Measuring a surface in squares

This lesson asks you to use the rows and columns of an array to measure the area of rectangles, and to show that the area of any rectangle is found by multiplying its side lengths. The museum offers one record on how an early book began its chapter on measuring area; arrays and rectangles are your class's to build. ¹

Fill a rectangle with unit squares in rows and columns. Count the rows and the columns, then show that multiplying them gives the same number as counting every square.

In the chapter on measuring in al-Khwarizmi's book, squares are told against the square of a cubit by a cubit. The record on mensuration says that an equal-sided square of side two is four times the cubit-by-cubit area, and a square of side half a cubit is a quarter of it ². So the squares are measured against one another, by how many times each one holds.

Al-Khwarizmi says a square of side two is four times the square of side one. Draw both on squared paper: why does the larger square need four small squares and not two?

• Mensuration: the rules of the chapter as the record gives them.
• The Book of al-Jabr and al-Muqabala: the book that holds the chapter.

Read the record Mensuration. Draw a square of side two and a rectangle of three by four on squared paper, and count the unit squares in each.