Free Distributive Property, Associative Property, and Commutative ... - Free Printable

Let’s go through each problem one by one. We’ll figure out which math law is being used in each case — either the commutative law or the associative law.

---

What are these laws?

- Commutative Law: You can switch the order of numbers when adding or multiplying, and the answer stays the same.
- Example: 2 + 3 = 3 + 2 → commutative for addition
- Example: 4 × 5 = 5 × 4 → commutative for multiplication

- Associative Law: You can group numbers differently (using parentheses) when adding or multiplying, and the answer stays the same.
- Example: (2 + 3) + 4 = 2 + (3 + 4) → associative for addition
- Example: (2 × 3) × 4 = 2 × (3 × 4) → associative for multiplication

Note: These laws only work for addition and multiplication — NOT for subtraction or division!

But wait — some problems here have subtraction or negative numbers. Let’s look carefully.

Actually, in problems like #2: (-8 -9) +8 = -8 +(-9+8), this is using the associative property of addition, even with negatives — because we’re regrouping terms that are being added (even if they’re negative).

Similarly, #6: (-3)×9 +7×9 = 9×(-3) +7×9 — here, we switched the order of multiplication: (-3)×9 became 9×(-3). That’s commutative law for multiplication.

So let’s solve each:

---

1. 2 × 3 = 3 × 2
→ Order switched → commutative law

2. (-8 -9) +8 = -8 +(-9+8)
→ Grouping changed → associative law (of addition)

3. 5 + 12 -12 = 5
→ This simplifies to 5 + 0 = 5 → but it’s not really a named “law” from the two options. Wait — actually, 12 -12 cancels out. But looking at structure: it’s just arithmetic. However, note that 5 + (12 -12) = 5 + 0 = 5. But the equation as written doesn’t show grouping change or order change. Hmm. Actually, maybe it’s showing that adding zero doesn’t change value? But that’s identity property. Since the worksheet only gives commutative or associative, perhaps this is miswritten? Or maybe it’s implying 5 + 12 -12 = 5 + (12 -12)? Still, no clear commutative/associative. Let’s skip and come back.

Wait — actually, looking again: 5 + 12 -12 = 5. This is just simplifying. But if we think of it as 5 + (12 -12) = 5, then it’s using the idea that 12 -12 = 0, and 5 + 0 = 5 — that’s additive identity. But since the worksheet only has commutative and associative, maybe this is a trick? Or perhaps it’s meant to be grouped as (5 + 12) -12 = 5 + (12 -12)? Then it would be associative. Let’s assume that’s what they mean → associative law

But actually, standard interpretation: this isn't clearly either. Let me check other sources — often in such worksheets, if you see something like a + b - b = a, it's illustrating inverse property, but since only commutative/associative are given, perhaps it's a mistake. Alternatively, maybe it's showing that you can rearrange? No. I think for consistency, let’s look at pattern.

Actually, let’s do all and see.

4. 6×8 -4×8 = (6-4)×8
→ This is distributive law! But again, not listed. The worksheet only says "commutative" or "associative". So maybe this is also not fitting? But wait — perhaps they consider factoring as part of associative? No, distributive is different.

This is a problem. The worksheet seems to include distributive and identity properties too, but labels only ask for commutative or associative.

Looking at the first one: it says “commutative law” already filled in.

Perhaps for this worksheet, they want us to pick between those two, even if it’s not perfect.

Let me try to interpret based on common textbook usage.

In many elementary worksheets:

- If order changes → commutative
- If grouping changes → associative
- If factoring out → distributive (but not listed)
- If adding zero or multiplying by 1 → identity

Since only commutative and associative are options, maybe for #4, it’s not applicable? But we have to choose.

Wait — look at #18: 6x + 5x = (6+5)x → that’s distributive.

Similarly, #4 is distributive.

But the worksheet might expect “distributive” but it’s not an option. The blank says “______ law”, and above it says “commutative law” for #1.

Perhaps the student is supposed to write “distributive” for some? But the instruction doesn’t say.

Looking back at user input: the image has blanks, and for #1 it’s filled as “commutative”.

For #2, it’s blank.

I think there might be a misunderstanding. Perhaps the worksheet includes more laws, but the user only showed part.

To resolve this, I’ll assume that for this exercise, we are to identify whether it’s commutative or associative, and if it’s neither, we might need to say so — but since it’s a fill-in, probably they expect one of the two.

Let’s analyze each carefully:

1. 2×3=3×2 → commutative (multiplication) → correct as given.

2. (-8-9)+8 = -8 + (-9+8) → left side: (-17)+8 = -9; right side: -8 + (-1) = -9. The grouping changed: originally ((-8)+(-9)) +8, now -8 + ((-9)+8). So associative law of addition.

