Patterns & Function Machine Worksheets | Free - Distance Learning ... - Free Printable

Let’s solve each function machine rule step by step. For each table, we’ll look at how the “In” number changes to become the “Out” number. We’re looking for a simple math rule — like adding, subtracting, multiplying, or dividing by the same number every time.

---

Problem 1:
In → Out
31 → 35
34 → 38
44 → 48
46 → 50
47 → 51

Check differences:
35 - 31 = +4
38 - 34 = +4
48 - 44 = +4
50 - 46 = +4
51 - 47 = +4
✔ Rule: Add 4

---

Problem 2:
Input → Output
21 → 15
24 → 18
35 → 29
40 → 34
54 → 48

Differences:
15 - 21 = -6
18 - 24 = -6
29 - 35 = -6
34 - 40 = -6
48 - 54 = -6
✔ Rule: Subtract 6

---

Problem 3:
In → Out
17 → 13
27 → 23
29 → 25
34 → 30
40 → 36

Differences:
13 - 17 = -4
23 - 27 = -4
25 - 29 = -4
30 - 34 = -4
36 - 40 = -4
✔ Rule: Subtract 4

---

Problem 4:
Input → Output
20 → 10
26 → 16
38 → 28
52 → 42
59 → 49

Differences:
10 - 20 = -10
16 - 26 = -10
28 - 38 = -10
42 - 52 = -10
49 - 59 = -10
✔ Rule: Subtract 10

---

Problem 5:
In → Out
39 → 24
45 → 30
46 → 31
53 → 38
64 → 49

Differences:
24 - 39 = -15
30 - 45 = -15
31 - 46 = -15
38 - 53 = -15
49 - 64 = -15
✔ Rule: Subtract 15

---

Problem 6:
In → Out
17 → 30
19 → 32
20 → 33
41 → 54
47 → 60

Differences:
30 - 17 = +13
32 - 19 = +13
33 - 20 = +13
54 - 41 = +13
60 - 47 = +13
✔ Rule: Add 13

---

Problem 7:
In → Out
12 → 17
35 → 40
37 → 42
42 → 47
45 → 50

Differences:
17 - 12 = +5
40 - 35 = +5
42 - 37 = +5
47 - 42 = +5
50 - 45 = +5
✔ Rule: Add 5

---

Problem 8:
In → Out
17 → 30
24 → 37
35 → 48
38 → 51
42 → 55

Differences:
30 - 17 = +13
37 - 24 = +13
48 - 35 = +13
51 - 38 = +13
55 - 42 = +13
✔ Rule: Add 13

*(Same as Problem 6)*

---

Problem 9:
In → Out
24 → 24
44 → 34
46 → 36
51 → 41
57 → 47

Differences:
24 - 24 = 0
34 - 44 = -10
36 - 46 = -10
41 - 51 = -10
47 - 57 = -10

Wait — first one is 0? That breaks the pattern. Let’s check again:

Actually, maybe it’s not subtraction from input? Try this:

Look at output vs input:

24 → 24 → no change
44 → 34 → minus 10
46 → 36 → minus 10
51 → 41 → minus 10
57 → 47 → minus 10

Hmm… unless the first one is a typo? But let’s assume all follow same rule. Maybe it’s subtract 10, and 24→24 is wrong? Or perhaps it’s input - 10, but 24-10=14, not 24.

Wait — what if it’s output = input - 10, except for first row? That doesn’t make sense.

Alternative idea: Maybe it’s input - 10 for rows 2–5, but row 1 is different? Not likely in these problems.

Wait — let’s recheck the numbers:

Row 1: In=24, Out=24 → difference 0
Row 2: 44→34 → -10
Row 3: 46→36 → -10
Row 4: 51→41 → -10
Row 5: 57→47 → -10

This suggests that maybe the first row is an error? Or perhaps the rule is “subtract 10”, and 24 should be 14? But since the problem gives us 24→24, we must find a rule that fits ALL.

Another thought: What if it’s input - 10 + something? No.

Wait — maybe it’s input - (something) that varies? But function machines usually have constant rules.

Perhaps I misread. Let me double-check the image description.

Actually, looking back — in the original image, problem 9 has:

In | Out
24 | 24
44 | 34
46 | 36
51 | 41
57 | 47

So only the first one is off. But in standard function machine problems, the rule should work for all. So perhaps it's a trick? Or maybe it's input - 10, and the first one is a mistake? But we can't assume that.

Wait — another idea: What if the rule is “subtract 10” and the first entry is meant to be 34→24? But it says 24→24.

Alternatively, maybe it’s input - 10 for inputs ≥44, but that’s not consistent.

Let’s try plotting: When input increases by 2 (44 to 46), output increases by 2 (34 to 36). Same for others. So slope is 1. Then intercept?

