algebra practice makes perfect

a ppt is here

algebra snakes and branches

a ppt is here

using a snake or a branching
emphasises the steps or relationships between things

these were used in the Shell centre project on decimals

algebra problems

rooftops

a ppt is here
teacher's guide here

then there are always 3D rooftops to explore...

front to back

you could start these tasks by exploring what happens to a 2-digit number when the front digit is put to the back and the two numbers are (a) added and (b) subtracted

and what happens with a 2-digit number if you square them first and then subtract?

establish that the result will always divide by 9 and by 11

puzzles that you could use algebra to solve

a variety of old puzzles, with thanks to David Wells and other puzzle collectors
a ppt is here

why x?

one story:
Rene Descartes (1596 to 1650) popularised the idea of using letters from the back end of the alphabet for variables 
and letters from the front end of the alphabet for known constants (or parameters)
that much is pretty definite

being French and writing in french, a typesetter would normally use many more y's and z's than x's, so they preferred to use the less frequently used letter: x      
or so it is said


another story:

is found as a TED talk by Terry Moore, who maintains that difficulties in translating from the Arabic al-shalan for 'the unknown thing' 




( "Shei" first letter of al-shalan)



led to using the Greek, Kai symbol

which later was adapted to the Latin, x
  











others try to link the use of this particular letter with the Greek word 'Xenos'
an imprecise word, indicating a stranger or alien or unknown

anyway, 
it is time to change this historical quirk to use 'n' as a variable ('n' standing for a'n'y number) if we want two commonly used variables we can use 'm' and 'n' and if we want three we can use k, m and n

this is a firm proposal
time to move on from x

algebra = a study of variables

the notion of a variable has several facets

students might find it helpful to consider the following products of two 'things'

using letters (as symbols) and the equals sign in ways that might seem fairly different:

• in an equation: 7 = 5n
• stating a relationship (general form of a direct proportion): y = mx
• as a formula: A = b.h or d = s.t
• in a (general rule) identity: 1 - n^2 = (1 – n)(1 + n)          (with a big equals, an equivalence)
• stating a general arithmetical (and algebraic) property/rule: 1 = n.(1/n) or k^8 = k^3 x k^5

swap times with sum

find some values for a, b and c so that

a x (b + c) = a + b x c

what is quite interesting about this problem is that there are several families of numbers (generalities) that fit the rule