3. 5 +12 -12 =5 → this is 5 + (12-12) =5+0=5. It’s using additive inverse and identity. Not commutative or associative directly. But if we consider the expression as 5 + 12 + (-12), then grouping: (5+12)+(-12) vs 5+(12+(-12)), which is associative. So perhaps associative.

4. 6×8 -4×8 = (6-4)×8 → this is distributive property: a*c - b*c = (a-b)*c. Not commutative or associative. But since only those two are options, maybe it's a mistake. However, in some contexts, people might mislabel. I think for accuracy, it should be distributive, but since not available, perhaps leave blank? But we must fill.

Looking at #18: 6x+5x=(6+5)x — same thing, distributive.

#19: 10x+9=9x+10 — this is not true unless x=1, so probably typo? 10x+9 vs 9x+10 — not equal generally. Should be 10x+9x or something. As written, it's incorrect. But if it were 10x + 9y = 9y + 10x, then commutative. Here it's 10x+9=9x+10, which implies 10x -9x =10-9, so x=1. Not a law. Probably typo, should be 10x + 9x = 9x + 10x or something. Assume it's meant to be commutative for addition: 10x + 9 = 9 + 10x? But it's written as 9x+10. Likely error. For sake of exercise, if it were 10x + 9 = 9 + 10x, then commutative. But as is, not valid. Perhaps it's 10* x + 9 = 9 + 10*x, but written poorly. I'll assume it's commutative for addition.

This is messy. To help the student, I'll provide the most reasonable answers based on standard interpretation.

Let me list them with reasoning:

1. Given: commutative — correct.

2. Regrouping addition: associative law.

3. Simplifying, but involves grouping: associative law (as explained).

4. Factoring: distributive law — but since not option, and worksheet might expect "distributive", but blank is for commutative/associative. Perhaps for this context, we skip or put distributive. But the user's image shows only those two mentioned. Looking at the top, it says "commutative law" for #1, implying that's the type of answer expected.

Perhaps the worksheet has a section where students learn multiple laws, but in this snippet, only commutative and associative are referenced.

Another idea: in some curricula, the distributive property is introduced later, and for now, they focus on commutative and associative.

For #4, it's clearly distributive, so maybe it's not to be answered with commutative/associative. But the blank is there.

Let's count the problems: 20 problems.

I recall that in some worksheets, they have:

- Commutative: order change
- Associative: grouping change
- Distributive: a(b+c)=ab+ac
- Identity: a+0=a, a*1=a
- Inverse: a+(-a)=0

So for this, I think the student is expected to know which law is illustrated, and write the name.

Since the first one is "commutative", and others are blank, likely they want the specific law name.

But the user's instruction is to solve the problem, so I'll provide the correct law for each.

For the response, since the user said "solve the problem accurately", I'll give the correct mathematical law for each.

And in the final answer, list them.

So let's do that.

1. 2×3=3×2 — commutative property of multiplication

2. (-8-9)+8 = -8 + (-9+8) — associative property of addition (since we're regrouping the addends)

3. 5 +12 -12 =5 — this is using the additive inverse (12 and -12 cancel) and additive identity (5+0=5). But if forced to choose from commutative/associative, it's not direct. However, the expression can be seen as 5 + (12 + (-12)) =5, and if originally it was (5+12)+(-12), then associative. So associative property of addition.

4. 6×8 -4×8 = (6-4)×8 — distributive property of multiplication over subtraction

5. 5x11 +12x11 +17x11 = ? — this is not an equality, it's an expression. Probably meant to be factored: 11*(5+12+17), so distributive property. But as written, it's incomplete. Assuming it's illustrating distributive law.

The problem is written as "5 x 11 + 12 x 11 + 17 x 11" and then blank. Likely, it's to recognize that this can be written as 11*(5+12+17), so distributive law.

6. (-3)×9 +7×9 = 9×(-3) +7×9 — here, (-3)×9 became 9×(-3), which is commutative property of multiplication.

7. (5 x 11) + 3 x (5 x 11) — again, expression, not equation. Probably to factor or something. If it's (5x11) + 3*(5x11) = (1+3)*(5x11), distributive. But as is, not clear. Perhaps it's showing that you can factor out (5x11), so distributive.

8. 10x12 -9x12 = (10-9)x12 — distributive property.

9. 4x2=2x4 — commutative property of multiplication.

10. 10x7 +9x7 = (10+9)x7 — distributive property.

11. (11x4)x8 = 11x(4x8) — associative property of multiplication.

12. 6÷3=3÷6 — this is false! 6/3=2, 3/6=0.5, not equal. Division is not commutative. So this is incorrect statement. Probably a trick question or typo. Should be for addition or multiplication. As is, no law applies since it's false. But if it were 6/3 and 3/6, it's not equal, so not commutative. Perhaps it's meant to be 6*3=3*6, but written as division. I think it's a mistake. For accuracy, we can say "not a valid law" or "incorrect". But for worksheet, perhaps they want "commutative" but it's wrong. I'll note it's incorrect.

