Resourceaholic: Substitution

John and I hosted a day’s on-line
workshop on ‘Substitution’ lately. Don attended our day workshops often
and at all times had one thing additional, fascinating and difficult to supply so we
co-opted him onto our crew. Therefore the workshops we now have hosted since Could 2020
have at all times reminded us of him.  So I used to be
questioning how Don used substitution in his duties – whether or not he made a Large Deal
of it or not.  We posed the query at our
workshop: ‘Is substitution a Factor?’  Our
reply within the day was ‘Sure’ as substitution crops up throughout arithmetic; the
purposeful substitution of 1 expression for an additional to simplify, acquire
perception, make clear, check, modify, make new issues attainable, and so on. and so on.

Shortly after this Richard
Perring posted a tweet asking the identical query however from a special
perspective. His query was about workouts in subbing numbers into algebraic
expressions in early algebra. 
the form of factor, the web is stuffed with them, e.g. ‘If p = 2; q = -6; r =
10, calculate -pq²r^3’. The train will not be about
algebra, it’s about calculating with unfavorable numbers upon getting understood
the syntax of the image system. It modifications algebra into numerical solutions. By
distinction this activity: ‘If p = 2; q = -6; r = 10, discover not less than 5 totally different algebraic
expressions whose worth is 4 utilizing as most of the letters and mathematical
indicators as you want’ focuses on turning arithmetical understandings into
algebraic expressions and launches ‘what if …?’ questions.  

So I started to go looking Don’s
assortment of duties to search out locations the place substitution provides helpful mathematical
views and handles.

I didn’t should look very far: https://donsteward.blogspot.com/2020/04/two-types-of-sum.html

Consider substituting expressions
for consecutive numbers into a,b, and c (that are
generalisations however not very useful ones) and you’ve got the beginnings of a
‘proof’ for conjectures that come up from doing the calculations. A dance begins
between generalisations, construction, examples and relationships that’s typical
of mathematical exploration and the related questioning: ‘What can I write
as a substitute of ….?’  ‘Can I check that with an
instance?’ is usually about substitutions which are useful in revealing or
expressing construction.

And a later slide of Don’s provides:

This offers a cause for changing into
extra fluent with such manipulations – ‘doing’ algebra with a function. There’s extra however you’ll should go to his
web site for that whereas I point out another issues I discovered as soon as I had ‘substitution’
in my sights.

I’ve to power myself to open
information with titles like ‘decimal subtraction’ however right here goes: https://donsteward.blogspot.com/search/label/decimalpercent20subtraction.

This doesn’t disappoint; I’ve
had some nice enjoyable exhibiting solely the primary two traces to folks and seeing what
occurs. I acquired myself hooked on questioning in regards to the cyclic nature of what
seems and rewriting numbers as sums of powers of ten, i.e substituting the
separate place values into the ‘solutions’. 
That is  the ‘undoing’ of what’s
executed in main college to construct up multi-digit numbers from the place worth
parts and is the premise for a lot of psychological strategies in Vedic arithmetic and
Trachtenberg strategies (see Google for these) and an previous methodology that was taught
in faculties within the 18^th and 19^th century known as ‘casting
out nines’. Some folks instantly substituted ‘9 tenths of…’ for the left
hand sides, which explains one thing in regards to the solutions you get however not (for me)
the cycles. The immediacy of this response was spectacular however they have been all
folks ‘of a sure age’ for whom expressing rational numbers as fractions
retains among the manipulability that may get misplaced in decimal notation. So
there are two sorts of substitution at work right here, each about equal numerical
buildings, used to discover after which clarify the generic behaviour seen from
the precise examples.

This jogs my memory of a function of
substitution that exposes its purposefulness. Firstly, it’s a two-way motion
wherein some issues are gained and a few are misplaced: generality/specificity;
approximation/accuracy; manipulability/visualisation and so on.  This final duo comes from enthusiastic about
modelling phenomena, but in addition from substitutions that change bases, akin to are
executed to be able to combine features.

Don’s web site holds many duties in
which substitution is extra specific than what I’ve provided up to now, see https://donsteward.blogspot.com/search/label/substitution.
Martin Wilson of Harrogate is credited with a few of  the concepts.  There are a number of of those that may be
explored by trial and adjustment, i.e. purposeful substitution to get a ‘really feel’
for what’s going on and likewise uncooked materials for later reflection – why these
numbers?  Each
the units beneath use buildings which we hope will develop into acquainted for learners
but in addition have additional options to consider. 
I may think about learners being requested to ‘make up a few of your personal like
these’ and therefore writing algebra for themselves, having used substitution to
check their innovations – some two-way quantity/algebra considering.

You could be questioning a couple of
place within the curriculum the place the phrase ‘substitution’ is used explicitly – the
answer of simultaneous equations. In our workshop, and likewise in a few of Don’s
duties, a strong use of substition seems to be the usage of equal
algebraic expressions, or quickly equal expressions to simplify the usage of
variables. A very easy model of this reasoning is: if a = b and a = c
then b = c, and a, b and c will be substituted for one another. Here’s a
improvement of that, the place expressions fairly than particular person letters will be
manipulated to ‘scale back’ the variety of variables in a scenario (https://donsteward.blogspot.com/2016/03/find-expressions.html
).  The duty is to precise a, b
and c when it comes to n. Slightly than utilizing the language of shifting
phrases from side to side over the equals signal like deranged chess items the language
of logical arithmetical reasoning can be utilized. For instance, within the first set:
‘if b – c = n, then I additionally know that c = b – n; does that assist?’ The usual
query: ‘if I do know …. then what else do I do know?’ kicks in huge time when
remodeling algebraic expressions. New expressions for a and b can
be substituted into the primary equation.

This realisation, that a lot
arithmetic is determined by substituting one expression for an additional when constructing expressions,
equations, mathematical fashions and so forth, appears to get misplaced in formulaic
approaches to simultaneous equations. The thought of substituting expressions into
different expressions has lately made extra sense to me than the normal
‘elimination and substitution’ language of strategies. For instance Don’s ‘the place do
the traces meet?’ duties (https://donsteward.blogspot.com/search/label/simultaneouspercent20equations)
cry out ‘substitute for y within the second equation’ fairly than the ‘rearrange
and match coefficients and subtract the equations’ that seems in some
textbooks.

I’m not saying that this sort of
substitution can be utilized to resolve all such issues, however the consciousness of the
energy of substituting one expression for a variable or one other expression
pervades arithmetic, so a standard pedagogic query might be: ‘is there a
substitution that may be constituted of the given info that provides
perception/simplifies/provides some traction?’. Being a bit fanciful – this might even be utilized in angle-chasing
conditions. I believe that if John and I wrote ‘Questions and Prompts for Mathematical
Considering’ at this time we’d embody a query of this sort that would flip many
procedural duties into one thing extra artistic.

In my subsequent Dose of Don I shall
return to discovering inspiration in his duties fairly than imposing my very own
perspective on them.