Times Tables – What Can You Do Once You’ve Learned Them

So What Can You Do Once You’ve Learned Your Times Tables?

The first thing is to answer questions you get asked at school by your teacher or in SATs

Next you’ve got a good start with division.  If you know 3 × 6 is 18

Then it’s not too hard to figure out that   18 ÷ by 6 is 3

and also 18 ÷ by 3 is 6.

For here you can go on to learn

    • remainders
    • fractions
    • long division

In science or economics or statistics although you may start off with some complicated equation, to get an answer you will have to put numbers in the equation.  Then what you’re left with is just arithmetic, times tables are a part of this.

To me knowing times tables and arithmetic is an important part of civil liberties, providing you are prepared to use your knowledge,  as it allows you to check the facts and figures you are given by politicians, businesses or bureaucrats.  Usually this involves

      • tax increases
      • price increases
      • putting up with shortages

Rainfall UK April 2012 – a real life example.

In April 2012 large parts of England were under a hosepipe ban because of a drought.  Almost immediately after the house pipe ban was announced torrential rain came – a boon to comics everywhere who could talk about

“This must be the wettest drought ever”

So how much rain fell?

How much water is this compared to what is used in England every day.

To find out we’ve got to

      • Look up some facts and figures
      • Do some multiplication and division
Fact
Source
Area of England is 130,000 sq km
or 130,000,000,000 square meters
as 1 sq km is 1,000,000 square meters
Rainfall on England in April 2012 was about 130mm
or 0.13 meters
Amount of water distributed a day is 16,000 million liters
or 16,000,000 cubic meters as 1 cubic meter is 1,000 liters

So the volume of water that fell during April 2012 was

0.13 × 130,000,000,000 = 16,9000,000,000 cubic meters

The amount of water used in UK in one year is
365 × 16,000,000 = 5,840,000,000 cubic meters

So more water fell in England during April than is pumped into water pipes of the whole UK in a year in fact 2.89 times as much (16,900,000,000 ÷ 5,840,000,000).

Now it is clearly unreasonable to expect all the water that falls on England to be captured.

But it is remarkable that in 1 month about 2.89 times as much rain fell on just England than is pumped into the water pipes of the entire UK over 1 year.

In addition figure 4i on page 14 of the Environment Agency document shows that 4000 million liters a day of water is lost through leaks.  So almost exactly one quarter of water that has been purified and pumped into pipes is wasted due to leaks.

These 2 facts prompt the following questions:

  • Why are water bills so high?
  • Why are water companies bonuses and dividends so high?
  • Has enough to been done to capture sufficient water to avoid standpipes in times of drought?

 

Scientific Notation –  a shorter way to write large numbers

There is another way of writing larger numbers which doesn’t use so many zeroes

100
is 102
as 100 is a 1 followed by 2 zeroes
1000
is 103
as 1000 is a 1 followed by 3 zeroes
1000000
is 106
as 1000000 is a 1 followed by 6 zeroes

As we will have 1011, this saves writing out a lot of zeroes.  The little number at the top right is called an exponent. These numbers with a 1 followed by a series of zeroes are called powers of ten.

Incidentally if you have numbers other than 100, 1000 etc they are written

200
2 × 102
4000
4 × 103
1600000
 1.6 × 106

Finally decimal fractions less than 1.0 have a negative exponent

0.1
1 × 10-1
0.01
1 × 10-2
0.13
1.3 × 10-1
0.04
4 × 10-2

As 1000 × 100 = 100,000
then 103 × 102 = × 105
so when multiplying powers of ten you just add the exponents.

As 1,000,000 ÷ 100 = 10,000
then 106 ÷ 102 = 104
so when dividing powers of ten you just subtract the exponents.

What about multiples of powers of 10?

3000 × 400 = 3 × 103 × 4 × 102 = 3 × 4 × 103 × 102 = 12 × 105 = 1.2 × 106

There are 3 steps

  • Add the exponents of the powers of ten
  • Multiply the multiples of the powers of ten
  • If necessary adjust so as to keep the number multiplying the power of ten in the answer between 1 and 10.

There are a similar three steps for division

  • Subtract the exponents of the powers of ten
  • Divide the multiples of the powers of ten
  • If necessary adjust so as to keep the number multiplying the power of ten in the answer between 1 and 10.

So 20,000 ÷ 500 = 2 × 104 ÷  5 × 102   = 2 ÷ 5 × 102 = 0.4 × 102 = 4 × 101 = 40

If you’ve used a spreadsheet such as Excel you may have seen a different version of this notation where 2 × 104 is written 2E+4.

A final change for brevity I will write

  • m for meters
  • m2 for square meters
  • m3 for cubic meters

then the table of facts can be rewritten as

Fact
Value
Units
Area of England is
1.3 × 1011
m2
Rainfall on England in April 2012 was about
1.3 × 10-1
m
Amount of water distributed a day is
1.6 × 107
m3

So the volume of water that rained on England in April was

1.3 × 1011 × 1.3 × 10-1 = 1.69 × 1010  m3

The amount of water distributed in UK in a year is

365 ×  1.6 × 107  = 3.65 × 102 ×  1.6 × 107= 3.65 ×  1.6 × 109 = 5.84 × 109 m3

So  1.69 × 1010 ÷ 5.84 × 10= 1.69 ÷ 5.84  × 101 = 0.289 × 10 = 2.89

Again 2.89 times more water fell on England during April 2012 than is pumped into water supply of whole UK in one year.

Same result, but neater shorter numbers and easier calculation.

If you have a spreadsheet why not use it to show your children this calculation?

This entry was posted in Calculation, Multiplication, Numbers, Times Tables and tagged , , . Bookmark the permalink.

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