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 identify properties of triangles and build different kinds, and to use area relationships to develop the area formula for a triangle. The museum offers one chapter that names three kinds of triangle and gives the area of a triangle with a worked example. Building triangles from measurements, parallelograms and your own development of the formulas are for your class to bring. 1
Curriculum expectations, with their official codes
- E1.1 identify geometric properties of triangles, and construct different types of triangles when given side or angle measurements
- E2.5 use the area relationships among rectangles, parallelograms, and triangles to develop the formulas for the area of a parallelogram and the area of a triangle, and solve related problems
2 The lesson
How do the three kinds of triangle differ?
Draw a triangle of each kind and look at its angles. Say what you see that sets each kind apart.
Why half of the base times the height?
Cut or fold a rectangle or parallelogram to make a triangle. Explain how the two areas are related.
Go deeper
The chapter on measuring in al-Khwarizmi's book says triangles are of three kinds, right, acute and obtuse. It works one example, an acute triangle of unequal sides of 13, 14 and 15 cubits: the height is found by al-jabr and al-muqabala, giving a height of twelve and an area of eighty-four 2 3. The record on mensuration gives the chapter's rule: the area of a triangle is its height times half its base 4.
Think about it
In al-Khwarizmi's example the height is twelve, the area is eighty-four, and the sides are 13, 14 and 15. Which side do you think was the base, and why?
Into the museum
- Mensuration: the chapter's triangles and their areas.
- The Book of al-Jabr and al-Muqabala: the book that holds the chapter.
- Al-Khwarizmi: the author of the book.
Try this
Read about mensuration in al-Khwarizmi's book. Draw a triangle, measure its base and height, and use the chapter's rule to find its area.
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 triangle's area in cubits. Mensuration is the science of measuring areas and volumes, and al-Khwarizmi gave it a chapter of his Book of al-Jabr and al-Muqabala. In it the area of a triangle is its height times half its base, and triangles are of three kinds: right, acute and obtuse. For an acute triangle with sides of 13, 14 and 15 cubits he gets a height of twelve and an area of eighty-four. Can you draw all three kinds?
I can now
Notes
- 1. Ontario Ministry of Education, The Ontario Curriculum: Mathematics
- 2. al-Khwarizmi, Kitab al-Jabr wal-Muqabala, ed. Mosharafa and Ahmad (Cairo, 1939), p.78
- 3. al-Khwarizmi, Kitab al-Jabr wal-Muqabala, ed. Mosharafa and Ahmad (Cairo, 1939), p.79
- 4. al-Khwarizmi, Kitab al-Jabr wal-Muqabala, ed. Mosharafa and Ahmad (Cairo, 1939), p.71