Higher Maths Anxiety – Is It Any Different From Fear And Anxiety

The Daily Mail today has a story about how maths CAN really make your head hurt, which reports on the rather more sober article in “Plos One” by Sian Beilock and Ian Lyons from University of Chicago.

It seems to me they’re really talking about panic, fear and embarrassment.  Just about everyone has had the experience of being in a class room when the teacher has just asked a question and is scanning round the class for someone to answer

Don’t pick me

Please don’t pick me

Oh Please don’t pick me

In many ways maths questions are worse than those from other subjects as there is not the option of giving your opinion, there is only one answer and you’ve either got it or you haven’t.

And if you don’t know the answer you’re worried about looking

  • foolish
  • ignorant
  • or stupid

Panic effects all of us, even me, even now.

The trick is learning to deal with it.

One way is to have a definite method to work out the answer

Another way is having enough experience of answering questions that you have the confidence you will be able to solve the problem, even though you may not have the answer right now.

Our brains can only cope with so many thoughts at anyone time, and if your brain is fully occupied with pictures and feelings of impending doom, there isn’t enough brain power left for thinking through to the answer.

Starting Arithmetic is not about rote learning, at least not in the usual sense.  When my children were growing up I showed them how it was much better to learn methods for solving problems, rather than just trying to memorise answers.  In Starting Arithmetic I just wrote down what I used to tell my children so you can tell your children or grandchildren.

Over the last year or so I have been campaigning over plans by the UK Environment Agency (EA) to knock down sea walls near where I live and let the sea flood farm land, parks and homes on the grounds that it is “uneconomic” to maintain sea walls.

I have written about this on another of my blogs, JeremyShiers.com/blog, and given talks.

After a recent talk someone asked me how I was doing with Starting Arithmetic and offered to help as she is a governor at a local school.

Then she said of course I’ve always been rubbish at maths, even times tables (strong positive statements always help!).  This is someone with grown children so well past school days.

So I explained there are 66 different answers in the times tables from 2×2 to 12×12

And if you use your imagination to see things written down in your mind some of the answers are easy.

Really really easy.

In the 10 table just see whatever number your are multiplying by 10 with a zero after it

So 5×10  is  5 0  or fifty

From 2×10 to 12×10 there are 11 answers, 1/6 of all there are to learn.

How hard was that?

Up to 9×11 things are almost as easy, whatever number you are multiplying 11 by you just see written down twice.

9×11 = 9 9 or ninety nine

3×11 = 3 3 or thirty three

So far she agreed things were easy but when we got to 9 times table there was a small panic attack.

There are 3 steps for numbers from 9×2 to 9×9 (we can already do 9×10 and 9×11)

And we make use of the fact that in the 9 times table when you add the digits they always come to 9.

  1. 0 + 9
  2. 1 + 8
  3. 2 + 7
  4. 3 + 6
  5. 4 + 5

There are only 5 pairs of digits, though they may come in either order for example

  • 18 in eighteen
  • 81 in eighty one

To multiply a number by 9 from 2 up to 9:

  • Whatever number you are multiplying by 9, multiply by 10
  • Make the number in the tens column 1 smaller
  • Choose whatever number for the units which makes 9 when you add it to what’s in the tens column

So for  7×9

  • 7×10 = 70
  • 70 has a 7 in the tens column, one less is 6 so 60
  • 6 + 3 = 9 so the answer to 7×9 is 63

Now I have to say at this point there was a little panic, a wobble if you like. But the moment passed and the correct answer was found.

Perhaps it was wishful thinking or just my imagination but I thought I detected a little pride in achieving what had been felt to be unachievable only minutes before.

Probably my only contribution was being there, if I hadn’t been there I feel that she would have just given up.  But as I was,  she thought a little longer, perhaps thirty seconds and got the right answer.

It is worth pointing out that the entire conversation from

  • I’ve always been rubbish at maths
  • To 7×9 is 63

was at the most 10 minutes.

