Direct logic
Can you work through these direct proofs, using our interactive proof sorters?
Age
16 to 18
Challenge level
Problem
To prove a theorem directly we start with something known to be true and then proceed, making small logical steps which are clearly correct, until we arrive at the desired result. So, because the starting point was true and each small step clearly correct, we know the result to be true.
Breaking down a mathematical argument into small steps requires patience and clear thinking.
In the following interactivities we have written out three proofs, broken them into small steps and then shuffled up the steps. Can you rearrange them into the correct logical order?
Proof of the formula for the roots of a quadratic equation
Proof of the formula for the sum of an arithmetic progression
Proof of the formula for the sum of a geometric progression
Teachers' Resources
Why do this problem?
This set of interactive problems will allow students to develop their understanding of clear mathematical proof. The interactivities provide a helpful scaffold to students just starting out with their understanding of proof. Students might be used to trying to do several algebraic steps in their heads at once. In these proof sorters, the logic is broken down into individual steps. This atomistic approach will help to train the minds of all students, even those who might already understand well the mathematical ideas involved in the interactitvities.Possible approach
This problem would work well in small groups or individually.
The proof sorters could be used when studying series or as a
refresher at a later point in the syllabus.
Each interactivity could also usefully be projected onto the
board at the start of a lesson. As students enter the room they
could try to work out which cards would come first in the
proof.
Key questions
Is there an obvious first line of the proof in each
case?
Which line follows immediately from the previous
line?
Possible extension
Can students create the proofs on paper directly without the
assistance of the indicator to the left of the interactivity? Can
they then recreate these proofs?
Perhaps students could create their own proof sorters?
Possible support
Encourage a trial and error to work out the order of some of
the trickier cards (the indicator on the left of each proof sorter
will go higher as a proof card is moved into the correct position).
Once the cards are in place, can students understand the flow of
logic?