Draft: this lesson was written with AI help and has not yet been read by a human editor.
1 What your class asks
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
Curriculum expectations, with their official codes
- E2.5 use the row and column structure of an array to measure the areas of rectangles and to show that the area of any rectangle can be found by multiplying its side lengths
2 The lesson
How many unit squares cover the rectangle?
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.
Go deeper
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.
Think about it
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?
Into the museum
- 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.
Try this
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.
3 Check yourself
Nothing here is marked or sent anywhere. It is for you.
In your own words: what does Mensuration (ilm al-misaha) have to do with this lesson?
Compare with the card
A cubit by a cubit. Mensuration, the record says, is the science of measuring surfaces and solids. Al-Khwarizmi gave it a chapter of his book on al-jabr and al-muqabala, which opens by saying that "one by one" means an area, that is, a cubit by a cubit. A square of side two is four times that area. How many such squares fit in a rectangle three units by four?
I can now