13. 5x2 +3x2 =8x2 — this is 10 +6 =16, and 8*2=16, so true, and it's because (5+3)*2=8*2, so distributive property.

14. 9+(4-10)=(9+4)-10 — left: 9+(-6)=3, right: 13-10=3, equal. Grouping changed: 9 + (4 + (-10)) vs (9+4) + (-10), so associative property of addition.

15. 18x6 +7x6 =11x6 — 108 +42 =150, 11*6=66, not equal! 18+7=25, 25*6=150, not 11*6. So probably typo, should be 25x6 or something. As written, false. If it were 18x6 -7x6 =11x6, then 108-42=66, 11*6=66, yes. So likely minus sign missing. Assume it's 18x6 -7x6 =11x6, then distributive property.

16. (12-10)x18 =12x18 -10x18 — distributive property of multiplication over subtraction.

17. 11x(4x3)=(11x4)x3 — associative property of multiplication.

18. 6x +5x = (6+5)x — distributive property.

19. 10x+9=9x+10 — as discussed, not generally true. If x=1, true, but not a law. Probably meant to be 10x + 9y = 9y + 10x or 10+9x=9x+10, which would be commutative for addition. Assume it's commutative property of addition: 10x + 9 = 9 + 10x, but written as 9x+10, which is different. Likely typo, should be 10x + 9 = 9 + 10x. So commutative.

20. 7-3=3-7 — 4 vs -4, not equal. Subtraction is not commutative. So incorrect statement. No law applies.

This is complicated. For the sake of providing a helpful answer to a student, I'll assume that the worksheet intends for us to identify the property being demonstrated, and for cases where it's distributive or identity, we still write the correct name, even if not commutative/associative.

But the user's image shows for #1 "commutative law", so perhaps the expected answers are among commutative, associative, distributive, etc.

Since the user didn't specify, I'll provide the accurate mathematical property for each.

In the final answer, I'll list the law for each number.

To make it simple for the student, I'll use:

- Commutative
- Associative
- Distributive
- And for incorrect ones, note it.

But for #12 and #20, they are false statements, so no law.

For #3, #5, #7, #15, #19, there are issues.

Let's create a clean list with best guess:

1. Commutative (given)

2. Associative

3. Associative (grouping)

4. Distributive

5. Distributive (factoring out 11)

6. Commutative (switched order of multiplication)

7. Distributive (can factor out 5x11)

8. Distributive

9. Commutative

10. Distributive

11. Associative

12. Incorrect (division not commutative)

13. Distributive

14. Associative

15. Distributive (assuming 18x6 -7x6 =11x6)

16. Distributive

17. Associative

18. Distributive

19. Commutative (assuming typo, meant 10x+9=9+10x)

20. Incorrect (subtraction not commutative)

For the response, since the student may not have learned distributive yet, but the problems include it, I think it's fine.

Now for the final answer, I'll list the law for each from 1 to 20.

But to match the format, perhaps write:

1. commutative

2. associative

3. associative

4. distributive

5. distributive

6. commutative

7. distributive

8. distributive

9. commutative

10. distributive

11. associative

12. not a law (or incorrect)

13. distributive

14. associative

15. distributive

16. distributive

17. associative

18. distributive

19. commutative

20. not a law

But for #12 and #20, since they are false, perhaps "none" or "invalid".

In many worksheets, they might expect "commutative" for #12 even though it's wrong, but that would be misleading.

I think for accuracy, I'll state the correct property.

So for the Final Answer, I'll provide the list.

Also, for #3, some might argue it's identity, but I think associative is acceptable.

Let's finalize.

After careful analysis, here are the laws for each problem:

1. commutative
2. associative
3. associative
4. distributive
5. distributive
6. commutative
7. distributive
8. distributive
9. commutative
10. distributive
11. associative
12. (incorrect statement - division is not commutative)
13. distributive
14. associative
15. distributive (assuming intended as 18×6 - 7×6 = 11×6)
16. distributive
17. associative
18. distributive
19. commutative (assuming intended as 10x + 9 = 9 + 10x)
20. (incorrect statement - subtraction is not commutative)

For the student, I'll simplify and just give the law names, noting errors briefly.

Since the user wants the final answer after reasoning, I'll box the list.

But to make it neat, I'll write:

Final Answer:
1. commutative
2. associative
3. associative
4. distributive
5. distributive
6. commutative
7. distributive
8. distributive
9. commutative
10. distributive
11. associative
12. invalid (not a law)
13. distributive
14. associative
15. distributive
16. distributive
17. associative
18. distributive
19. commutative
20. invalid (not a law)

For #12 and #20, "invalid" means the equation is false, so no property applies.

I think this is accurate.