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Home / Science, Engineering & Maths / Professional Development for Teachers / Maths Subject Knowledge: Graphs, Functions and Solving Equations Graphically / Inputs and outputs
Inputs and outputs
Paula and Michael demonstrate how a table of values can be generated using a function machine.
PAULA KELLY: So we’re going to start, first of all, by looking at some function machines. OK? So by a function machine, we just mean we start with an input, an operation happens, and then we have an output.
MICHAEL ANDERSON: OK.
PAULA KELLY: OK. So we’ll start nice and easy. Let’s start with our inputs. And then with our input, we’ll start with a multiply by 3.
MICHAEL ANDERSON: OK.
PAULA KELLY: And then we’re going to lead that into our output.
OK. So if we have our input– First of all, we’ll do a table of values. So our input and our output.
If I start with an input, we’ll start with really easy. If I just have 1–
MICHAEL ANDERSON: OK.
PAULA KELLY: –our output is going to be 3.
MICHAEL ANDERSON: OK. So you put 1 into the function machine, the function machine multiplies it by 3, and 3 is the output.
PAULA KELLY: Fantastic. Good.
MICHAEL ANDERSON: OK.
PAULA KELLY: OK. So, similarly, if we start with 2, multiply it by 3, our output is 6.
MICHAEL ANDERSON: OK.
PAULA KELLY: 3 would amount to 9, 4 to 12, and so on.
MICHAEL ANDERSON: OK. And looking at those inputs, you’ve kind of chosen 1, 2, 3, 4. Relatively straightforward, and they’re all going up by 1. The outputs look to be going up by 3 each time. In fact, it looks like the 3 times table.
PAULA KELLY: It’s exactly what it is. Yeah. We multiply by 3, it’s going to map on to a 3 times table.
MICHAEL ANDERSON: Mm-hmm.
PAULA KELLY: OK. However, rather than have consecutive numbers, we could try a different number. Which number should we have?
MICHAEL ANDERSON: Well, what happens if you put 100 in?
PAULA KELLY: OK. So same again. We’re still multiplying by 3. But because we haven’t chosen consecutive numbers, our 3 times table isn’t consecutive either.
MICHAEL ANDERSON: And can we put any number into this function machine?
PAULA KELLY: Any you like.
MICHAEL ANDERSON: OK. Well what happens with 0?
PAULA KELLY: So again, so we still do the same function. But we know that three lots of 0, is still going to be 0.
SPEAKER 2: OK, Yeah. So nothing in–
BOTH: Nothing comes out.
PAULA KELLY: Yes. Lovely.
MICHAEL ANDERSON: Negatives? Can you put negatives in?
PAULA KELLY: What negative shall we have?
MICHAEL ANDERSON: We’ll start with negative 1.
PAULA KELLY: OK. So negative 1. We multiply 3 times as much, is negative 3.
MICHAEL ANDERSON: OK. And I presume it’s the same. So negative 2?
PAULA KELLY: Negative 2 would be the same.
MICHAEL ANDERSON: And what about different types of numbers? So could you put decimals in?
PAULA KELLY: Absolutely, yes. What decimal should we have?
MICHAEL ANDERSON: 0.5
PAULA KELLY: 0.5. OK. And same again. We multiply it by 3 to give us 1.5.
MICHAEL ANDERSON: Nice. And can you put fractions in as well?
PAULA KELLY: Yes. Decimals, fractions, positive integers, negative integers. Anything would work.
MICHAEL ANDERSON: Anything at all. Brill, nice.
PAULA KELLY: So this time, we going to have a slightly different function machine. We’re still going to have our input. This time, rather than one operation, we’re going have two operations.
MICHAEL ANDERSON: OK. So the previous one, we had a function machine with one step. This is a two-step function machine?
PAULA KELLY: Absolutely.
MICHAEL ANDERSON: OK.
PAULA KELLY: OK. At this time we’ll have– similar to before, we’ll multiply by 3.
MICHAEL ANDERSON: OK.
PAULA KELLY: And then we’ll add on 2.
MICHAEL ANDERSON: All right. OK.
PAULA KELLY: OK. So we’ll have our table of values again. They’ve got our input and our output. OK. Choose another 1 again?
MICHAEL ANDERSON: Yeah, why not?
PAULA KELLY: Keep it simple.
MICHAEL ANDERSON: Yeah.
PAULA KELLY: OK. So this time, you multiply by 3, we add on 2. 1 multiplied by 3 gives us 3, 2 higher is going to give us 5.
MICHAEL ANDERSON: OK. So 1 goes in, and 5 comes out.
PAULA KELLY: Absolutely. If we had 2, multiply by 3 to give 6, 2 higher is going to be 8.
MICHAEL ANDERSON: Nice.
PAULA KELLY: I’m going to do one more. So 3, three 3’s we know are 9, add on 2, we have 11.
MICHAEL ANDERSON: And just looking, again, at the patterns in the outputs, they seem to be going up by 3 again each time. So 5, to 8, to an 11. So I’d guess that if we put 4 in, we’d get 14 out.
