This is a single variable algebra puzzle in which students try to find squares of different sizes to tile a rectangle. I prefer the two variable algebra variant on the right.
Get students to create their own history puzzles.
Algebra on Rectangles
(R. L. Brooks, C. A. B. Smith, A. H. Stone and W. T. Tutte, 1930s)
Students use linear algebra to build rectangles out of squares. The results are aesthetically pleasing and difficult to reproduce by trial and error.
Lessons should start on graph paper - giving students the challenge of trying to "build a rectangle using squares of all different sizes." Students will find it extremely difficult. After several minutes hold a vote asking if it is an impossible problem.
Download puzzle-sheets here.
PS. These puzzles are based on the work of Brooks, Smith, Stone and Tutte at Cambridge in the 1930s. Although this technique failed to produce a result, it can be retroactively used to create the best imaginable way to introduce algebra.
PPS. The power of introducing algebra by way of a problem so difficult that students cannot solve it through non-algebraic means is not to be underestimated.
Standards for Mathematical Practice
MathPickle puzzle and game designs engage a wide spectrum of student abilities while targeting the following Standards for Mathematical Practice:
MP1 Toughen up!
This is problem solving where our students develop grit and resiliency in the face of nasty, thorny problems. It is the most sought after skill for our students.
MP3 Work together!
This is collaborative problem solving in which students discuss their strategies to solve a problem and identify missteps in a failed solution. MathPickle recommends pairing up students for all its puzzles.
MP6 Be precise!
This is where our students learn to communicate using precise terminology. MathPickle encourages students not only to use the precise terms of others, but to invent and rigorously define their own terms.
MP7 Be observant!
One of the things that the human brain does very well is identify pattern. We sometimes do this too well and identify patterns that don't really exist.