I notice that only one (area) measurement is given on the flag.
I notice that the areas of three other red stripes and three of the white stripes are the same as the given area.
I notice that the given area looks to be a little greater than half of the area of each longer red or white stripe.
I notice that the vertical length of the blue rectangle is 7 times the vertical length of each stripe.
I notice that the 50 stars are arranged in alternating rows of five and six stars.
I wonder if it is possible to determine all of the length and area measurements of the flag just by looking at the picture.
I wonder what fraction of the width of the flag the small stripes are.
I wonder if the flag is drawn to scale.
I notice (from measuring) that the width of the blue rectangle is 2/5 of the width of the flag.
I wonder what the width:length ratio is for this flag.
I wonder if all U.S. flags have the same official width:length ratio.
I wonder how much fabric it would take to make the whole flag (and how much of each color).
I wonder what the length and width of the flag are.
As they notice and wonder, students may create their own questions and strategies related to ratios, percentages, lengths, and areas. Many of their strategies may flow out of things that they have wondered about. They may also
Create their own accurate drawing of this flag.
Create drawings of other flags and analyze the ratios, percentages, lengths, and areas.
Create methods to estimate or calculate measurements related to the stars on the flags.
Reflecting and Extending
I notice that there are many ways to calculate the measurements for the flag.
I notice that a percentage is a special kind of ratio.
I notice that I can figure out a lot about the flag from just from the one measurement given.
I notice that exploring the mathematical measurements of the flag helps me to appreciate it in new and different ways.
I wonder if the colors of the flag have meanings.
I wonder how hard it would be to redesign the stars on the flag if more states joined the union.
I wonder if there are reasons for the ratios that people chose when they designed the flag.
You can find a more structured problem built around this image in in Exploration 4: Perplexing Percentages from my book, Advanced Common Core Math Explorations: Ratios, Proportions, and Similarity.
Students may make measurements of the picture to determine ratios of different lengths. A key fact is that the (horizontal) width of the blue rectangle is 2/5 or 40% of the width of the entire flag. Also, the width : length ratio for the flag is 1.9. (Ratios of the flag dimensions vary somewhat [from about 1.6 to 2], but 1.9 is typical, and it applies to this picture.)
Given this information, students can calculate all of the relevant lengths and areas except for those of the stars. They can also determine percentages or other ratios relating these measurements. Calculating or estimating the areas of the stars would be an interesting challenge. Some students may also be curious about the scale factor between the picture and the actual flag that it represents.
A few sample results:
The red stripes cover about 41.5% of the total area of the flag.
The red stripes cover an area of 19.8 square feet.
The white stripes cover about 36.9% of the total area of the flag.
The white stripes cover an area of 17.6 square feet.
The total area of the flag is 47 and 2/3 square feet.
The vertical length of the flag is very close to 5 feet (about 5.01).
The horizontal length of the flag is very close to 9.5 feet (about 9.52).
Some students may enjoy doing similar investigations for other flags, especially if they have another country of origin or if they special connections to other countries.