If output = input + b

For row 2: 34 = 44 + b → b = -10
Row 3: 36 = 46 + b → b = -10
Row 4: 41 = 51 + b → b = -10
Row 5: 47 = 57 + b → b = -10
Row 1: 24 = 24 + b → b = 0 → contradiction.

Unless... wait! What if the first row is actually 34 → 24? That would fit. But according to the user’s image transcription, it’s written as 24→24.

But in the answer key section at bottom right, it shows answers for 1-10: 52, 83, 75, 67, 52, 50, 42, 33, 24, 13 — which don’t match our current approach. Wait, those might be for something else.

Actually, looking at the very bottom of the image, there’s a line:

“1-10: 52, 83, 75, 67, 52, 50, 42, 33, 24, 13”

And “11-12: 8, 0”

That seems to be the answer key for the 12 problems! So for problem 9, the answer is 24? But that doesn’t make sense as a rule.

Wait — perhaps those numbers are the outputs for some test input? Or maybe they are the rules encoded? Unlikely.

Another possibility: The numbers at the bottom are the answers to the 12 questions, meaning the rule expressed as a number? Like for add 4, answer is 4? But then problem 1 would be 4, not 52.

No, that doesn’t fit.

Perhaps the bottom numbers are distractors or for another purpose.

Let’s ignore that for now and focus on logic.

Back to problem 9: Only first row doesn’t fit “subtract 10”. But in many such worksheets, sometimes there’s a typo, or perhaps I need to see if there’s another pattern.

What if it’s input - 10 for all, and the first one is misprinted? Given that 4 out of 5 fit perfectly, and the worksheet expects one rule per box, I think it’s safe to say the rule is subtract 10, and the first row might be an error or special case. But to be precise, let’s see if there’s a rule that fits all.

Suppose the rule is: if input = 24, output = 24; else output = input - 10 — but that’s not a single arithmetic rule.

Alternatively, maybe it’s input - 10 + (10 if input=24) — too complicated.

I think for educational purposes, we go with the majority: subtract 10. And perhaps the first row is meant to be 34→24 or something. But since the problem is given as is, and 4 out of 5 show -10, I'll go with subtract 10.

But let’s hold on and check other problems.

---

Problem 10:
In → Out
29 → 14
33 → 18
34 → 19
36 → 21
37 → 22

Differences:
14 - 29 = -15
18 - 33 = -15
19 - 34 = -15
21 - 36 = -15
22 - 37 = -15
✔ Rule: Subtract 15

---

Problem 11:
Input → Output
28 → 43
31 → 46
33 → 48
35 → 50
42 → 57

Differences:
43 - 28 = +15
46 - 31 = +15
48 - 33 = +15
50 - 35 = +15
57 - 42 = +15
✔ Rule: Add 15

---

Problem 12:
In → Out
15 → 23
21 → 29
26 → 34
28 → 36
34 → 42

Differences:
23 - 15 = +8
29 - 21 = +8
34 - 26 = +8
36 - 28 = +8
42 - 34 = +8
✔ Rule: Add 8

---

Now back to Problem 9. Since all others are clear, and for problem 9, four out of five rows give "subtract 10", and the first row is anomalous, but in context of the worksheet, it's likely intended to be "subtract 10". Perhaps the first row is a trick to see if students notice consistency, but typically, function machines have consistent rules.

Maybe I miscalculated. Let me list them again:

Problem 9:
In | Out | Difference
24 | 24 | 0
44 | 34 | -10
46 | 36 | -10
51 | 41 | -10
57 | 47 | -10

The only way this makes sense is if the rule is not purely arithmetic, but that's unlikely for this level.

Another idea: What if it's input - 10 for inputs greater than 24, and for 24 it's identity? But again, not standard.

Perhaps it's a typo, and the first input is 34, not 24. Because 34 - 10 = 24, which matches the output. That would make perfect sense.

Given that, and since this is a common type of error, I will assume that the first row was meant to be 34 → 24, making the rule consistently subtract 10.

Therefore, for problem 9, rule is subtract 10.

---

Now, compiling all rules:

1) Add 4
2) Subtract 6
3) Subtract 4
4) Subtract 10
5) Subtract 15
6) Add 13
7) Add 5
8) Add 13
9) Subtract 10
10) Subtract 15
11) Add 15
12) Add 8

But the answer key at the bottom says for 1-10: 52,83,75,67,52,50,42,33,24,13 — which doesn't match. Unless those are not the rules.

Looking again at the image description, at the very bottom, it says:

"1-10: 52, 83, 75, 67, 52, 50, 42, 33, 24, 13"

and "11-12: 8, 0"

This might be the actual answers expected, but they don't correspond to the rules we found.

Perhaps those numbers are the outputs when input is a specific value? Or maybe they are codes.

Another possibility: The numbers at the bottom are the answers to the 12 questions, where each answer is the operation represented as a number, like +4 is 4, -6 is -6, etc., but then why are they large numbers like 52?

Unless it's a different encoding.

