In this activity students apply proportional reasoning to solve problems about the speed of a train. This activity serves as an intermediate step building towards unit rates involving compound fractions because the distance the train travels is fractional but the given time is a whole number. As students create representations to help them find equivalent ratios, they are making use of structure (MP7). 

Monitor for students who:

• Divide  by 6 to find the distance traveled in 1 minute, then multiply it by 100.
• Draw a double number line.
• Create a table of equivalent ratios.
• Write an equation to represent the relationship between distance traveled and elapsed time.

The key connection is how each method finds and uses the unit rate.

The goal of this discussion is to review different strategies for solving problems involving constant rates. Display 2–3 approaches from previously selected students for all to see. If time allows, invite students to briefly describe their approach. Use Compare and Connect to help students compare, contrast, and connect the different approaches. Here are some questions for discussion:

• “How do these different representations show the same information?”
• “How does the distance traveled in 1 minute show up in each representation?”
• “Did anyone solve the problem the same way but would explain it differently?”

The key takeaway is that there are many different ways to represent the situation, but it is often helpful to find the unit rate.

In this activity students solve a problem involving constant speed. Both the distance and the time are given as fractions, giving students the opportunity to divide a fraction by a fraction to calculate the unit rate. As students calculate equivalent rates and compare the rates, they are reasoning quantitatively and abstractly (MP2).

Monitor for students who use these different strategies to compare the two runners’ speeds:

• Find the amount of time it would take each person to run the same distance.
• Find the distance each person could run in the same amount of time.
• Calculate a unit rate for each person. (This is a special case of one of the two previous bullets, depending on whether the unit rate represents the person’s speed or pace.)

The Warm-up was intended to remind students of some strategies for dividing fractions by fractions, but students may need additional support working with the numbers in this task.

Invite students to share how they determined whether Noah or Lin ran faster. Ask students to describe similarities and differences between the different strategies and representations used.

To involve more students in the discussion, 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?”

Highlight any places in the solution process where students divided fractions. Ask students to describe how they found the quotient.

In this activity, students determine a scale factor that will create a scaled copy that fits within given maximum dimensions. The ratio between the maximum length and width is not the same as the ratio between the original figure’s length and width. Students must choose how much extra space to leave so that the scaled copy keeps the same aspect ratio as the original figure.

There is more than one reasonable solution to this problem, depending on what other aspects students choose to consider. For example, some students might try to find the largest possible scaled copy that will fit on the cover while others may choose to leave extra room on the cover for a title. Another possible approach would be to strategically choose a value that fits inside the maximum and can be calculated efficiently, such as 10 inches by 7 inches. As students reason about the situation and its constraints, they are making sense of the problem (MP1).

In the digital version of the activity, students use an applet to resize and reposition an illustration of the Mona Lisa. The applet allows students to visually compare the size of the painting with the size of the notebook. This activity works best when students have access to the applet because they will benefit from seeing the relationship in a dynamic way. If students don't have access, displaying the applet for all to see would be helpful during the Launch.

The purpose of this discussion is to highlight the different considerations that students took into account when deciding on an appropriate size for the scaled image. Invite students to share their reasoning for choosing the scale factor they did.

Consider asking students:

• “Were there any similarities between the methods you and your partner used? Were there any differences?”
• “Is it possible to scale down the Mona Lisa so that it perfectly covers the notebook? Why or why not?” (No, if you made it fit both the width and height, it would look squished vertically or stretched horizontally compared to the painting.)