An algebra pyramid
The algebraic expressions in the cells of the pyramid are found by multiplying the expressions of two adjacent cells directly below that cell, as shown in the left pyramid:
The right pyramid asks for strategies that are not at all clear from the beginning.
Potential for PD
Practicing algebraic skills is mainly seen as working
through a set of reproduction exercises based
on worked out examples in the textbook and/or
examples we give. These often go from simple to
more complex. A quite simple variation on these
standard exercises can change a classroom setting
from doing standard exercises with prescribed,
correct answers into a dynamic environment in
which investigations, discussions, problem solving
activities and students’ creativity come up very
naturally. These problems are productive rather
than non-productive exercises.
A collection of these kinds of productive algebra tasks we can use for practicing many aspects of algebraic skills can be found in a unit designed by Martin Kindt (Freudenthal Institute, 2004): Positive Algebra: a collection of productive exercises.
The activity as presented above was used in a classroom after a session on Tackling unstructured problems, the Professional Development Module 2 on the PRIMAS website. The second task in the activity is an example that illustrates how students’ own productions can help to make clear what level of conceptual and procedural knowledge students have already obtained and build on what they know. Professional Development Modules 3 and 6 aim at stimulating teachers’ ability to design and try out similar tasks in their classrooms. (See Professional Development Modules 3: Learning concepts through IBL, and 6: Building on what students already know.)
Apart from practicing algebra skills, this kind of task requires the student skills of problem solving, choosing own strategies and communicating ideas and results. As part of the inquiry process, students need (individually or in small group settings) to explore, investigate and communicate their ways of working on the given task.
A Mathematics teacher took two textbook exercises that were simple versions of the algebra pyramid and devised an unstructured version of them. She first asked her students to analyse the following pyramid and try to discover how it is constructed:
written in the square above. They had recently
been working on multiplying variables, so this was
After five minutes and some discussion, she challenged the students by asking them to design such pyramids themselves that were to be solved by fellow students. One ‘easy’ (‘makkelijk’ below) and one more ‘complex’ pyramid that came up in the classroom are presented here:
First of all, the teacher was amazed by the students’ engagement while working with algebraic expressions for more than 30 minutes. During an intermediate classroom discussion, one pair posed the problem of the minimal amount of information that must be supplied for such a pyramid to be solved. This inquiry question was taken up by other groups. The teacher was enthusiastic, and reported that the students improved their understanding of algebraic skills more than usual. She commented that the students ‘owned’ the mathematical problems they had solved.
- Download: Worksheet: Algebra pyramid
- Download: Kindt, M. (2004).Positive algebra: a collection of productive exercises