This Math Talk focuses on multiplication involving unit fractions. It encourages students to rely on the meaning of fractions and the properties of operations to find the product of a unit fraction and a whole number or a decimal.

In grade 4, students learned that a non-unit fraction can be expressed as a product of a whole number and a fraction. For instance, can be expressed as . In grade 5, they interpreted a fraction, such as , as a quotient, , and connected the two interpretations of (as and ). They also observed the commutative property of multiplication and saw that and have the same value. In both grades, students relied on contexts to reason about and represent problems involving multiplication of a whole number and a fraction.

Two ideas that build on these prior understandings will be relevant to future work in the unit and are important to emphasize during discussions:

• Dividing by a number is the same as multiplying by its reciprocal.
• The commutative property of multiplication can help us solve a problem regardless of the context.

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 purpose of this activity is to encourage students to use a table to find the price for one taco for two different situations. Students are likely to divide the cost of the tacos by the number of tacos to find the cost for one taco, which is appropriate. Use the opportunity to remind students that dividing by a whole number is the same as multiplying by its reciprocal (a unit fraction). This insight will come in handy in future activities and lessons.

This activity introduces students to the strategy of using an equivalent ratio with one quantity having a value of 1 to find other equivalent ratios. Students look at a worked-out example of the strategy, make sense of how it works, and later apply it to solve other problems.

There are two key insights to uncover here:

• The ratios we deal with do not always have corresponding quantities that are multiples of each other. (For example, in the activity, 5 is not a multiple of 8, or vice versa).
• In those situations, finding an equivalent ratio where one of the quantities is 1 can be a helpful intermediate step.

Also reinforced here is an idea from grade 5, that dividing by a whole number is equivalent to multiplying by its reciprocal. (For instance, dividing by 5 is the same as multiplying by .)

Expect some students to initially overlook the benefit of using a ratio involving a “1,” to rely on methods from previous work, and to potentially get stuck (especially when dealing with a decimal value in the last row). For example, since the table shows an arrow and a multiplication from the first to the second row and from the second to third, students may try to do the same to find the missing value in the fourth row. While finding a factor that can be multiplied to 8 to obtain 3 is valid, encourage students to consider an alternative, given what they already know about the situation (namely, how much the person earned in 1 hour). If needed, support their thinking by asking how much Lin would earn in 2 hours and then in 3 hours.

Monitor for students who can:

• Articulate why is used as a multiplier.
• Reason why using a ratio with one of the values being 1 helps to find other equivalent ratios.
• Reason about equivalent ratios in other ways.

Invite these students to share later.

Previously, students explored the limitation of a double number line when dealing with greatly scaled-up ratios. They saw that extending the number lines can be impractical. Here, they encounter a situation involving significantly scaled-down ratios, in which a double number line is likewise impractical (that is, there is not enough room to fit relevant information) and see that a table is clearly preferable.

The given table deliberately includes more rows than necessary to answer the question. Some students may realize that it is not necessary to fill in all the rows if they use a different factor in finding equivalent ratios. Monitor for students who take such shortcuts so they can share later. Their reasoning can further highlight the flexibility of a table.