Download depth task - Mathematics Mastery

Autumn 2 Example
Base nine number system
Normally we count in 10s  we call this a base 10 number system.
So 2563 is short for 2 × 1000 + 5 × 100 + 6 × 10 + 3 × 1 … almost like we say it
Earlier in the year you might also have explored binary (base 2) and base 5 number
systems. If we worked in base 9, our columns would be 1s, 9s, 81s, 729s…
2563 would be 3 × 729 + 4 × 81 + 5 × 9 + 7 × 1
2563base 10 = 3457base 9
Find these numbers in base 9
865
1054
2054
We can also multiply and divide numbers in base 9, see the examples below—see if you
can understand the process and answer the following questions:
3457
×1 1 1 2 3
11483
3457
1 1 1 2
3 11483
Remember that 3 × 7 is
23 base 9 (2 x 9 + 3 x 1)
Remember that 11base 9 ÷ 3
is (1 × 9 + 1 × 1) ÷ 3 so it is
3 remainder 1
Calculate:
i)
426base 9 × 3
iv)
1060base 9 ÷ 3
ii)
155base 9 × 7
v)
3075base 9 ÷ 5
iii)
8432base 9 × 6
vi)
3885base 9 ÷ 8
Investigate multiplication and division in other bases  try base 3, base 7 or base 11!
Associative and Distributive Bases
(6 × 8) × 3 = 6 × (8 × 3)
3 × (4 + 7) = (3 × 4) + (3 × 7)
These calculations
are in base 10
Is multiplication associative and distributive in base 9? Or in binary, or base 5?
Autumn 2 Example
More Multiplication Methods
What if you only knew how to multiply and divide by two... you couldn’t multiply
much could you...?
… You could if you knew the Russian peasant method:
Calculate 17 × 243
1) You half the
numbers in this
column each time,
discounting any
remainders
3) You cross out any
lines with an even
number in this
column
17
243
8
486
4
972
2
1944
1
3888
= 4131
2) You double the
numbers in this
column each time
4) You add the
remaining numbers
in this column
Follow the same 4 steps for these calculations:
i) 13 × 345
ii) 24 × 301
iii) 31 × 197
Explore the method  why can we just discount the remainders, and why do we only add
the odd columns? Are the two connected?
Investigate the link between the Russian Peasant Method and Binary Multiplication
An Interesting Rule for Divisibility
General Rule:
Example: Is 7 a factor of 623?
1)
Write 623 in base 8:
= 1157base 8
2)
Add all the digits:
= 14
3)
Are the digits divisible by 7? YES
Write the dividend in base one greater
than the divisor.
If the sum of the digits is a multiple of
the divisor then the divisor is a factor.
So 7 is a factor of 623!
This is just like the normal test for divisibility by 9… why?
Explore other rules for divisibility in different bases, for example how can we tell if a
number in base 9 is divisible by 9, and what about divisible by 3?