And in that 10 minutes we covered

  1. 11 answers in 10 times table
  2.  8 answers in 11 times table
  3.  8 answers in  9 times table

So 27 answers in all out of a total of 66!!! Over 40% of all there is to learn.

As Franklin D. Roosevelt said

We have nothing to fear except fear itself.

Learning at home should be a much safer environment for your children as they can make mistakes without fear of being judged as stupid by the peers or teachers.

And after all a mistake isn’t a mistake if you learn from it.

In helping your children learn, even simple things like times tables, you are giving the the confidence they can learn, that they can be right.  Such confidence is essential.  In the words of Henry Ford.

If you think you can

Or think you can’t

You’re probably right

 

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Learning Times Tables – Rote Learning Understanding Or Do You Need Both

Recently in UK Nick Herbet, the schools minister, said pupils should learn their tables by rote at primary school.

A few days later it was reported

teaching children maths by making them learn times tables by rote could worsen exam results because they risk failing to properly understand the subject, according to an Oxford University study.

So whose right?  In my view

  • they are both right
  • they are both wrong
  • they are both being silly
  • and by far the most important they both miss the following
      1. interest and motivation are vital for learning
      2. it is better to learn a method to calculate answers to times tables questions than to just memorise facts
      3. with practice at multiplication the answers get memorised automatically

Let’s deal with the first three points. They are both being silly as nobody is saying either

  1. Pupils must learn entirely by memorisation without understanding
  2. Pupils must only learn by developing understanding without any element or practice or repetition

If anything the two ‘different’ positions are actually two different aspects of learning,
so the whole debate is a pointless waste of time.

Instead of bickering and point scoring why not focus on the important things about learning.

Firstly to learn, train or practice anything you need a reason, something which will fill you with a desire to keep working and learning. What keeps athletes and musicians (or doctors, lawyers and scientists) training and practicing for hours every day, for years and years.

They must have some sort of goal which they are aiming for.

If you can inspire children (or anyone) to want to learn, your job is pretty much done.

There are many goals such as such as.

  • To be good at science, maths or engineering
  • To be a great scientist and make new discoveries
  • To understand how the world works
  • To teach others
  • To invent something great like the iPad
  • To help build a rocket to journey to the stars
  • Become the world’s greatest investor like Warren Buffett
  • Become a finance whizz on Wall Street or City of London

The second important fact about learning tables is it is essential to learn a method for working out answers if you can’t remember them. And if you have a method for working out the answer it’s a good thing to check any answer that pops into your head to see if it is actually right.

The next time someone tells you rote learning is bad ask them,

How did you learn

  • your name
  • the letters of the alphabet
  • the names of the numbers from 1 to 10, from 1 to 100
  • the names of colours
  • the names of days of the week
  • the name of months in the year
  • you stop at a red light
  • 2 + 2 = 4
  • how did you learn to count from 1 to 100, this is just the one times table.
    The other tables step forwards (from zero) in steps of 2 to 12
    (or 2 to 10 if you think 12 is too challenging)

And by the way that’s a method for working out an answer in the times tables,
just step forward from 0.

So for 4 × 5 we take 4 steps forward from 0 in steps of size 5

0, 5, 10, 15, 20

But as 4 × 5 = 5 × 4 we could take 5 steps forward from 0 in steps of size 4

0, 4, 8, 12, 16, 20

Which way do you think is easier?

Why not choose the easiest way.

We learn by building associations, which sometimes just come from saying things, or hearing things over and over again. So although learning by rote is used as a term of abuse by some people it is actually part of how we learn.

A very important point is you don’t just learn by memorising (say as lines in a play or a poem) you also learn by answering questions.

Each time you answer 6 × 6 = 36, (or hear someone else answer it) you cement that knowledge a little bit more in your mind.

But what happens if the answer doesn’t come?

Well there’s not much point in waiting more than a few seconds as you can usually work the answers in less than 30 seconds just by stepping through.

And the best thing about this is it helps learn both

  1. both 6 × 6 = 36
  2. and how to work out 6 × 6 = 36 by stepping through numbers 6 at a time

There are certain tricks or short cuts which I explain in Starting Arithmetic
but stepping through always works.