PAULA KELLY: Absolutely. Yeah. You can predict our next numbers by going up in 3’s. Adding another 3 to make it 14, and so on. OK.
MICHAEL ANDERSON: But these aren’t the 3 times tables coming out this time.
PAULA KELLY: No we’re looking for– trying to find this rule. We know it’s the 3 times table, but, 2 higher. That’s why it’s multiply by 3, add on 2.
SPEAKER 2: Plus 2. And again, we can put any numbers into here? So–
PAULA KELLY: Any at all, yeah.
MICHAEL ANDERSON: If we put 100 in again?
PAULA KELLY: Let’s have 100. OK. So 100, we multiply by 3, add on 2, get 302.
MICHAEL ANDERSON: OK.
PAULA KELLY: Same again. If we had positive integers, negative integers, decimals, fractions, any number would work.
MICHAEL ANDERSON: So go on then. What happens if we put, say, negative 5 in?
PAULA KELLY: OK. So negative 5. We multiply that by 3, negative 15. Be careful. Negative 15, add on 2, we’re going to have negative 13.
MICHAEL ANDERSON: Oh, OK. Nice.
PAULA KELLY: OK.
MICHAEL ANDERSON: Brill. And again, you can put any numbers you like in. And we just multiply them by 3, and add 2, to get the output.
PAULA KELLY: Lovely.
SPEAKER 2: OK.
PAULA KELLY: What about, though, if we just have the output?
MICHAEL ANDERSON: OK. So if the output was, say– I don’t know, 35?
PAULA KELLY: OK.
MICHAEL ANDERSON: Would we be able to work backwards?
PAULA KELLY: We absolutely would. We’re going to do the inverse this time. So if our output was 35, we would subtract 2.
MICHAEL ANDERSON: OK.
PAULA KELLY: And then divide by 3.
MICHAEL ANDERSON: So kind of go the opposite way around, and do the inverse of whatever we did to get the output from the input.
PAULA KELLY: Absolutely. So we start with 35, we subtract 2.
MICHAEL ANDERSON: OK. 33.
PAULA KELLY: 33. And then we divide that by 3 to end up with 11.
MICHAEL ANDERSON: OK. So if 35 came out of the function machine, 11 must have been the number that went in.
PAULA KELLY: Yes. We can check. We multiply by 3, 33, add on 2, we got it right.
MICHAEL ANDERSON: Brill.
PAULA KELLY: So this time, we’re still going to have our function machine. So we’re going to have our input.
We can have– let’s have two operations. And we have our output.
This time, in our table of values– So our input and our output. This time, if I tell you the inputs and outputs, you’re going to tell me what’s happening in our function machine.
MICHAEL ANDERSON: OK. So just by knowing what goes in to the function machine, and what goes out of the function machine, I hopefully will be able to figure out the actual values. The operation of the function machine itself.
PAULA KELLY: We’ll see, yeah.
MICHAEL ANDERSON: We’ll give it a go.
PAULA KELLY: An easy one to start. We’ll start with 1. OK? So if we had an input of 1, my output would be negative 7.
MICHAEL ANDERSON: OK. So it’s not just timesing by negative 7?
PAULA KELLY: It could be. We’ll see if it matches our next value.
MICHAEL ANDERSON: OK.
PAULA KELLY: What should we have next?
MICHAEL ANDERSON: Let’s try 10.
PAULA KELLY: OK. If we had 10, our output would be 20. Would that match your rule?
MICHAEL ANDERSON: Not really. Seems to have doubled this time. But I suppose there is two steps. So it’s a little bit more complicated than maybe just timesing one thing by another. What happens if we put in, say, 4?
PAULA KELLY: 4. OK. That gives an output of 2.
MICHAEL ANDERSON: OK. Appears to have halved it. I suppose we had a bit of success doing consecutive values as inputs last time. So let’s say, 5?
PAULA KELLY: 5. OK. So our output would be 5.
MICHAEL ANDERSON: OK. What happens if you put 6 in?
PAULA KELLY: So, 6.
That would give us 8.
MICHAEL ANDERSON: OK. So 4, 5, 6, go up by 1 each time.
PAULA KELLY: Yeah.
MICHAEL ANDERSON: And then on the outputs, it goes 2, 5, 8. So it seems to be going up by 3, which is the same as some of the other examples. Like the 3 times tables. So I’m going to guess the first bit is multiply by 3.
PAULA KELLY: Absolutely. You can see a common difference. We got some consecutive numbers here, our common difference is 3, we’re going to compare this to our 3 times table.
MICHAEL ANDERSON: OK.
PAULA KELLY: Our first step is going to be multiply by 3.
MICHAEL ANDERSON: OK. So if we take that 4 example, 4 multiplied by 3 gives us 12. But we ended up with a value of 2. To get from 12 to 2, we have to take away 10. So is it take away 10 as the final box?
PAULA KELLY: It could be. It is for this one. But we want to make sure it’s for all of them. So would it work for our next one?
MICHAEL ANDERSON: Well 5 times 3 is 15, 15 take away 10 is 5. So it seems to work.
PAULA KELLY: Seems to work.
MICHAEL ANDERSON: Have I got it right?
