don steward
maths teaching 10 ~ 16

Sunday, 1 March 2015

what's the question?

I very much like this idea
but I'm not sure where it came from and would like to credit the originator(s)

it looks like some of the material developed by the Shell Centre (e.g. in their MAP division task)?

it was used as a fine example of how reversing the question can often lead to a more challenging task

pythagorean quadratics

without looking up pythagorean triples to see which work

division cycling

I recently revisited this work for a session (end of Feb 2015) in Huddersfield
it was offered as a task that made long division a bit more interesting
(a long divided by a short anyway)

I'm afraid that I misinformed the attendees... all divisors do actually work

here is the main task:

 what do you notice?

dividing by 4 is the best place to start
can you create other numbers so that when you divide by 4 they 'cycle'
[i.e. the lead digit goes to the end]

as was found in the session, you can work backwards or forwards to create these numbers

with division by 4, all the lead digits will work
there are some families: those starting with 2, 5 and 8 for example

dividing by other numbers is also interesting:

unfortunately the lengths of the numbers for other divisors are rather long:
  • dividing by 2 needs a number that is 18 digits long
  • 3 needs 28
  • 4 needs 6
  • 5 needs 42 apart from the one example above
  • 6 needs 58 (not for the faint hearted)
  • 7 needs 22
  • 8 all need 13
  • 9 needs 44

however, these are the best tables practice ever

it's interesting, if peculiar
that there is only one six long example for dividing by 5 and the division yields the repeating block for 1/7th as a decimal...

when dividing by 4
if you chop up the six digit numbers into two blocks of 3 
and add them e.g. 205 + 128 you get some interesting results

as you do if you chop them into three blocks of 2 and add them

all reminiscent of turning fractions into decimals with prime divisors

Ed Southall has kindly posted the slides from this session on his blog

and here's the T shirt:

Wednesday, 25 February 2015

loop cards

the clever design of these sets of 6 'cards' is due to Adrian Pinel at Chichester

this set of resources practices substitution but also a with particular multiplication table

the 'cards' do not need to be cut out
students could just link the six expressions in a line

the number at the left is the input value
the arrow should connect with the correct output value
which will be the input value for the next card
and so on...

each student could be asked to work on one set of six 'cards' e.g. set 'c' this being a different set to their neighbour

then they could work with their neighbour e.g. with the two sets 'c' and 'f' etc.

the next set of loop card resources is intended to practice substitution with simple fractions of an amount

linear relationships

there is an argument for working with relationships before equations
the variables properly vary rather than being 'as-yet-unknown' numbers - that can be found

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 substitute numbers to 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 and you might choose to use a box and a triangle (or some other symbols) in place of x and y

Thursday, 19 February 2015

decimal multiplication targets

question 20(c) is from the 'Mathematical Education on Merseyside Challenges' (Peter Giblin and Ian Porteous)

Tuesday, 17 February 2015

4 less than a square

having obtained the two solutions, by substituting values for 'n'
the two expressions could be equated and then rearranged to form a quadratic equation = 0

the two roots can then be related to the equations

breeze blocks

sculpture by Sol LeWitt

how many breeze blocks are there?

each one has dimensions in the ratio 1 : 1 : 2 (as can be ascertained from the pictures)

Monday, 16 February 2015


these resources are based on Dave Hewitt's erudite work

there are youtube videos demonstrating uses of his software Grid Algebra:
the unplugged, 'tapping' version of this task is used here for work on expressions and simplification
it can also be used for quick number and  algebra practice and is simple enough to set up

some work needs to be done on analysing why a large expression can be reduced to a much simpler one

Sunday, 15 February 2015

il machina

students introduce some digit values for 'a' and 'b', with 'a' bigger than 'b'
what happens with the outcomes for various input pairs, (a, b)?

why does this work?

students could make up their own 'machinas' to have the same effect

consider what happens when  a = b
when 'a' is less than 'b' 

find various pairs of numbers (a, b) that give an output of 23 for question (1)
not just digits
with 'b' negative

Wednesday, 4 February 2015

dog, cat, rabbit

a fairly common puzzle (throughout history)
this version from riddleministry