Why bother to learn tables if we can learn a method so we can always work out the answers?

You may remember first learning to drive there seemed so many different things to do all at the same time, the number of tasks seemed overwhelming. With practice changing gear, accelerating, steering, braking, signaling becomes automatic freeing your mind to concentrate on watching the road.

Your mind can only cope with so much at any one time.
When trying to solve more advanced problems if tables have been learned so answers are automatic this frees up the brain to focus on bigger picture.

Regular practice means answers get memorised and so we can answer quickly.
Without regular use, what has been learned fades away.

Use need not only happen at school. Why not encourage your children to enjoy a hobby which involves some maths and tables.

  • cooking
  • carpentry
  • navigation
  • electronics
  • astronomy

Most of what is called maths (or math) at primary school is actually arithmetic.
Although science and engineering may involve more complicated maths
if you are going to work out any definite answer at some point you have to
put numbers in, in which case you are back to arithmetic.

There are some complications

    • there are usually some formula and algebra
    • it is necessary to work with units such as kilograms, meters, seconds
    • many of the calculations involve very large or very small numbers. To make things easier numbers are written using powers of ten, so called scientific notation.

But after all this is just plain arithmetic.

April 2012 was unusually wet in England.
In this post  Times tables what can yo do once you’ve learned them I show you can use arithmetic to estimate the amount of water that fell on England in April 2012 and compare this with the amount of water pumped into water pipes in a year for all UK.

It seems nearly 3 times more water fell on just England in just April 2012 than is used by whole UK in a year!

Now why are our water bills so high?

In the pdf version of this post I show how you can use arithmetic, with a little help from Newton, to calculate how long the moon takes to go once around the earth (27.4 days).

Posted in Maths, Times Tables | Tagged , , , | 1 Comment

Summer Holidays Games and Maths

School summer holidays are a great time for children to play free from the routine of having to go to school every weekday.

Some children drop back during the long break, whilst others do not.  A gap begins to open which widens with each holiday (especially the long summer breaks).

Can any thing be done?

Yes

And it’s very easy, just spend a few minutes a day,  a few days a week doing things were your children have to think a bit.

  • Read a book
  • Tell a story
  • Play card games or board games (snakes and ladders or monopoly)
  • Write a story
  • Let your children work out how much money they need to buy some ice creams and what the change will be from £5

If your children don’t know all their times tables why not try and learn some or all of them!

Even if they just learn one new times table over the holiday that’s progress.

I like using physical things, for example

  • If there are 5  stools and each has 3 legs on how many legs are there altogether?
  • If there are 4 horses pulling a coach how many legs are there altogether?

Going to the beach gives more great opportunities

Collect pebbles in to groups, each with the same number in  – then count the total number

Draw out a board of 10 x 10 squares (or 12 x 12) and  number the squares  1 to 100 (or 1 to 144) and then let your children jump forwards 2, 3, 4 at a time playing a sort of numbers hopscotch.

Of course you don’t have to make square you could draw the numbers in a long line, on the back of a snake for example.
At SevenLittleMistakes.com there are more ideas in this post Are School Holidays Too Long.

To encourage you to spend some time with your children learning maths I’ve reduced the price of Starting Arithmetic and Times Tables From Starting Arithmetic by two/thirds during the summer holidays.

It’s not hard

It’s not complicated

It just needs doing regularly

After all if you never spoke to your children how good would they be at speaking?

And if you children already know all their times tables (well done!) there’s always more, for example

  • Division
  • Division with remainders
  • Decimals and percentages

Have fun!

 

Posted in Board Games, Counting, Games, Multiplication, Numbers, Times Tables | Tagged , , , , , | Leave a comment

11+ And KS2 Maths – A Common Type Of Question Has Divide And Multiply For Scaling

In basic arithmetic there are four different operations

+ – × ÷

There is a type of question which turns up in various forms, all basically the same, which involve divide and multiply (division and multiplication if you prefer). I call this scaling.

