Practice worksheet for understanding function machines in math.

Let’s solve this step by step.

We are told that the machine takes an IN number, does something to it (adds or subtracts), and gives an OUT number. The operation is shown in the middle box — either “+” or “–” with a number.

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Example 1:

- IN = 4 → +3 → OUT = 7 ✔
- IN = 3 → +2 → OUT = 5 ✔
So we just add the number in the middle to the IN number to get OUT.

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Example 2:

- IN = 7 → +5 → OUT = 12 ✔
- IN = 8 → +8 → OUT = 16 ✔
Again, adding works.

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Example 3:

- IN = 6 → –2 → OUT = 4 ✔
- IN = 5 → –3 → OUT = 2? Wait — no! It says OUT = 14? That doesn’t match.

Wait — let me check again.

Actually, looking at Example 3:

First row:
IN = 6 → –2 → OUT = 4 → 6 - 2 = 4 ✔

Second row:
IN = 5 → –3 → OUT = ? But it says 14? That can’t be right unless I misread.

Wait — look again at the image description.

Actually, in Example 3, second row:
It shows:
[5] → [–3] → [ ] and then next to it says “14”? No — wait, maybe the 14 is for another part?

Hold on — let me re-read the examples carefully from your text.

You wrote:

> Example 3
> 6 → –2 → 4
> 5 → –3 → ? but then it says “14” — that must be a typo in my reading.

Actually, looking back at your original problem statement:

In Example 3, you have:

Row 1: 6 → –2 → 4
Row 2: 5 → –3 → ??? and then it says “14” — but that doesn’t make sense because 5 - 3 = 2, not 14.

Wait — perhaps the “14” is actually the OUT for a different example? Let me check the structure.

Looking at your input:

After Example 3, you list:

A.
4 → +2 → ?
3 → +2 → ?

B.
? → +7 → 14
7 → +3 → 10
9 → +4 → ?

C.
6 → –4 → ?
8 → –7 → ?
? → –4 → 11

D.
5 → –3 → ?
6 → –3 → 13? Wait — 6 - 3 = 3, not 13. That can’t be.

Wait — there's confusion here. Let me go back to the actual examples given in the worksheet as described.

From your text:

Example 1:
4 → +3 → 7
3 → +2 → 5

Example 2:
7 → +5 → 12
8 → +8 → 16

Example 3:
6 → –2 → 4
5 → –3 → ??? and then it says “14” — but that must be a mistake in transcription.

Actually, looking at your full text:

You wrote:

> Example 3
> 6 → –2 → 4
> 5 → –3 → [blank] and then next to it says “14” — but that 14 is probably for the next section.

Wait — no, in your message, after Example 3, you have:

“A.” which has two rows:
4 → +2 → ?
3 → +2 → ?

Then “B.” has three rows:
? → +7 → 14
7 → +3 → 10
9 → +4 → ?

Ah! So the “14” in B is the OUT for the first row of B, where IN is missing.

Similarly, in C:
6 → –4 → ?
8 → –7 → ?
? → –4 → 11

And D:
5 → –3 → ?
6 → –3 → 13? Wait — 6 - 3 = 3, but it says 13? That can't be.

Wait — let me read your original input again carefully.

You wrote:

> D.
> 5 → –3 → ?
> 6 → –3 → 13
> 4 → –3 → 10

That last one: 4 - 3 = 1, not 10. Something is wrong.

Unless... oh! Maybe the operation is not what it seems? Or maybe I misread the signs?

Wait — let’s look at Example 3 again as written by you:

> Example 3
> 6 → –2 → 4
> 5 → –3 → [blank] and then it says “14” — but that 14 is likely not part of Example 3.

Actually, in your initial description, after Example 3, you have sections A, B, C, D — each with their own problems.

Let me ignore the "14" in Example 3 for now and focus on the pattern.

From Examples 1, 2, 3:

- When it’s “+”, we add the number to IN to get OUT.
- When it’s “–”, we subtract the number from IN to get OUT.

Check Example 3 first row: 6 - 2 = 4 ✔

Now, for the problems:

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Section A:

First row:
IN = 4, operation = +2 → OUT = 4 + 2 = 6

Second row:
IN = 3, operation = +2 → OUT = 3 + 2 = 5

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Section B:

First row:
OUT = 14, operation = +7 → so IN + 7 = 14 → IN = 14 - 7 = 7

Second row:
IN = 7, operation = +3 → OUT = 7 + 3 = 10 ✔ matches given

Third row:
IN = 9, operation = +4 → OUT = 9 + 4 = 13

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Section C:

First row:
IN = 6, operation = –4 → OUT = 6 - 4 = 2

Second row:
IN = 8, operation = –7 → OUT = 8 - 7 = 1

Third row:
OUT = 11, operation = –4 → so IN - 4 = 11 → IN = 11 + 4 = 15

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Section D:

First row:
IN = 5, operation = –3 → OUT = 5 - 3 = 2

Second row:
IN = 6, operation = –3 → OUT = 6 - 3 = 3 — but you wrote “13”? That must be a typo in your transcription.

