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.

While there are different ways to reason about the costs of tacos, focus the discussion on how students found the cost of a single taco in each situation. Remind students that dividing by a whole number is the same as multiplying by its reciprocal (a unit fraction). If students use a table and division to find the cost of one taco, consider annotating the table to show the number being multiplied to the values in one row to find the values in the next row.

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Conclude the discussion by inviting 2–3 students to share a norm they identified in action. Provide this sentence frame to help students organize their thoughts in a clear, precise way:

• “I noticed our norm “” in action today and it really helped me/my group because .”

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.

Select a few students to share about the use of as a multiplier and to explain the reasoning process shown in the table. If different approaches are used, take the opportunity to compare and contrast the efficacy of each.

If students had trouble reasoning to find the pay for 2.1 hours of work, help them articulate what they have done in each preceding case and urge them to think about the 2.1 the same way. If they are unsure whether multiplying 18 by 2.1 would work, encourage them to check whether the answer makes sense. (For 2 hours of work, Lin would earn \$36, so it stands to reason that she would earn a bit more than \$36 for 2.1 hours.) In doing so, students practice decontextualizing and contextualizing their reasoning and solutions (MP2).

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.

Watch out for students being overly precise or wildly imprecise with drawing tick marks on their double number line diagram. We want them to eyeball approximately half the distance, but it would be too time-consuming to measure precisely.

The discussion should center around why the table was easier to use for this problem: the numbers we started with were so large that there wasn’t enough room to locate 1 gigabyte on the number line.

If any students multiplied the ratios by a fraction other than so that they did not have to fill all the rows, consider highlighting this shortcut. (They could even divide 128 and 16 by 128 to arrive at an answer directly, using what they have learned about unit price.) It shows how the table enables reasoning with numbers (rather than with lengths) and is more flexible.