Question

label: equivalence
1
$4\times3=12$
$6\times2=12$
2 $2(3+2) = 2\times3+2\times2$
3 $4\times3 = 3\times4$
4 $2(4\times3) = 4(3\times2)$
commutative property 1. ______
associative property 2. ______
equivalent 3. ______
distributive property 4. ______

Explanation:

Step1: Analyze Item 1

Item 1 has \(4\times3 = 12\) and \(6\times2=12\). Both equal 12, so they are equivalent. So match "equivalent" to 1.

Step2: Analyze Item 2

Item 2: \(2(3 + 2)=2\times3+2\times2\). This is the distributive property (multiplying a sum by a number is same as multiplying each addend and adding). So match "distributive property" to 2.

Step3: Analyze Item 3

Item 3: \(4\times3 = 3\times4\). This is commutative property (order of multiplication doesn't matter). So match "commutative property" to 3.

Step4: Analyze Item 4

Item 4: \(2(4\times3)=4(3\times2)\). This involves regrouping (associative property, since multiplication is associative: \((a\times b)\times c=a\times(b\times c)\), and here we also use commutative). So match "associative property" to 4.

Answer:

1. equivalent
2. distributive property
3. commutative property
4. associative property