Question

1. isaac says that you can create three different rectangles with different side lengths using 16 unit squares. use pictures, numbers, and words to show what isaac is saying.

Explanation:

Step 1: Recall Area of Rectangle

The area of a rectangle is \( A = \text{length} \times \text{width} \). Here, the area is 16 (since we use 16 unit squares), so we need to find pairs of positive integers \((l, w)\) such that \( l \times w = 16 \).

Step 2: Find Factor Pairs of 16

• Factor pair 1: \( 1 \times 16 = 16 \). So a rectangle with length 16 units and width 1 unit.
• Factor pair 2: \( 2 \times 8 = 16 \). So a rectangle with length 8 units and width 2 units.
• Factor pair 3: \( 4 \times 4 = 16 \). Wait, but this is a square (a special case of a rectangle). Wait, maybe the problem considers non - square rectangles? Wait, no, a square is a rectangle. But let's check again. Wait, 16 can be factored as:
• \( 1\times16\): rectangle with length 16, width 1.
• \( 2\times8\): rectangle with length 8, width 2.
• \( 4\times4\): square (rectangle) with length 4, width 4.

To show with pictures:

• For \( 1\times16 \): Draw a row of 16 unit squares (length 16, width 1).
• For \( 2\times8 \): Draw 2 rows of 8 unit squares each (length 8, width 2).
• For \( 4\times4 \): Draw a 4 - by - 4 square (which is a rectangle with length 4, width 4).

In words: We know that the area of a rectangle is the product of its length and width. When we have 16 unit squares, the total area of the rectangle we form is 16 square units. We find all the possible pairs of positive integers (length, width) whose product is 16. The pairs are (1,16), (2,8), and (4,4). Each pair represents the side lengths of a different rectangle (the last one is a square, which is a type of rectangle).

Answer:

We can create three rectangles with side - length pairs \((1,16)\), \((2,8)\), and \((4,4)\) using 16 unit squares. The first rectangle has length 16 units and width 1 unit, the second has length 8 units and width 2 units, and the third (a square) has length 4 units and width 4 units.