don steward
mathematics teaching 10 ~ 16

Thursday, 15 March 2018

growing squares under the stairs

these resources are based upon an article by Daniel Pearcy in Maths Teaching 247 (July 2015)

starting from a coordinate sequence
to powers (the y coordinates)
to sums of powers (the x coordinates)

the powerpoint looks at squares drawn underneath e.g. y = 2x + 1

what would the next coordinates be?
and the next?

coordinate sequences

what are the coordinates of e.g. the lowest left hand points of the Ms
what would the next ones be?

try some with your initials

coordinate practice

checking that students can plot coordinates in all four quadrants

the powerpoint begins by involving negative coordinates to plot squares

and goes on to involve points on straight line graphs

probably you can let us know some other points that will lie on this line

maybe there's a rule that says whether or not a point lies on this line

coordinates CBSE questions

it is interesting to see how an approach that can involve vectors and properties of quadrilaterals develops from the KS2 based SAT questions (here) to these from the CBSE exam (India) for Y10 students

the powerpoint goes through some of the ways to solve question 5(b)

printable version

quadrilaterals and coordinates

problems based on the KS2 SAT questions
the ppt is here

 same questions printable version

Sunday, 11 March 2018

smaller radiating directed numbers

still not easy - but simpler
thanks to Tom Francome
there should be no duplicates of numbers in the loops for each task

students make their own - then send them to me...

Friday, 2 March 2018

Thursday, 1 March 2018

from one fraction to another

@pbruce maths and @misswillismaths indicated a wish for such a resource
it seemed like a good idea
with some generalities unearthed

a proof involves expanding brackets to form a quadratic expression
a proof is reasonably straightforward

how are these two general forms related?

Sunday, 25 February 2018

Leicester ATM/MA presentation

many thanks to all those who attended

thanks to Rob and the other organisers: the ATM/MA East Midlands local branch

the powerpoints for each of the three sessions need to be downloaded for the animations to work

session 1:
  • a number task
  • pythagoras put in context(s) with some practice tasks
  • Babylonian tablets
  • factorising quadratic families
  • coordinate questions from KS2 SAT papers
  • Van Hiele's interesting view that geometry is best done with a square grid
  • coordinate and perpendicular bisector questions from CBSE (10) India past papers
  • linking area and straight line graph equations (thanks to Simon Mazumder)
  • linking squares drawn underneath straight line graphs to summing geometric series
  • 3D geometry
  • Euler's relationship
  • total angle sum (TAS)
  • total angle deficiency (TAD)
  • truncating polyhedra

Saturday, 17 February 2018

directed number arithmogons

the word 'arithmogons' (rather than 'arithmagons') seems to stem from an article by Alistair McIntosh and Douglas Quadling in Maths Teaching number 70 (in 1975)

amongst many other things Leo Moser (1921 to 1970) studied pairs of numbers adding up to totals, including the work in the third resource: pairs of numbers always summing to a square number

the powerpoint goes through various algebraic solution steps - one good reason for studying arithmogons, as well as (in this case) practice with directed numbers

Craig Barton details the reasons he enjoys working with arithmogons and has various tasks based on their structure here

Monday, 5 February 2018

tangents to circles

these are based on problems from the CBSE exams (India) Y10

simultaneous equations generalising

these resources follow a theme of providing practice questions with some pattern built in

that way the 'depth' to a task can involve generalisation and proof

some of the proofs are demanding
you might go through steps with students with them trying to explain what is happening
the powerpoint goes through the proof steps (but it might be better to go through this on the board)

Saturday, 3 February 2018

similar triangles

these questions are from or are similar to CBSE (India) Y10 papers

(5) and (6) also need circle theorems

Tuesday, 30 January 2018

radiating equations

maybe curiously, maybe not, a single operation changes both sides of an equation

the powerpoint introduces this notion (with animations if downloaded)

introducing the idea of the transformation

Sunday, 28 January 2018

pythagorean triples

studying square numbers with two of them summing to a third (for integers) has interested various ancient and hopefully modern civilisations

this work develops some of the extensive ideas provided by Dr Ron Knott at Surrey University (many thanks to him)
Hannah Jones used these and other resources to devise classroom tasks for an EPQ project

pythagorean triple introduction ppt

various patterns can be used to find pythagorean triples, those starting with an odd number being more commonly known

finding pythagorean triples ppt

seeking pythagorean triples with a selected shortest side uses the difference of two squares and factor pairs

triples for a shortest side ppt

a longer list of pythagorean triples is available at TSM resources

there are various formulae for finding pythagorean triples from a parameter (or two)
these provide primitive triples (without common factors) but not usually multiples of these

triple generators ppt

the graphical work of Adam Cunningham and John Ringland (in the Wikipedia entry) on primitive pythagorean triples is interesting

graphing primitive triples ppt

there are also some novel methods for generating triples from fractions, identified in Dr Ron Knott's work

fraction generators ppt

some problems with lengths in triangles and rectangles, involving pythagorean triples

triangle lengths ppt

a variety of problems, all involving the 3, 4, 5 triangle

3, 4, 5, problems ppt

some problems on quadratic equations set up as pythagorean triples with various expressions for their lengths

expressions triples ppt

the perimeters of pythagorean triple triangles have some interesting patterns

perimeters of triple triangles ppt