Friday, 31 January 2014

Happy Year of the Wooden Horse!

Today is the first day of the Chinese New Year.  Like the Western New Year, Chinese people usher in the Chinese New Year with a countdown and fireworks on new year's eve.

It is certainly much more festive and done a much bigger scale in the Far East than in the UK.  The Chinese New Year lights and the number of decorations in the Far East where Chinese is the dominant culture reminds me very much of the same level of festivities during Christmas time here.

There is also the Chinese New Year equivalent of the Winter Wonderland.

Tiger and I are joining many Chinese people around the world to celebrate this ancient tradition.  I will be taking a few days off to celebrate it with my family, but I hope to be back soon to share what we have done.

To all who celebrate Chinese New Year:

This post is linked up to Chinese Activities Link Up.

Wednesday, 29 January 2014

A Chinese New Year Taster

Being the last-minuter that I am, I must be asking for trouble by declaring on Monday that we plan to learn more about Chinese New Year this year, when the big day is only days away.  That is a sure way to work myself into a frenzy!

Luckily the V&A came to my rescue yet again, with a whole day of Chinese New Year related activities!  Woo hoo!  With many varied cultural activities prepared by professionals, all we had to do was to show up and work our way through them -- such an arrangement is this homeschool mum's best-case scenario.

The museum's central courtyard has a large landscape installation by a modern Chinese artist.  The landscape is of major significance in Chinese art, as seen in the art of bonsai and traditional Chinese landscape paintings.  While the pre-Impressionistic Western landscape paintings' emphasis is on realistic representation, the Chinese landscape paintings have always been non-representative, with its emphasis being on conveying certain philosophical and spiritual aspirations (most notably due to the influence of Laozi's philosophy as expressed in Taoism).

The first craft that Tiger made at the museum was the Chinese Opera mask.

The mask is made entirely out of paper and looks like this when completed:

Traditional Chinese Opera masks are painted directly onto the actors' faces, using specific colours and patterns to represent well-known operatic characters.

After the craft session, the museum even had a short "Introduction to Chinese Opera" workshop to take the children through a few of the uniquely-identifiable stage gestures and movements of each character.

Other activities Tiger was to do that day included:
(1) Making a mini kite out of lollisticks, craft tissues and coloured cards.

(2) Watching a paper cutting demonstration.

(3) Practising Chinese calligraphy.  The practice in Chinese New Year is to write certain auspicious wordings on a piece of red paper and stick them around the house to signify bringing good luck and prosperity into the home.  A very common word that is used is "fu" (福), that embodies the meaning of good luck and good fortune.  There are many different styles or schools of Chinese calligraphy, hence there are numerous ways to write the word "fu", as you can see Tiger practising below.

The day ended with us attending a traditional Chinese instrumental concert, performed by the musicians from the UK Chinese Ensemble.

The style, sounds, and atmosphere of traditional Chinese music is very different from those of classical Western music.  The concert gave the audience an excellent experience with a wide repertoire of solo pieces, duets, as well as ensemble pieces.  For example:
(1) a zither or guqin (古琴) ensemble:

(2) a yangqin (扬琴) solo:

(3) an erhu (二胡) and yangqin (扬琴) duet:

(4) a guzheng (古筝) solo:

(5) a xiao (萧)solo:

(6) a pipa (琵琶) solo:

(7) an ensemble piece:

The finale was a traditional Chinese New Year piece titled <<喜洋洋>>, loosely translated as "Beaming with Joy":

The score for this piece of traditional Chinese New Year music was rearranged for the ensemble performance at the V&A.  It was originally written for a full Chinese orchestra:

The day has been a good introductory overview to the Chinese culture.  To do justice to the depth and breadth of the Chinese culture (as it is with any culture), we will have to spend more time to study the individual components of this one-day "crash course" individually.

