Sam Loyd's dissection puzzle

By joining corners of a square to the mid-points you can create a square in the middle.
This is the basis of the dissection:

The pieces can be rearranged by students (Tangram like) to form various shapes:

similar triangles

the yellow triangle has sides of lengths 9cm and 12cm

the three triangles are similar

together they form a rectangle

what is the length of the rectangle?

what are the other lengths involved in the diagram?

note that you do not need to use pythagoras
knowing the ratio of the two shorter sides in the central triangle and
that the areas of the 3 triangles = a rectangle will suffice

multiplying decimals

what are

0.3 x 0.4
0.3 x 0.6
0.7 x 0.4
0.7 x 0.6 ?

and how does the diagram help?

such diagrams may well be helpful...

however, it seems that an attempt to explain decimal multiplication in terms of 'decimal places' might also benefit from an explanation involving fraction multiplication
[3 tenths x 4 tenths is 12 hundredths]

there are also possible anxieties about this model because the 'unit' shifts  from being in relation to a length (the side of the square) to a relationship with an area...

just as well that 1 x 1 is still 1...

fractions to decimals

in England, KS3 SAT questions were posed (2000 ~ 2010)

this is one of the questions from those tests:

and a variation on this idea:

diamonds carat size

the numbers of carats are related to the diameter of the diamond







the relationship is close to a power rule
what is it?

decimal multiplication and division

what numbers go in the spaces to complete these multiplication tables?

(intended to be done without a calculator)

to extend/develop this task:
find the sum of the four numbers in the main part (not the outside) of the multiplication grid for each of the questions

and try to explain the result
or give further examples

taking away

using each of the numbers 2 , 8 , 10 and 14,
can you replace the symbols to make this statement true?

there are four different solutions

choose any four numbers for a, b , c and d so that they increase by a regular amount (i.e. are in an arithmetic sequence)

• what is the value of (a - b) - (c - d) in general?
• what is the value of (b - a) - (c - d) in general?
• other expressions e.g (d - a) - (b - c)

simultaneous equations with patterns

based on 'the Moscow Puzzles' (originally published in 1956) by Boris Kordemsky
his number 240 (is question number 6 below)

looking carefully at the coefficients...
what do you notice?
how does it help?

what happens if you add the two equations?

what happens if you subtract them?

simultaneous linear relations

find values for a , b and c that fit these equations simultaneously

the intention is to use trial and improvement, with some thought
especially where multiples or near multiples are involved

tree diagrams and some tasks

maybe tree diagrams are better presented this way up?
(although it's harder to write the probability fractions in the gaps sideways)

what is the probability of getting exactly two, 3s when three dice are rolled?

some tree diagram blanks, round the usual way
a ppt is here

dominoes

a domino is twice as long as it is wide

if the perimeter of the whole shape is 80 cm
what is the area of each domino?

if the perimeter of this whole shape is 600 mm
what is the area of each domino?

[ from a UK KS3 SAT question ]

mean, median and mode - which is bigger?

for a symmetrical distribution, mean = median = mode
(one simple way of checking for a normal distribution but you need more information)

for a (simple, not overly lumpy) negatively skewed distribution
[mean - mode] is negative (since mode is bigger then mean)
the median is greater than the mean

(averages are the x-values)














for a positively skewed distribution
the mean is greater than the median

for a small data set of five numbers
the data needs to be kind of
negatively skewed for the median to be bigger than the mean
e.g. 1 , 1 , 3 , 4 , 5
e.g. 2 , 2 , 8 , 9 , 10

how much bigger can the median be when compared with the mean (and also be bigger than the mode)?

for: a , a , b , b + 1 , b + 2
show that b needs to be bigger than a + 1.5 for the median to be bigger than the mean

money exchange

Jesse decides to sell the 198 dollars he had left over from a holiday
comparing rates in a few places, he opts for the highest rate of 1.8 (dollars to the pound) rather than the lowest , of 1.5

too late, he realises his mistake
how much did he lose out?

nuts

the need to turn £10.40 into a whole number could be a clue
or set up an equation with the two variables...

all options

four squares, each shaded in half
join together to form a block

sometimes there are four orientations and sometimes there are only two, or one
when?

all patterns have a negative

all of the options (thanks to Puntmat for their corrections)

why are there 4^4 options altogether?

Puntmat suggest using all of the options to decide the probability of obtaining
(i) a block with a line of  symmetry (the fraction is a cube/ 7th power)
(ii) a block with reflection or rotational symmetry (of order 2 or 4)

the latter (ii) is a much neater fraction...

students could consider the ways to split a square in half and then combine these to form a 'block'
this can be coded (e.g. using: top left, top right, bottom left, bottom right)

for example...

ratio in car parks

a consideration of the compound measure of density, in a context of ascertaining which car park seems (and is) fuller

relating the number of cars to the number of spaces

a powerpoint is here

order of calculating

what is quite tricky to appreciate with bidmas or bodmas or pemdas is that there is no preference for division over multiplication or addition over subtraction (so it should really be b i d/m a/s)

Matt Dunbar (Trinity Maths) proposes using a trapezium picture:

in a similar vein (from the NCETM site)















when there is any ambiguity you follow a left to right procedure

using all of the digits 1 to 9, once only, can students see how the numbers at the ends are created?

in newspaper versions, these puzzles do not follow the bidmas rules

these puzzles do follow the bidmas rules

you use 1 to 9, once only, to complete the puzzles