Solving One- and Two-Step Problems Using Graphs - Third Grade Mathematics

Data displays are not just pretty pictures to look at; they are powerful tools designed to help us solve real-world problems and answer complex questions. In third grade, you will move beyond simply reading single values from a graph to solving one-step and two-step word problems involving "how many more," "how many fewer," and finding combined totals. In this chapter, you will master the step-by-step problem-solving strategies for bar graphs and pictographs.

Types of Graph Questions

Questions based on graphs fall into three primary categories:

1. Read-Off Questions (Zero calculations):
   "How many students chose soccer?" -> Simply read the bar or count symbols.

2. One-Step Comparison Questions (One calculation):
   "How many more students chose soccer than baseball?" -> Subtract!
   "How many students chose soccer or baseball combined?" -> Add!

3. Two-Step Problem Questions (Two calculations):
   "How many more students chose outdoor sports (soccer + baseball)
   than indoor sports (swimming + gymnastics)?" -> Add, add, then subtract!

Worked Examples Using a Scaled Bar Graph

Let us analyze this bar graph showing favorite school lunch choices:

Favorite Lunch Choices:
Scale of 5:
   30 |
   25 |          +----+
   20 | +----+   |    |
   15 | |    |   |    |   +----+
   10 | |    |   |    |   |    |   +----+
    5 | |    |   |    |   |    |   |    |
    0 +-+----+- -+----+- -+----+- -+----+-
        Pizza    Tacos   Burger   Salad

Data Values:
- Pizza: 20 students

- Tacos: 25 students

- Burgers: 15 students

- Salad: 10 students


Example 1: One-Step Comparison Problem

Question: How many more students voted for Tacos than for Burgers?
- Look up Tacos: 25

- Look up Burgers: 15

- Operation: Subtraction ("How many more" means find the difference)
25 - 15 = 10 more students voted for Tacos.

Example 2: Two-Step Comparison Problem

Question: How many more students voted for the top two choices combined than for the bottom two choices combined?
- Step 1: Find the sum of the top two choices (Tacos + Pizza):
25 + 20 = 45 students.
- Step 2: Find the sum of the bottom two choices (Burgers + Salad):
15 + 10 = 25 students.
- Step 3: Find the difference between the two combined groups:
45 - 25 = 20 students! Answer: 20 more students voted for the top two choices.

Example 3: Two-Step Total and Difference Problem

Question: The cafeteria manager prepared 80 lunches in total. How many students did not vote for any of these four choices?
- Step 1: Find the total number of students who voted:
20 + 25 + 15 + 10 = 70 students.
- Step 2: Subtract from the total lunches prepared:
80 - 70 = 10 students. Answer: 10 students did not vote for these choices.

Watch Out for the Scaled Key!

When solving problems from a pictograph, always remember to convert symbols to real quantities before performing your calculations!

Pictograph:
Apples:   [ * ] [ * ] [ * ]
Oranges:  [ * ]
Key: Each [ * ] = 4 fruit.

Method A (Convert then subtract):
Apples: 3 x 4 = 12
Oranges: 1 x 4 = 4
Difference: 12 - 4 = 8 more apples.

Method B (Subtract symbols then multiply):
Apples has 3 symbols; Oranges has 1 symbol.
Difference in symbols = 3 - 1 = 2 extra symbols.
Multiply by key: 2 x 4 = 8 more apples!

Both methods yield the exact same answer: 8 more apples. Method B is a great shortcut for mental math!

Chapter Practice Exercises

Use the Favorite Lunch Choices graph (Pizza: 20, Tacos: 25, Burgers: 15, Salad: 10) for questions 1 through 3:
1. How many students voted in all?

2. How many fewer students voted for Salad than for Pizza?

3. Did more students vote for Tacos alone, or for Burgers and Salad combined? Explain by showing your math.

4. A pictograph shows books read:
Grade 3: 6 book symbols Grade 4: 9 book symbols Key: Each book symbol = 5 books. a. How many books did Grade 4 read? b. How many more books did Grade 4 read than Grade 3?
5. A school has a goal of collecting 100 toys for charity. A bar graph shows they have collected 35 board games, 25 stuffed animals, and 20 puzzles. How many more toys do they need to collect to reach their goal?


Solutions and Step-by-Step Answers

  1. Total students: 20 + 25 + 15 + 10 = 70 students voted.
  2. Salad vs. Pizza: Pizza = 20, Salad = 10. 20 - 10 = 10 fewer students voted for Salad.
  3. Compare Tacos (25) to Burgers + Salad: Burgers + Salad = 15 + 10 = 25. Tacos alone has 25 votes. They are exactly equal! Neither group has more.
  4. Book pictograph: a. Grade 4: 9 symbols x 5 books per symbol = 45 books. b. Difference in symbols = 9 - 6 = 3 symbols. 3 x 5 = 15 more books read by Grade 4 (or 45 - 30 = 15).
  5. Toy drive goal:
  6. Step 1 (Find toys collected so far): 35 + 25 + 20 = 80 toys.
  7. Step 2 (Find remaining to reach goal): 100 - 80 = 20 more toys needed.