Reading and Drawing Scaled Single and Multi-Bar Graphs - Third Grade Mathematics

A bar graph uses solid rectangular bars of varying lengths or heights to display and compare data visually. In third grade, you transition from simple single-unit bar graphs to scaled bar graphs, where the axis counts by 2s, 5s, 10s, or other intervals. In this chapter, you will master reading vertical and horizontal scaled bar graphs, interpreting values between scale lines, and analyzing double-bar graphs that compare two sets of data side by side.

The Anatomy of a Scaled Bar Graph

A bar graph has several essential components that work together to communicate information:

Title: Explains what data the graph is displaying.
Scale (Axis): A numbered line with equal intervals (counting by 2s, 5s, 10s...).
Categories: The labels for each group being measured.
Bars: Rectangular strips whose length corresponds directly to the data value.

Vertical Scaled Bar Graph (Scale of 5):

Number of
Books Read
   20 |                +----+
   15 |       +----+   |    |
   10 |       |    |   |    |   +----+
    5 | +---+ |    |   |    |   |    |
    0 +-+---+-+----+- -+----+- -+----+-
        Liam   Maya    Noah     Zoe
                  Students

Reading Values on Scaled Axes

The vertical axis on a vertical bar graph (or horizontal axis on a horizontal bar graph) uses a regular scale.

Reading Exact Grid Lines

Look at the graph above:
- Liam's bar reaches the 5 line -> Liam read 5 books.

- Maya's bar reaches the 15 line -> Maya read 15 books.

- Noah's bar reaches the 20 line -> Noah read 20 books.


Reading Bars Between Grid Lines

What if a bar stops halfway between two grid lines? Look at Zoe's bar: it stops halfway between 5 and 10!
- What number is exactly halfway between 5 and 10?

- The difference between 10 and 5 is 5. Half of 5 is 2.5 (or in an even scale like 0, 10, 20: halfway between 10 and 20 is 15).
On a scale counting by 2s (0, 2, 4, 6, 8, 10): A bar stopping halfway between 6 and 8 represents exactly 7!

Scale of 2s Example:
   8 |
   6 |       +----+  <- Bar stops halfway between 6 and 8!
   4 |       |    |     Value = 7!
   2 |       |    |
   0 +-------+----+--

Horizontal Bar Graphs

A bar graph can also be oriented horizontally, with categories on the vertical axis and the numbered scale running along the bottom.

Favorite Fruit:
Apples   | [====================] (20)
Bananas  | [===============] (15)
Berries  | [==========] (10)
         +---------+---------+---------+
         0         10        20        30

Horizontal bar graphs are read the exact same way as vertical bar graphs: look at where the right edge of each bar lines up with the scale below.

Multi-Bar Graphs (Double Bar Graphs)

A double bar graph displays two side-by-side bars for every category, allowing you to compare two different groups (such as boys vs. girls, or Saturday vs. Sunday sales).

Weekend Movie Attendance:
Category: Morning vs. Afternoon
Scale of 10:

        [B1] = Saturday    [B2] = Sunday

Comedy:      Sat: 40 | Sun: 50
Action:      Sat: 60 | Sun: 30
Animated:    Sat: 70 | Sun: 80

A legend or key identifies which color or shading pattern belongs to each group.

Chapter Practice Exercises

  1. On a bar graph with a scale counting by 5s (0, 5, 10, 15, 20, 25): a. What value is represented by a bar that reaches the 15 line? b. What value is represented by a bar that reaches the 25 line? c. If a bar stops halfway between 10 and 15, what number does it represent?
  2. On a bar graph with a scale counting by 10s: a. What number is halfway between 30 and 40? b. What number is halfway between 70 and 80?
  3. A bar graph of favorite ice cream flavors shows: Vanilla: 18 votes, Chocolate: 24 votes, Strawberry: 12 votes, Mint: 9 votes. a. Which flavor received the most votes? b. How many more votes did Chocolate receive than Strawberry? c. How many total votes were cast?
  4. What scale (counting by 1s, 2s, 5s, or 10s) would be best to display data where the values are 50, 75, 40, and 90? Explain why.
  5. Draw or describe how you would set up a bar graph showing the number of pets in a classroom: 8 dogs, 6 cats, 2 birds, and 4 fish.

Solutions and Step-by-Step Answers

  1. Reading scale of 5s: a. The 15 line = 15. b. The 25 line = 25. c. Halfway between 10 and 15 represents 12.5 (or about 12 or 13 in whole-number contexts).
  2. Scale of 10s: a. Halfway between 30 and 40 is 35. b. Halfway between 70 and 80 is 75.
  3. Ice cream graph: a. Chocolate received the most votes (24 votes). b. Chocolate vs. Strawberry: 24 - 12 = 12 more votes. c. Total votes: 18 + 24 + 12 + 9 = 63 votes.
  4. A scale of 5s or 10s would be best. Counting by 10s or 5s keeps the graph compact and clean because 40, 50, 75, and 90 are all multiples of 5, and counting by 1s would require 90 lines, which would be far too tall!
  5. Setup:
  6. Horizontal axis: Categories (Dog, Cat, Bird, Fish).
  7. Vertical axis: Scale counting by 2s from 0 to 10 (0, 2, 4, 6, 8, 10).
  8. Draw bars: Dog bar reaches 8; Cat bar reaches 6; Bird bar reaches 2; Fish bar reaches 4.