The post How To Find The Slope Of A Line appeared first on StartingArithmetic.com Blog.
]]>Children are taught the equation of a line at primary school, which includes the slope. It is much less common for them to be taught how to calculate the slope of a line from some data, which is strange as they are taught how to use spreadsheets and spreadsheets make this easy.
This post will show you how. I reckon children around 9 or 10 could reproduce the spreadsheets in this post, perhaps with a little help. My 16 year old daughter whizzed through the post creating all the spreadsheets in about 15 minutes.
If 1 ice cream costs £3 then
2 ice creams cost £6 and
3 ice creams cost £9
The cost is 3 times the number of ice creams, which could be written as the equation
y = 3 x (which means 3 × x – the × is usually not written in equations)
where y is the cost of the ice creams and x is the number of ice creams.
We could draw up a table of number of ice creams and how much they cost.
Spreadsheets are ideal for this, for this post I’ve used the free Open Office Calc
The 3.0 in cell B16 is the slope of the graph, as calculated by the slope function.
In the next cell, C16, I’ve entered the formula as text. I find it is useful to do this so when I look at the spreadsheet a few months later I’ve got a note to help me remember what is going on.
The spreadsheet will even plot a graph of the line which shows how the cost varies with number of ice creams. And it will even write the equation of the line as in the graph below
y = 3 x
The slope of this line is 3, the number which multiplies x.
The spreadsheet can calculate the slope of a line from a two columns of numbers using a spreadsheet function called slope (as in the example above).
But we knew the number that multiplied x was 3 (each ice cream cost £3).
This is just a graphical way at looking at what was covered in this post
11+ And KS2 Maths – A Common Type Of Question Has Divide And Multiply For Scaling
What about when we have a graph but we don’t know the slope of the line?
Here is a table of values, like the one above, except a small random amount has been added to or taken away from each y value (the cost of the ice creams), the random amount is in column C.
The slope is not exactly three and the line through the data points is not exactly straight, thick dark blue line. There is also a thin light blue straight line (mostly covered by the dark blue line) which is the line of the equation.
This line is called the trend line, it is the straight line which the spreadsheet has calculated is the best straight line ‘through’ the points given no straight line actually goes through all the points.
A real example.
Permanent Service for Mean Sea Level (PSMSL) hold measurements of sea levels from around the world. Here is a graph of sea levels at Newlyn, Cornwall from 1916 to 2014, sea levels are measured in mille metres(mm).
The slope of the trend line is 1.82, which tells us the average rate of sea level rise at Newlyn is 1.82 mm/year.
There are some periods (e.g. 1970 to 2000) when sea levels have been mainly below the trend line, and others (e.g. 1950 to 1970) when sea levels have been largely above.
There is no sign of a recent acceleration in rate of sea level rise. It is claimed by those who support Climate Change/Global Warming the rate of sea level rise is now above 3mm/year, there is no sign of this in the graph.
Indeed you could even ask is Global Warming happening at all given the lack of acceleration in the rate of sea level rise, given this is a predicted consequence of Global Warming.
So here is an example of how simple arithmetic, the use of freely available data and software tools allows you and your children to check if what you are being told by politicians and the media is true. In this case it would seem not.
Now I’m going to show you,how you and your children can create the spreadsheet results shown above. For this first chart I’m going to go through the steps in great detail. There may seem a lot of things to do but if anyone gets involved in presenting numeric data (at school, college or at work) they will do this sort of thing so often it becomes second nature.
But if you’d rather not do that you can download slopeOfLine.xls
To create the 2 columns of number of ice creams you could just type the numbers in, which wouldn’t be to bad with just 20 numbers. But there are 2 short cuts.
Enter 1 in cell A4, then drag the cursor over cells A4 to A13
From menus select: Edit/Fill/Series
This will open the Fill Series dialog box
click ok and the numbers 2 to 10 will be entered in cells A5 to A13
Click cell B4 and enter the formula =a4*3
I’ve entered the formula as text in cell C4 as a reminder, just put a ” first (“=a4*3)
Once you hit return the spreadsheet will display result of formula
Click on cell B4 again and copy (ctrl + c)
drag the cursor over cells B4 to B13
and paste (ctrl + v)
the spread sheet will enter formula in each cell after adjusting the references, so in cell B5 the formula becomes =A5*3.
