ratio and right angled triangles

the powerpoint is here

I think it might be easier to recognise similarity/enlargement in right angled triangles rather than rectangles

reasons for similarity can be articulated by considering the scale factors of enlargements

those who seek to use additive methods for ratios might appreciate that this doesn't work out too well

the link between equivalent ratios and straight line graphs can be explored
(as can tan (base angle), probably at a later stage)

it's not usually a good idea to ask which ratio is bigger - because they could be turned around

but, this question can be posed in the spirit of asking which is the better deal, £3 for 5 articles or £4 for 7 articles?

the steeper the line the bigger the ratio

which can also be appreciated by scaling the ratios

simplest ratio pairs

intended to be done orally
but the questions are provided
the powerpoint is here

simplest forms of ratios

 the powerpoint is here

you get, I get (in a ratio)

the idea of:
when I get £2
you get £3
can serve as a context for considering ratios

this work is probably best done mentally
but here is a powerpoint

KS2 SAT items

in England, all 11 year olds do a national KS2 SAT test

these six questions are adapted test questions (making some of the second resource slightly harder)

CSMS ratio items

the CSMS (concepts in secondary maths and science) project was extensive (thousands of students) and longitudinal - looking across the three central years of secondary education in England

it was led by Kath Hart
whose thesis on this is available as a pdf
work was started in 1976/1977 and published in 1981

these are four of the eight diagnostic test items (they were questions 1, 3, 5 and 7) used to appraise cognitive growth

the work drew on the research of Piaget and the 'eel feeding' test was based on one of his experiments

this work has been repeated and built on recently by Jeremy Hodgen et al, for the ICCAMS project (first phase 2008 to 2012)
this project looks at two aspects: multiplicative reasoning and algebra, before and after interventions

'onion soup' and some wording is adapted

the main student error involved was the use of additive rather than multiplicative methods
students were more confident to use doubling and halving rather than additive methods - sometimes over using doubling

data for 'Mr Short' shows a propensity to use addition, difficult to dislodge
(Y8 = 12 to 13 years old, etc)

ratio and recipes

maybe a helpful context for ratio
try to adapt the given recipe for different numbers of people

'onion soup' is adapted from the CSMS (Chelsea) work done by Kath Hart et al, a longitudinal study started in 1976 and published in 1981

powerpoint is here

ratio and rectangles

'lengths and areas are the most natural means to visualise magnitudes with respect to teaching ratio'

Hans Freudenthal

linking ratio to enlargement, as Freudenthal advocates

the powerpoint is here

I'm not sure whether this is better with all the dots or not...

one with, one without

what is the ratio of one of the stripes in simplest form?

in the form 1 : n ?

equivalent ratios

scaling a ratio
forming equivalent pairs for 3 : 5
the powerpoint is here

exploring the notion of equivalent ratios

 rectangles in proportion

 adopting different units

 a window on a multiplication grid

right angled triangles underneath a line