# Algebra Charts – Inspired by Gattengo

This Starting Point was inspired by Caleb Gattengo’s famous tens chart.

Alf Coles has written an excellent article on how the conventional tens chart might be used in the classroom.  https://nrich.maths.org/10314

I particularly like the points listed about creativity.
• Students asking their own questions
• Students following their own lines of enquiry
• Students noticing patterns
• Students making predictions and conjectures
• Students choosing and sharing methods of representation
Something I’ve been thinking a lot about over the past couple of years it the relationship between number and formal algebra.  More precisely I’ve been thinking about how algebra in early secondary involves teaching all of the number operations again, but in base x instead of base 10.  I’ve noticed in many cases that what is good pedagogy in the teaching of number, may also be good pedagogy in the teaching of algebra.

The three algebra variants I have included are not designed to be used as ‘worksheets’ in the conventional sense.  The grids are the stimulus for some mathematical activity.  The creativity comes in the questions we pose pupils.

We might also show pupils just the first couple of rows or columns and ask them complete the remainder of the grid.

I have included 3 different algebra variants on the chart, to serve different potential  purposes.

1.  Basic multiplication and division of algebraic terms, with integer coefficients.

2.  More multiplication and division of algebraic terms, but this time moving into negative powers. (1/x etc). I had considered using the explicit x^-1 but I felt the representation chosen would connect more obviously with fraction teaching and previous place value work.

3.  The final chart is more complex as I have included both fractional powers and fractional coefficients.  These are the two things I want pupils to consider here.  I am hoping that they attend to how these fractions mean different things.

There are many other ways this could be expanded upon.  For instance, I was contemplating writing a version without fractional powers – everything written in root form.

So for this resource, it is down to the teacher to plan their questions and to be reactive in the moment.  I suggest this is used as the focus of a whole class discussion and, perhaps, in conjunction with pupils working and pairs and feeding back via a mini whiteboard.

It would also be nice having pupils create questions for each other and challenge each other.

What other directions can you extend the grids?

What about using power increments of 1/3 instead of 1/2?  Etc etc…

Credit: @chrismcgrane84