Now we can draw graph with trendline
drag the cursor over cells A4 to B13
From menus select Insert/Chart
In the Chart dialogue select Chart Type/XY Scatter/Lines Only
then click Finish
The chart will be displayed with a blue border. When the chart has a blue border this means the chart has been selected and the spreadsheet menus are changed to allow you to alter the properties to be edited. If the chart is not selected (i.e. it doesn’t have blue border, just double click on it and the blue border will come back).
(Note there will be some text displayed to right of graph saying something like
Column B, you can click on this (to select) then press delete to get rid of it).
When the chart is selected you can use the Insert/Trend Lines to insert a trendline.
From the Trend Lines dialog box select Linear and Show Equation then click ok
and we’re done
Here I’ve entered the slope function in cell B16 and displayed what I’ve entered as text in C16.
The second graph is very similar to the first so I will briefly describe the steps but point out the differences.
Enter 1 to 10 in column A as before. I used the same spreadsheet and cells A27 to A36
In B27 enter the forumla =A27*3 + C27
In C27 enter the formula = rand()0.5
There are a lot of numbers after the decimal point.
Select menu Format/Cells
Select Numbers tab of the Format Cells dialog, then the entry which only displays 2 numbers after the decimal point, then click OK.
Then drag cursor over cells B27 and C27, to select them
Copy (ctrl + c)
Drag cursor over cells B28 to B36
Paste (ctrl + v)
The spreadsheet will recalculate different random numbers each time it recalculates the sheet. In Open Office Calc, and in most spreadsheets, pressing F9 will cause the sheet to be recalculated.
Now you can select cells A27 to B26 and insert the chart and display the trendline as before.
Despite having set the format of numbers in cells B27 to C36 to only display to numbers after the decimal point, this will not necessarily be the case with the trendline equation.
If the chart isn’t selected (have blue boudary) double click on it.
Then double click on trendline equation to select it.
Right click on trendline equation
Select Format Trend Line Equation
Select Number tab
Select to only display 2 numbers after decimal point (as before)
and we’re done
For the next graph we need to get the annual sea level data for Newlyn from PSMSL.
If you’re like to get straight to the point click on the link below.
http://www.psmsl.org/data/obtaining/rlr.annual.data/202.rlrdata
Otherwise we’ll head over to
If you click on graphic next to the one labelled ALL (i’ve written click here on the image above).
You’ll be presented with a long table which seems to start with Scandinavian countries, then Baltic, Germany, Holland and Britain. British stations seem to start from north east Scotland and proceed clockwise around the coast. Newlyn is in Cornwall and numbered 202 by PSMSL.
If you click on the 202 you will be presented with a page specific to Newlyn
And if you scroll down a bit there is a link to download annual sea level data
When you click on this link you will be presented with a page like this.
Click on the page
select all (ctrl a)
copy (ctrl c)
Head back to a spreadsheet,
click on a cell (e.g. A1)
paste the data for Newlyn (ctrl v)
And you get the data but it’s all in one column.
Select all the rows with data (for me this was A1:A99)
From menus select Data/Text to Columns
Ensure semicolon is ticked in Text to Columns Dialog
you even get a preview of your data.
Click OK
And we’re done
Almost
When PSMSL is missing data for a year they enter 99999.
At the time of writing this only happens in 2007 and 2010
You can either delete the rows or simpler, click in each cell in turn and press backspace key. This will cause the spreadsheet to think these cells have no data, which means it will ignore these rows if they are included when using slope function or drawing a graph. You could always leave 99999 in draw graph, use slope function, the get rid of 99999s and see the difference it makes.
Now you can select the cells (A1:B99)
Insert Chart
Add Trendline
And do formatting as before and you will end up with something like
There is a mass of free data available on the internet (all the other stations on PSMSL for a start) so you can grab the data and fit trend lines.
How about this from the wonderful woodfortrees.org
http://woodfortrees.org/data/hadcrut4gl
Though you will have to remove the extra lines before and after the data.
When you do a plot using woodfortrees the url is something like this
http://woodfortrees.org/plot/hadcrut4gl
Edit the url changing plot for data
and it gives you the data the plot is based on.
Have a go and let me know how you get on.