Note there is a pdf version of this post created with latex and tikz which I recommend you read as

  1. latex and tikz enables the maths and graphs to be formatted better
  2. there are more example questions

On with post.

It makes sense to learn to recognise this type of problem and how to solve it as there are often several questions of this type in an 11+ or KS2 maths tests. Part of the art of succeeding at tests and exams is to be able to answer as many questions as you can as quickly as possible, in order to

a)   secure as many marks as possible as early as possible

A key point in answering questions in exams and tests is to answer the easy question first.  You do not (usually) have to answer the questions in the order they are set.

b)   leave more time for harder questions

So if you know there is a type of question which may crop up more than once in various different forms it makes sense to learn how to recognise and solve these problems quickly.

Here is a simple example

If 2 ice creams cost £2.40 how much would 3 cost?

To solve this you have to first find out how much 1 ice cream costs
Divide the £2.40 (the price of 2 ice creams) by 2
So 1 ice cream costs £1.20
Then multiply by £1.20 by 3 to get £3.60 for the price of 3 ice creams.

I usually start explaining how to solve these problems  like this, doing the divide first and then the multiply to stress why these steps are done, i.e. to find the price of one and once you know this you can multiply to find the price of any number.

If you really want to stress the steps it’s as well to say these questions

  • only involve divide and multiply
  • never add and subtract

Later you can explain for sums involving only multiplication and division they can be worked out in any order, so each of the following all have the same value

eqn_1Note the sums in brackets are worked out first

and it make sense to choose the one that’s easiest to work out. Here there’s not much in it, but if the numbers we had to multiply and divide by where larger, for example

If 22 ice creams cost £2.40 how much would 33 cost?

It makes sense to see the problem as

eqn_2And then it’s probably easiest to switch to

eqn_3In KS2 the numbers are usually easy, in 11+ the numbers can involve 3 decimal places.

There is a variation of this type of problem where you start with the price of 1, for example

Butter costs 1.56 a kilogram how much would 1.6 kg cost?

There are 900 words in each chapter of a book. In 12 chapters how many words are there?

Sometimes children get confused as they don’t have to do the divide.

In 11+ maths exam often starts with questions that just tell you to calculate something for example

Calculate 1.2 × 1.43

Later on the questions are usually asked in words which mean whoever is taking the test has to translate the words into a series of arithmetic calculations.

Here are some example questions (there are more in the pdf version of this post) which show some of the different ways this type of scaling problem can come up, occasionally there are variations with also include + and -.

Question 1

1/7 of a number is 9.

What is 1/3?

Question 2

Butter costs £1.47 a kilogram how much does 1.92 kilograms cost.

Question 3

piechart_1

The pie chart gives the percentages of different types of fruit in pupils lunch boxes at a school.

If there are 11 lunch boxes with apples how many lunch boxes have bananas?

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Number Bonds Of 20 Worksheets

Every time this post is displayed a new set of questions about number bonds of 20 will be randomly generated in the table below.

There may be some duplicates, that can happen with random numbers.  Anyway it is always a good idea to ask the same question more than once as you may not get the same answer each time.

What are number bonds?

This post explains number bonds in more detail but basically the number bonds of a particular number are all the pairs of smaller numbers that add up to that number.  The questions in the table below should make things clear.

The worksheets use the following alternatives for adding

  • +
  • plus
  • add
  • and

and the following alternatives for subtracting

  • minus
  • subtract
  • takeaway
  • less
Number bounds of 20
20
less
4
=
?
?
and
16
=
20
20
subtract
?
=
8
6
plus
?
=
20
20
minus
2
=
?
0
add
?
=
20
?
add
20
=
20
2
plus
?
=
20
?
+
6
=
20
20
less
?
=
4
20
?
=
10
18
+
?
=
20
?
and
10
=
20
20
minus
2
=
?
20
minus
?
=
6

 

Number Bonds Worksheets Spreadsheet

Here is an Excel spreadsheet which will generate Number Bonds Worksheets
There are separate posts for number bonds of 9, 10 and 20 but all link to the same spreadsheet which has 3 worksheets in it.

Why Bother To Learn Number Bonds?

One reason is that children are taught number bonds and so will be expected to answer questions on them.

It’s just another way to practice addition and subtraction.

Learning number bonds of 10 and 20 help learn how to to arithmetic with more than one column.

 

 

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Number Bonds Of 9 Worksheets

Every time this post is displayed a new set of questions about number bonds of 9 will be randomly generated in the table below.

Note there may be some duplicates, that can happen with random numbers.  It doesn’t hurt to ask the same question more than once so you can check you get the same answer each time.

What are number bonds?

This post explains number bonds in more detail but basically the number bonds of a particular number are all the pairs of smaller numbers that add up to that number.  The questions in the table below should make things clear.

The worksheets use the following alternatives for adding

  • +
  • plus
  • add
  • and

and the following alternatives for subtracting

  • minus
  • subtract
  • takeaway
  • less

Number bounds of 9

9
minus
9
=
?
9
less
?
=
0
3
plus
?
=
9
?
and
5
=
9
?
add
9
=
9
9
subtract
1
=
?
7
and
?
=
9
?
and
8
=
9
?
add
0
=
9
6
+
?
=
9
6
+
?
=
9
9
takeaway
?
=
0
?
plus
8
=
9
9
minus
3
=
?
9
minus
0
=
?

Number Bonds Worksheets Spreadsheet

Here is an Excel spreadsheet which will generate Number Bonds Worksheets
There are separate posts for number bonds of 9, 10 and 20 but all link to the same spreadsheet which has 3 worksheets in it.

Why Bother To Learn Number Bonds Of 9?

It is particularly useful to learn the number bonds of 9 as it just so happens for every number in the 9 times table the digits (the numbers in the different columns)  always add up to 9.  Though for 99 you have to add twice (9 + 9 = 18, 1 + 8 = 9)

This gives an easy way to work out the answers to the 9 times tables from 9 × 2 to 9 × 9

  1. Find 10 times
  2. Make the tens column of the answer one smaller
  3. Put the number bond of 9 in the units column

For example 9 7

  1. 10 × 7 = 70
  2. 70 changes to 60
  3. The 6 + 3 = 9 so the answer is 63

 
Note it is easier to work out 9 ×10 and 9 × 11 from 10 and 11 times tables
 
9 × 10 – write a 0 after the 9 – so 9 × 10 = 90
 
9 × 9 – write the 9 down twice – so 9 × 11 = 99
 

 

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Number Bonds Of 10 Worksheets

Every time this post is displayed a new set of questions about number bonds of 10 will be randomly generated in the table below.

Note there may be some duplicates, that can happen with random numbers.  Anyway it is always a good idea to ask the same question more than once as you may not get the same answer each time.

 What are number bonds?

This post explains number bonds in more detail but basically the number bonds of a particular number are all the pairs of smaller numbers that add up to that number.  The questions in the table below should make things clear.

The worksheets use the following alternatives for adding

  • +
  • plus
  • add
  • and

and the following alternatives for subtracting

  • minus
  • subtract
  • takeaway
  • less

Number bounds of 10

0
plus
?
=
10
10
?
=
9
10
subtract
?
=
6
10
?
=
3
2
+
?
=
10
10
?
=
4
10
takeaway
?
=
6
?
plus
0
=
10
6
plus
?
=
10
10
subtract
?
=
10
10
less
3
=
?
10
minus
?
=
8
1
add
?
=
10
10
takeaway
?
=
1
?
+
0
=
10

 

Number Bonds Worksheets Spreadsheet

Here is an Excel spreadsheet which will generate Number Bonds Worksheets
There are separate posts for number bonds of 9, 10 and 20 but all link to the same spreadsheet which has 3 worksheets in it.

Why Bother To Learn Number Bonds?

One reason is that children are taught number bonds and so will be expected to answer questions on them.

It’s just another way to practice addition and subtraction.

Learning number bonds of 10 and 20 help learn how to to arithmetic with more than one column.

Suppose you had to add 8 and 4.

If you knew 8 + 2 = 10, and 2  +  2 = 4

then you could work out 8 add 4 is the same as 10 add 2 = 12.

 

 

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Snakes and Ladders Board Printable Template

Snakes and ladders (or chutes and ladders) printable template for you to make your own board for free.

Click here to in exchange for leaving your email address.

It doesn’t actually have any snakes or ladders on! You can draw those as you wish.

Either print out the template on A4 cardboard, or print on A4 paper and stick to board. There are 2 pages so you will have to tape the 2 parts together, just like a real snakes and ladders board. And just about every other game board.

Here is a if you wish to draw the numbers in as well.

This post shows the squares Snakes and Ladders go between and has a picture of a completed board.

Of course you can always save time and buy one from Amazon!

 

[Update 25 June 2012]
It’s not to hard to draw a basic ladder here’s one I did earlier
Hand Drawn Ladder

Drawing snakes is bit more difficult but there are plenty of web sites showing you how
howtodrawsnakes.com/
does what it say on the tin and shows you how to draw realistic looking cartoon snakes.

Here is a more detailed video instruction.
drawingnow.com/videos/id_45517-cartoon-snake.html

Here is an example of simplistic snakes on a snakes and ladders board.
craftscope.com/snakes-and-ladders-game/
But I’m not knocking it by saying it’s simplistic – it works!

[End Update 25 June 2012]

 

[update 3 September 2017]

If you’d like to draw better snakes&ladders, or anything else,
have a look at this blog post www.jenreviews.com/how-to-draw-better/  a 21 page post written by Jenn who produced the drawingnow.com/videos/id_45517-cartoon-snake.html

[end update 3 September 2017]

 

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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?

Posted in Calculation, Multiplication, Numbers, Times Tables | Tagged , , | Leave a comment

Free Times Tables Worksheets

There are lots of people searching Times Tables worksheets on the internet, according to Google.

And there are lots of sites offering such worksheets.

So here is a small contribution. Every time this post is displayed a new set of questions will be randomly generated in the table below.  Note there may be some  duplicates, that can happen with random numbers.

9
×
11
12
×
10
2
×
2
7
×
4
3
×
2
1
×
9
12
×
5
2
×
1
1
×
10
11
×
7

Times Tables Worksheets Spreadsheet

Here is an Excel spreadsheet which will generate times tables worksheets.

There are two worksheets

  • One generates questions from all tables from 0 × 0 to 12 × 12. Whilst it’s not usual to learn 0 × table I like to include it as it’s really easy – all the answers are 0. The 1 x table is not much harder.
  • The other allows you to select a number and all the questions will be specific to that table

Personally I am not a big fan of such worksheets, particularly when your children are starting to learn tables.

The biggest advantage you have when working with someone directly is you can respond to their answers.

So if some questions are easy you can move on.

If some questions are hard or impossible to answer you can explore why.

Worksheets do not offer this flexibility.

To ask a times table question all that is needed is to choose 2 numbers between 0 and 12, for example

3 × 7

In Starting Arithmetic I provide times tables templates which I recommend your children fill in to explore and create their own personal copies of the Times Tables.  These templates are shown in learning times tables and in tips for helping your children learn times tables

Once your children have worked out their own copies of the tables you can at first ask them questions whilst they are actually looking at what they have worked out.

Then you can ask whilst their tables are in sight but they are not actually looking at them.

But what numbers to multiply?

If you concentrating on learning one table at a time that fixes one of the numbers,
for the other you just have to choose a number between 0 and 12.

You could

  • step forwards from 0, 1 at a time
  • step backwards from 12, 1 step at a time
  • step forwards 2,3,4… at a time
  • step backwards from 12 in steps of 2,3,4… at a time
  • choose a number in somewhere in the middle, say 6, and then alternately choose numbers higher and lower than this
  • throw to 2 dice to get a number between 2 and 12
  • use the RandBetween spreadsheet function which is available in Excel, OpenOffice and Google Docs. This is how I created the spreadsheet which generates times tables worksheets.

Whatever method you use

  • ask the same question several times in a session, although you should get the same answer each time it is common to get different answers.
  • ask the questions both ways around so if you ask 2 × 3 also ask 3 × 2.
    Stress both these questions have the same answer as times tables answers are ALWAYS the same which ever way around the numbers are.

Perhaps by now you will agree that coming up with questions is not that hard, it’s just a matter of doing it. A bit like going for a run or going to the gym. And like going to the gym, short regular sessions are better than a long session every now and then, when you try and make up for the sessions you’ve missed.

Also it is important to not worry about how well things go on any particular day, but trust that by doing the work, over time improvement will come naturally.

I have left the most important thing to last. Although answering questions helps build knowledge and understanding there is something more important. And that is to learn a definite method for working out answers if you get stuck.

In the worst case you can just step forward 1 step at a time in groups of whatever number you are multiplying.  So 3 × 4 would be counted

  • 1 2 3
  • 4 5 6
  • 7 8 9
  • 10 11 12

This might seem like a lot of work, but it is guaranteed to work, and this is how some children handle questions when they first start. Also it probably takes less than 10 seconds to do. This the crux of the matter.  Children can wait a minute or two for the answer to just pop into their head. This is pointless, if they haven’t got the answer after 2 or 3 seconds they should start working it out as this will almost certainly be quicker than just waiting.

Later when they get to know their tables a bit, and/or get better at adding they may just go

3 6 9 12

Both these methods will always work, so there is never any reason to just wait in silence for the answer to arrive. If they don’t know the answer they should work it out and you should encourage them to do so.  Out loud is best, that way you know how they are tacking the problem.

Now it would be be possible to work out numbers further up the times table using either of these methods but it would be a lot of work and hence error prone. Imagine working out 12 × 12 by counting forward 1 at a time!

Fortunately there is a short cut. Some of the tables are much easier than others. I would count

  • 2 x
  • 5 x
  • 9 x
  • 10 x
  • 11 x

as easy, though 9 x and 11 x are harder when the answers are over 100.

Many questions may have one or both numbers from the ‘easy list’.

But there will be some, such as 6 x 7, where this is not so. The thing to do here is to go to the nearest ‘easy’ table
and then either add on, or step forward 1 at a time.

For 6 x 7, the closest ‘easy table’ is 5x.

5 x 7 is 35 and 6 x 7 will be 7 more than this.

The simplest way is to step forward 1 at a time 7 steps – 36, 37, 38, 39, 40, 41 and 42.

Usually children find stepping forward easier than adding. But by the time they are learning 6x and 7x times tables they will be able to add two numbers that sum to less than 20. So they will know that

5 + 7 = 12

So you could encourage them along the adding path by saying something like

As 5 + 7 = 12  has a 2 in units
the answer for 35 + 7 must also have a 2 in the units
As 5 + 7 = 12  has a 1 in tens
the answer for 35 + 7 must also be 1 more in the tens than 35,
so a 4 in the tens and 2 in the units is 42

This may seem like more effort than stepping forward 1 at a time but with practice it will be come easier and more fluent until finally it is automatic.

The worst case is 8 x 8 which might involve adding on from 5 x 8 3 times.

It is simpler to say
5 x 8 = 40
3 x 8 = 24
so 8 x 8 = 40 + 24 = 64

Alternatively 8 x 9 is 72 so 8 x 8 is 8 less than 72
Then work the answer out by subtraction or stepping backwards.

However children tend to find subtraction harder, which ever method they have used to do it. Often by the time they’ve become comfortable with subtracting 8 from 72 they already know what 8 x 8 is!

In summary if your children learn a method they can always work the answer out.

If they rely on memory, or the answer just popping into their heads when memory fails or nothing pops they are stuck.

Enjoy your times tables worksheets.

Posted in Multiplication, Numbers, Times Tables, Times Tables Worksheets | Tagged , , , , , | 1 Comment