About the Mathematics and Structure
This string is intended to build intuition for inverse variation, which can be described as "y varying inversely to x. This kind of functional behavior is like the rational parent function y = k/x. If students have prior experience reasoning in ratio tables by scaling in tandem, this Problem String may be challenging. Keep students in context, reasoning about the number of workers it takes to clear the whole field and the time that takes.
If students start to talk about "clearing half the field" or in other ways chopping up the field, remind them that this Problem String is all about clearing the whole field. The number of workers and the time is always based on clearing the whole field, getting the whole job done.
The first two problems give students a nice factor and product of 20 workers, half (10) and double (40). The next problem, 24 hours, is a multiple of each of the previous number of hours, 6, 12, 3, so students can reason from any/all of them.
The fourth problem, 15 workers, is most directly figured from the previous 5 workers (times 3). Students may try to "split the difference" because 15 workers is halfway between 20 and 10 workers, so students may reason that the corresponding number of hours is halfway between their corresponding 6 and 12. This "split the difference" strategy only works in linear situations, where the rate of change is constant. The rate of change for inverse variation is not constant.
The last two problems are given to help nudge students to think about the total number of worker-hours in this situation, which is 120 worker-hours.
To facilitate the string:
- Set the stage for the context. You have an orchard full of fruit. You know that 20 workers can clear the orchard, pick all of the fruit, in 6 hours. If that is true, how long would it take 10 workers?
- Ask students to justify their answer. Notice if students are thinking additively or multiplicatively. Encourage students to stay in context. "10 what? If you halve number of hours? What is your 6, the 12? For those numbers, what's happening in the orchard?"
- Model the doubling and halving using arrows. Use multiplicative language like, "So if you halve the number of workers, how long would it take?" "If you halve the number of workers, it would take double the amount of time?"
- Repeat for the other questions. For the 40 workers, encourage students to look for relationships between both the 20 and 10 to 40 workers.
- When you get to the 24 hours, ask, "What if you had a whole day, 24 hours, how many people would you need? You don't want to get it done any faster. You have all 24 hours. You want to hire just the right number of people.
- Before you ask the number of hours it will take for 1 worker, ask students if they notice a number happening all through the table. Wave back and forth from left to right to nudge students to look between the number of workers and the number of hours. Help students notice the 120 "worker-hours". The product of each of the row entries is (number of workers)(number of hours) = number of worker-hours (120).
- If it takes 20 workers 6 hours to clear the orchard, pick all of the fruit, how long would it take 10 workers to clear the orchard?
- What if you had a whole day, 24 hours, how many people would you need? You don't want to get it done any faster. You have all 24 hours. You want to hire just the right number of people. How many people would that be?
- What is happening in this table? Is this a ratio table? In a ratio table, the ratios y/x are all equivalent. What is equivalent in this table? The product xy? What is that product? Can that help you find the time it would take one person? The number of people you need to clear the orchard in one hour?
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