Two-column proofs

Concept summary and connections

To do a high-quality proof or other complex problem, it's important to put some comments in your math-code, so to speak. The way we're going to do that is with a two-column proof format.

Writing a proof

  1. Start by reading the problem and identifying things that will probably need to be variables. You'll definitely want variables for the desired unknown, and you'll want more variables for anything else that's either unknown or changing as part of the problem.
  2. Write your variable names down along with a specific explanation of what they represent (e.g. x: The number of jelly donuts sold)
  3. In the first rows of the table, write down any equations you can glean from the statement of the problem and label them given. These would be things like x+y=90 for the jelly donuts problem below.
  4. Now you have to think about how to make use of all of the data in the problem, and for a proof in particular you need to look for extra relationships you might be able to exploit. These will typically be in the form of equations the connect parts of the problem together. Write these into the table along with their justifications.
  5. Start doing the math. If you end up needing other results, just add them to the table right where you're at and make a little notation to the side.
  6. Every step that invokes a non-trivial concept needs to have a comment to the right explaining why you're allowed to do it.

Worked examples

A simple example

This is a prealgebra-level problem that demonstrates the concept using properties of basic algebra.

At the last fundraising event, you sold 90 donuts. Jelly donuts were $7, and plain were $4. If you made $399, how many of each type did you sell?

Let x = the number of jelly donuts soldLet y = the number of plain donuts soldx+y=90given7x=money from jelly donutsgiven4y=money from plain donutsgiven7x+4y=399given, total is sum of parts from each kindy=90−xsubtraction Property of Equality, from (2)7x+4(90−x)=399substitution property of equality7x+360−4x=399distributive property3x=39subtraction property of equalityx=13 jelly donutsdivision property of equality13+y=90substitution property of equalityy=73 plain donutssubtraction property of equality

Media resources

Guided practice

Homework