two, 2-digit multiplications

four tasks with a common theme:

1.   (10n + 5) squared
2.   from a square number, go a bit more and a bit less and multiply them
3.   31 squared and 29 squared
4.   units sum to 10, in a long multiplication sum

these involve (a + b) squared, the difference of two squares and (a - b) squared

they could be a starting place for a study of these expansions

all start with number calculations (long multiplication practice) and move on to consider 

related images (an array) and then 

expanding quadratic brackets examples (which play automatically)

ppts are here:

1.  (10n + 5)^2

2.  24 x 16 

3.  31 ^ 2

4.  42 x 28

task 1

task 2

task 3

task 4

long multiplication teaching

Deborah Loewenberg Ball (Michigan, USA) suggests that teaching involves content knowledge (having the skill) and pedagogic knowledge (having an awareness of how to teach it effectively and how to respond to student misconceptions)

the first resource is for teachers
what misconception might students have, to do calculations in this manner?
what might a teacher do in response?

one strategy to aid skill development is to provide examples
it seems helpful for these to be dynamic - either filmed or in a powerpoint loop

the intention is that you have the powerpoint playing whilst students are working
i.e. set the powerpoint to run and leave it to continue, in a loop

loop 1 is here
the shortened algorithm
58 times 73
35 times 67

loop 2 is here
the expanded algorithm
38 times 46
238 times 46

not looped, long multiplication examples are here
dotty grids, grid method, expanded and shortened algorithms

also not looped, lattice ('gelosia') algorithm examples are here
maybe better left until later in a student's life
especially where other methods are fragile
used historically, with good reason

mirror multiplications

first, a few examples indicating that 'mirror' multiplications do not usually have the same answers

some of these are close results

and they can be the same ...
253 x 64 = 46 x 352

further examples of mirror multiplications with the same answers

a powerpoint for this task includes a step by step calculation for 253 times 64

considering how mirror multiplications can have the same solutions:

as an initial step:
how can the lead and final digits of the two multiplications be made to be the same when they cannot be the same numbers?

for the number in the centre of the 3-digit number:
a grid method (expanded) representation shows how equal answers are obtained for the examples above
what is their common property?

longer multiplication with patterns

consecutive digits in a multiplication sum

thanks to Ken Milton, University of Tasmania

non-calculator multiplication practice

see long multiplication teaching here

a long multiplication puzzle from counton
what are the missing numbers?

[questions: why aren't there three lines in the solution?
what number must the orange triangle be, and why?
what number (other than 1) has the same last digit when it is squared?
why must the yellow circle be the purple circle + 1? ]

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:

long multiplication pattern answers

multiplications and long multiplications
without a calculator
pattern answers (to enable correctness to be checked)

a ppt is here

the product product

a special number

loooooong multiplication - grid method

the 'Gelosia' or 'lattice method' for long multiplication was a method seemingly in common use centuries ago

it can be a helpful method for some students

using 37 multiplied by a two digit multiple of 3 (not bigger than 27) provides some easy to check answers
e.g. 37 x 15 ,  24 x 37 ,  37 x 27

the method was used historically in Europe (i.e. by their great, great ancestors)

two pictures from Italy of long multiplication sums, supposedly from 1478 CE
in the Treviso arithmetic:

the sum is 934  x  314
(a problem from over 540 years ago)

the sum is
56789 x 1234

















from an older manuscript, around 1300 CE:

the sum is

4 569 202  x  502 403

with an answer of 2,295,570,802,406