accurately. The purpose is not to generate a large amount of work. It is to connect objects, symbols, equal groups and language.
On a later day, change the objects and the key. Ask the child to teach the new graph to you. An explanation such as “You cannot just count the squares, because each square means three” shows more than a correct answer by itself.
Where this sits in Primary Mathematics
MOE’s current Primary Mathematics syllabus includes reading picture graphs with scales in P2. The official scope reference is the Primary Mathematics Syllabus, updated October 2025, page 34.
The teaching route is simple: secure one-to-one picture reading, introduce one-to-many representation, and then connect the same scale awareness to bar graphs. A child who understands that a symbol’s value comes from a key has a useful starting point for reading other data representations.
For the earlier step, use Reading Picture Graphs One Object at a Time. For the later connection, continue to Reading Bar Graphs Without Confusing Scale and Frequency. The Mathematics Learning Hub keeps these ideas connected to the wider learning route.
For level-specific teaching support, the existing Primary 2 Mathematics Tuition guide remains the service route. This article’s job is narrower: help a learner read the key, preserve its meaning, and answer the question in the correct units.