What do you notice? What do you wonder?

1. Here is a square with side lengths of 1 inch.

   1. How many squares with side lengths of inch can fit in a square with side lengths of 1 inch?

   2. What is the area of a square with side lengths of inch? Explain or show your reasoning.

   

2. Here is a rectangle that is inches by inches. The side length of each grid square is inch.

   

   Show that a rectangle that is inches by inches has an area of square inches. You can use the drawing if you find it helpful.

Each of these multiplication expressions represents the area of a rectangle.

1. All regions shaded in light blue have the same area. Match each diagram to the expression that you think represents its area. Be prepared to explain your reasoning.

   

   A rectangle with a horizontal dotted line about three quarters of the way down its width. The top portion is shaded blue.

   

   A rectangle with a horizontal dotted line about three quarters of the way down its width and a vertical dotted line about three quarters of the way to the right of its length. The top left portion of the rectangle is shaded blue.

   

   A vertically oriented rectangle shaded blue.

   

   A rectangle with a vertical dotted line about three quarters of the way to the right of its length. The left portion of the rectangle is shaded blue.
2. Use the diagram that matches  to show that the value of  is .

Noah would like to cover a rectangular tray with rectangular tiles. The tray has a width of inches and an area of square inches.

1. Find the length of the tray in inches. Show your reasoning.
2. The tiles are inch by inch. Draw a diagram to show one way Noah could lay the tiles. Your diagram does not need to show every tile but should show known measurements.
3. How many tiles would Noah need to cover the tray completely, without gaps or overlaps? Explain or show your reasoning.