This post is linked up to:
  1. Creative Kids Culture Blog Hop #12
  2. Hip Homeschool Hop - 1/28/2014
  3. History and Geography Meme: Ancient Greece activities
  4. Entertaining and Educational - Energy
  5. Collage Friday: Thankful for Homeschool
  6. Weekly Wrap-up: The One with all the Snow
  7. The (New) Homeschool Mother's Journal (2/1/14) 
  8. Chinese Activities Link Up

Tuesday, 28 January 2014

Eye on the Ball, Please! Part 3

This is the third part of a series of my termly plans.  The first two parts are:
  1. Ball #1: Language Arts
  2. Ball #2: Mandarin

Ball #3: Mathematics - Addition and Area
The way that we have been learning mathematics here seems to be increasingly topic- or themed-based rather than curriculum-based, which suits us fine.  We started off following a maths curriculum, which worked very well, but was starting to get a little tedious for Tiger.  Once we started experimenting with learning-what-we-want-to-learn-when-we-want-to in maths, the approach feels a lot more logical and natural to us and is what we do now.

This term we are going to focus on two topics:
  1. addition
  2. area

What About Gaps?
It seems that many homeschoolers are uncomfortable with the idea of not following a curriculum in maths.  Usually their biggest concern is gaps in their children's maths skills or knowledge.  Following an established curriculum certainly gives some people -- notably adult and children who are naturally predisposed to a systematic, highly organised way of learning -- a sense of security that they have "covered everything" that a child of a certain age/level needs to know.

The way that I handle the question of gaps is very simple:  I make a mental note of it when I notice it, then I find an opportune time to introduce it.  The next question then is, what is an opportune time?  Well, my definition of an opportune time is:
  • when I have gathered the necessary teaching materials/resources; or
  • when the topic ties in with another area that we are studying, e.g. there are many cross-curricula opportunities to be created around the topic of geometry, symmetry, tessallations, Islamic art, etc; or
  • immediately - we are prepared to drop everything else if it is a matter of urgency to learn a certain topic.  So far, we haven't come across any situation that requires us to adress our mathematical deficiencies immediately.

My confidence in our approach comes from the realisation that:
  1. Time is on our side.  Tiger is only nine years old so he still has quite a few years to go before he has to worry about standardised tests or entrance exams.  I might start panicking if we were still on the topic of addition when he is 19 years old.
  2. Tiger is capable of buckling down to do the work when it becomes necessary.

Once I got over my initial fear of gaps (yes, I experienced much doubts and fears in the early days as well), I have come to view the learning of mathematics to be more than the acquisition of multiple sets of facts, short-cuts, and formulae.  It is a language of its own.  As soon as I realised that mathematics is a language, I decided that, as with any language-learning, establishing a strong foundation is very important.  Just as a strong language user needs to have a contextual base to understanding and mastering the subtlties of any language, a strong maths user needs to have be able to apply mathematical skills and knowledge for them to become meaningful.  Merely memorising mathematical formulae without the experience of applying them to solve real problems (as opposed to clinically designed textbook problems) is akin to memorising English idioms without knowing how to use them in the right context.

What About Practice and Mastery?
Having explained (I hope) above why I'm ok with Tiger's maths progress not following exactly in the same order or at the same pace as determined by any curriculum, you can see why I can tolerate us dwelling on any topic -- addition comes to mind immediately -- forever.  However, the time that we spend on any one topic is not used on repetitive drills or learning the same set of skills at the same level over and over.  An example is the latest addition game that we played.   Different maths skills were used at the same time to solve that set of problems -- addition, long multiplication, counting, number sense.  When we play maths games or riddles like the example that I have just referred to, both Tiger and I can easily and clearly determine whether he has any gaps in his knowledge (in which case we will work at closing those) or where his gaps are (so that we can direct our effort more effectively)

A tool that I find to be effective for practice is the Khan Academy's maths section.  I've mentioned before our sporadic use of the website whenever we feel Tiger needs to have more focused practice on any topic.  Most recently, we discovered a new feature on the website called Mastery Challenge, which is very similar to a series of short composite tests on various maths topics.  Tiger doesn't mind working on a few rounds of these each day.

