progression in teaching ratio

my (current) views on ratio units for Y7 (11/12 yr olds), Y8 (12/13 yr olds) and Y9 (13/14 yr olds)
and some suggested diagrams that seem helpful when teaching ratio topics
a powerpoint is here

yogurt ratios and percentages

an updated version

using fairly realistic data (in 2019 anyway)

a powerpoint is here

there are many other applications of Pearson's square technique (e.g. feed proportions) on the net
this is a weighted mean calculation
and it is reasonably straightforward to see how/why it works by working through a 'normal' calculation technique

I don't think students should learn it but it is used widely and it is of interest because (a) it's neat (b) it works and (c) it can be related to a 'normal' calculation

density with grids

John Mason commented that the topic of compound measures might be better started with 'density' than speed - because it can be more visual

a powerpoint is here

part : whole ratios, maybe conveniently expressed as a fraction
some involving cancelling
some involving the lcm (or both)

see also population density of grids
and ratio in car parks

integer intersection points

the problem is adapted from that posed by user 526427 on MathStack Exchange, March 2nd 2018

using straight line graphs, find two intersection points for various ratios DN : NC (in the form 1 : k where k is an integer)

what if these intersection points have integer x-values?

a powerpoint

the width doesn't change the x values of the intersection points
(a stretch)

harder GCSE ratio questions

a collection
the powerpoint is here
Jo Morgan (resourceaholic) presents a good overview of ratio questions and how to approach them here

see also the collections of ratio questions for three of the GCSE exam boards
AQA
Edexcel
OCR

sum, difference, product ratios

from a question posed by Tony Gardiner

trial and improvement is an expected strategy, initially at least, involving equivalent ratios (i.e. scaling)

one method to do these questions

ratio of difference to sum

these resources are based on an idea from the NCETM mastery materials
you are told the ratio of the difference (a - b) to the sum (a + b)
a powerpoint is here

ratio possibly with algebra questions

some demanding ratio questions that are used or adapted from recent GCSE papers and some from the additional/further maths papers

opinions differ on how to teach approaches to these sorts of questions
all can involve algebra (a relatively unthinking and so, for me, safer way) but can also be done by considering e.g. lowest common multiples with the part (or whole) that does not change and then appropriate scaling (equivalent ratios)

the powerpoint is here

you can solve these questions by listing equivalent ratios (scaling)

and seeing which 'works'















these considerations can be sped up by considering the lcm of what does not change, in this case the number of blue beads (thanks to Richard Trimble)

a bar model might help (even if only retrospectively...)

and then consider what fraction the extra 5 must be of the original red bar

or students might prefer to use an algebraic technique

involving simultaneous equations:

or, setting up an equation and cross-multiplying:

the course AQA 8360 has involved questions like ratio (iii)

show that the second statement follows from the first