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]]>The post All Children Will Be Expected To Learn Times Tables By 11 appeared first on StartingArithmetic.com Blog.
]]>The problem with memorising times tables is to answer a question all you can do is to wait until the answer pops into your head. What do you do if the answer doesn’t pop?
Worse if the wrong answer pops into your head then you’re wrong without knowing it.
The only defence there is against being wrong in maths and science is to check.
In order to check, you have to know a method to calculate the result.
This is why I say it is better to learn how to calculate a result this way you’re never stuck. In practising using the methods to find the answers to times tables questions people will naturally begin to memorise the answers.
But by learning the methods they learn how to check their answers or to calculate the answer if they fail to remember.
After all what is needed is not to learn the times tables but to be able to recall or calculate the results when you need to.
It has been known for over 100 years the best way to learn is to use short sessions which are repeated regularly.
Why not explore the posts on this blog, for example
one way improve maths sats results better teaching
tips for helping your children learn times tables
Even better register your email and receive a free pdf guide explaining how you can help your children or grandchildren learn arithmetic in just a few minutes a day.
After all something must be going wrong at primary schools for some children for Nicky Morgan to make this announcement.
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]]>The post Common Themes In Recent Posts About Succesful Learning appeared first on StartingArithmetic.com Blog.
]]>Arithmetic with Richard Dunne
One Way To Improve Sats Results – Better Teaching
French with Michel Thomas
Getting Failing Students To Learn – Michel Thomas Shows How
Maths and English at Bolton College
How To Get People Who Fail Maths And English GCSE To Pass
There seem to be some common themes
I wonder if it’s that simple how come everyone isn’t doing it?
Just out of curiosity what approach does Starting Arithmetic follow? Why not sign up and get your free copy of What Is Starting Arithmetic And Why Should You Care to find out more.
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]]>The post Primary School Arithmetic Is Essential To Pass Many GCSEs appeared first on StartingArithmetic.com Blog.
]]>Here are some of them
You or your children may not be interested in these subjects but ignoring them makes it hard to get the expected 5 or more passes at GCSE.
Often the arithmetic which would have been all of a question at primary school is now a tool to help solve a problem. Here is an example from chemistry
2 grams of calcium react with hydrogen chloride to give calcium chloride and hydrogen.
Ca + HCl > CaCl_{2} + H_{2}
What mass of hydrogen is produced?
You need to know
So in this chemical reaction the mass of hydrogen produced will be 2/40 the mass of calcium.
2 grams x 2/40 = 1/10 gram hydrogen.
That’s it – pretty simple huh?
But if you can’t do or struggle with the arithmetic the question will be difficult and confusing.
If you become a chemist, or use chemistry in your job, this sort of calculation will be part of your life.
It is an example of what I call scaling
If 2 ice creams cost £2.40 how much do 7 cost?
This sort of question is much beloved of television interviewers trying to show adults can’t do maths (by which they mean arithmetic) when they ambush shoppers and ask them to work out if multibuy offers are value for money
Is it better to buy 2 cans of tomatoes for £3 or 4 for £3?
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]]>The post One Way To Improve Maths Sats Results – Better Teaching appeared first on StartingArithmetic.com Blog.
]]>Kids Don’t Count was aired in 2 parts in February 2010. Here is part 2.
The year before Richard came nearly 1/2 of year 6 failed to get the expected level 4 in maths KS2 sats.
Worse 20% only got a level 2.
It would be easy to just criticise the school for poor maths teaching but they definitely deserve credit for asking for help.
All of the year 6 which Richard taught got at least a level 3 in KS2 sats. Remember the previous year 20% only got a level 2. Some of the pupils got level 4 or 5, even those who had been heading for level 2 at start of the year.
Richard did not have a full year with the children
So I reckon Richard only had about 20 weeks before sats.
How much better would the results have been if the children had been taught by Richard (or someone like him) from year1. Say 215 weeks instead of 20 ( 5 x 39 + 20= 215).
Richard was confident, lively and enthusiastic. He involved and motivated the whole class. He talked about things they could understand (the number of paper cups on a table), and introduced them to how maths is written.
You act the real story
I’ll write the maths story
A large part of learning maths (if not all of it) is learning what the symbols mean and how to manipulate them.
The program pointed out you could become a primary school teacher with only a grade C at GCSE. How can someone who does not properly understand maths confidently teach it.
Richard spent almost all his time involving the class. The other teachers spent a lot of time talking to the class, about what they were about to do or what they had just done. Many of the children looked bored.
When Richard taught the class the children looked, involved, happy and enthusiastic.
So I guess they weren’t stressed then.
If sats are so important how come Wales and Northern Ireland have given them up and Scotland never had them?
Maths is a subject which builds on what has come before. So if you don’t understand what has come before it’s going to be difficult if not impossible to move on. The National Curriculum seems to be based on children learning certain things at certain times during their schooling. This might be fine if all children were the same, but they are not.
If children don’t understand a piece of maths it would seem to make more sense to repeat until they do. At Barton Hill this lack of understanding was clearly due to the teaching as children started to understand maths, and improve rapidly once Richard arrived.
In addition one teacher, Carol White, went back over material the children should have already learned rather than teaching new material and (surprise surprise) the children’s maths improved.
Dispatches asked 155 primary school teachers from schools across the country to take a test of primary school maths
Only one teacher got all answers right.
The Times reported that only
20 per cent of the teachers correctly calculated that 4 + 2 x 5 is 14
a third of the teachers solved the calculation 1.4 divided by 0.1.
Finally what an earth is the point of spending 11 weeks, nearly 1/3 of school year, practising taking an exam rather than learning how to answer the questions in the exam!
Comparing Richard’s approach to teaching with that described by
Pat Harrison from Bolton College and Michel Thomas – the Language Master
Personally I think learning (or cramming) to pass an exam is of no use compared to learning something so well you can use it for the rest of your life.
After all isn’t the point of school learning that you can use it outside school?
Links to Richard Dunne and Maths Makes Sense – Richard’s method of teaching maths.
Some articles about Richard Dunne at Barton Hill
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]]>The post Get Failing Students To Learn – Michel Thomas Shows How appeared first on StartingArithmetic.com Blog.
]]>Here are some key points of Michel’s method
The learner must relax and be free of stress. The teacher is responsible for learning and remembering. 
The learner must think through the answer, using rules and methods supplied by the teacher 
Guessing is not allowed 
Learning comes from answering questions, not from trying to remember 
Learning is achieved in many small steps 
Michel believed, as do I, his method was applicable to all learning, eg maths. 
Success in learning is it’s own reward not least as it raises the learners confidence and aspirations 
Michel expects all his students will succeed 
BF Skinner, leading US behavioural psychologist, said of teaching
If you want to be a good teacher you have to have the student teach you what to do.
The student is always right.
If the student isn’t learning it isn’t the student’s fault. It is the teacher’s fault.
If the student isn’t learning the teacher has not created an environment in which students learn. You must let the student specify the conditions under which the student will actually learn.
This book, which sadly seems out of print, describes Michel’s approach to teaching and learning in detail.
Following Language Master Hodder and Stoughton commissioned Michel to produce commercial versions of his courses. The Start versions allow you to sample Michel’s method for just a few £s.
Here is a review from amazon of Start German
I have never studied German nor any similiar languages, or so I thought. After explaining the ground rules for learning (don’t try to memorise and don’t get stressed out about it), I was told that English and German have a lot in common. Some of our words have AngloSaxon roots and this was demonstrated, bringen, comen, finden etc are pretty much what they look like. Within a short time very basic phrases and rules were embedded and understood. Overall this is an introduction, but it does a very good job of instilling a few “aha” moments.
When I studied languages at school there was an awful amount of learning of grammatical rules, cramming in meaningless vocabulary and general complexity. The Michel Thomas method is not arcane nor a gimmick, it makes a language accessible and the learning much more straightforward. If my schoolday languages were taught this way, I am sure we would be a multilingual nation.
And here is a review of Start French
After hearing a lot of very good reviews on this product, I decided to purchase the Start French product. I was a bit hesitant at first because of the “No Books, No Writing” part and that it was 100% Audio (being able to read and write things I learn is a big thing for me because I’m more of a Visual Learner), however, to my surprise I managed to understand how the French Language works the more and more I listen to the individual tracks on the CD.. along the way, the Teacher provides brief but on point and EXTREMELY important information regarding pronunciation, it is especially helpful because a lot of French words have the same spelling in English but have different pronunciations!
