arithmetic sequences meets equations

this task was devised by Martin Wilson, of Harrogate
the powerpoint is here

students could substitute numbers (trial and improvement)

or, if they appreciate that there are equal 'steps' between adjacent (and other) pairs of terms,
they can set up and solve linear equations with the unknown on both sides

for the quadratic version there may appear to be two solutions
but checking will show that, actually, only one of these works

quadratic nth term using factors

this is an idea for working on quadratic sequences, developed with Martin Wilson in Harrogate

we hope that students will be able to make a link between factorising numbers and factorising a quadratic expression

the powerpoint is here

the technique involves

• transforming a given sequence by adding or subtracting a number to/from all of the terms, this number chosen so that all of the resulting terms have factor pairs
• seeing if this provides a regular growing (or decreasing) pattern
• if not, looking at another option

this approach might precede an algebraic one (outlined here)

three GCSE exam board questions set this summer (2017) are explored, one that can be factored without transformation

cuboid with a shell

the difference (i.e. volume of the shell)
between a cuboid (with 3 consecutive numbers as the dimensions) and
another cuboid where the dimensions are all 1 unit bigger

a 2 by 3 by 4 cuboid
with a shell around it

the difference is always 6 times a square number

quadratic nth term

quadratric sequences

intending to familiarise students with the properties of a growing quadratic sequence
rather than finding a quadratic rule from a (given) procedure

the second resource was suggested by David Wells
intending to explore the differences - possibly by students looking at a few quadratic rules to discover generalities for themselves

you might choose to just use the second resource
hopefully students will appreciate the symmetry of some sequences

possibly exploring when a growing quadratic sequence has symmetry and when it has not

counting hexagons

this task was suggested by David Wells in his problem solving books

it may be helpful to look at the centred hexagonal numbers first

how were these patterns formed?
what would/could the next one look like?
how many of the unit hexagons (i.e. the grid hexagons) are there inside each shape?
how does the pattern continue?

for the first pattern, it might help to appreciate a general rule by counting the number of unit hexagons in:

and, for the second pattern,  maybe consider several of these together:

quadratic matchstick patterns

du

the idea for using the two different lengths between dots ( 1 and root 2) is from a KS3 SAT question (2003, level 4 to 6, paper 2, question 19)

number trees

corner polygonal numbers

geogebra growing from a corner

using growing quadratic sequence techniques