The purpose of this activity is to help students understand that when a quantity is measured in two different units, the two measurements form a ratio. We can convert from one unit to the other by using what we know about equivalent ratios, including the strategies and representations for reasoning about them. 

Though the term “unit rate” is not yet used in this lesson, students interact with the idea that any ratio  has two associated unit rates: and , each with a particular meaning in the context. Here, students reason that since there are 8 kilometers in approximately 5 miles, there are kilometers in 1 mile, and there are of a mile in one kilometer.

Students can then choose to use familiar reasoning strategies or either of the unit rates to solve a problem about constant speed. Monitor for students who use the following strategies to determine whether 75 miles per hour is faster than 80 kilometers per hour:

• Finding a ratio that is equivalent to 5 miles to 8 kilometers and has 75 for the distance in miles, and then comparing the corresponding distance in kilometers (120) to 80 kilometers.
• Finding a ratio that is equivalent to 5 miles to 8 kilometers and has 80 for the distance in kilometers, and then comparing the corresponding distance in miles (50) to 75 miles.
• Converting 80 kilometers to miles by multiplying 80 by , the amount of miles in 1 kilometer, and comparing the result (50) to 75 miles.
• Converting 75 miles to kilometers by multiplying 75 by , the amount of kilometers in 1 mile, and comparing the result (120) to 80 kilometers.

Students may use a double number line diagram, a table, or equations with any of these strategies.

Focus discussion on different approaches to the second question. The goal is to highlight that the same distance measured in miles and in kilometers form a ratio, and that we can use equivalent ratios to convert from one unit to the other. This includes using associated rates per 1 mile and per 1 kilometer. Display 2–3 approaches from previously selected students for all to see. If time allows, invite students to briefly describe their approaches.

Then use Compare and Connect to help students compare, contrast, and connect the different approaches. Here are some questions for discussion:

• “What do the approaches have in common? How are they different?”

• “How does the ratio of miles to kilometers show up in each method?”

During discussions, encourage students to explain the meaning of any numbers used and the reason for using particular operations. For example, if students multiplied 80 by , ask them to explain what means in this situation and why they decided to multiply 80 by it. Consider displaying and using representations such as double number line diagrams or tables to facilitate these explanations.

In this activity, students convert between pounds and kilograms. It gives students another opportunity to see that the strategies for reasoning with equivalent ratios can be used to convert from one measurement unit to another.

Some students may apply insights from the activity about distance in miles or kilometers and use unit rates ( kilogram per 1 pound and pounds per 1 kilogram) to perform the conversions. This is also fine, but note that unit rates are not the focus here and will be studied more closely in upcoming lessons.

Monitor for students who use different representations and strategies, and select them to share later.

Invite previously selected students to share. Highlight approaches that involve reasoning about equivalent ratios. Consider displaying a double number line diagram or a table of equivalent ratios to help students attend to the meaning of the numbers and the operations used.

For example, when discussing the last question, consider using a table to show that multiplying both 22 pounds and 10 kilograms by (or dividing both values by 10) gives the number of pounds in 1 kilogram. Multiplying that number by 8 gives the number of pounds in 8 kilograms.

If any student used the unit rate kilogram per pound, consider pointing out that this value is a common approximation of the conversion factor from pounds to kilograms and not the true conversion factor.

In this activity, students practice converting between two units in a cooking context. The conversion factor between cups and tablespoons is given in the form of a unit rate, 16 tablespoons for 1 cup. Students will need to decide whether to multiply or divide by the unit rate, and might choose to create a double number line diagram or a table to support their reasoning. Several of the measurements include fractions, giving students an opportunity to practice grade 5 operations with fractions, such as multiplying mixed numbers by whole numbers and dividing whole numbers that result in fractions.

Select students to share based on their strategies, sequencing from less efficient to more efficient. Highlight approaches using multiplication by 16 when converting from cups to tablespoons and division by 16 (or multiplication by ) when converting from tablespoons to cups. Record the representations or strategies that students shared, and display them for all to see.

After the strategies have been shared, ask students how they would know whether to multiply or to divide. Emphasize that we multiply or divide depending on the information we have. Since 1 cup equals 16 tablespoons, if we know a quantity in cups, we can multiply it by 16 to find the number of tablespoons. On the other hand, if we know a quantity in tablespoons, we can divide it by 16 (or multiply by ) to find the number of cups.