Question

to find the area of the composite figure, two students divided the shape in different ways. use the tiles to show the expression that could be used to find the area for each partial shape. images of composite figures and expression tiles: \\(\frac{3+5}{2} cdot 4\\), \\(3 cdot 6\\), \\(2 cdot 3\\), \\(\frac{1}{2} (2 cdot 4)\\)

Explanation:

Step1: Analyze the first division (left figure)

The left composite figure can be divided into a trapezoid and a rectangle? Wait, no, looking at the tiles, maybe a triangle - like part and a rectangle. Wait, the expression \(\frac{3 + 5}{2}\cdot4\) is the area of a trapezoid (since area of trapezoid is \(\frac{(a + b)}{2}\cdot h\)), and \(3\cdot6\) or \(2\cdot3\)? Wait, let's check the second division (right figure). The right figure might be divided into a triangle and a larger rectangle? Wait, the expression \(\frac{1}{2}(2\cdot4)\) is the area of a triangle (area of triangle is \(\frac{1}{2}\cdot base\cdot height\)).

Wait, the problem is to match the expressions to the partial shapes. Let's take the first student's division (left): the lower part is a rectangle, maybe with dimensions 3 and 6? No, wait the tiles: let's assume each tile is 1x1. So for the trapezoid - like part in the left, the two parallel sides are 3 and 5, height 4, so area \(\frac{3 + 5}{2}\cdot4\). The rectangular part? Wait, maybe the other part is a rectangle with \(2\cdot3\)? Wait, no, let's think again.

Wait, the key is to identify which expression corresponds to which partial shape. Let's take the expression \(\frac{3 + 5}{2}\cdot4\): this is a trapezoid with bases 3 and 5, height 4. The expression \(\frac{1}{2}(2\cdot4)\) is a triangle with base 2 and height 4. The \(3\cdot6\) is a rectangle with length 3 and width 6? No, maybe 3 and 6? Wait, maybe the first division (left) has a trapezoid (\(\frac{3 + 5}{2}\cdot4\)) and a rectangle (\(2\cdot3\))? Wait, no, let's check the second division (right). The right figure has a triangle (\(\frac{1}{2}(2\cdot4)\)) and a larger rectangle (\(3\cdot6\))? Wait, maybe:

For the left figure (first student's division):

• The upper slanted part: trapezoid with bases 3 and 5, height 4: area \(\frac{3 + 5}{2}\cdot4\)
• The lower rectangular part: maybe \(2\cdot3\) (since 2 and 3 tiles)

For the right figure (second student's division):

• The upper triangular part: triangle with base 2 and height 4: area \(\frac{1}{2}(2\cdot4)\)
• The lower rectangular part: \(3\cdot6\) (3 and 6 tiles)

So we need to match each partial shape to the expression. But the problem says "Use the tiles to show the expression that could be used to find the area for each partial shape." So let's confirm each expression:

• \(\frac{3 + 5}{2}\cdot4\): Trapezoid area formula \(\frac{(a + b)}{2}h\), so this is for a trapezoid with \(a = 3\), \(b = 5\), \(h = 4\)
• \(3\cdot6\): Rectangle area \(length\times width = 3\times6\)
• \(2\cdot3\): Rectangle area \(2\times3\)
• \(\frac{1}{2}(2\cdot4)\): Triangle area \(\frac{1}{2}\times base\times height=\frac{1}{2}\times2\times4\)

So depending on how the students divided the shape, each partial shape's area is calculated by one of these expressions. For example, the triangular part in the right figure would use \(\frac{1}{2}(2\cdot4)\), the trapezoidal part in the left would use \(\frac{3 + 5}{2}\cdot4\), the smaller rectangle (dotted) might use \(2\cdot3\), and the larger rectangle might use \(3\cdot6\).

But since the problem is to show the expression for each partial shape, we can identify that:

• Trapezoid: \(\frac{3 + 5}{2}\cdot4\) (because trapezoid area is \(\frac{(base1 + base2)}{2}\times height\))
• Triangle: \(\frac{1}{2}(2\cdot4)\) (because triangle area is \(\frac{1}{2}\times base\times height\))
• Small rectangle: \(2\cdot3\) (area of rectangle is length×width)
• Large rectangle: \(3\cdot6\) (area of rectangle is length×width)

Answer:

To determine the expression for each partial shape:

• Trapezoidal partial shape: Use \(\boldsymbol{\frac{3 + 5}{2}\cdot4}\) (area of trapezoid: \(\frac{(a + b)}{2}\cdot h\), where \(a = 3\), \(b = 5\), \(h = 4\)).
• Triangular partial shape: Use \(\boldsymbol{\frac{1}{2}(2\cdot4)}\) (area of triangle: \(\frac{1}{2}\cdot base\cdot height\), where \(base = 2\), \(height = 4\)).
• Small rectangular partial shape: Use \(\boldsymbol{2\cdot3}\) (area of rectangle: \(length\times width\), where \(length = 2\), \(width = 3\)).
• Large rectangular partial shape: Use \(\boldsymbol{3\cdot6}\) (area of rectangle: \(length\times width\), where \(length = 3\), \(width = 6\)).

(Note: Match each expression to the corresponding partial shape based on the division method shown in the figures.)