After listening to just the first 5 Tracks, I have realised that I have managed to absorb EVERYTHING learnt so far without even realising it for myself (if that makes sense), I just somehow KNEW what to say out loud; but the bigger picture is I didn’t even need to “memorise” everything the Teacher on the CD teaches me, he just gives me the right tools (some really useful Vocabulary) and I feel that I was given the opportunity to use my head and seriously think about how to say this and that using the Vocabulary that was provided. Gradually, as the Tracks are played one by one, the Vocabulary builds and thus allows me to make longer and more detailed sentences in French!
I can admit that I have not yet listened to the whole CD yet and people may think it a bit too early to write a review, but I wanted to write this particular review in order to stress the fact that only by listening to the first 5 Tracks on the CD, already I felt confident to play around with the Vocabulary and say sentences in French that wasn’t actually taught to me on the CD.
When I first encountered Michel Thomas I was struck by how similar his methods were to those I had already independently developed for StartingArithmetic which I wrote to show how I had helped by children learn arithmetic in just a few minutes a day.
StartingArithmetic talks about arithmetic, naturally, but the main focus is how parents (or grandparents) can help children learn.
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]]>The post How To Get People Who Fail GCSE Maths and English To Pass appeared first on StartingArithmetic.com Blog.
]]>Here is the interview.
What do you think of the interview? Here is a transcript
I felt a mixture of encouragement and depression.
Encouraged there are people who are successfully getting students to pass maths and english GCSEs, despite previous failure.
Depression as if this can be achieved in further education colleges why not in schools? Surely it should be a matter of urgency whatever techniques being successfully used at FE colleges be immediately adopted in schools, if only for those who are not succeeding under the current school regime.
Or is it better for students to spend a few years (2? 5? 10?) not understanding what they are being taught?
It seems to me many of the points made by Pat, I have made in StartingArithmetic using different words.
Many others have also made similar points (e.g. John Taylor Gatto, Michel Thomas).
Here are some examples.
Relevance


Pat  Contextualisation
Once they understand the relevance of english and maths in the subjects 
John Gatto  Where the colonists geniuses? No, the truth is reading, writing and arithmetic only take about 100 hours to transmit as long as the audience is eager and willing to learn. 
Me  It’s curious how speed and accuracy increase along with the “Whats in it for me” factor.
you need to know where they are at (as the californians say) 
Expectation and Aspiration


Pat  But a lot of the priority is to ensure that we do raise their aspirations and they are clear that there is an expectation they will achieve their GCSE. So it’s the positive attitudes and role models of our staff are really important.We raise aspirations, we make sure the learners do know they will achieve but clearly what the main thing is to make the learning relevant to the subject that they’re studying. Once they understand the relevance of english and maths in the subjects they want to learn, then the learning will take place somewhat quickly. 
Michel Thomas  Give students an opportunity to experience success that is all the motivation that’s needed. 
Me  As Henry Ford said
If you think you can 
Small Steps – Working One To One


Pat  ..so we work at the pace of the learner, but we make sure that it’s small steps of achievement. And once they get these small steps of achievement in the vocational area of their choice, and the english and maths is contextualised within that learning then there is a positive achievement and pathway towards the GCSE. 
Me  Little and Often
In the long run it can be more productive to have short sessions where just a few facts are learned at each session. Followed later by some short revision sessions to ensure the knowledge is locked in. 
Some of John Gatto’s ideas are explained in
Jonathan Solity describes Michel Thomas method
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]]>The post Learning Times Tables – How Folding A Tape Measure Can Help appeared first on StartingArithmetic.com Blog.
]]>In this post (and associated video) I use 3 × 5 = 15 as an example.
Here is a tape measure folded in fives, the folds are on 5, 10 and 15.
So 3 folds of size 5 get to 15.
The numbers run 1 to 5 left to right, 6 to 10 right to left and 11 to 15 left to right.
Here the folds are on 3, 6, 9, 12 and 15.
So 5 folds of size 3 also get to 15.
On a snakes and ladders board the numbers run from left to right and then right to left in alternate rows, just like the folded tape measure above.
