sin2x

graph snaps

two root

babylonian quadratic equation solving

without using algebraic symbols, some Babylonians devised a procedure for solving a type of problem that would normally be solved by creating a quadratic equation


the problem was likely to be expressed in terms of the semi-perimeter and area of a rectangle being given

a specific questions would model a method that enables the lengths of the two sides to be calculated

a harder example:
the perimeter of a rectangle is 20 and the area is 23
what are the lengths of the two sides?

David Wells addresses this and other methods of solving similar problems in his excellent book,  'Hidden Connections, Double Meanings' (pages 113 to 116)

he highlights the economical (and lovely) technique of calling the roots 5 + b and 5 - b                        (because they sum to 10 this must be so)
then multiplying these roots gives a convenient rearranged form, that is easily solved

weight balancing

ignore moments for these questions

a powerpoint is here

triangular number pictures

these fine diagrams are presented in Brent Yorgey's blog, 'The Math Less Travelled'

what do they show about triangular numbers?

and this is a slightly different diagram to one posted elsewhere

what does it show?

dissection

show how to dissect this shape into 4 congruent pieces:

HCF

looking at the hcf (sometimes called the greatest common divisor) of two numbers
student: "Why didn't you tell us that you can just subtract the two numbers to get the hcf?"
teacher: "Because it doesn't always work"
student: "It does on the two examples we've looked at"
teacher: "Oh, so how about 5 and 15?"
student: "10"
teacher: "It didn't work"
student: "Well when does it work then?"

a tricky calculation

ask someone to think of any prime number bigger than 3

you should be able to predict what the remainder is...

cancelling down

tap top

a problem solving approach to factorising harder quadratic expressions  - by stealth, as he says, comes from the interesting work of Jason Dyer
 

go diagonally across the 'tap tops' to get the two factors
(top left number x with bottom right) times (top right number x with bottom left)

more questions than you will probably (ever) need:

largest square

what is the largest square that can be drawn inside a  3 , 4 , 5 right angled triangle?

UK stopping distances

the UK Highway Code provides the following data (as a guide):

a rule for the 'thinking distance' is linear

a rule for the 'braking distance' can be found using a spreadsheet and fitting a (quadratic) polynomial and displaying the equation of the trendline

magic triangle

there are a total of 6 solutions to this problem

it is reasonably easy to establish that the three corner numbers sum to 15

there are:
2 with 2 , 5 , 8
2 with 4 , 5 , 6
1 with 3 , 5 , 7
1 with 1 , 5 , 9 in the corner positions

what is interesting is that for just one of these solutions, if you square all the digits the line totals are still all the same (discovered by David Collison)

a factoring rule

This was one of Joseph Liouville's simpler theorems (1809 - 1882)

write out all the factors of a number
then write the numbers of factors for each of these factors
the sum of these squared = the sum of the cubes of these

triangular numbers squared

the fact that a triangular number (Tn) of triangular numbers (Tn) is the sum of the first n cubes
can be tested numerically

a form of diagram for this result can be quite hard to generalise (for me anyway)

the result is also shown with this slightly easier to appreciate (as a generality) diagram:

triangular number building blocks

(1) it is easy to show that a triangular number plus the next triangular number make a square number

(2) can students show that 8 of a triangular number plus 1 makes a square number? (a result credited to Plutarch (100CE))

the algebraic justification for this is reasonably straightforward
a diagram is probably more (immediately) compelling:

(3) can students show that a triangular number plus 6 times the next triangular number plus the next triangular number is a square number?
using a diagram?
using algebra?

(4) can students show with a diagram that 9 of a triangular number plus 1 makes another triangular number?

the algebraic justification for this is a little more demanding
however, there is a neat associated diagram:

HCF and LCM problems

based on a KS2 SAT question
















the following problems are slightly adapted from Ms Sia's Realm of Maths blog
many thanks for these