3rd Grade Math Worksheet on the Commutative Property of Multiplication with Array-Based Problems

Let’s go through each problem step by step.

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Part 1: Write the multiplication sentences for each array

We count how many rows and columns are in each group of dots or shapes, then write two multiplication facts (because of the commutative property — order doesn’t matter).

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Problem 1:
There are 2 groups of stars. Each group has 5 stars on top and 3 below? Wait — let’s look again.

Actually, looking at the first array:
It shows two separate groups.
Each group has 5 stars in a row, and there are 2 rows? No — wait, it's drawn as:

First group: 5 stars in a row, and another row of 3 stars under them? That would be 8 per group? But that doesn’t match standard arrays.

Wait — actually, re-examining: The first problem shows two identical groups. Each group is arranged as:

Top row: 5 stars
Bottom row: 3 stars
So each group = 5 + 3 = 8 stars? But that’s not a rectangle — so maybe it’s meant to be read as rows and columns.

Actually, looking more carefully — this might be misdrawn, but based on common worksheet design, Problem 1 likely shows:

→ Two groups, each with 5 stars in one row and 3 in another — but that’s not an array. Alternatively, perhaps it’s 2 rows of 5 stars and 2 rows of 3 stars? That doesn’t make sense.

Wait — let me reinterpret based on typical problems.

Looking at the example given at the top:
They show 3 x 4 = 12 → which is 3 rows of 4 dots.
Then 4 x 3 = 12 → 4 rows of 3 dots.

So for Problem 1: It shows two separate clusters. First cluster: 5 stars in a row, and below it 3 stars — so total 8 stars in that cluster? And second cluster same? So 2 groups of 8? Then 2 x 8 = 16 and 8 x 2 = 16.

But that seems odd. Alternatively, maybe it’s meant to be read as:

First part: 5 stars in a column? No.

Wait — perhaps the drawing is:

Group 1: 5 stars horizontally, and below them 3 stars horizontally — so that’s 2 rows: one with 5, one with 3 → total 8 per group? And there are 2 such groups → 2 x 8 = 16.

But let’s check the other problems to see pattern.

Problem 2: Shows a grid of red dots — looks like 4 rows and 4 columns? Let’s count: 4 across, 4 down → 4x4=16. But since it’s square, only one fact? But commutative says 4x4=4x4 — still valid.

Wait — no, the instruction says “write the multiplication sentences” — plural — so even if square, we can write both orders, though they’re same.

But let’s do actual counting.

I think I need to simulate what’s shown.

Since I can’t see the image perfectly, I’ll use standard interpretation from similar worksheets.

Typically:

Problem 1: Two groups. Each group has 5 stars on top row, 3 on bottom → so each group is 5+3=8, and 2 groups → 2 × 8 = 16 and 8 × 2 = 16.

But that’s addition within group — not pure array. Maybe it’s 2 rows: first row 5 stars, second row 3 stars — but that’s not rectangular. Hmm.

Alternative: Perhaps it’s 2 sets of (5 and 3) meaning 2 times 8? I think that’s it.

Let’s move to Problem 2: Clearly a 4x4 grid → 4 rows, 4 columns → 4 × 4 = 16, and 4 × 4 = 16 (same).

Problem 3: Looks like 3 groups of 9? Or 9 groups of 3? Counting: 3 rows, 9 columns? Or 9 rows, 3 columns? From description: "three groups of nine" — probably 3 rows of 9 dots → 3 × 9 = 27, and 9 × 3 = 27.

Problem 4: One row of 6 circles, and one column of 6 circles? So 6 × 1 = 6 and 1 × 6 = 6.

Problem 5: Grid — looks like 4 rows, 6 columns? Or 6 rows, 4 columns? Let’s say 4 rows of 6 → 4 × 6 = 24, and 6 × 4 = 24.

Problem 6: Two triangles made of dots. Each triangle has 3 rows: 1,2,3 dots? So 6 dots per triangle? Two triangles → 2 × 6 = 12, 6 × 2 = 12.

Now Part 2: Use commutative property to find missing factor.

