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Triangle Area

Pre-test 01/21/04 and Post-test 02/02/04

In both tests, students responded to the following: Your 12-year-old nephew has come across the formula A=bh/2 for the area A of a triangle with base b and height h. He cannot make any sense of it. How do you explain the use, meaning and reasons for this formula to him? Your 17-year-old nephew wants to know why the formula is true and if it always works. What do you say to him?

Between the pre- and post-test, students were asked to read the references on the Triangle Area Page. I provided an overview of the material in the reading in (informal) lecture format on January 26. Students were asked to think about the material at home and write up answers to some specific questions (see assignment due Jan. 28 on Triangle Area Page). They discussed their responses in class, January 28. The post-test took place at the beginning of the February 2 class.

Test Results

Summary comments: pre-test | post-test

Details: In the table below, I have arranged the pre- and post-tests in rough order based on a hasty estimate of the amount of new information appearing on the post-test and how well it's organized. This ordering is not exact, and papers that occur earlier are not necessarily superior to papers that appear later. (Students viewing their work here should be aware that the statements made below are not intended to reflect grading or to communicate approval/disapproval. Students will receive course credit for participating in the pre-post test system if their work shows serious intent. All of the papers below show serious intent.)

I do not yet have a meaningful, objective and replicable way of coding responses, but in attempting to write comments on each paper, I find certain recurring themes or motifs. In the future, it may be possible to use this phenomenon--which seems to be a common feature of collections of class work--as a basis for a more rigorous system for describing/classifying responses. The number of themes/motifs and the detail and complexity with which they are treated clearly changes from pre-test to post-test. There are also variations from student to student. The linking of themes (e.g., through the coherent development of a story, through deductive arguments, or by comparisons or analogies) varies, too.

Click on the thumbnail to see a larger image. Papers that are hard to read on screen may be easier to read if printed.
Code
Name
Pre-test
Comments on pre-test
Post-test
Comments on post-test
lapim
 Right triangle & rectangle area compared. No discussion of definition of area. Area explained with grids (pictured) and linked to rectangle area formula. All cases of triangles considered and triangle area formula fully justified.
ihlew
 Right triangle & rectangle area compared. No discussion of definition of area. Area explained with grids (pictured) and linked to rectangle area formula. Area formula for right triangles considered. Diagrams show extensions to general triangles. Brief reference to parallelograms.
horse
 Base and height discussed. Right triangle & rectangle area compared. No discussion of definition of area. Area explained with grids (pictured). Base and height discussed. Brief reference to parallelograms.
dipit
 Right triangle & rectangle area compared. No discussion of definition of area. Area explained with grids (not pictured) and linked to rectangle area formula. Base and height considered carefully. Parallelograms defined and related to triangles and to rectangles via dissections. (But dissections not demonstrated.)
kevoc
 Area considered briefly. Base and height considered. Right triangle & square area compared. Area explained with grids (pictured). Triangles related to parallelograms. Area formula for parallelograms explained in simple case.
kitty
 Area explained in terms of grids. Triangle area formula "is faster," but author does not explain why it's true. Area explained with grids (pictured) and linked to rectangle area formula. Area formula for right triangles considered. Extensions to general triangles explained and diagrammed. Connections to parallelograms mentioned.
bytep
 Right (isoceles) triangles and squares related. Mentions need to understand area, but no discussion of definition of area. Area explained with grids (not pictured). Jump to parallelogram area. Areas of triangles and parallelograms linked. Jump to formula. Areas of rectangles and parallelograms linked by Euclid 35. Euclid 35 diagrammed.
acegi
 Rectangle area formula.Rght triangles related to rectangles, giving triangle formula (for right triangles). Mentions need to understand area, but no discussion of definition of area. Area explained with grids (pictured) and linked to rectangle area formula. Area formula for right triangles derived. Formulae for arbitrary triangles derived with graphical and algebraic descriptions/support.
lbugy
 Symbols A, b and h defined. Area explained with grids (pictured), and linked to rectangle area formula. Rectangles related to right triangles, giving triangle formula (for right triangles). Area explained with grids (pictured). Two congruent triangles assemble into a parallelogram. Parallelogram defined. Base and height of a parallelogram. Euclid 35 stated. Jump to area formula for parallelogram. Triangle area formula derived.
zapdy
 Rectangles related to right triangles, giving triangle formula (for right triangles). No discussion of definition of area. Area explained with grids (pictured) and linked to parallelogram area formula. Parallelograms related to triangles via dissections. More on parallelogram area and area formula.
zoink
 Base and height discussed. Right triangle & rectangle area compared. No discussion of definition of area. Area explained with grids (pictured) and carefully linked to rectangle area formula. A parallelogram is a rectangle "shifted", and has same area as rectangle. Any triangle can be combined with a copy of itself to make a parallelogram, giving formula. Base and height
basum
 Area "is a measurement of the in side.." Base and height considered. Grids illustrated. Dissection of rectangle. Area explained with grids (pictured) and linked to rectangle area formula. Area formula for right triangles derived. Two copies of a non-right triangles make a parallelogram. Base and height of triangles. Euclid 35 stated and related to the context.
alemo
 Area is like "floor of ...[a] room." Rectangle area formula.Rght triangles related to rectangles, giving triangle formula (for right triangles). "Area is the space enclosed.." A, b, and h explained carefully and illustrated. Parallelogram area formula gives triangle area by cutting on diagonal. Area explained with grids (pictured) and linked to rectangle area formula. Area formula for right triangles derived. Parallelograms dissect and reassembled to rectangles (wide base case illustrated, and recognized as special).
ennae
 Base and height diagrammed. Area is "amount of space inside". Cutting a square on the diagonal explains the formula.ight triangle & square compared. Area explained with grids (pictured) and linked to parallelogram area formula. Parallelograms related to triangles via dissections. Euclid 35 diagrammed and used to show that for any parallelogram there is a rectangle with same base, height and area. Proof of Euclid 35.
zxvtr
 "The formula is true because a triangle is half a rectangle." Area explained with grids (not pictured) and linked to rectangle area formula. Area formula for right triangles derived. Possibility of generalization to other triangles asserted.
catil
 Formula for area of square lead to triangle area formula by cutting square on diagonal. Area is like "amount of grass it would take ... to cover [a] field." Rectangle area formula leads to triangle area formula by diagonal cut.
cepod
 Rectangles related to right triangles, giving triangle formula (for right triangles). No discussion of definition of area. Area explained with grids (pictured) and linked to square and rectangle area formulae. Rectangle area formula leads to triangle area formula by diagonal cut. Euclid 35 quoted. for any parallelogram there is a rectangle with same base, height and area.
mbrir
 Area is "space enclosed." Base and height explained and pictured. Rectangle and parallelogram area formulae mentioned. Parallelograms related to triangles via dissections, explaining triangle area formula.
moron
 Area is "space inside." Base of a triangle is "longest side." A square is made of 2 triangles. Other comments. Area is "space enclosed." Any side of a triangle may be base. Triangles relate to parallelograms. Diagram from Euclid 35.
smile
 Area explained with grids (pictured) and very carefully linked to rectangle area formula. A triangle is 1/2 of a rectangle (by folding). Triangle area formula comes from parallelograms. Area is amount of space inside. Graph paper. Every triangle is half of a parallelogram. Parallelograms dissect to rectangles by various means.
cojoc
 Area explained with grids (pictured) and carefully linked to square area formula. A triangle is 1/2 of a square. Every triangle is half a "quadrilateral." Area explained with grids (pictured) and carefully linked to square area formula. A triangle is 1/2 of a square. Every triangle is half a "quadrilateral."