Commutative, Associative or Distributive? | 3rd Grade Math - Free Printable

I will solve this problem by identifying whether each equation demonstrates the Commutative, Associative, or Distributive property. I'll go through each one step-by-step.

First, let me recall the definitions of these properties:

- Commutative Property: The order of numbers can be changed without affecting the result (e.g., a + b = b + a or a × b = b × a).
- Associative Property: The grouping of numbers can be changed without affecting the result (e.g., (a + b) + c = a + (b + c) or (a × b) × c = a × (b × c)).
- Distributive Property: Multiplication distributes over addition (e.g., a × (b + c) = a × b + a × c).

Now, I will examine each equation in the image and determine which property it illustrates.

A: 2 × (7+4) = (2×7) + (2×4)

This shows multiplication distributing over addition. It's the Distributive Property.

B: (5+4) + 6 = 5 + (4+6)

This shows that the grouping of numbers in addition can be changed. It's the Associative Property.

C: (6×9) + 6 = 5 × (9+6)

Let me check if this is mathematically correct first. Left side: (6×9) + 6 = 54 + 6 = 60. Right side: 5 × (9+6) = 5 × 15 = 75. Since 60 ≠ 75, this equation is incorrect. However, if we ignore the error and look at the structure, it appears to be attempting to show the Distributive Property, but incorrectly.

D: 8 + (2+4) = 8 + (4+2)

This shows that within the parentheses, the order of addition is changed. This is the Commutative Property applied to the numbers inside the parentheses.

E: (7×12) + (7×6) = 7 × (12+6)

This is the reverse of the Distributive Property, showing that a common factor (7) can be factored out. It's still the Distributive Property.

F: 10 × (3×2) = (10×3) × 2

This shows that the grouping of numbers in multiplication can be changed. It's the Associative Property.

G: (2+3) + 6 = 6 + (2+3)

This shows that the order of the addends is changed. It's the Commutative Property.

H: 14 × (2×7) = (2×7) × 14

This shows that the order of multiplication is changed. It's the Commutative Property.

I: (8×15) + (8×5) = 8 × (15+5)

This is another example of the Distributive Property, where 8 is factored out.

J: (18×1) + 10 = 4 + (18×10)

Let me check if this is mathematically correct. Left side: (18×1) + 10 = 18 + 10 = 28. Right side: 4 + (18×10) = 4 + 180 = 184. Since 28 ≠ 184, this equation is incorrect. Looking at the structure, it doesn't clearly demonstrate any of the three properties due to the error.

Based on my analysis, here are the answers for each part:

A: D (Distributive)
B: A (Associative)
C: (Incorrect equation, but attempting Distributive)
D: C (Commutative)
E: D (Distributive)
F: A (Associative)
G: C (Commutative)
H: C (Commutative)
I: D (Distributive)
J: (Incorrect equation, not applicable)

Note: Parts C and J contain mathematical errors and do not correctly illustrate any property.