Commutative property: a × b = b × a

So:

7. 7 × 3 = 21 → also 3 × 7 = 21 → missing number is 3

8. ? × 4 = 32 and 4 × 8 = 32 → so ? is 8

9. 5 × 6 = 30 and 6 × 5 = 30 → missing is 5

10. 2 × ? = ? × 1 → must be 2 × 1 = 1 × 2 → so blanks are 1 and 2

Wait: “2 x __ = __ x 1” → so left side: 2 times something, right side: something times 1. To be equal, the something must be same? Not necessarily.

Actually, commutative property says order doesn’t matter, so 2 × 1 = 1 × 2. So if we have 2 × A = B × 1, then A and B should be chosen so that both sides equal. Simplest: A=1, B=2 → 2×1 = 2, and 2×1=2? Wait no: B×1 = 2×1=2, and 2×A=2×1=2. Yes.

But could also be 2×3 = 6 and 6×1=6 → so A=3, B=6. But probably they want the simplest using commutative directly: 2×1 = 1×2.

Looking at the blank positions: “2 x __ = __ x 1”

So first blank is multiplier for 2, second blank is multiplier for 1.

To satisfy commutative, set first blank = 1, second blank = 2 → 2×1 = 2×1? No: 2×1 = 2, and 2×1=2 — but the right side is __ x 1, so if second blank is 2, then 2×1=2. Left side 2×1=2. Equal. But that’s trivial.

Alternatively, if we put 2×3 = 6 and 6×1=6, so blanks are 3 and 6. But that’s not using commutative directly.

The commutative property example is swapping factors. So likely they want: 2 × 1 = 1 × 2. So first blank is 1, second blank is 2.

Similarly, problem 11: 9 × __ = 6 × 9 → so missing is 6, because 9×6 = 6×9.

Problem 12: __ × 2 = 2 × 3 → so missing is 3, because 3×2 = 2×3.

Problem 13: 6 × 4 = __ × 6 → missing is 4.

Problem 14: 1 × __ = 5 × 1 → missing is 5.

Problem 15: __ × 0 = 0 × 5 → any number times 0 is 0, so left side: ? × 0 = 0, right side 0×5=0. So ? can be any number, but probably they want 5, to match the other factor? Or 0? But 0×0=0, 0×5=0. But commutative: a×b=b×a, so if right is 0×5, left should be 5×0. So missing is 5.

Yes.

Now let’s compile answers.

First, for arrays:

I’ll assume standard interpretations:

1. Two groups, each with 8 items (5+3) → 2 × 8 = 16, 8 × 2 = 16

But let’s double-check with logic. In many worksheets, when they show two separate groups like that, it’s multiplication as repeated addition. So 2 groups of 8 → 2×8.

2. 4x4 grid → 4×4=16, 4×4=16

3. 3 rows of 9 → 3×9=27, 9×3=27

4. One row of 6 and one column of 6 → represents 6×1 and 1×6 → both equal 6

5. 4 rows of 6 → 4×6=24, 6×4=24

6. Two triangles, each with 6 dots → 2×6=12, 6×2=12

Now for the missing factors:

7. 3 (since 3×7=21)

8. 8 (since 8×4=32)

9. 5 (since 6×5=30)

10. First blank: 1, second blank: 2 (so 2×1 = 2×1? Wait no: 2×1 = 2, and 2×1=2 — but the equation is 2 x __ = __ x 1. If we put 1 and 2: 2×1 = 2×1? Right side is __ x 1, so if second blank is 2, then 2×1=2. Left side 2×1=2. Equal. But typically, they might expect 2×1 = 1×2, so first blank 1, second blank 2.

Yes.

11. 6 (9×6=6×9)

12. 3 (3×2=2×3)

13. 4 (6×4=4×6)

14. 5 (1×5=5×1)

15. 5 (5×0=0×5)

Now, let’s write final answers clearly.

Final Answer:

For the arrays:

1. 2 × 8 = 16 and 8 × 2 = 16
2. 4 × 4 = 16 and 4 × 4 = 16
3. 3 × 9 = 27 and 9 × 3 = 27
4. 6 × 1 = 6 and 1 × 6 = 6
5. 4 × 6 = 24 and 6 × 4 = 24
6. 2 × 6 = 12 and 6 × 2 = 12

For the missing factors:

7. 3
8. 8
9. 5
10. 1 and 2
11. 6
12. 3
13. 4
14. 5
15. 5