Wait — in your original message, you said:

> D.
> 5 → –3 → ?
> 6 → –3 → 13
> 4 → –3 → 10

But 6 - 3 = 3, not 13. And 4 - 3 = 1, not 10.

This suggests that perhaps the operation is not subtraction? Or maybe it’s addition?

Wait — let’s check if it could be addition.

If in D, operation is “–3”, but maybe it’s meant to be “+3”? Let’s test:

If 6 + 3 = 9, not 13.

Or maybe the operation is applied differently?

Another possibility: perhaps the number in the middle is not the operand, but something else? But the examples show otherwise.

Wait — look at Example 3 again. You have:

> Example 3
> 6 → –2 → 4
> 5 → –3 → [blank] and then it says “14” — but that 14 is probably for section B.

I think there was a formatting issue in your message.

Let me assume that in section D, the operations are correct, and the given OUT numbers are typos, or perhaps I misread.

But in your text, for D, you have:

> 5 → –3 → ?
> 6 → –3 → 13
> 4 → –3 → 10

This is inconsistent with subtraction.

Unless... oh! What if the operation is “minus” but the number is negative? Like –(-3)? But that would be +3.

Let’s try that.

Suppose in D, the operation is “– (-3)” which is +3.

Then:

5 + 3 = 8
6 + 3 = 9 — still not 13.

Not matching.

Perhaps it’s multiplication? But examples don’t show that.

Another idea: maybe the “–3” means subtract 3, but the OUT is given incorrectly in your transcription.

Let me check the third row of D: 4 → –3 → 10

4 - 3 = 1, not 10.

4 + 6 = 10? Not related.

Perhaps the operation is not on the IN number directly, but something else.

Wait — let’s look back at the directions:

“Make the Function Machine work by filling in the empty circles with the correct numbers that are missing. You’ll see an IN number, an INSIDE number, and their sum, the OUT number.”

“Sum” — that implies addition!

In the directions, it says “their sum, the OUT number”.

But in Example 3, it’s subtraction: 6 - 2 = 4, which is not a sum.

Contradiction.

Let me read the directions again from your text:

“Directions: Make the Function Machine work by filling in the empty circles with the correct numbers that are missing. You’ll see an IN number, an INSIDE number, and their sum, the OUT number.”

It says “sum”, but in Example 3, it’s clearly subtraction.

Perhaps “sum” is used loosely, or it’s a mistake.

In Example 1 and 2, it’s addition, and OUT is sum.

In Example 3, it’s subtraction, and OUT is difference.

So probably, “sum” in the directions is a misnomer; it should be “result” or “output”.

So we should follow the operation shown: if it’s “+”, add; if it’s “–”, subtract.

For section D, if the operation is “–3”, then:

5 - 3 = 2
6 - 3 = 3
4 - 3 = 1

But you have “13” and “10” as given OUT for the second and third rows, which don't match.

Unless the operation is “+3” instead of “–3”.

Let me check if that works.

If operation is +3:

5 + 3 = 8
6 + 3 = 9 — not 13
4 + 3 = 7 — not 10

Still not.

What if the operation is “×3” or something? But examples don't show multiplication.

Another possibility: perhaps the “INSIDE number” is not the operand, but the result of some other operation.

But the examples are clear: in Example 1, 4 + 3 = 7, etc.

Let’s look at section D as given in your text:

> D.
> 5 → –3 → ?
> 6 → –3 → 13
> 4 → –3 → 10

This is impossible with subtraction.

Unless the “–3” is a label, and the actual operation is different.

Perhaps the number in the middle is the amount to add or subtract, but the sign indicates direction, and for D, it's always subtracting 3, but the OUT is given, so we need to find IN or something.

In the first row of D, IN is given as 5, operation –3, OUT is blank — so we calculate 5 - 3 = 2.

In the second row, IN is 6, operation –3, OUT is given as 13 — but 6 - 3 = 3 ≠ 13, so contradiction.

Unless the operation is not applied to IN, but to something else.

Perhaps the "INSIDE number" is the output of a previous step, but the diagram shows IN -> inside -> OUT.

I think there might be a typo in your transcription of the problem.

Let me assume that in section D, the operation is “+3” instead of “–3”, and see if it fits.

If +3:

5 + 3 = 8
6 + 3 = 9 — not 13
4 + 3 = 7 — not 10

No.

What if it's “+8” for the second row? 6 + 8 = 14, close to 13.

Not exact.

Another idea: perhaps the operation is “minus” but the number is the OUT minus IN or something.

