The purpose of this activity is to prepare students to create their own scale drawing. First, they calculate a scale given one actual distance and its corresponding scaled distance. Then, they use the scale to calculate other scaled distances.

The discussion highlights that a scale can be expressed in different ways. Some wordings convey the meaning of the numbers more clearly than others do. Also, different pairs of numbers may be used to express the same relationship.

Monitor for students who:

• Use ratio language in their scale, such as “16 cm to 4 m” or “4 cm for every 1 m.”
• Find equivalent scales with smaller numbers, such as “1 cm represents 0.25 m.”
• Use words to clarify the meaning of the numbers, such as “4 cm on the drawing represents 1 m in the actual room.”
• Oversimplify the relationship between actual and scaled lengths, such as “4 cm = 1 m” or “The scale factor is 4.”

If the optional activities “Scaling More Pattern Blocks” and “Biking through Kansas” were not completed in previous lessons, then this is the first time Math Language Routine 7: Compare and Connect is suggested in this course. In this routine, students are given a problem that can be approached using multiple strategies or representations, and they record their method for all to see. They then compare and identify correspondences across strategies by means of a teacher-led gallery walk with commentary or teacher think-aloud (such as “I notice . . . I wonder . . .”). A typical discussion prompt is “What is the same and what is different?”, comparing their own strategy to the others. The purpose of this routine is to allow students to make sense of mathematical strategies and, through constructive conversations, develop awareness of the language used as they compare, contrast, and connect other ways of thinking to their own.

The goal of this discussion is to compare different ways to express the same relationship between scaled and actual lengths. Display 3–4 representations from previously selected students for all to see. Use Compare and Connect to help students compare, contrast, and connect the different ways to express the scale. Here are some questions for discussion:

• “What do the scales have in common? How are they different?”
• “How do these different representations show the same information?”
• “Are there any benefits or drawbacks to one representation compared to another?”

The key takeaways are:

• Because of equivalent ratios, there are multiple ways to express the same scale.
• Scales are often simplified to show the actual distance for 1 scaled unit (or a benchmark fraction, such as 0.25, 0.5, 0.75).
• A good scale makes it clear which value refers to the scale drawing and which refers to the actual object.

Given their work on scaled copies, students may be inclined to say that the scaled and actual lengths are related by a scale factor of 4. Ask: “Are the actual lengths four times the lengths on the drawing? Why or why not?” Point out that because the units for the two quantities are different, multiplying a scaled length in centimeters (for example, 2.5 cm) by 4 will yield another length in centimeters (10 cm), which is not the actual length. However, it is not essential for students to know that the scale factor here is actually 250. That idea will be explored in an upcoming lesson.

In this activity, students create two different scale drawings of the state of Utah and notice how the scale impacts the size of the drawing. As students notice how the different scales impact the size of the scale drawings, they are making use of structure (MP7).

This is the first time Math Language Routine 3: Critique, Correct, Clarify is suggested in this course. In this routine, students are given a “first draft” statement or response to a question that is intentionally unclear, incorrect, or incomplete. Students analyze and improve the written work by first identifying what parts of the writing need clarification, correction, or details, and then writing a second draft (individually or with a partner). Finally, the teacher scribes as a selected second draft is read aloud by its author(s), and the whole class is invited to help edit this third draft by clarifying meaning and adding details to make the writing as convincing as possible to everyone in the room. Typical prompts are: “Is anything unclear?” and “Are there any reasoning errors?” The purpose of this routine is to engage students in analyzing mathematical writing and reasoning that is not their own, and to solidify their knowledge and use of language.

Ask students what the two scale drawings share in common. (They both represent Utah, they both have the same shape, and they both can be used to measure distances in the actual state of Utah.)

Ask students how the two scale drawings differ. (The one at a scale of 1 centimeter to 50 mi is larger than the one at a scale of 1 centimeter to 75 mi.)

Some students may notice that the scale drawing at a scale of 1 centimeter to 75 mi is actually a scaled copy of the other drawing, with a scale factor of 1.5. If so, ask them to share their observation that links scale drawings with scaled copies.

Use Critique, Correct, Clarify to give students an opportunity to improve a sample response for the first scale drawing by correcting errors, clarifying meaning, and adding details. 

• Display this first draft and image:
  “Since 1 cm represents 50 mi, I divided 110 and 70 each by 50, and got 2.2 and 1.4. The small rectangle that is missing is 2.2 cm tall and 1.4 cm wide.”

  

• Ask, “What parts of this response are unclear, incorrect, or incomplete?” As students respond, annotate the display with 2–3 ideas to indicate the parts of the writing that could use improvement.
• Give students 2–4 minutes to work with a partner to revise the first draft. 
• Select 1–2 individuals or groups to read their revised draft aloud slowly enough to record for all to see. Scribe as each student shares, and then invite the whole class to contribute additional language and edits to make the final draft even more clear and more convincing.