median
don steward
mathematics teaching 10 ~ 16

Thursday, 6 December 2018

trapezium area and quadratic

based on
'exercises in algebra'
part three
H.F.Browne
1936

surface area and factorisable quadratrics

set up a quadratic equation
and solve it

trial and improvement may well be speedier...
(students shouldn't do that
because they might neglect to find further solutions,
- even though there aren't any)




Wednesday, 5 December 2018

percentage increase/decrease questions

a powerpoint for this is here

relating % increase and decrease to a picture





















a number, increased by N% is N




















a number, decreased by N% is N



Monday, 3 December 2018

search for unusual cuboids

this involves work on the surface area and volume of a cuboid
the powerpoint is here

there are three tasks
  • adding 1 to each of the (integer) dimensions, doubles the volume
  • different cuboids with the same surface area and volume
  • cuboids with the same value for the surface area and the volume (see lovely cuboids for a slightly different approach)

task 1

this task was offered by James Tanton





















cuboids that double in volume when 1 is added to each of the (three) integer dimensions





















task 2

different cuboids with the same surface area and volume

factors of the volume can be used to narrow the search





















task 3

same value for surface area as volume

there are 10 altogether



see also lovely cuboids for a different approach to task 3


rule to points 3

two methods:

  • the first involving finding one pair and then seeing how the patterns develop
  • the second involving a transformation of the linear relationship until a simpler form is reached

the powerpoint is here







rules to points 2

further work on finding integer points/pairs that fit linear relationships
then, the reverse, finding a linear rule that fits a sequence of points

the powerpoint is here







these are also of the form
ax - by = 1

rules to points 1

considering linear rules/relationships (with some simple quadratic relationships later)
finding integer points that fit rules
the powerpoint is here











Saturday, 1 December 2018

similar simultaneous equations

it is possibly interesting that fairly similar looking pairs of simultaneous equations (like (1) and (2)) have quite different solutions

it might also be interesting to explore why pairs of simultaneous equations have the same solution




















maybe invite students to solve question (4) a different way

and also maybe question (6)


equal tops pyramids

these tasks follow an idea of Martin Wilson, of Harrogate, England

the purpose is for students to practice simplifying expressions, within a context
a powerpoint is here

all of the equal top block numbers can be found by trial and improvement and that is the intention, especially for the harder tasks

other techniques can be used

sheet 1 leads to a simple linear equation
sheet 2 leads to a linear relationship that generalises to c = 2n, d = 3n - 4 with top numbers: 6n - 5
sheet 3 can be solved with simultaneous equations, with one solution
sheet 4 leads to a relationship that generalises to a = 2n + 1, b = 3n - 1 with top numbers: 12n - 5
sheet 5 with 3 variables can be generalised to a = 4n - 2, b = 7n - 2, c = 14n - 6
sheet 6 has 4 variables, the smallest equal top number is a triangular number (and you can also make 81 and ...?)
sheet 7 is one that Martin used with his students
they found the smallest possible equal number by finding common terms in the four sequences (which is less than 50) you can also make 209 and ...?









simple linear relationships

there is an argument for working with (linear) relationships before equations
because the variables properly vary rather than being 'as-yet-unknown' numbers - that can be found
such work could precede straight line graphs
maybe lending further insight into the gradient (equal steps)

I'm told that in Hungary the maths curriculum starts from this more general appreciation before moving to the simpler, equation, cases (Paul Andrews' various articles with Gillian Hatch when he was at Manchester Metropolitan, Cambridge, now at Stockholm e.g. for BSRLM

when one of the variables is fixed you then have a linear equation

the intention of these tasks is that students find integer pairs that fit the rules, positive integers initially

they may well notice patterns that enable other pairs to be more easily found and lead this work into negative numbers

the Cuisenaire rod resources 'rod relationships' might be one way to begin such an exploration