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: 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,

1. the area of a rectangle is 42 square units. one side has a length of 7 units. what is the other side length? explain how you know using pictures, equations, and words.

Explanation:

Problem 2 (Mathematics - Geometry/Algebra)

Step 1: Understand the Area of Rectangle

The area of a rectangle is given by \( A = l \times w \), where \( l \) is length and \( w \) is width. Here, we have 16 unit squares, so the area \( A = 16 \) square units. 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: \( 4 \times 4 = 16 \). So, a square (which is a special rectangle) with length 4 units and width 4 units.

Step 3: Represent with Pictures (Brief)

• For \( 1 \times 16 \): Draw a row of 16 unit squares.
• For \( 2 \times 8 \): Draw 2 rows of 8 unit squares each.
• For \( 4 \times 4 \): Draw a 4x4 grid of unit squares.

Step 1: Recall the Area Formula for a Rectangle

The area of a rectangle is given by the formula \( A=l\times w \), where \( A \) is the area, \( l \) is the length of one side, and \( w \) is the length of the other side.

Step 2: Substitute the Known Values

We know that \( A = 42 \) square units and \( l = 7 \) units. We need to find \( w \). Substituting the known values into the formula, we get \( 42=7\times w \).

Step 3: Solve for \( w \)

To solve for \( w \), we can use the inverse operation of multiplication, which is division. We divide both sides of the equation \( 42 = 7\times w \) by 7. So, \( w=\frac{42}{7}=6 \) units.

Step 4: Explain with Pictures and Words

• Words: The area of a rectangle is the product of its length and width. If we know the area and one side length, we can find the other side length by dividing the area by the known side length. Since the area is 42 and one side is 7, dividing 42 by 7 gives us the length of the other side.
• Equation: \( w=\frac{A}{l}=\frac{42}{7} = 6 \)
• Picture (Brief): Draw a rectangle. Label one side as 7 units. The total area of the rectangle is made up of 42 unit squares. If we divide the 42 unit squares into groups of 7 (since one side is 7 units), we will have 6 groups, which represents the length of the other side.

Answer:

(for Problem 2):
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 (area \(1\times16 = 16\) square units). The second has length 8 units and width 2 units (area \(2\times8=16\) square units). The third is a square (a special rectangle) with length and width 4 units (area \(4\times4 = 16\) square units). We can represent them by drawing the corresponding arrangements of unit squares.

Problem 3 (Mathematics - Algebra/Geometry)