Unit 2, Lesson 12
Navigating a Table of Equivalent Ratios
Grade 6
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Learn how to use instructional routines to facilitate problem-based learning!
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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,
Two ideas that build on these prior understandings will be relevant to future work in the unit and are important to emphasize during discussions:
Tell students to close their books or devices (or to keep them closed). Reveal one problem at a time. For each problem:
Keep all previous problems and work displayed throughout the talk.
Find the value of each product mentally.
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To involve more students in the conversation, consider asking:
Learn how to use instructional routines to facilitate problem-based learning!
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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:
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:
Invite these students to share later.
This may be some students’ first time reasoning about money earned by the hour. Take a minute to ensure everyone understands the concept. Ask if anyone has earned money based on the number of hours doing a job. Some students may have experience being paid by the hour for helping with house cleaning, a family business, babysitting, dog walking, or doing other jobs.
Give students quiet think time to complete the activity and a minute to share their responses (especially to the last two questions) with a partner before discussing as a class.
Lin is paid $90 for 5 hours of work. She used the table to calculate how much she would be paid at this rate for 8 hours of work.
A 2-column table with 5 rows of data. The first column is labeled "amount earned, in dollars" and the second column is labeled "time worked, in hours." Row 1: 90, 5; Row 2: 18, 1; Row 3: 144, 8; Row 4 and 5: blank. An arrow pointing from row 1 to row 2 is labeled "times one-fifth." Another arrow pointing from row 2 to row 3 is labeled "times 8."
At this rate, how much would Lin be paid for 3 hours of work? For 2.1 hours of work?
Select a few students to share about the use of
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).
Optional
MLR8: Discussion Supports
Learn how to use instructional routines to facilitate problem-based learning!
None
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.
Ask students if they have heard of memory cards for electronic devices and invite students to share what they know.
Explain that digital devices such as phones, tablets, and computers use memory cards to store data, such as documents, pictures, and videos. The data can be stored temporarily or permanently, and then retrieved as needed—not unlike how we hold on to information and then retrieve it from our memory.
Tell students that the data that a memory can store is often measured in megabytes (MB) or gigabytes (GB). One byte is the amount of memory needed to store one letter of the English alphabet. (More than 1 byte may be needed for a letter or character of another language.)
In 2022, 128 gigabytes (GB) of portable computer memory cost $16.
One set of tick marks has already been drawn to show the result of multiplying 128 and 16 each by
Next, keep multiplying by
Here is a table that represents the situation. Find the cost of 1 gigabyte. You can use as many rows as you need.
| memory (gigabytes) | cost (dollars) |
|---|---|
| 128 | 16 |
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
Finding a row containing a “1” is often a good way to work with tables of equivalent ratios. For example, the price for 4 lbs of granola is $5. At that rate, what would be the price for 62 lbs of granola?
Here are tables showing two different approaches to solving this problem. Both of these approaches are correct. However, one approach is more efficient.
Less efficient
More efficient
Notice how the more efficient approach starts by finding the price for 1 lb of granola.
Remember that dividing by a whole number is the same as multiplying by a unit fraction. In this example, we can divide by 4 or multiply by
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| number of tacos | price in dollars |
|---|---|
Use the table to help you solve these problems. Explain or show your reasoning.
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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.
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