In this activity, students extend their recent work with scale drawings to scale figures with fractional side lengths. As they explore connections between the dimensions of the different sizes of flags and the ratios of the side lengths, students make use of repeated reasoning (MP8). This prepares students for upcoming work with ratios and rates involving fractions.

Monitor for groups who choose to work on each of the different options for flag uses: classroom flag, postage stamp, space shuttle, shoulder patch, or an original idea.

A note about flag dimensions: The official United States government flag has sides with ratio , that is the width of the flag is 1.9 times its height. However, there are many commercially sold flags that use different ratios. This activity is working with the official ratio. If there is a flag displayed in the classroom, it would be interesting to check if it uses the official ratio or one of the other common commercial ratios, such as or or .

The goal of this discussion is to highlight equivalent ratios between the flags’ measurements. Display 2–3 approaches from previously selected students for all to see. If time allows, invite students to briefly describe their approach. Use Compare and Connect to help students compare, contrast, and connect the different approaches.

Here are some questions for discussion:

• “What do the approaches have in common? How are they different?”
• “Is this flag a scaled copy of the original 19-by-10-foot flag? How do we know?”
• “How does the scale factor show up in each representation?”

The key takeaway is that the ratios between the lengths in the scaled copy are equivalent to the ratios between lengths in the original figure. Even when lengths have fractional values, they can still be scaled by the same scale factor.

In this partner activity, students take turns matching images of flags to verbal descriptions of what percentage of the flag’s total area is red, white, and blue. As students trade roles explaining their thinking and listening, they have opportunities to explain their reasoning and critique the reasoning of others (MP3).

This activity reminds students of previous work they did with percentages. It prepares students for later in this unit when they will calculate the percentage that one number is of another number.

The purpose of this discussion is to remind students of strategies for determining the percentage that one number is of another number. Much of the discussion takes place between partners. First, invite students to share how they matched the flags with the descriptions of the percentages.

If not mentioned in students’ explanations, highlight the use of benchmark percentages, such as:

• One-half of Chile’s flag is red, which is 50%.
• Three-fourths of Samoa’s flag is red, which is 75%.
• On the Czech Republic’s flag, the red area is equal to the white area.

If not mentioned in students’ explanations, ask students what the percentages of each color add up to. Key takeaways include:

• For the flags that are only red, white, and blue, the three percentages add up to 100%.
• On the Malaysia flag, there is also some yellow, so the red, white, and blue regions do not add up to 100%.

Next, invite students to share their reasoning for the flags from France and Laos. To involve more students in the conversation, consider asking:

• “Who can restate ’s reasoning in a different way?”
• “Did anyone use the same strategy but would explain it differently?”
• “Did anyone solve the problem in a different way?”
• “Does anyone want to add on to ’s strategy?”
• “Do you agree or disagree? Why?”

The key takeaways are:

• Percentages do not have to be whole numbers. For the three equally sized regions of the French flag to add up to 100%, they each need to be %.
• We can find the percentage that one number is of another number by dividing. On Laos’s flag, the red area is 300 square units, while the whole flag is 600 sq. units. , so the red area is 50% of the flag.

In this activity students calculate percentages of areas on the United States flag.

Knowing the side lengths of the flag and of the union allows students to compute the area of the flag and of the union. They can then compute what percentage of the flag is taken up by the union. Finding out what percentage of the flag is red requires additional reasoning. Students can either compute the area of the red stripes or they can see what fraction of the non-union part of the flag is red.

This is a good opportunity for students to estimate their answers and get a visual idea of the size of different percentages.

The purpose of this discussion is to emphasize the use of proportional reasoning when working with percentages. Ask students to share how their estimated percentages compared with their calculated percentages. Consider asking questions such as:

• “How did you use the image of the flag to guide your estimate?”
• “Where did you use rounding in your estimations or calculations?”