This Math Talk focuses on the connections between division, fractions, and decimals. It encourages students to think about the relationship between a fraction and division (an idea from grade 5) and to rely on what they know about equivalent fractions and decimals to mentally solve problems. The understanding elicited here will be helpful later in the lesson when students interpret or find unit rates in a situation given a ratio of two quantities.

To mentally determine if the statements involving division, fractions, and decimals are true, students need to look for and make use of structure (MP7).

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?”

Highlight that a fraction can be seen as division of the numerator by the denominator: . By the same token, the result of dividing a number by another number, , can be expressed as a fraction .

In this activity, students explore two unit rates associated with the ratio, think about their meanings, and use both to solve problems. The goals are to:

• Help students see that for every situation that can be represented with a ratio and an associated unit rate , there is another unit rate that also has meaning and purpose within the situation.
• Encourage students to choose a unit rate flexibly depending on the question at hand.

Students begin by reasoning whether the two rates per 1 (cups of wheat per 1 cup of water, or cups of water per 1 cup of wheat) accurately convey a given recipe. After students conclude that both rates are valid, they use the rates to determine unknown amounts of wheat or water.

As students work and discuss, monitor for those who use different representations (a table or a double number line diagram) and those who construct logical arguments in different ways.

In this activity, students are introduced to the term "unit rate" as a general name for describing the amount of one quantity in a ratio per 1 of the other quantity. Given a ratio , they calculate and and interpret both unit rates in context. Students also practice selecting the better unit rate to use ( or ) based on the question posed.

Though the activity prompts students to think in terms of unit rate, some students may still reason in other ways that are more intuitive to them. For example, to find out how many weeks it would take to use 3 gallons of detergent, students may:

• Triple the ratio of gallons to weeks to arrive at , and then divide each 30 and 18 by 10 to get 3 gallons and 1.8 weeks.
• Divide both 10 gallons and 6 weeks by 5 to get 2 gallons in 1.2 weeks, and then add half of each quantity (1 gallon and 0.6 week) to get 3 gallons in 1.8 weeks.

Such lines of reasoning show depth of understanding and should be celebrated. Students should also see, however, that some problems (such as in the context of detergent use and ticket price) may be more efficiently solved using unit rates. 

As they decide which unit rates to use, students have an opportunity to interpret numbers in context and practice reasoning quantitatively and abstractly (MP2). For example, they may comment that of a ticket costing a dollar doesn’t make sense since it is not possible to purchase a fraction of a ticket. Encourage such observations and use them as entry points for discussing when or how each unit rate could make sense.

As students work, identify 1–2 students per question to share their choice of unit rate and how it was used to answer the question.