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. the expressions are \\(\frac{1}{2}(3 \cdot 3)\\), \\(\frac{2 + 5}{2} \cdot 3\\), \\(2 \cdot 2\\), \\(2 \cdot 5\\)

Explanation:

Step1: Analyze the first student's division

The first student's shape has a rectangle, a triangle, and two squares? Wait, no, looking at the tiles: the expressions are $\frac{1}{2}(3\cdot3)$, $\frac{2 + 5}{2}\cdot3$, $2\cdot2$, $2\cdot5$. Let's match each expression to the partial shapes.

For the square - like dotted part: area of a square with side 2 is $2\cdot2$. For the rectangle part: length 5? Wait, no, the tile $2\cdot5$: maybe a rectangle with length 5 and width 2? Wait, the triangular part: area of a triangle is $\frac{1}{2}(base\cdot height)$. If base is 3 and height is 3, then $\frac{1}{2}(3\cdot3)$. The trapezoid - like part: area of trapezoid is $\frac{(a + b)}{2}\cdot h$, so $\frac{2+5}{2}\cdot3$ (where a = 2, b = 5, h = 3).

Wait, the problem is to match each expression to the partial shape. Let's assume the first student's division:

• The square (dotted) has area $2\cdot2$ (since side is 2, area of square is side×side = 2×2).
• The rectangle: $2\cdot5$ (length 5, width 2).
• The triangle: $\frac{1}{2}(3\cdot3)$ (base 3, height 3, area of triangle is $\frac{1}{2}bh$).
• The trapezoid: $\frac{2 + 5}{2}\cdot3$ (bases 2 and 5, height 3, area of trapezoid is $\frac{(a + b)}{2}h$).

So for each partial shape, we match the expression:

• Square (dotted): $2\cdot2$
• Rectangle: $2\cdot5$
• Triangle: $\frac{1}{2}(3\cdot3)$
• Trapezoid: $\frac{2 + 5}{2}\cdot3$

(But since the problem says "use the tiles to show the expression that could be used to find the area for each partial shape", we need to assign each expression to its shape. Let's confirm:

• Area of square: side = 2, so $2\times2 = 4$.
• Area of rectangle: length = 5, width = 2, so $2\times5 = 10$.
• Area of triangle: base = 3, height = 3, so $\frac{1}{2}\times3\times3=\frac{9}{2}=4.5$.
• Area of trapezoid: bases 2 and 5, height 3, so $\frac{(2 + 5)}{2}\times3=\frac{7}{2}\times3 = 10.5$.

So each expression corresponds to a partial shape. For example, the square - dotted part: $2\cdot2$; the rectangle part: $2\cdot5$; the triangle part: $\frac{1}{2}(3\cdot3)$; the trapezoid part: $\frac{2 + 5}{2}\cdot3$.

Answer:

To match each expression to the partial shape:

• Square (dotted): $2 \cdot 2$ (area of square: side×side, side = 2).
• Rectangle: $2 \cdot 5$ (area of rectangle: length×width, length = 5, width = 2).
• Triangle: $\boldsymbol{\frac{1}{2}(3 \cdot 3)}$ (area of triangle: $\frac{1}{2} \times$ base×height, base = 3, height = 3).
• Trapezoid: $\boldsymbol{\frac{2 + 5}{2} \cdot 3}$ (area of trapezoid: $\frac{(\text{base}_1 + \text{base}_2)}{2} \times$ height, bases = 2, 5; height = 3).

(If the question expects assigning each tile to its shape, the key is recognizing the formula for each shape: square ($s^2$), rectangle ($l\times w$), triangle ($\frac{1}{2}bh$), trapezoid ($\frac{(a + b)}{2}h$) and matching to the expressions.)