To me, it works in the same way as what we have been doing at home to identify gaps and areas of improvement. For example, one of the recent challenges shows that Tiger needs more work on the long division (which we had touched upon briefly in November but not formally worked on yet).  With this information, Tiger watched the two-digit division video:

and worked through the example alongside the video, before attempting several rounds of two-digit divisions on the site.  Working on these exercises enables Tiger to solve a recent maths challenge, but it is not the way in which I would like him to learn long division (even though he certainly can do them mechanically now), so we will spend some time later in the year to learn division in a more visual, hands-on way.

I don't claim that our method of learning maths is superior to any other method, but it is one that suits Tiger's need for variety, applicability, and minimal repetition.

Monday, 27 January 2014

Eye on the Ball, Please! Part 2

This is the second part of a series of my termly plans.  The first part is Ball #1: Language Arts.

Ball #2: Mandarin - Language and Culture
Tiger's Chinese class is working out very well for him.  He has been able to keep up with the speed of the lessons (by putting in a fair bit of hard work) and it is paying off.  His interest and confidence in learning the language has increased substantially because: (1) he feels encouraged and recognised by his teacher when she gave him a classroom assistant role to teach the rest of the class a small part of the new lessons, and (2) he is motivated to maintain his recent exam success.

We are all encouraged by Tiger's positive experience at the Chinese school so he will continue with the course there.  Staying on top of the weekly homework and revisions will be our focus -- there is very little time or need for us to do much else in terms of studying Mandarin at home. 

Our commitment to learning Mandarin also means that we will have to forego Latin.  Boo.  Sed valefaciens, et lingua decora!

Learning a language is best supplemented by an understanding of its culture of origin.  Getting to know a culture makes the language come alive, so while the Chinese school is now taking on the bulk of language training for Tiger, we are supplementing his learning and experience by watching a few cultural-related clips online.

We find the individual lessons at Growing up with Chinese to be very lively and interesting.  Each lesson focuses on a certain aspect of social engagement, as well as a few tips on understanding the Chinese characters, all set within a story.

For a more contemporary understanding of modern China, we are currently watching the 10-part BBC TV series, Real Chinese:

After we have watched the series above, I plan to continue with our armchair cultural immersion by working our way through the documentaries here.

A new project that I'd like to take on this year is to spend two weeks focusing on Chinese New Year.  It will be very similar to how we usually spend the month of December focusing on Christmas-related activities.  I am getting a lot of ideas for Chinese New Year from Marie's Pastiche, which has a whole year's worth of China-related ideas and activities to learn from.

This post is linked up to Chinese Activities Link Up.

Friday, 24 January 2014

Water, Water Everywhere

It is still raining quite a bit where we are.  We have been measuring the daily rainfall using our homemade rain gauge.

The amount of rainfall that we get is nothing compared to some parts of the country.  It started with heavy rainfall in December, building up to floods occuring in many parts of the country by early January.  We have been looking at the photographs here, here, and here to see the damage and inconveniences caused by having too much water.

To get to the bottom of things, we decided to find out what might have caused the extreme weather, specifically the factors that might have led to flooding:

Naturally, being near a river is going to be a concern at a time like this.  However, river flooding doesn't necessarily have to become hazardous, if proper planning and management have been done to make use of the sediments deposited by the flood water, as we have learnt two years ago in world history -- where the flooding of the River Nile was a major source of the fertiility of the lands around it.  Below are some phots of the experiment we did to illustrate the flooding of the Nile leading to fertile land.  We used planted grass seeds in an aluminium tray, flooded the "river", and waited for a week for all the water to be absorbed and the grass to grow.

