At Elmwood Middle School, sixth graders spend three hours every night studying. Seventh graders spend four hours. Eighth graders spend five hours. Let the students’ grade be the input 𝑥. What is the function rule between the students’ grade and the amount of time the students spend on homework every night?
Now, when we’re faced with a wordy problem like this, it makes sense to read it several times slowly and carefully. Let’s go through it again more slowly and try to work out what it’s asking us to do. Well, the question seems to be about the amount of time that different children spend studying. We’re given three pieces of information about three different grades of sixth grade, seventh grade, and eighth grade. Now at the moment, these facts are almost lost within all the words. So why don’t we pull them out and perhaps write them in a table?
Our first column could show the grade that the students are in. And in our second column, we could show that the time that they spend working on their homework every night. And this is in hours. So the first fact we can see is that sixth graders spend three hours every night. So in grade six, the time spent on homework is three hours. Next, we’re told that seventh graders spend four hours. So in grade seven, time taken, four hours. And finally, eighth graders spend five hours. And we can see patterns here already. The grade number is increasing by one each time, as we would expect. But also the amount of time that students spend in their homework is also increasing by one hour every time.
The question then says, “Let the students’ grade be the input 𝑥.” Sounds like we now need to start thinking about our table as if it was an input-output table. And we’re told that the input is the students’ grade. And we can call this 𝑥. Let’s change the title of our input column to actually show the letter that we’re told to use. Now it’ll also be a good idea to use a letter for the output. We’re not told which letter to use. But because we’re measuring time, we often use the letter 𝑡 to represent time when we’re dealing with functions. Now we’re asked to find the function rule between the students’ grade, which we’re calling 𝑥. And the amount of time the students spend on homework every night, which we’re calling 𝑡. To help us find the answer, we need to think about what we do to the input to find the output. What happens to the numbers in the left-hand column so that we get the answers in the right-hand column.
Let’s begin by looking at the first pair of numbers. Here, we can see that the output 𝑡 is worth three. So what do we do to six to give us three? Well, one answer could be that we divide it by two. Three is half of six. So perhaps our function rule is 𝑡. The length of time that it takes for students to do their homework is 𝑥, the grade that they’re in, divided by two. Does this function rule work? Let’s test it with another grade. Let’s try the seventh grade.
When we’re thinking about the seventh grade, 𝑥 is going to be worth seven. And if we divide seven by two, we get three and a half. And so the length of time that the seventh graders should spend on their homework should be three and a half hours. Is that right? No, we can see in the table that they’re supposed to spend four hours on homework. The rule can’t be divide by two. Let’s go back again to our first pair of numbers. And think again. What do we do to six to give us three? Well, another way to get from six to three is to take away three. So if we wrote this as a function rule, we’d write 𝑡. The length of time that the students took to do their homework should be the grade that they’re in, which is 𝑥, take away three.
Let’s try this rule with the other grade and see if it works. If we’re in grade seven, then 𝑥 is going to be worth seven. And if we take away three from seven, we’re left with four. This is correct. Seventh graders spend four hours doing their homework. And finally, if we’re in the eighth grade, then 𝑥, which is the input, is going to be eight. We need to subtract three from this. This will give us an answer for 𝑡 of five. In other words, the length of time that eighth graders are supposed to do their homework for is five hours. And if we look at our table, we can see that this is correct. We found a function rule that works for all of the grades.
If we call the students’ grade 𝑥 and the amount of time they spend on homework 𝑡. Then the function rule between the students’ grade and the amount of time they spend on homework every night can be written as 𝑡 equals 𝑥 minus three.