3A · Chapter 7 · Lesson 2
Bar Graphs and Tables
Build and interpret bar graphs and tables with scales and category labels.
Today's challenge
Compare the two outer bars: what is the difference in items between Paper and Glass? Try your own method, then compare it with a model.
What you'll learn
- Build and interpret bar graphs and tables with scales and category labels.
- Explain the connection between a diagram and a number sentence.
- Solve a fresh application without step-by-step hints.
Before this lesson
Equal intervals on number lines and reading tables.
Prerequisite competency: 3A-C07-L01
Entry · 2 questions
Warm-up: prerequisite check
Try on paper before continuing. If the questions are hard, review the prerequisite concepts.
- E1
Read the pictograph below (each symbol represents 5 fruits). How many fruits are shown altogether?
Pictograph: read the key and scale carefully before solving. Show answer and reasoning
Answer: 85
There are 17 symbols across all categories. The key makes each symbol worth 5, so 17×5=85 fruits.
- E2
Read the pictograph below (each symbol represents 1 fruits). How many fruits are shown altogether?
Pictograph: read the key and scale carefully before solving. Show answer and reasoning
Answer: 16
There are 16 symbols across all categories. The key makes each symbol worth 1, so 16×1=16 fruits.
Learn the concept
Words to know
- bar graph
- rectangles showing amounts by a numerical scale
- scale
- the value of one equal axis interval
Build and interpret bar graphs and tables with scales and category labels.
Label categories and consistent numerical intervals. Read each bar against the axis marks and convert heights into counts.
Why this works
A bar graph uses a consistent numbered axis. If the axis increases by five each grid interval, a bar reaching the fourth mark after zero represents 20. Read the actual scale before comparing bars. Equal distances on the axis must mean equal changes in the values.
Check the meaning
Do not confuse the number of grid boxes with the numerical quantity when the scale is 2, 5, or 10 per division.
Concrete → Pictorial → Abstract
- Build: Use counters, grid paper, or labeled bundles to represent the story or quantity.
- Draw: Sketch an array, place-value exchange, or equal-unit bar; label what is known.
- Calculate: Write an equation matching the representation and solve.
- Verify: Use an inverse operation or a second strategy to check.
For advanced students: if you can already explain this concept, start with the independent or challenge set.
Worked examples and immediate checks
Explain each step aloud or in writing. Work the fresh question before revealing its solution.
Worked example 1
Reason and verify
Compare the two outer bars: what is the difference in items between Paper and Glass?
- Label categories and consistent numerical intervals. Read each bar against the axis marks and convert heights into counts.
- The bars represent 40 and 40 items. Their difference is 0.
- Do not confuse the number of grid boxes with the numerical quantity when the scale is 2, 5, or 10 per division.
Result: 0
Understanding check 1 — your turn
Read the bar graph below. What is the difference between the largest and smallest categories?
Reveal reasoning
Answer: 10 — Greatest value 14; least value 4; their difference is 10 fruits.
Worked example 2
Reason and verify
Compare the two outer bars: what is the difference in items between Glass and Paper?
- Label categories and consistent numerical intervals. Read each bar against the axis marks and convert heights into counts.
- The bars represent 4 and 3 items. Their difference is 1.
- Do not confuse the number of grid boxes with the numerical quantity when the scale is 2, 5, or 10 per division.
Result: 1
Understanding check 2 — your turn
Read the bar graph below. What is the difference between the largest and smallest categories?
Reveal reasoning
Answer: 30 — Greatest value 50; least value 20; their difference is 30 fruits.
Worked example 3
Reason and verify
Compare the two outer bars: what is the difference in items between Glass and Paper?
- Label categories and consistent numerical intervals. Read each bar against the axis marks and convert heights into counts.
- The bars represent 14 and 8 items. Their difference is 6.
- Do not confuse the number of grid boxes with the numerical quantity when the scale is 2, 5, or 10 per division.
Result: 6
Understanding check 3 — your turn
Read the bar graph below. What is the difference between the largest and smallest categories?
Reveal reasoning
Answer: 1 — Greatest value 7; least value 6; their difference is 1 vehicles.
Guided · 4 questions
Guided practice
Practice with the model visible. Explain the units at each step.
- G1
Compare the two outer bars: what is the difference in vehicles between Bikes and Buses?