The Thames is a tidal river, hence very susceptible to flooding.  Having London flooded would be extremely costly, hence the Thames Barrier has been very much in use these few weeks to make sure that doesn't happen:

Apart from rivers being a potential source of flooding, coastal flooding can be lethal too:

The force of water can be illustrated by making a simple water wheel using plastic cups and paper plates.  It works but isn't very robust, being made out of paper plates and all, so the rain made it very soggy after a few hours.  Nonetheless, it served its purpose before it disintegrated.

It seems that the trouble is going to persist for a while yet...

For anyone who is living in a high flood-risk area, it's probably a very good idea to learn from here all about how to prepare for a potential flood, what to do during and after a flood, and how to manage flood risks.  Alternatively, much can be learned from books too.

This post is linked up to:
  1. Geography and History Meme: Ancient Egypt activities for kids
  2. Entertaining and Educational - Hasty Pudding
  3. Collage Friday: The Key to Successful Homeschool
  4. Weekly Wrap-up: The One with the Love Languages, Too Much Castle, and New Curriculum
  5. Homeschool Mother's Journal {January 25, 2014}
  6. Science Sunday: What Do Your Students Need to Know Before High School - Biology
  7. Hip Homeschool Hop - 1/28/2014

Thursday, 23 January 2014

It's Not Very Clear...

A few mornings ago we woke up to a thick fog.

It wasn't just in our area.  It seemed that most of the country was shrouded in the fog.  Since it was everywhere, we decided to find out more about it.  We were surprised to find there to be so many different classifications of fogs, and determined that what we had must have been a freezing fog (as opposed to a coastal fog), which is related to the supercooled water concept that we learnt last week.  It might well have been a radiation fog, had it not been for the accompany feathery ice crystals that we saw on the window pane in the morning.

We have driven in thick fogs before so we remember what the hazard that the fog can present.  Therefore, it is good to heed the advice of the Met Office, even though what they're telling us is very much a matter of common sense.

There are two types of warnings for a fog:
(1) yellow

(2) amber

This post is linked up to:
  1. Nature Study Monday: Izula-Bullet Ant-Paraponera Clavata, aka: Scary bugs!
  2. Hip Homeschool Hop - 1/21/14
  3. Entertaining and Educational - Hasty Pudding
  4. Collage Friday: The Key to Successful Homeschool
  5. Weekly Wrap-up: The One with the Love Languages, Too Much Castle, and New Curriculum
  6. Homeschool Mother's Journal {January 25, 2014}
  7. Science Sunday: What Do Your Students Need to Know Before High School - Biology

Wednesday, 22 January 2014

Eye on the Ball, Please! Part 1

This is a bit of a planning post for this term, mainly to help me keep our focus for the new few months.  I realise that it is now near the end of January 2014, but a few of Tiger's classes have not settled until very recently so it is until now that I have a better idea of what our term will look like from now until the Easter holidays in mid April.  Over the next few days, I will be posting up the rest of our plan for this term.

Ball #1: Language Arts - Poetry and Shakespeare
This term our focus for language arts is poetry.  This is driven primarily by the happy coincidence that both his drama teachers, without knowledge of what the other is doing, want to focus on poetry.  Tiger has two drama teachers -- one focusing primarily on literary analysis and acting skills, the other focusing on speech and recitation.  Since such a serendipity doesn't happen all the time, I shall take my cue from it and use our time at home to brush up on our poetry skills through our weekly poetry tea where we shall continue to bake and read to each other.  Nobody here will object to more tea and cakes!

I hope to add a little bit more meat to our weekly tea sessions by using the Arrow Poetry Guide, and to focus on two specific poems:
  1. Jabberwocky by Lewis Carroll
  2. The Highwayman by Alfred Noyes

Somehow this is turning out to be a rather Shakespearean term for us.  To begin with, his drama/literary analysis class will be doing five sessions of The Winter's Tale.

