In this introductory activity, students explore the meaning of scale. They begin to see that a scale communicates the relationship between lengths on a drawing and corresponding lengths in the objects that they represent, and they learn some ways to express this relationship:

• “ units on the drawing represent  units of actual length”
• “at a scale of units (on the drawing) to units (actual)”
• “ units (on the drawing) for every units (actual)”

Students measure lengths on a scale drawing and use a given scale to find corresponding lengths on a basketball court (MP2). Because students are measuring to the nearest tenth of a centimeter, some of the actual measurements they calculate will not have the precision of the official measurements. For example, the official measurement for is 0.9 m.

Before debriefing as a class, display the table showing only the scaled distances so students can do a quick check of their measurements (which they may round differently). Explain to students that the distances on a scale drawing are often referred to as “scaled distances.” The distances on the basketball court, in this case, are called actual distances.

Direct students’ attention to the reference created using Collect and Display. Ask students to share how they used the given scale to find actual distances. Invite students to borrow language from the display as needed and update the reference to include additional phrases as they respond.
To further students’ understanding of scale, discuss:

• “Does a scale of ‘1 cm for every 2 m’ mean that the actual distance is twice the distance on the drawing? Why or why not?”
• “Which parts of the court can be drawn using the ‘1 cm for every 2 m’ rule?” (Everything that is flat on the court can be drawn using this scale, but the heights of things like the basketball hoop or the bleachers cannot be shown on this drawing.)
• “Can we reverse the order in which we list the scaled and actual distances? For example, could we say ‘2 m of actual distance to 1 cm on the drawing’ or just ‘2 m to 1 cm’ for the scale?” (The scaled distance is conventionally stated first, but the actual distance represented could also come first as long as the meaning is clear from the context.)

If needed, a short discussion about accuracy of measurements on the scale drawing might highlight some possible sources of measurement error. For example, the lines on the scale drawing have width, and this could contribute a small error depending on whether the measurement is from the inside or the outside of the lines.

This activity introduces students to graphic scales. Students interpret a scale drawing that has a graphic scale. They use it to find actual measurements and express it non-graphically. Students choose what strategy and tools they will use to compare the scaled heights of several buildings with the graphic scale (MP5).

Monitor for students who use these strategies to determine the actual heights:

• Use their fingers to estimate the length of the graphic scale and compare it to the heights of the buildings.
• Use an index card to copy the length of the graphic scale once and use that benchmark to measure the heights of the buildings.
• Use an index card to copy the length of the graphic scale multiple times, end-to-end, creating an informal ruler that can be used to measure the heights of the buildings.
• Draw multiple copies of the graphic scale end-to-end on the image, creating a vertical axis that can be used to measure the heights of the buildings.
• Use a ruler to measure both the graphic scale and the heights of the buildings and then use ratio reasoning to calculate the actual heights.

Plan to have students present in this order, from less precise to more precise.

The purpose of this discussion is to familiarize students with using a scale drawing with a graphic scale to calculate actual distances. Invite previously selected students to share how they estimated the heights of the buildings. Sequence the discussion of the strategies in the order listed in the Activity Narrative. (It is not necessary to demonstrate every possible method. The goal is for students to see some methods that are less precise and some methods that are more precise.) If possible, record and display their work for all to see.

Connect the different responses to the learning goals by asking questions such as:

• “How was the scale used in each method?”
• “What worked well in _____’s approach? What did not work well?”
• “How did each approach deal with potential measurement error?”

After several methods have been shared, discuss the scale drawing of the towers more broadly. Consider asking:

• "Besides height information, what other information about the towers does the drawing show?" (Widths of the buildings and their overall shapes.)
• "What information does it not show? (The depth of each building, any projections or protrusions, shapes of different parts of the buildings, distances between the different buildings, etc.)
• “How is this scale drawing the same as that of the basketball court? How are the two scale drawings different?” (They both show information on a single plane and are drawn using a scale. The basketball court is a flat surface, like the drawing of the court. The drawing of the towers is a side view or front view. The actual objects represented are not actually flat objects.)