parallel line angle relationships and proofs

(i) angles on parallel lines questions and (ii) relationships and proofs

the powerpoint is here

getting a positive whole number

largely a substitution exercise
involving trial and perseverance, with some thought maybe

script:
"I'm thinking of a number and you don't know what it is (or do you...?)
here it is [write b]
whatever it is, I'll multiply it by 2 [put a 2 in front of the b]
then I'll add 5 to that
then I'll divide all this by 3 [maybe emphasise by drawing the line slowly, making sure that students are looking at the board]

what does this expression say happened to my number?

OK, all of this here [putting a hand over the expression] is 9
so what was my number?

good, now I'm interested in what other positive whole numbers that this expression could be..."

find a general form for the letters
possibly explaining why these work

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

pattern answers, multiplying by a single digit

adding fractions procedure

the hope and intention of looping a powerpoint presentation is for students to attend to the structure of solution steps

the powerpoint needs to be downloaded for the animations to work

having introduced the procedure and indicated reasons for the steps, you might leave the powerpoint playing on the board
or you might use it to introduce the skill - asking students to discuss with their neighbour what is going on (and why)

equations with the as-yet-unknown on both sides

initial work, deciding what the steps are that have been used
the powerpoint is here

the slides below are on a powerpoint that loops
hopefully highlighting steps to a solution
(if it is downloaded)
the intention is that it can be left looping - as a worked example

some quiz questions

bead patterns

continue and complete a pattern:

see frieze patterns

pattern experiments

Lucy Walker's pattern experiments

given the sides, make a polygon

powerpoint: sides

and another, that goes off the grid