Compare line graphs that use different gridline scales

5.MD.B.2 · adapt · grade 5

Archetype: Read and Scale a Data Graph · step in a 21-type progression

Representative Problem

The line graphs show how many chocolate cookies and how many strawberry cookies a bakery sold each day. For which kind of cookie is the difference between its best-selling day and its worst-selling day larger?

(Figure) Two line graphs shown side by side. On both graphs the horizontal axis is the date — day $4$ , day $5$ , day $6$ , and day $7$ .

Left graph, "Chocolate Cookies Sold": the vertical axis is the number sold, with major gridlines at $0$ , $100$ , $200$ and each small grid square worth $50$ . The numbers sold are $100$ on day 4, $150$ on day 5, $200$ on day 6, and $150$ on day 7.

Right graph, "Strawberry Cookies Sold": the vertical axis is the number sold, with major gridlines at $150$ , $200$ and each small grid square worth $10$ (the lower part of the axis is cut off with a wavy line). The numbers sold are $190$ on day 4, $150$ on day 5, $170$ on day 6, and $230$ on day 7.

Show solution

Understand

Two line graphs show daily cookie sales. Chocolate days 4-7 are 100, 150, 200, 150 (each small square = 50). Strawberry days 4-7 are 190, 150, 170, 230 (each small square = 10, with a wavy break below). For each cookie I find the difference between its highest and lowest day, then say which cookie has the larger difference.

Givens

Chocolate sold: day4 = 100, day5 = 150, day6 = 200, day7 = 150
Strawberry sold: day4 = 190, day5 = 150, day6 = 170, day7 = 230
Chocolate graph small square = 50; Strawberry graph small square = 10
The two graphs use different vertical scales

Unknowns

Which cookie has the larger gap between its best-selling and worst-selling day

Constraints

Values must be read using each graph's own square size, not the apparent height

Plan

#15 Organize Information in More Ways · also uses: #7 Identify Subproblems

Because the graphs use different square sizes, comparing bar/point heights is misleading. I re-organize the data as actual numbers from each scale, then break the question into two subproblems: the chocolate range and the strawberry range, and compare those two numbers.

Execute

#7 Identify Subproblems 5.MD.B.2

Reading the chocolate graph (each square = 50), the most was 200 on day 6 and the least was 100 on day 4. The difference is 200 - 100.

200 - 100 = 100

Highest minus lowest gives the spread of one data set.

#7 Identify Subproblems 5.MD.B.2

Reading the strawberry graph (each square = 10, axis cut by a wavy line), the most was 230 on day 7 and the least was 150 on day 5. The difference is 230 - 150.

230 - 150 = 80

I use the actual numbers, not how tall the line looks, because the squares are worth different amounts.

#15 Organize Information in More Ways 5.MD.B.2

Chocolate's range is 100 and strawberry's range is 80. Since 100 > 80, chocolate has the larger difference between its best and worst day.

100 > 80

Once both data sets are in real numbers, a plain comparison answers the question.

Answer: Chocolate cookies (range 100, vs strawberry's 80)

Review

Strawberry's line looks taller and steeper, but that is only because its squares are worth 10 each; in true numbers its swing is just 80, while chocolate swings a full 100. Reading by scale, not by appearance, gives the correct winner.

Guess and check by appearance would be tool 6, but it would fail here; instead one can convert each swing into squares (chocolate 2 squares, strawberry 8 squares) and multiply by each square value to confirm 100 vs 80.

Standards · min grade 5

5.MD.B.2 Make a line plot to display a data set and solve problems using the data — Reading values from each graph's scale and comparing the ranges

💡 When two graphs have different square sizes, read the real numbers first - a steeper line isn't always a bigger change!