summing the cubes

Nicomachus appreciated this rule (~60 to 120 CE)

thanks to David Wells for the numerical approach

a ppt is here

this task can begin with an exploration of numbers
with a result being easy to find

it can be justifed using numbers

there is a novel diagrammatic representation

algebra can be tackled (if you can expand a cubic bracket) to justify the uniqueness of the first few results

a result that is needed later

see youtube

or nrich 325

how can you be sure that this process continues?

an interesting alternative diagram
possibly using Cuisenaire rods

algebra used to justify that you need to start at 1
(for the first two, three and four consecutive cube cases anyway)

the three consecutive cube case is easiest:

find the shape puzzles

these are based on an idea by Naoki Inaba
he produced the 'area mazes' as well as many other puzzles
he appears to refer to these shape tasks as Zukei puzzles (42 of them)

there are translations into the English/American names for shapes, done by Sarah Carter (who introduced them to a wider audience via her Math = Love blog)
she posted her translation from Japanese on Sat Dec 17th, 2016
one version of these is here

for those with tablet/laptop availability, Desmos have an activity builder (tool) that can be used when looking for the shapes in the original (Naoki Inaba) tasks, here
students can either construct lines or draw freehand and these steps can be reversed, or erased

many thanks to all of the above people

for my versions, there almost certainly needs to be a presentation/reminder about how you know lengths are the same - using the same or rotated vectors (or by involving pythagoras)
and how you can tell that lines are parallel or perpendicular
this is highly likely to involve vectors (maybe calling it a journey between two points)

join four dots to make a shape, as specified for a particular row
other dots are there to create the puzzle
some of the shapes are tilted

I've tried to create puzzles that have just one solution but I may well have overlooked some options...
(please let me know via the email address, bottom right)

for my shapes, in this task I have chosen not to involve special examples of e.g. a parallelogram should be a general example rather than a rhombus, rectangle or square

a ppt is here

congruent parts and similar parts

dissect the shapes into two congruent pieces
with one, connected line that goes dot to dot

fairly easy when one (half) shape is translated to the other:

involving a rotation of 180 degrees:

two identical halfs:

this time one of the halfs is rotated through 90 degrees to the other half

two similar shapes
one of the parts is an enlargement of the other (not scale factor 1)

powerpoints:

• congruent halfs
• similar parts

new way of solving quadratic equations

Dr Po-Shen Loh's reported 'not commonly known' method for solving quadratic equations is a good one. I posted a set of resources detailing an identical method, involving solving a quadratic equation by transformation on 11th Nov 2016 (here).

This was introduced to me a long time ago by David Wells who has published various detailed elaborations of this method (as early as 1981) and then in his book, 'Hidden Connections Double Meanings' (chapter 16, pages 114 to 116), published in 1988 by CUP.

It is a technique that undoubtedly warrants serious attention and it is not found in many texts that I have looked at.

The technique may well predate David Well's published works. The ancient Babylonians (4000 years ago) certainly could solve quadratic equations, maybe using a similar technique - although they appear to have found results by other methods.

It is admirable that Dr Po-Shen Loh has rediscovered and been able to achieve considerable attention for this method. He is clearly open to people suggesting other discoverers. Maybe David Wells should get a feature in newspapers?

frequency trees and percentages

a ppt is here

HKMO problem

note that a and b are complex numbers

students are supposed to use algebraic manipulation rather than finding a (= 3 + 2i) and b (= 3 - 2i)

it is not necessary to know the factorisations for the sum or difference of two cubes

not HKMO problems
but the same ilk

some square number patterns

all due to David Wells

proofs could involve expanding brackets and simplifying expressions
or use the difference of two squares equivalence
or could be considered by using diagrams

regular polygons with algebra

question 3 is from a Hong Kong Maths Olympiad paper