lengthy equations

forming equations

length comic strips

what are the two original lengths, 'a' and 'b' ?

1.

2.

3.

4.

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

squares in squares

find relationships between the (integer) lengths of the squares

what is the smallest (integer) length of side of the big square?

puzzles that you could use algebra to solve

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

simultaneous stairs

[ I was reminded of these neat results following an ATM/MA session in Birmingham recently ]

hollow square

in Napoleonic battles a hollow square was a popular formation for an infantry battalion e.g. Wellington's army at Waterloo, to cope with cavalry charges

not Wellington's army...

a recreation of Wellington's army hollow square formations












for a battalion of 960 people, how many possible hollow square arrangements are there?
state the widths for each

if you want to start off with an easier number of people, find the 3 options for each of:

• 48
• 45
• 80 

and the 4 options for 96

making squares

two integer numbers, 'a' and 'b' sum to a square number
and
double the first, add the second makes another square number

what (integer) values could 'a' and 'b' have?

for example, a = 9, b = 7 works

try to find several values

and maybe general rules
e.g. for when the two square numbers are consecutive

where lines meet

geometry steering algebra

Hans Freudenthal advocates utilising the compelling image of a straight line when teaching directed numbers (e.g. in chapter 15 of 'Didactical Phenomenology of Mathematical Structures')

in Freudenthal's words:
'the justification of the numerical operations and their laws by the simplicity of the algebraic description of geometrical figures and relations'
a rule e.g. y = 5  -  x generates some points by inputting some positive x values
if these points are joined and this part of the line is then extended into the negative quadrants
some (directed number) calculation results emerge

this work is better managed with talk but the resource intends to nudge in the right direction

mega quadratic equations

these questions are very similar to one in the NCTM (US) 1988 'Ideas of Algebra' yearbook posed by Terry Goodman and Martin P Cohen

with or without algebra?

there is an interesting article by Prof. Andrew Hacker in the New York Times (July 28th 2012) questioning the value of algebra teaching

this seems to be a Romans v Greeks debate

the Romans, apparently, interested themselves (initially at least) in the practical and everyday uses of the subject: utilitarianism


the Greeks, seemingly, viewed the subject more aesthetically and as a subject of considerable beauty

other numbers of nets

it is fairly easy to prove that there are only 2 different nets for a tetrahedron, by considering where edges can be broken

e.g. using Polydron

the octahedron has 11 different nets, the same as the cube:

and a square based pyramid has 6:

loci

lost in motion
on Vimeo

1 off a cube

equivalent things

the intention of this first resource is that students plug in the given input values
and then, maybe with some nudging, appreciate that there are shorter rules that give the output numbers from the input numbers

and that these simpler expressions can be obtained by cancelling the algebraic fraction

these resources involve justifying then creating equivalent (same only look different) expressions to the one in the middle

there is at least one solution to these: