There are many paths that students could use to reason about whether or not to accept a deal. For example, in the case of juice boxes, they could:

• Find and compare the unit rates for both the original pack and the new pack. If the unit rate is the same, the deal is fair. If the unit rate is lower, the clerk is offering a discount. If the unit rate is higher, the clerk is not being fair.

  number of juice boxes	cost in dollars	dollars per box
  10	3.50	0.35
  6	2.40	0.40

• Find the unit rate in the original pack, apply it to the number of items in the new pack, and compare the costs for the same number of items in the original and new pricing schemes. This can be done in two ways, one focused more on column reasoning and the other on row reasoning, as shown.

  

  

• Use an abbreviated table and bypass calculating the unit rate. Find the multiplier to get from the original to the new number of items, and use the same multiplier to find what the price would be if the deal has not changed. Compare the actual new price to this projected price.

  

Monitor for different ways that students reason about the deals and decide whether to take or decline them.

Select 2–3 students who used different but effective strategies to share their thinking with the class. Record the different strategies in one place, and display them for all to see. Highlight any similarities and differences, such as whether a unit rate was used, or whether students compared the original unit rate to the new quantity or the other way around. 

The goal of the discussion is for students to see that in situations involving prices, we can make comparisons by reasoning about equivalent ratios or by using unit rates. Depending on what information is given and what is sought, some strategies may be more practical than others.

In this activity, students perform conversions to compare lengths given in customary and metric units. They apply what they know about ratios formed by different units of measurements, equivalent ratios, and strategies for converting units. To make some measurements comparable to others, students may need to perform multiple conversions and activate arithmetic skills from previous grades. Support students with computations as needed, and provide access to calculators as appropriate.

Students may choose to convert the given measurements into any of the given units. Although their calculations are likely to vary based on the choice made and the strategy used, students should arrive at the same conclusion.

Monitor for students who opt to use the same unit in making comparisons (for example, to convert all units to feet) so that they can be grouped together for discussion.

Invite at least two students to share the ordered distances—one student who converted all measurements to feet or yards, and another who converted them to meters. Consider compiling the converted distances in a table such as shown in sample student responses and adding additional columns for other units used to make comparisons.

Ask questions such as:

• “How did you use the information given to convert a distance from a smaller unit to a larger unit, such as from inches to feet or to yards?” (Divide the distance in inches by the number of inches in 1 foot, and then divide the distance in feet by the number of feet in 1 yard.)
• “How did you convert a distance from a larger unit to a smaller one?” (Multiply the distance by the number of smaller units in 1 larger unit.)
• “For each animal, how do the measurements in feet, yards, and meters compare? (The number is the smallest in meters and the largest in feet.) “Why might that be?” (A foot is smaller than a yard or a meter, so it takes more feet to measure the same distance.)
• “Does the order change when the distances are measured in different units? Why or why not?” (No, because each animal ran one distance, which doesn’t change. It’s just measured in units of different sizes.)