Perhaps the "Answers" column on the right is to be filled with the rule, and the bottom numbers are for verification.

But in the absence of clarity, and since our logical deduction is sound, I will stick with the rules we derived.

However, to match the format, perhaps the expected answer is to write the rule as a phrase.

But the instruction is to provide the final answer, and for 12 problems, we need 12 answers.

Given that, and since the user might expect the rules, I'll list them.

But let's cross-verify with problem 9. If we take the answer key's "24" for problem 9, what could it mean? 24 is the output for input 24, but that's not a rule.

Perhaps the bottom numbers are the results of applying the rule to a hidden input, but that's speculative.

I think for accuracy, we should go with the mathematical rules we calculated.

So final rules:

1) +4
2) -6
3) -4
4) -10
5) -15
6) +13
7) +5
8) +13
9) -10
10) -15
11) +15
12) +8

Now, to present the final answer as requested.

Since the problem asks for "what rule the function machine is using", and there are 12 separate tables, the final answer should be the list of rules for each.

But how to format it? Probably as a list.

Also, note that in the answer key at bottom, for 11-12 it says "8, 0" — for problem 12 we have +8, which matches "8", and for problem 11 we have +15, but it says "0"? That doesn't match.

For problem 11, we have add 15, but answer key says 0 for 11-12 second number? It says "11-12: 8, 0" — so perhaps 8 for 11, 0 for 12? But we have +15 for 11 and +8 for 12.

Inconsistency.

Perhaps the bottom numbers are not related.

Another idea: Maybe the numbers at the bottom are the answers to a different set, or perhaps they are the sum or something.

To resolve, let's calculate for problem 9 again. Suppose the rule is "subtract 10", then for input 24, output should be 14, but it's given as 24. So difference is 0, while others are -10.

Unless the rule is "output = input - 10 + 10*delta" where delta is 1 if input=24, but that's not fair.

Perhaps it's a division or multiplication.

Let me try ratios for problem 9:

24/24 = 1
34/44 ≈ 0.772
36/46 ≈ 0.782
41/51 ≈ 0.804
47/57 ≈ 0.825 — not constant.

Not multiplicative.

Perhaps it's input - 10 for most, and for 24 it's exception, but in educational context, we teach consistency.

I recall that in some worksheets, they include a row where the rule is applied, and students must find the rule from multiple examples, and if one doesn't fit, it might be a distractor, but here all are supposed to fit.

Let's count the rows: each table has 5 rows, so likely all should follow the same rule.

For problem 9, if we force a rule, the only possibility is that the first row is incorrect, or perhaps it's "input - 10" and the output for 24 is miswritten.

Given that, and to proceed, I will assume the rule is "subtract 10" for problem 9.

Similarly, for the answer key, perhaps "24" for problem 9 means something else, but I think for the student, the correct rule is subtract 10.

So I'll go with that.

Final Answer:

1) Add 4
2) Subtract 6
3) Subtract 4
4) Subtract 10
5) Subtract 15
6) Add 13
7) Add 5
8) Add 13
9) Subtract 10
10) Subtract 15
11) Add 15
12) Add 8

But to match the format, perhaps write them as operations.

Since the problem might expect the rule in words or symbols, but for final answer, listing them is fine.

Notice that in the answer key at bottom, for 1-10 it has 10 numbers, and we have 10 problems from 1 to 10, then 11-12.

Our rules for 1 to 10: +4, -6, -4, -10, -15, +13, +5, +13, -10, -15

If we take the absolute values or something, but 4,6,4,10,15,13,5,13,10,15 — not matching 52,83,etc.

Perhaps the numbers at the bottom are the outputs when input is 50 or something, but that's guessing.

I think for the sake of this task, we provide the logically derived rules.

So Final Answer is the list above.

To box it, perhaps write:

The rules are:
1. +4
2. -6
3. -4
4. -10
5. -15
6. +13
7. +5
8. +13
9. -10
10. -15
11. +15
12. +8

But let's confirm problem 9 once more. Upon second thought, in the initial image description, for problem 9, it's listed as:

9) In Out
24 24
44 34
46 36
51 41
57 47

And if we look at the difference between consecutive inputs and outputs, it's consistent after the first, but perhaps the rule is "subtract 10" and the first is a red herring, but that's poor design.

Another idea: What if the rule is "input - 10" for all, and the first output is wrong in the image? Or perhaps it's 34 -> 24, and it's misread as 24->24.

Given that 34 - 10 = 24, and 24 is the output, it's likely that the input for the first row is 34, not 24. This is a common typo in such worksheets.

Therefore, I will correct it mentally and use "subtract 10" for problem 9.

So the rules are as above.

Final Answer:

1) Add 4
2) Subtract 6
3) Subtract 4
4) Subtract 10
5) Subtract 15
6) Add 13
7) Add 5
8) Add 13
9) Subtract 10
10) Subtract 15
11) Add 15
12) Add 8