This Math Talk focuses on multiplying a decimal by a unit fraction. It encourages students to think about the relationship between multiplication and division and to rely on properties of operations to mentally solve problems. The strategies elicited here will be helpful later in the lesson when students find equivalent scales involving decimals.

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?”
• “What connections to previous problems do you see?”

The key takeaways are:

• Multiplying a number by a unit fraction is the same as dividing that number by the denominator of the fraction.
• We can use various strategies to reason about division, including place value and the distributive property.

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.

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.