Areas of Parallelograms

Why do this problem?

The problem looks as though it is about vectors and area formulae. It diverts via alternative methods for finding area, into observing, then systematically hunting down a numerical relationship, proving this algebraically and geometrically, and testing where the proofs are valid. It also relates to determinants of matrices which students may encounter in later years.

Possible approach

You may be interested in our collection Dotty Grids - an Opportunity for Exploration, which offers a variety of starting points that can lead to geometric insights.

As students enter, they can work on the two introductory parallelograms, and any other examples with one vector along an axis.You could have a permanent prompt written in the corner of the board "do you know what the next question ought to be?"

Stop the group after most students begin to realise how thenumbers in the vectors are linked to the areas. Ask for clear statements of observations/conjectures or, if appropriate, theorems+proofs.
Probably someone in the class will establish the next thing to do - but if not: "it seems that the numbers in the vectors are linked to the area of the parallelograms, is it true for all vectors? If so what is the link? Can we prove it?
What do you think we need to do next? Does anyone have any questions?"

"Try these two examples and see what happens." Expect that some students will realise that they do have questions - how to find the height and base (or the area) in this tilted context. If this is a general concern, make the question public, and ask the group to divert and work on finding a good technique.

Allow time for pairs to generate plenty of examples, searching for the rule.

As individual pairs become convinced, ask what the next question should be. It's hard to control 4 variables, can they fix three and let one vary?Can they go straight to analysing the general situation?

As a teacher, when you overhear good questions between students, write these on the board, as prompts/ models of good practice for the others.

Bring the group together with sufficient time to work together understanding the diagram in the solution, generating the algebra, and manipulating the terms. Ideally, the teacher need only show the diagram, and then let students suggest the next question, and the next question as the group find answers. The teacher's role here is to keep everyone together, make helpful suggestions where necessary and make sure the group doesn't go off track.

Key questions

How did you get that area?

What should the next question be?

Possible support

Alternative problems with areas on an integer grid, are isosceles triangles, and tilted squares. Both of these require investigative, systematic, shape work, and relate directly into familiar mathematics.
 

Possible extension

The diagram used above doesn't seem to apply to a situation with a, b, c, or d negative. Take a numerical example with at least one negative, adapt the area diagram to summarise a general case. Will the algebra still work out? How many cases would you need to prove it geometrically? How many cases would you need to solve it algebraically?