Let’s calculate for the second row of D: IN=6, OUT=13, so if operation is x, then 6 op x = 13.

If op is +, then x = 7.

If op is -, then 6 - x = 13 => x = -7, which is possible, but the inside shows “–3”, not “–(-7)”.

This is messy.

Perhaps in section D, the operation is consistently “–3”, and the given OUT for the second and third rows are mistakes, and we should calculate based on the operation.

But in the problem, for section D, the second and third rows have OUT given, so we might need to find the operation or something, but the operation is given as “–3” for all.

Let’s look at the third row of D: IN=4, operation=–3, OUT=10.

4 - 3 = 1, not 10.

4 + 6 = 10, so if operation was +6, but it's written as –3.

I think there is a error in the problem transcription.

Perhaps the “–3” is not the operation, but the IN or something, but the diagram shows IN on left, operation in middle, OUT on right.

Another thought: in some function machines, the operation might be applied in reverse, but the examples show forward.

Let’s double-check Example 3.

You have:

> Example 3
> 6 → –2 → 4
> 5 → –3 → [blank] and then it says “14” — but that 14 is likely for section B.

In your text, after Example 3, you have:

"A." with two rows, then "B." with three rows, including "? → +7 → 14", so the 14 is for B, not for Example 3.

Similarly, for D, the "13" and "10" are given as OUT for those rows, so we must use them to find something, but what is missing?

In section D, for the first row, IN and operation are given, OUT is blank — so we calculate OUT.

For the second row, IN and operation are given, OUT is given as 13 — but 6 - 3 = 3 ≠ 13, so perhaps the operation is not "–3", but something else, but it's written as "–3".

Unless the "–3" is a red herring, but that doesn't make sense.

Perhaps the number in the middle is not the operand, but the result of IN and OUT or something.

Let’s think differently.

In the directions, it says "you'll see an IN number, an INSIDE number, and their sum, the OUT number."

"Sum" suggests addition, so perhaps even when it shows "–", it's not subtraction, but the inside number is to be added, and the "–" is a distractor or something.

But in Example 3, 6 and 2, if we add, 6+2=8, but OUT is 4, which is 6-2.

So it must be subtraction.

Perhaps for section D, the operation is "–3", but the OUT is given, so for the second row, 6 - 3 = 3, but it's written as 13, which might be a typo, and it should be 3.

Similarly, 4 - 3 = 1, not 10.

Or perhaps it's 6 + 7 = 13, but why 7?

Another idea: perhaps the "INSIDE number" is the amount to add or subtract, but for D, it's always 3, and the sign is minus, but the OUT is calculated as IN + inside or something.

Let’s calculate the difference between OUT and IN for the given rows in D.

For second row: IN=6, OUT=13, difference = 7

For third row: IN=4, OUT=10, difference = 6

Not constant.

Ratio: 13/6 ≈ 2.16, 10/4 = 2.5, not constant.

Perhaps the operation is "multiply by 2 and add 1" or something, but that's complicated, and examples are simple add/subtract.

I think the most reasonable explanation is that there is a typo in your transcription of section D.

Perhaps in section D, the operation is "+3" for all, and the OUT for second row is 9, not 13, and for third row is 7, not 10.

But you have "13" and "10", which are close to 9 and 7? Not really.

13 and 10 could be for different operations.

Let’s look at the first row of D: 5 → –3 → ? , so if we calculate 5 - 3 = 2, that should be the answer for the first blank.

For the second row, if IN=6, operation=–3, OUT should be 3, but it's given as 13, which is likely a typo, and it should be 3.

Similarly, for third row, 4 - 3 = 1, not 10.

Perhaps the "–3" is " +3" , and 6+3=9, 4+3=7, still not 13 and 10.

Another possibility: perhaps the operation is "–" but the number is the OUT, and we need to find IN, but in the diagram, IN is on left, so for second row, if OUT=13, operation=–3, then IN - 3 = 13, so IN = 16, but IN is given as 6.

Conflict.

I think I need to proceed with the calculation based on the operation given, and assume that the given OUT for second and third rows of D are errors, or perhaps they are for different purposes.

But in the problem, for section D, the first row has OUT blank, so we fill it.

The second and third rows have OUT given, so perhaps we don't need to fill anything for them, or perhaps we need to verify, but the task is to fill empty circles.

In section D, only the first row has an empty OUT circle; the other two have OUT given, so perhaps we only need to fill the first one.

Similarly, in other sections, only the blanks are to be filled.

Let me check the structure.

In your text, for each section, there are rows with some blanks.

For example, in A: both rows have OUT blank.

In B: first row has IN blank, third row has OUT blank.

In C: first and second have OUT blank, third has IN blank.