Bar graph: read the key and scale carefully before solving. Show answer and reasoning
Answer: 25
The bars represent 50 and 25 vehicles. Their difference is 25.
- G2
Compare the two outer bars: what is the difference in vehicles between Buses and Bikes?
Bar graph: read the key and scale carefully before solving. Show answer and reasoning
Answer: 0
The bars represent 6 and 6 vehicles. Their difference is 0.
- G3
If Buses increases by 4 vehicles, what will its NEW count be?
Bar graph: read the key and scale carefully before solving. Show answer and reasoning
Answer: 18
Its current count is 14; adding 4 makes 18 vehicles. The axis scale represents quantities, not number of bars.
- G4
If Buses increases by 10 vehicles, what will its NEW count be?
Bar graph: read the key and scale carefully before solving. Show answer and reasoning
Answer: 50
Its current count is 40; adding 10 makes 50 vehicles. The axis scale represents quantities, not number of bars.
Independent · 8 questions
Independent practice
Work without relying on the model or checking answers prematurely.
- I1
Use the bar graph scale: how many books are shown for Science?
Bar graph: read the key and scale carefully before solving. Show answer and reasoning
Answer: 16
The Science bar reaches 16 on the numeric scale, meaning 16 books.
- I2
Use the bar graph scale: how many books are shown for Science?
Bar graph: read the key and scale carefully before solving. Show answer and reasoning
Answer: 15
The Science bar reaches 15 on the numeric scale, meaning 15 books.
- I3
Use the bar graph scale: how many books are shown for Science?
Bar graph: read the key and scale carefully before solving. Show answer and reasoning
Answer: 5
The Science bar reaches 5 on the numeric scale, meaning 5 books.
- I4
Use the bar graph scale: how many books are shown for Science?
Bar graph: read the key and scale carefully before solving. Show answer and reasoning
Answer: 14
The Science bar reaches 14 on the numeric scale, meaning 14 books.
- I5
Compare the two outer bars: what is the difference in books between History and Mystery?
Bar graph: read the key and scale carefully before solving. Show answer and reasoning
Answer: 15
The bars represent 40 and 25 books. Their difference is 15.
- I6
Compare the two outer bars: what is the difference in books between Mystery and History?
Bar graph: read the key and scale carefully before solving. Show answer and reasoning
Answer: 2
The bars represent 6 and 4 books. Their difference is 2.
- I7
Compare the two outer bars: what is the difference in items between Paper and Glass?
Bar graph: read the key and scale carefully before solving. Show answer and reasoning
Answer: 0
The bars represent 14 and 14 items. Their difference is 0.
- I8
Compare the two outer bars: what is the difference in items between Glass and Paper?
Bar graph: read the key and scale carefully before solving. Show answer and reasoning
Answer: 10
The bars represent 50 and 40 items. Their difference is 10.
Challenge · 2 questions
Challenge practice
Solve original application and explanation problems. Draw and defend a useful model.
- C1
Two bars have heights 4 and 7 grid spaces and the scale is 5 books per space. What is their difference?
Show answer and reasoning
Answer: 15 books
The height difference is 3 grid spaces, and 3×5=15 books.
- C2
A bar graph begins at 20 instead of 0. Two bars end at 23 and 28. What is their actual difference, and why might the picture exaggerate it?
Show answer and reasoning
Answer: 5
28−23=5. A truncated axis makes the bars’ visible heights 3 and 8, exaggerating their relative-looking difference.
Exit · 3 questions
Exit check
Solve independently; revisit the skill if an important explanation is missing.
- E1
Read the bar graph below. What is the difference between the largest and smallest categories?
Bar graph: read the key and scale carefully before solving. Show answer and reasoning
Answer: 15
Greatest value 35; least value 20; their difference is 15 items.
- E2
Read the bar graph below. What is the difference between the largest and smallest categories?
Bar graph: read the key and scale carefully before solving. Show answer and reasoning
Answer: 4
Greatest value 10; least value 6; their difference is 4 items.
- E3
Read the bar graph below. What is the difference between the largest and smallest categories?
Bar graph: read the key and scale carefully before solving. Show answer and reasoning
Answer: 2
Greatest value 16; least value 14; their difference is 2 items.