these are the two powerpoints from a London ATM/MA session that I led on Sat 16th March 2019

first half

second half

thanks to all those who braved a squally day and gave up a Saturday morning to attend - this was very much appreciated

particular thanks to:

the person who travelled up from Gloucester and helped me distribute sheets before the session

the person who kindly bought me a coffee in the break

the various insights (opposite angles of a kite by subtraction rather than addition, a larger arrowhead for question 17)

particularly good to meet:

the 80 year old who had been taught Euclidean geometry

the three musketeers from Essex

two students that I had taught who turned up: Kieron and Alice, great to meet you again

it is admirable that these ATM/MA local sessions happen

thanks to Mark Horley and Peter Wright for organising the event and for helping me to set up

## Sunday, 17 March 2019

## Saturday, 23 February 2019

### given the lcm and hcf

reversing the question

given the lcm and hcf of a pair of numbers

what could they be?

the powerpoint is here

from an idea by David Wells (question 1)

given the lcm and hcf of a pair of numbers

what could they be?

the powerpoint is here

from an idea by David Wells (question 1)

### lcm and hcf generalising

deducing the hcf and lcm of two numbers from their prime factorisations

the powerpoint is here

type 1

type 2

the powerpoint is here

type 1

type 2

## Wednesday, 20 February 2019

### directed number addition and subtraction 2 (of 2)

the powerpoint is here

with subtraction, viewed as a 'gap', you can always shift it along to a more convenient location on a number line

add (or subtract) the same amount to both keeps the sum the same

looking at sequences/patterns to decide what the results should be

with subtraction, viewed as a 'gap', you can always shift it along to a more convenient location on a number line

add (or subtract) the same amount to both keeps the sum the same

looking at sequences/patterns to decide what the results should be

### directed number addition and subtraction 1 (of 2)

the powerpoint is here

uses directed numbers as vectors

involving the start and end (like stations) and the journey

these journeys can be described in two different (but the same) ways

Simon Razvi, in Birmingham (England) suggested that the positions on a number line are like nouns (e.g. 'negative 4')

and the journeys are like verbs (e.g. 'subtract 4')

usually these get muddled up

and usually I think there's value in muddling them up

[technically I guess you can't add a position on a number line to anything!]

it seems to me that it's (one dimensional) vectors that can be added and subtracted

thanks to Worcester Uni (PGCE) students and Jane Moreton for their comments (in Jan 2018)

uses directed numbers as vectors

involving the start and end (like stations) and the journey

these journeys can be described in two different (but the same) ways

Simon Razvi, in Birmingham (England) suggested that the positions on a number line are like nouns (e.g. 'negative 4')

and the journeys are like verbs (e.g. 'subtract 4')

usually these get muddled up

and usually I think there's value in muddling them up

[technically I guess you can't add a position on a number line to anything!]

it seems to me that it's (one dimensional) vectors that can be added and subtracted

thanks to Worcester Uni (PGCE) students and Jane Moreton for their comments (in Jan 2018)

### directed number teaching four articles

four articles on the teaching of directed number

(1) some history

(2) three effective models

(3) a 'vector'approach

(4) further considerations

(1) some history

(2) three effective models

(3) a 'vector'approach

(4) further considerations

### simultaneous equations generalising 4 (out of 6)

using the first two terms with the 4th of a fibonacci sequence (question 1)

and then the first two terms with the 5th of a fibonacci sequence (question 2)

can be extended...

powerpoint is here

and then the first two terms with the 5th of a fibonacci sequence (question 2)

can be extended...

powerpoint is here

### simultaneous equations generalising 3 (out of 6)

moving to wider and wider generality

to (eventually) any three corresponding terms in two different linear sequences

a bit of an epic foray...

powerpoint is here

### simultaneous equations generalising 1 (out of 6)

part of a series

simultaneous equation solving

with generalisations that can be explored and proved

(David Wells: put a pattern in and you'll probably get a pattern out)

powerpoint is here

justifying this pattern involves the difference of two cubes

simultaneous equation solving

with generalisations that can be explored and proved

(David Wells: put a pattern in and you'll probably get a pattern out)

powerpoint is here

justifying this pattern involves the difference of two cubes

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