PAULA KELLY: Yes. One more for good luck.
MICHAEL ANDERSON: So 6 times 3 is 18, take away 10 is 8.
PAULA KELLY: Absolutely. Lovely. So we’re happy our rule is multiplied by 3, subtract 10.
MICHAEL ANDERSON: Excellent.
PAULA KELLY: OK. We’ll try one more. OK? So we’ll have– same again. Our function machine. Got our input. We’ll keep our two functions. Or two operations, rather.
MICHAEL ANDERSON: But I suppose a function machine could be any number of operations, as complicated as we’d like.
PAULA KELLY: Absolutely, yes. Don’t want to be too mean to you though.
MICHAEL ANDERSON: OK. So two– we’ll stick with two.
PAULA KELLY: OK. So we got our input, and we’ve got our output. OK.
MICHAEL ANDERSON: OK. So I’ve learned from the last step. Let’s just put the numbers of 1, 2, 3, and the consecutive integers. So what happens when we put 1 in?
PAULA KELLY: A really sensible choice. So a number of 1 would give us an output of 98.
MICHAEL ANDERSON: Well I’m not sure I like this very much. OK. So 2 going in? What happens to 2?
PAULA KELLY: That would give us 96.
MICHAEL ANDERSON: OK. So it seems to be going down this time, no? What happens when we put 3 into it?
PAULA KELLY: Good, sensible choice. If we put 3 into there, we’re going to end up with 94.
MICHAEL ANDERSON: OK. So it seems to be going down by the same amount each time, that kind of linear sequence. So if it’s going down, and by 2 each time, that’s almost a little bit like the 2 times tables. Because the 2 times tables goes up by 2 every time. This is going down. I’m going to guess it’s like, the negative 2. So do you multiply by negative 2 to start with?
PAULA KELLY: Absolutely. Yeah.
MICHAEL ANDERSON: OK.
PAULA KELLY: Our common difference is 2. It’s decreasing, so our negative 2 times table.
MICHAEL ANDERSON: OK.
PAULA KELLY: OK. So we start by timesing by negative 2.
MICHAEL ANDERSON: OK. Now if we took that number 1, the first one, and times it by– multiplied it by negative 2, you get negative 2. And to get all the way up to 98 from negative 2, all the way to 98 would be adding 100.
PAULA KELLY: Absolutely.
MICHAEL ANDERSON: Is that right?
PAULA KELLY: Yeah.
MICHAEL ANDERSON: So that next box is add 100.
PAULA KELLY: We think it is.
MICHAEL ANDERSON: OK.
PAULA KELLY: But it’s best to be clear. Just double check this one.
MICHAEL ANDERSON: So when 2 goes in, multiply that by negative 2, you get negative 4. And then when you add 100 to that, or 100 take away 4 as well is a different way of working it out, you get 96. OK. So it seems to work.
PAULA KELLY: Seems pretty good.
MICHAEL ANDERSON: 3 times negative 2 is negative 6. Add 100 to that, and yeah, 94. So I think if you were to put 4 in, your answer would be 92?
PAULA KELLY: 92.
MICHAEL ANDERSON: Yeah.
PAULA KELLY: What would be a really quick way of working that out, thinking about the numbers that we’ve chosen here?
MICHAEL ANDERSON: So instead of multiplying by negative 2 and adding 100, I suppose it’s just going down by 2 each time. So it’d be quite easy to predict the next term if we put 5 in, or 6 in, or 7 in, et cetera. Brill. So have I got it right?
PAULA KELLY: You have. So we know our function is multiply by negative 2, and then we’re going to add on 100.
MICHAEL ANDERSON: Perfect.
PAULA KELLY: Great.
In this step Paula and Michael demonstrate how a table of values can be generated using a function machine. Starting with a recap of a simple function machine they progress to consider a two-step function machine.
A connection with solving equations is made when Paula and Michael want to find what input is required to produce a given output.
In the final part of the video Paula and Michael explore how function machines can be linked with finding the position to term rule in a sequence by exploring what function machine produces a given table of inputs and outputs. More on this can be found in the course Understanding Numbers.
Task
This task will help you think about the connection between function machines and sequences. For each of the following tables, decide what function will produce the following inputs and outputs? Share your thoughts in the comments section below.
Function machine A
| INPUT | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
|---|---|---|---|---|---|---|---|---|---|---|
| OUTPUT | 7 | 14 | 21 | 28 | 35 | 42 | 49 | 56 | 63 | 70 |
Function machine B
| INPUT | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
|---|---|---|---|---|---|---|---|---|---|---|
| OUTPUT | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
Function machine C
| INPUT | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
|---|---|---|---|---|---|---|---|---|---|---|
| OUTPUT | 1 | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 |
Teaching resources
There are a variety of activities to develop these skills in these collections of SMILE resources on the STEM Learning website.
The following resources on the Nrich Website are also useful
Worksheet questions
Now do question two from this week’s worksheet.
This article is from the online course:
Maths Subject Knowledge: Graphs, Functions and Solving Equations Graphically
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Maths Subject Knowledge: Graphs, Functions and Solving Equations Graphically
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