71

use any two numbers

• multiply them
• then add them
• then total these

what if I want the total to be:

(a) 11
(b) 23
(c) 71?

 find (and establish that there are only) five integer solutions, ignoring reversals, to

hexagon areas

from any point inside a regular hexagon, lines are joined to each vertex

prove that the two triangles coloured pink have the same total area as those coloured red (and also orange)

prove that this result will be true for any regular polygon with an even number of sides










[ idea from MEI resources ]

multi-divideable numbers

the numbers below are particularly great because
(i) they each involve all of the digits (0 to 9) and
(b) they divide by all the numbers from 1 up to 18

so students can be set tasks involving a long number divided by a much shorter one - of your or their choice

what is the hcf of these numbers?

eight 8s

use eight 8s in an addition sum making 1000

eight odds

add 8 odd numbers to get a total of 20

six of the 11 solutions are shown

you might consider the numbers of 1s or the options for the high numbers, e.g. you can't use 19, 17 or 15

try to find the other solutions

[Moscow puzzles 49]

add to times

for some addition sums you can replace the addition signs by multiplication signs and still get the same result

the (single) choices for 2 numbers, 3 numbers and 4 numbers are shown



what (3) choices are there for five numbers?

[Moscow puzzles 52]

area by adding/subtracting

work out the areas of these shapes

cubic graph

what are the coordinates of the various points on this cubic graph?

what would the y-value be
when x = -12?
when x = 16?

maybe use a spreadsheet to ascertain the function






David Wells points out that:
IC is a tangent to the curve at point C and BG is a tangent at G

all cubic curves have rotational symmetry about their point of inflection (E)
and the sum of the x-distances from the point of inflection (E) of any straight line cutting the curve at three points is always zero (for a cubic)

in the case of a tangent, two of the points are coincident

 what is the equation of this cubic?

substitution 'cards'

the idea is to give students one of the (six) cards each

in pairs they then try to work out which of the listed  n-values makes their expressions the same

plotting a quadratic

James Tanton has several interesting quadratic graph plotting techniques
based on the vertical line of symmetry
he introduces these amongst his various you tube offerings

when sketching a quadratic, instead of completing the square he suggests factorising the non-constant terms - to easily locate two values with the same y-ordinate:

x = 0 or 5 both give y-values = 3

so the curve must be symmetrical about a vertical line mid-way between these two  x values
(at x = 2.5), enabling the lowest point (vertex) of the quadratic to be calculated

this lowest point (vertex) can also be located by considering the lowest point of  y = x(x - 5) and then translating this 3 spaces up

solving un-nice quadratics

several U.S. blogs (James Tanton in particular) advocate multiplying by 4 in order to more easily complete the square when the x-coefficient is an odd number:

which seems a helpful technique

my preference, as mentioned previously, is to transform the equation using a difference of two squares technique (as advocated by David Wells):

this preference is because it's straightforward
and also because I quite like students to be able to mentally square something.5

maxprod

use the digits 1 to 5 once only

form a multiplication sum





in an attempt to find the largest product (result) that can be obtained

an intention is that students try out (on paper) one 4 by 1 digit multiplication and also one 3 by 2 digit multiplication

then they could use a calculator to speed up an exploration of other options - but this might miss out on an understanding of how the 'partial products' contribute

the various (close) options can be considered in an expanded form to see where there are advantages.

the largest is 22, 412 for the first five digits








larger (and smaller) versions of the task can also be explored:

(thanks to Colin Foster for his convenient collection)





and patterns/general formats explored:

getting three

place four digits in the boxes
multiply across and add the two products
multiply down and add the two products
how can you get a difference of 3 between these two totals?









3 has been chosen as an initial target because any four consecutive numbers give this difference (but there are plenty of other ways to obtain 3)

proving that four consecutive numbers give a difference of 3 is good practice in expanding brackets

how can you get a difference of 0?

explore how you can predict the difference from the original chosen digits without doing too much work...

students might be able to appreciate what is going on from a numerical perspective:
6 x 5 - 6 x 2 = 6 x 3
7 x 5 - 7 x 2 = 7 x 3

the '3' is present from 5 - 2
the result is 1 x 3 because 6 and 7 differ by '1'

timestable

place the digits 2 to 7 (used once only) in the circles

multiply (in pairs) to get the nine products in the table

add these nine products

how do you place the six digits to get the highest possible total?

the maximum is 182
how is this maximum achieved?

all of the row (and column) totals divide by a certain number (other than 1)
what is it and why is this?

Kaprekar's constant

choose any 4 digits, possibly with repeats
arrange them largest down to smallest (descending order)
reverse this
subtract

keep doing this, until you have a good reason to stop
(i.e. not exhaustion or boredom - this is a fairly tedious task to do entirely without a calculator...).

note that if you obtain just 3 digits e.g. the reverse of 8820 is 0288.

students could build up some form of overview of what happens for a series of 4 digit numbers (this diagram is a subset of the options)
there is a fuller picture on Wiki





it should take at most 7 iterations (steps) to arrive at Kaprekar's 4-digit constant: 6174 (or 7641 etc)

students could work with 3-digit numbers instead
this time the 'constant' is ???

which 3-digit number takes most steps to reach this?

division by 37

each of the numbers below will divide exactly by 37

the results have a fairly obvious connection with the starting numbers

the intention is that students write down the 37 times table to work out the results, without using a calculator

can other five digit numbers with the same property?

yes, there are other examples,
starting e.g. 6 0 _ _ _ _

what fraction?

what fraction is the red square of the whole square?

going 3D,

what fraction (volume) is the smallest cube of the largest one?

a model for powers of 8

seven 20ths

two 2-digit subtraction

a ppt is here

same as those above
hmmm... that's curious

impossibilities

two digadds making 200

place any four digits in the cells (there can be repeated digits)

• read the two 2-digit numbers across and add them
• read the two 2-digit numbers down and add them
• then add these two totals together

you are trying to get a target of 200

students might try to obtain 100 by adding across and 100 by adding 'down'

e.g. with

3 6

 6 4 

having found several solutions, probably by adjusting solutions that are close to 200,

they might notice that the four digits in all of the (several) solutions sum to 19

they can go on to explore this algebraically:

a b

c d

20a + 11(b + c) + 2d = 200

when a = 8,  (b + c)  can only be 2 and d = 9

when a = 7,  (b + c)  can only be 4 and d = 8

and so on...

you can deduce that d - a = 1  if  a + b + c + d = 19

and that 2a +(b + c) = 18