median
don steward
mathematics teaching 10 ~ 16

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

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)





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

simultaneous equation generalising 6 (out of 6)

some reasonable ones..
involving a linear nth term

powerpoint is here




simultaneous equations generalising 5 (out of 6)

three patterns

powerpoint is here





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



simultaneous equations generalising 3 (out of 6)

starting off with consecutive terms in a linear growing sequence (i.e. the same difference)
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 2 (out of 6)

matching y coefficients
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

solving simultaneous equations article

some suggested steps for teaching this topic

is here

thanks to Dave Hewitt

three tasks involving simple simultaneous equations

three contexts
involving a + b = a number and a - b = another number
the powerpoint is here

task 1
where simple linear graphs meet






















task 2
a fibonacci sequence with two numbers missing























task 3
number shacks
the next 'shack' is formed from the previous one
working backwards...?