1 off a cube

factorising n^3 - 1
a powerpoint is here

equivalent things

the intention of this first resource is that students substitute the given input values
and then, maybe with some nudging, appreciate that there are shorter rules that give the output numbers from the input numbers

and that these simpler expressions can be obtained by cancelling the algebraic fraction

the following resources involve justifying then creating equivalent ('same only look different') expressions to the one in the middle

students can be asked to create their own expressions or versions

fraction approximations to square roots

there are patterns to be found in the fraction approximations

this work links with Heron's method for calculating square roots (which seems to have been known to the Babylonians)

see also the very helpful CTK (Cut the Knot) insights blog where the use of the 'Regula Falsi' method is explored

equivalent nth terms

how can you 'see' this regularly growing pattern?

cube coordinates

how have the vertices of the cube and inbetween points been numbered?

what happens with the coordinates that are vertices of an equilateral triangle?

what coordinates could you have for a regular hexagon?

this diagram shows what seems to be a common convention for 3D coordinates:

the positive z axis coming out of the page rather than going into it, which is 'right-handed'

[the diagram with the coordinates above follows this convention - albeit in an unusual orientation]

growing cubelets

what will the next block look like?

and the next?

and the next?

gives (n + 1)^3

cuboid transformation

what happens to the

• volume
• surface area

when the cuboid is folded?

what happens for other, similar, cuboid 'foldings'?

growing two ways

what is happening here?

patterns?

rules?

in general?

nets of a cuboid

how many different nets of a 2cm by 3cm by 4cm cuboid can be found?

students could find a few and maybe draw accurate diagrams (with or without a square cm grid)










you could consider the edges that could be 'unjoined' at various stages (to leave 5 joined)

some sources have argued that there are the same as for a cube (which seems to be quite incorrect)

there appear to be 54 (with a fairly high degree of confidence...)

here's a 'justification' involving an attempt at a systematic approach (enhanced by Puntmat pointing out that I'd missed two of the options ...)

the fine people at Puntmat found two more, to complete trios in the picture above:

so that makes 54, a number that the Puntmat team and I think is a very neat answer....

suggesting that there might be another way of deducing this total (unless someone finds any more options of course...)

anyway, our answer is 54

many thanks to the Puntmat team

nets of a cube

nets are 'different' when they are not a reflection or a rotation of another net

how many edges have been 'unjoined' to make the nets?

geogebra versions from azb

nth terms of a continuing, regular pattern

if you counted pencils from the start (as 1)
and when you finish a row you keep counting, in the next row
and so on (for ever)

in what number positions will there be a dark blue pencil?

how many would you need to count beyond a million to a position where the pencil would be dark blue?

same thing, with coloured eggs
this is a 'window' of a regular pattern that continues downwards for ever and ever...
what positions do the blue eggs occupy?
the yellow ones?

same thing, with stripes
which continue to the right (or left) for ever and ever...

what positions do the dark blue stripes occupy?






links with modular arithmetic

pencil art by Jennifer Maestre

Jennifer Maestre

pyramids and prisms and Euler

these are representations of the pyramid family:

what do they all have in common?

how would you describe a pyramid to someone who wasn't clear (or over the phone)?













why are there the same number of faces as there are vertices for any pyramid?

what is the relationship between the number of arcs and the number of vertices for any pyramid?
why?

Euler's relationship is F + V = E + 2
faces + vertices = edges + 2

why must this be true for any pyramid?


these are representations of some of the prism family:

what is a common feature of this family of solids?

















can you explain why for every prism, the numbers of vertices and the numbers of edges are in the same multiplication table?

what is the relationship between the number of faces and the number of vertices for any prism?
why?

why must Euler's relationship (F + V = E + 2)
be true for any prism?

angles in polygons by chopping

what do you see here?

rhombuses?
rectangles?
equilateral triangles?
(bigger) right angled triangles?
trapezia?
kites?
arrowheads?
regular hexagons?

students could create their own shapes using four of the 30, 60, 90 triangles

an intention is that they then sum the angles at the corners to explore the angle sums of polygons (with various numbers of sides)
 

in the above resources the triangles' vertices do not all meet at the vertices of the polygons

so students will need to work out the angle sums of the polygons by deciding which of the 30, 60, 90 angles to use and which not to use

for the following resources, the vertices of the 30, 60, 90 triangles are all at the vertices of the polygons 

I think these are all the options for quadrilaterals and pentagons

I don't think this is all the options for hexagons (I gave up)