Squares - all up to 30²should be learned;
The NEXT square is made from adding on to the last known square the number itself and the next number,
(N+1)² = N² + N + N+1
eg 41² = 40² + 40 + 41 = 1681
eg 13² = 12² + 12 + 13 = 144+25 = 169
and going down, subtract both numbers;
19² = 20² – 20 – 19 = 400 - 39 = 361
Squares of numbers ending in 1/2 are made in a similar way:
(N + ½² = N * (N+1) + 1/4
so (51/2 ) ² = 5 x 6 + 1/4 = 30 1/4
(10 1/2 ) ² = 10 x 11 + 1/4 = 110 1/4
… and numbers ending in 5 go much the same … (25)² = 20*30 + 25 = 625
- take the 2, multiply by 3 and stick a 25 on the end. Similarly 652 is made of 6 * 7 and a 25 on the end, 42 & 25, 4225…
Fractions: equivalents
3/5 = 6/10 = 12/20.. Multiply the top by the same as the bottom. Cancelling is the reverse: 36/100 = 18/50 = 9/25… keep going until both numbers have no common factor. Factors are an important part of Yr8 work.
Fractions: arithmetic
Multiplication is the ONLY easy process
a x b = ab numerator
c d cd denominator
Adding, and subtracting are much harder:
First decide what units to work in; at worst, multiply the denominators…
a + b = ad + bc and then do
c d cd cancelling
Subtraction is almost identical:
a - b = ad - bc and then do
c d cd cancelling
Division requires an extra move: dividing by 2 is the same as multiplying by 1/2, similarly dividing by 2/3 is the same as multiplying by 3/2, so
a ÷ b = a x d = ad
c d c b bc
Series:
Adding terms to a series by writing successive differences and seeing a pattern. Other series, eg Fibonacci [1,1,2,3,5,8,13,21,34…..add previous two], Factorial [1,2,6,24,120,720,…. N!], primes are not a series but a sequence (no formula).
Series with names: odds, evens, triangle Nos, Pyramid Nos, Squares, cubes
Finding the formula of a series: Find the differences which are constant: First is 1, second is N, third is1/2 N^2, fourth is 1/6 N^3, fifth is 1/24 N^4,….. so R+1 th is 1/R! N^R. Subtract each successfully found biggest term and treat the remainder as a new problem. Excel spreadsheet on internal network solves these for you.
Numeric bases.
Counting on fingers in binary. Adding, subtracting and multiplying in Binary, ternary, duodecimal, hexadecimal……. Working up to using any base. Converting numbers between bases, particularly any combination of bases 2,4,8,16 and in/out of denary.
