Lesson 1

This warm-up prompts students to reason about proportional relationships in geometric objects as a review of work done earlier in grade 7. As students discuss their answers with their partner, select students to share their answers to the second question during the whole-class discussion. Select students so that different sets of objects and their scale factors are represented in the discussion.

Arrange students in groups of 2. Give students 1 minute of quiet work time and another 1–2 minutes to share their solutions with their partner. Ask students to make sure they have the same objects identified in the first question. If one partner is missing a set of scaled objects, they should add them to their list during their partner discussion.

Students might think H is a scaled version of A or B. Suggest that they consider possible scale factors to get from, for example, A to H.

Select a couple of students, with a variety of answers to the second question, to share their answers to the second question. Ask students, “Did you encounter any objects that you initially believed were scaled versions of one another? How did you decide that they weren't?”

In this introductory lesson students get a chance to recall what they have previously learned about ratios and proportional relationships. They will build on these ideas in the next few lessons where they will work with ratios and rates involving fractions. In this activity, students can leverage their recent work on creating scale drawings to make connections between the dimensions of a different sized flag and the ratio of the side lengths.

A note about flag dimensions: The official government flag has sides with ratio \(1:1.9\), i.e., the width of the flag is 1.9 times its height. However, many commercially sold flags 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 \(2 : 3\) or \(5 : 8\) or \(6 : 10\).

Another interesting note about the flag: The current 50 star version of the flag was designed by a 17 year old HS student as a class project.

Arrange students in groups of 3–4. If the class room has a display of the flag which can be reached, ask a student to measure the dimensions of the flag.

Tell students, "The United States flag is displayed in many different sizes and for different purposes. One standard size is 19 feet by 10 feet. What would be a possible use for a flag of this size?"

Ask student where else they have seen flags displayed. 

Record the groups' measurements in a table to show that there is a constant of proportionality.

This activity continues to look at the U.S. flag by asking questions about percentages, which students studied in grade 6. Later in this unit, students will continue working with percentages, including percent increase and decrease.

Knowing the side lengths of the flag and of the union allows you to compute the area of the flag and of the union. Students 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. 

Arrange students in groups of 3–4. Tell students that they will continue to examine the United States flag, this time looking at area.

The purpose of this discussion is to emphasize the use of proportion when working with areas of rectangles. Ask students to compare their estimated percentage with their calculated percentage. Ask students such as:

• “Did you use the image of the flag—a rectangle—to guide your estimate?”
• “Did you round the given dimensions to find the estimated percentage?”