If students are unsure how to begin finding the actual length of the landing gear or actual height of the spacecraft, suggest that they first find out the length on the drawing.

Students may measure the height of the spacecraft in centimeters and then simply convert it to meters without using the scale. Ask students to consider the reasonableness of their answer (which is likely around 0.14 m) and remind them to take the scale into account.

The goal of this discussion is to highlight how units matter in problems involving a scale without units. First, poll the class on their answers to the first question. Highlight the multiplication of scaled measurements by 50 to find actual measurements.

Next, display 2–3 approaches to the second question from previously selected students for all to see. Use Compare and Connect to help students compare, contrast, and connect the different approaches. Here are some questions for discussion:

• “How is the scale ‘1 to 50’ used in each method?”
• “Why do the different approaches lead to the same outcome?”
• “Are there any benefits or drawbacks to one approach compared to another?”

The key takeaways are that when we have a scale without units:

• We can use any unit we want to measure the scale drawing.
• We will get an actual measurement in that same unit.

• We may choose what unit we use to measure the drawing based on how we want to express the final answer.

  (Since the question asks for a height in meters, using centimeters as the unit would be more efficient than using inches because fewer conversions would be required. If the question asked for actual height in feet, then inches would be a more strategic unit to use.)

Next, poll the class on their answers to the third question. Highlight that finding the length on the scale drawing involves dividing the actual measurement by 50 (or multiplying by ).

If time permits, display the image from the blackline master. Sketch a stick figure that is 1.4 inches tall to represent Neil Armstrong standing next to the Apollo Lunar Module. Select students who gave their heights in different units to share their solutions to the last problem. Sketch a stick figure with its height to scale for each student who shares an answer. Consider displaying a photograph of one of the astronauts next to the Lunar Module, such as shown here, as a way to visually check the reasonableness of students’ solutions.

Students may be confused about whether to multiply or divide by 2,000 (or to multiply by 2,000 or by ) when finding the missing lengths. Encourage students to articulate what a scale of 1 to 2,000 means, or remind them that it is a shorthand for saying “1 unit on a scale drawing represents 2,000 of the same units in the object it represents.” Ask them to now think about which of the two—actual or scaled lengths—is 2,000 times the other and which is of the other.

For the third question relating the area of the real flag to the scale model, if students are stuck, encourage them to work out the dimensions of each explicitly and to use this to calculate the scale factor between the areas.

Select a few students with differing solution paths to share their responses to the first question. Record and display their reasoning for all to see. Highlight two different ways for dealing with unit conversions. For example, in finding scaled lengths, one can either first convert the actual length in meters to centimeters and then multiply by , or multiply by first, and then convert the quotient to centimeters.

Poll the class on their responses for the second question. If there is disagreement about any of the answers, invite a few students to share their reasoning. Emphasize that all of the scales are equivalent because in each scale, a factor of 2,000 relates scaled distances to actual distances. Reiterate the fact that a scale does not have to be expressed in terms of 1 scaled unit, as is shown in the last three sub-questions, but that 1 is often chosen because it makes the scale factor easier to see and can make calculations more efficient.

If time permits, help students recognize that the areas of the two flags are related by a factor of 4,000,000. Both the length and the height of the large flag are 2,000 times the corresponding side of the small flag, so the area of the large flag is times the area of the small flag. Another way to look at it is that there are 10,000 square centimeters in a square meter, so in square centimeters, the area of the large flag is 1,045,440,000. Dividing this by the area of the small flag in square centimeters, 261.36, also gives 4,000,000.