Question

another look!
you can use the distributive property to break an area into smaller rectangles to find more familiar facts to multiply.
you can separate a rectangle into two smaller rectangles with the same total area.
you can write the multiplication fact that represents the area of the large rectangle.
$a = l×w$
$4×5 = 20$
you can write the multiplication facts that represent the area of each of the smaller rectangles.
$4×5 = 4×(3 + 2)$
$4×5 = (4×3)+(4×2)$
$4×5 = 12 + 8 = 20$
in 1 - 4, complete the equations that represent the picture.
1.
$3×\square=\square×(3 + \square)$
$3×\square=(3×\square)+(\square×2)$
$3×\square=\square+\square = 15$
2.
$\square×7=\square×(\square+4)$
$\square×7=(\square×3)+(4×\square)$
$\square×7=\square+\square = 28$
3.
$3×\square=\square×(2 + \square)$
$3×\square=(3×\square)+(\square×4)$
$3×\square=\square+\square = 18$
4.
$\square×6=\square×(\square+3)$
$\square×6=(\square×3)+(5×\square)$
$\square×6=\square+\square = 30$

Explanation:

Step1: Analyze the first equation

For the first problem, the large rectangle has a width of \(5\) (since \(3 + 2=5\)) and a height of \(3\). Using the distributive property \(a\times(b + c)=a\times b+a\times c\). Here \(a = 3\), \(b = 3\), \(c = 2\).
\(3\times5=3\times(3 + 2)\)
\(3\times5=(3\times3)+(3\times2)\)
\(3\times5 = 9+6=15\)

Step2: Analyze the second equation

For the second problem, assume the height is \(4\) (because of the distributive property application). The large rectangle has a width of \(7\) (split into \(3+4\)). Using the formula \(a\times(b + c)=a\times b+a\times c\), where \(a = 4\), \(b = 3\), \(c = 4\)
\(4\times7=4\times(3 + 4)\)
\(4\times7=(4\times3)+(4\times4)\)
\(4\times7=12 + 16=28\)

Step3: Analyze the third equation

For the third problem, the large rectangle has a width of \(6\) (split into \(2+4\)) and height of \(3\). Using the formula \(a\times(b + c)=a\times b+a\times c\), where \(a = 3\), \(b = 2\), \(c = 4\)
\(3\times6=3\times(2 + 4)\)
\(3\times6=(3\times2)+(3\times4)\)
\(3\times6=6+12=18\)

Step4: Analyze the fourth equation

For the fourth problem, the large rectangle has a width of \(6\) (split into \(3+3\)) and height of \(5\). Using the formula \(a\times(b + c)=a\times b+a\times c\), where \(a = 5\), \(b = 3\), \(c = 3\)
\(5\times6=5\times(3 + 3)\)
\(5\times6=(5\times3)+(5\times3)\)
\(5\times6=15+15=30\)

Answer:

1. \(3\times5 = 3\times(3 + 2)\), \(3\times5=(3\times3)+(3\times2)\), \(3\times5=9 + 6=15\)
2. \(4\times7=4\times(3 + 4)\), \(4\times7=(4\times3)+(4\times4)\), \(4\times7=12+16 = 28\)
3. \(3\times6=3\times(2 + 4)\), \(3\times6=(3\times2)+(3\times4)\), \(3\times6=6 + 12=18\)
4. \(5\times6=5\times(3 + 3)\), \(5\times6=(5\times3)+(5\times3)\), \(5\times6=15+15=30\)