At startingarithmetic.com/arithmetic/timesTables/timesTableGrid.html there is a times tables calculator grid which helps to explore times tables and can simulate the left to right, right to left “snake effect” seen when folding a tape measure.
The initial screen looks like this
Here is a video where I talk about tape measures, learning times tables and demonstrate this grid. Note the grid features in the second part of the video.
The key to understanding how to use the grid is the text on the buttons shows what happens
WHEN YOU CLICK THE BUTTON
So, if a button says “Snake On” it means the snake effect will be turned on when you click on the button. Which implies the snake effect is currently switched off.
The following table shows the text which may be shown on the different buttons.
This button always stays the same as there is only one option

The “Count to” button shows 3, 5 and 15 as 3 × 5 = 15 is being used in this example
and is the default when you visit the web page.
The text on the “Count to” button cycles through 3 possible options.
In this example it starts with
You can type different numbers in the Rows and Cols boxes at the top left of the page, the grid will change when you
Why not have a go, show your children and let me know how you got on.
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]]>Studies in US have found when you compare the influence teachers and parents have on primary school children one group clearly has more influence.
PARENTS.
I think I will say that again
Studies in US have found parents have a bigger influence on their children when at primary school than teachers.
It’s what I have been saying for years!
It is my belief, based on observing fellow parents, that they drastically underestimate the huge effect they have on their childrens’ development. Two years ago I wrote a report which said exactly this. You can download the report,Seven Little Mistakes in exchange for leaving your email address.
The primary school day in UK is from about 8:45 to 15:00.
Allow 1 ¼ hours for lunch, playtime etc; this leaves 5 hours or 300 minutes for lessons.
Suppose there are 20 pupils in a class and the teacher shared their time equally between pupils that is only 15 minutes a day each. And if there are 30 pupils only 10 minutes.
In practice their will be times when a teacher addresses the whole class, which means a pupil gets more than the 10 or 15 minutes. On the other had their will be assemblies, PE and other activities which mean school time isn’t used for core learning.
What I have tried to show is if you can give your child 10 or 15 minutes help with maths or english although this may seem not very much it is likely to be as much or MORE attention than they are getting at school.
More important it is individual attention from someone important to your child – YOU.
Much of what Carol had to say was about maths (or more accurately arithmetic as most of primary school “maths” is actually arithmetic).
It seems either
What ever the reason maths is a subject which parents feel particularly blocked from helping their children.
My message is have just have a go, after all 2+2 still = 4.
In particular times tables, a common bug bear, are still exactly the same.
I only spent a few minutes googling and didn’t get past the first page of the search results but how about these.
education.stateuniversity.com/pages/2309/ParentalInvolvementinEducation.html
www.happinessonline.org/LoveAndHelpChildren/p16.htm
The message is clear – parental involvement and expectations are crucial.
As a parent you can offer
compare learning maths with learning to speak or learning to read or write.
I doubt it.
Much more likely you children learn to speak naturally by interacting with you and others as you go about your daily lives. If it works for speaking why not for everything else?
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]]>The post Learn 3 And 4 Times Tables Easily Using Your Fingers appeared first on StartingArithmetic.com Blog.
]]>The video explains it all but for those who don’t like videos.
We count moving through numbers one at a time.
Times tables are just stepping through numbers in steps of 2, 3, 4 or whatever.
If you turn your hand so the palm is facing you, you will see 3 creases on each of the 4 fingers.
So if we count along the fingers we move in steps of 3 along each finger.
If we count along a crease across the fingers we move in steps of 4.
That’s basically it.
Once you’ve reached the twelfth crease (i.e. counted from 1 to 12) you can just go back to where you started and count on again (13 to 24, 25 to 36 and so on).
As an added bonus as there are 12 creases on all the fingers of one hand you can count through all the creases to step through the 12 times table. It’s a bit long winded but it gets you the answer.
Finally you can use pairs of fingers to step through the 2 times tables.
If you have children or grandchildren who are learning their times tables why not share this with them.
And more practice with them until they become completely confident.
Starting Arithmetic aims to give parents the tools to help their children gain such familiarity with primary maths that it is second nature to them. Then exams and tests are easy in the way if someone asked you your name you would answer without difficultly or hesitation.
Finally if you would like a copy of the number grid mentioned in the video click here to download the number grid.
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