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

## Sunday, 1 March 2015

### 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:

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:

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:

Labels:
division

## 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

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

Labels:
fractions of,
multiplication simple,
substitution

### 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

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

Labels:
linear relationships

## Thursday, 19 February 2015

### decimal multiplication targets

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

Labels:
decimal multiplication

## 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

Labels:
factorising quadratics,
substitution

### 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)

how many breeze blocks are there?

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

Labels:
cube numbers,
volume cuboid

## Monday, 16 February 2015

### tapping

these resources are based on Dave Hewitt's erudite work

there are youtube videos demonstrating uses of his software Grid Algebra:

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

there are youtube videos demonstrating uses of his software Grid Algebra:

- grid algebra use with younger students
- grid algebra 1 getting to know the grid
- grid algebra 2 movements
- grid algebra 3 introducing and using letters
- grid algebra 4 inverse operations and solving equations

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

Labels:
expressions,
simplification

## Sunday, 15 February 2015

### il machina

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'

not just digits

with 'b' negative

Labels:
expressions,
symbols

## Wednesday, 4 February 2015

## Tuesday, 3 February 2015

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