In D: first row has OUT blank, second and third have OUT given, so perhaps only the first row needs to be filled for D.

And for the second and third rows of D, the OUT is given as 13 and 10, which may be correct for a different operation, but since the operation is given as "–3", and IN is given, we should calculate OUT as IN - 3.

But 6 - 3 = 3, not 13, so perhaps the operation is not "–3", but the number is 3, and the sign is part of the operation, but for D, it's consistently "–3", so likely a typo in the given OUT.

Perhaps in D, the operation is " +3" , and the "–" is a mistake.

Let me assume that for D, the operation is "+3", then:

5 + 3 = 8
6 + 3 = 9
4 + 3 = 7

But you have 13 and 10, which are not 9 and 7.

13 - 6 = 7, 10 - 4 = 6, not consistent.

Another idea: perhaps the "INSIDE number" is the amount to add, and the "–" means something else, but in examples, "–" means subtract.

I recall that in some worksheets, the function machine might have the operation as "add or subtract the number", and for D, it might be that the number is 3, and we add or subtract based on context, but the sign is given.

Let’s calculate what operation would give the given OUT.

For second row of D: IN=6, OUT=13, so if operation is +x, then 6 + x = 13, x=7

If operation is -x, 6 - x = 13, x= -7

But the inside shows "–3", so not matching.

Perhaps the "3" is not the operand, but the result or something.

I think I have to conclude that for section D, with the given information, for the first row, IN=5, operation=–3, so OUT=5-3=2.

For the other rows, since OUT is given, and it doesn't match, perhaps it's a different problem, or perhaps we ignore the given OUT for calculation, but that doesn't make sense.

Let’s look back at your original message.

You wrote for D:

> D.
> 5 → –3 → ?
> 6 → –3 → 13
> 4 → –3 → 10

But in the context, perhaps the "13" and "10" are not the OUT for those rows, but part of the section label or something, but that doesn't make sense.

Perhaps it's a list, and for D, there are three rows, and for the second and third, the OUT is given, so we don't need to fill them, only the first row's OUT is to be filled.

And the given OUT for second and third are correct for the operation, but 6-3=3≠13, so unless the operation is different.

Another possibility: perhaps the "–3" means "subtract 3 from the OUT" or something, but that would be unusual.

Let’s try that.

Suppose for a row, IN -> operation -> OUT, but the operation is applied to OUT to get IN or something.

For example, in second row of D: if OUT=13, and operation is "–3", then IN = OUT - 3 = 13 - 3 = 10, but IN is given as 6, not 10.

Not matching.

If IN = OUT + 3, then for OUT=13, IN=16, not 6.

No.

Perhaps the operation is between IN and OUT.

I think I need to accept that for section D, with the given, for the first row, OUT = 5 - 3 = 2.

For the other rows, since OUT is given, and it's inconsistent, perhaps it's a typo, and we should proceed with the calculation for the blanks.

In section D, only the first row has a blank OUT, so we fill that.

Similarly, in other sections, we fill the blanks.

So let's do that.

Summary of what to fill:

A.

- First row: OUT = 4 + 2 = 6
- Second row: OUT = 3 + 2 = 5

B.

- First row: IN = 14 - 7 = 7 (since IN + 7 = 14)
- Third row: OUT = 9 + 4 = 13

C.

- First row: OUT = 6 - 4 = 2
- Second row: OUT = 8 - 7 = 1
- Third row: IN = 11 + 4 = 15 (since IN - 4 = 11)

D.

- First row: OUT = 5 - 3 = 2

For the other rows in D, since OUT is given, and it's not matching, but perhaps in the actual worksheet, it's correct, or perhaps we don't need to worry about them for filling blanks.

So the answers to fill are:

A: 6, 5
B: 7, 13
C: 2, 1, 15
D: 2

Now, to present the final answer.

Since the user asked to solve the problem, and the problem is to fill the empty circles, I'll provide the values for the blanks.

In the response, I should list the answers for each section.

So for A: the two OUT values are 6 and 5
For B: the IN for first row is 7, and OUT for third row is 13
For C: OUT for first row is 2, OUT for second row is 1, IN for third row is 15
For D: OUT for first row is 2

So the final answer can be listed as:

A: 6, 5
B: 7, 13
C: 2, 1, 15
D: 2

To box it.

But let's confirm with the examples.

In Example 3, you have 6 -> -2 -> 4, which is correct.

For C, third row: ? -> -4 -> 11, so IN - 4 = 11, IN = 15, correct.

For B, first row: ? -> +7 -> 14, IN = 7, correct.

For D, first row: 5 -> -3 -> 2, correct.

The given OUT for other rows in D may be errors in transcription, but for the blanks, we have the answers.

So I'll go with that.

Final Answer:
A: 6, 5
B: 7, 13
C: 2, 1, 15
D: 2