When rewriting our scheme of work for years 7 to 11 I was conscious that there were some perennial problems I wanted to try and solve.  One such problem was that of algebraic misconceptions that arise year-in, year-out.  I decided that I would try to address these with what I can best describe as prognostic practice, which is practice that directly prepares the students for the algebraic work earlier on in the year.  Here are some of the problems, and the associated practice we will do:

1.  The problem of indices, specifically when students write that

$(2x)^5 = 2x^5$

When we do Unit N4 (Powers, Roots and Primes) students will do an exercise that involves things like this:

$(2 \times 6)^5 = 2^5 \times 6^5$

2. The problems of expanding brackets, specifically when students write that

$3(x + 2) = 3x + 2$

$\frac{x + 10}{5} = x + 2$

$(x + 2)^2 = x^2 + 4$

When we do Unit N3 (Multiplication and Division) students will do exercises that involve things like this:

$3 \times 24 = 3 \times (20 + 4) = 3 \times 20 + 3 \times 4$

$\frac{168}{4} = \frac{160}{4} + \frac{8}{4}$

$(38)^2$ written as a grid multiplication.

3.  The problem of writing fractions, specifically when students do not understand that

$\frac{x}{5} \times 3 = \frac{3x}{5} = \frac{3}{5} \times {x}$

$\frac{x}{5} \div 3 = \frac{x}{5 \times 3} = \frac{x}{3} \div {5}$

When we do Unit N7 (Fractions) students will do exercises that involve things like this:

Rewrite the following calculations in as many ways as possible:

$\frac{2}{5} \times 3$

$\frac{2}{5} \div 3$

Of course, all of these techniques will be clearly modelled by the teacher first. I don’t expect this to completely solve the problem, but in giving students a numerical counterpart to the algebra, explicitly practised at the appropriate time, it is my hope that mistakes and misunderstandings are reduced.

This is post 3 in a series.  The other posts are:

Adventures in Mastery 1: Starting Our Journey

Adventures in Mastery 2: Writing a Scheme of Work