addition, target 200

a ppt is here

plenty of practice at adding two. 2-digit numbers
students will hopefully have time to generate quite a few of their own examples, following their own inclinations
before considering analytical aspects
the questions intend

so, relations

making one

rule to points 3

two methods:

• the first involving finding one pair and then seeing how the patterns develop
• the second involving a transformation of the linear relationship until a simpler form is reached

the powerpoint is here

rules to points 2

further work on finding integer points/pairs that fit linear relationships
then, the reverse, finding a linear rule that fits a sequence of points

a powerpoint is here

these are also of the form
ax - by = 1

rules to points 1

considering linear rules/relationships (with some simple quadratic relationships later)
finding integer points that fit rules
the powerpoint is here

equal tops pyramids

these tasks follow an idea of Martin Wilson, of Harrogate, England

the purpose is for students to practice simplifying expressions, within a context
a powerpoint is here

all of the equal top block numbers can be found by trial and improvement and that is the intention, especially for the harder tasks

other techniques can be used

sheet 1 leads to a simple linear equation
sheet 2 leads to a linear relationship that generalises to c = 2n, d = 3n - 4 with top numbers: 6n - 5
sheet 3 can be solved with simultaneous equations, with one solution
sheet 4 leads to a relationship that generalises to a = 2n + 1, b = 3n - 1 with top numbers: 12n - 5
sheet 5 with 3 variables can be generalised to a = 4n - 2, b = 7n - 2, c = 14n - 6
sheet 6 has 4 variables, the smallest equal top number is a triangular number (and you can also make 81 and ...?)
sheet 7 is one that Martin used with his students
they found the smallest possible equal number by finding common terms in the four sequences (which is less than 50) you can also make 209 and ...?

simple linear relationships

there is an argument for working with (linear) relationships before equations
because the variables properly vary rather than being 'as-yet-unknown' numbers - that can be found
such work could precede straight line graphs
maybe lending further insight into the gradient (equal steps)

I'm told that in Hungary the maths curriculum starts from this more general appreciation before moving to the simpler, equation, cases (Paul Andrews' various articles with Gillian Hatch when he was at Manchester Metropolitan, Cambridge, now at Stockholm e.g. for BSRLM) 

when one of the variables is fixed you then have a linear equation

the intention of these tasks is that students find integer pairs that fit the rules, positive integers initially

they may well notice patterns that enable other pairs to be more easily found and lead this work into negative numbers

the Cuisenaire rod resources 'rod relationships' might be one way to begin such an exploration

directed number arithmogons

the word 'arithmogons' (rather than 'arithmagons') seems to stem from an article by Alistair McIntosh and Douglas Quadling in Maths Teaching number 70 (in 1975)

amongst many other things Leo Moser (1921 to 1970) studied pairs of numbers adding up to totals, including the work in the third resource: pairs of numbers always summing to a square number

the powerpoint goes through various algebraic solution steps - one good reason for studying arithmogons, as well as (in this case) practice with directed numbers

Craig Barton details the reasons he enjoys working with arithmogons and has various tasks based on their structure here

equable orthogonal hexagons

an 'equable' shape has the same value for the perimeter as the value for the area
this work can involve setting up and solving linear equations and forming linear relationships
and generalisations (that can be proved)

an orthogonal polygon just has right angles and 270 degree angles (that I like to call 'left' angles) inside it

[sometimes these are called 'rectilinear polygons'
but 'rectilinear' idetifies shapes that are 'bounded by straight lines' rather than those involving multiples of 90 angles]

find the linear rule

given 7 points that fit a (linear) rule
but 2 of them are incorrect

from an idea by David Wells

plotting them as points seems to be cheating...

however, putting them in order is eminently sensible

the powerpoint goes through a couple of examples