In addition, I found out about a MOOC looking at Shakespeare's Hamlet, which I am personally very interested to attend since I have not read Hamlet before.  I asked Tiger whether he would be interested to study this particular play with me and he said yes, so we are attending this MOOC together while working at our own levels.

Shortly after this course ends is the start of another Shakepearean course -- Shakespeare and His World, which Tiger and I will again attend together.

Grammar, Spelling, Writing
Handwriting will continue to be practised, because I think it is important to have beautiful handwriting. We will start with a review of print handwriting, followed by cursive handwriting.

Most of writing will be covered through our study in poetry and literature, as well as the monthly book club which has a creative writing component in addition to book discussion.

At home, we will change our strategy with regards to grammar study.  Instead of a touch-and-go approach to the different parts of speech, we will focus on just one or two aspects of it so that Tiger can have a good understanding of the different components of grammar.  As such, this term we will focus solely on studying nouns.

For spelling, we will resume where we had left off with All About Spelling level 2.

This post is linked up to:
  1. Hip Homeschool Hop - 1/21/14 
  2. Entertaining and Educational - Hasty Pudding
  3. Collage Friday: The Key to Successful Homeschool
  4. Weekly Wrap-up: The One with the Love Languages, Too Much Castle, and New Curriculum
  5. Homeschool Mother's Journal {January 25, 2014}

Friday, 17 January 2014

Water is Super Cool!

Since there has been so much rain, we continue our rain study by using a science kit that we had picked up at a charity shop that is relevant for our purpose.

There are 25 experiments to do with this kit, but not all provide new learning experience for us so we only did those what looked interesting and taught us something new.

1) Density
The first experiment we did was to find out whether all liquids weigh the same.  To do that, we prepared two jars of warm water -- the jar on the left has several tablespoons of salt added to it.

We then gently put each egg into the jar.  The egg in the salt water jar floats because salt water is denser than plain water, so the weight of the egg does not have to push away or displace as much water to make space for itself (compared to the egg in the plain water), therefore it floats.


I think this can also be explained by the differences in molecular structures between salt water and plain water, but we will come to that when we learn more of chemistry.  For the moment, the above simple explanation is enough for us.

The next experiment has to do with mixing oil and water.  First, Tiger half filled a glass with cooking oil, then topped it up with plain water.  He observed that the oil found its way up the glass and floated on top of the water.  This is because water is less dense than water.

2) Temperature
We wanted to find out whether salt affects the freezing point of water, so we filled two containers with cold water and added a teaspoon of salt to the one labelled with "S".


Both containers were put in the freezer and checked every 15 minutes for their stages of turning into solid.

Ice crystals started forming in the plain water container after 30 minutes, while nothing happened with the salt water in the same duration.  After 90 minutes, the plain water had become solid ice while the salt water was still in liquid form.

3) Supercooled Water
The coolest thing we found out was about supercooling.  The gist of it is to bring water to a temperature below freezing point while maintaining its liquid form, then turn it into frozen ice instantly.  In the natural world, this phenomenom happens in the clouds in the northern hemisphere during winter for rain to form.

It is too cool to miss, so we decided to try it out for ouselves.

Alas, the cool factor eluded us.  We couldn't make the experiment work despite numerous attempts.

Apparently, achieving the desired result isn't so easy after all, according to the clip below.  Phew, that made us feel less incompetent.  The clip also explains what is happening to the molecules during the process:

4) Surface Tension
This is an experiment where we made a cascade of slow flowing water using the attraction of water molecules and surface tension.

We wet a piece of cotton string then tied one end of it to the jug handle.  The jug was filled with water.  The other end of the string was held taut against a small jar.  Tiger then lifted the jug so that the string passed over the spout.  The water was poured gently.  If you look carefully in the clip here, you can see the water flowing along the length of the string into the jar.

The reason for this observation is because the wet string attracts the molecules in the water while the surface tension creates a skin on the outside of the water.

5) Absorbency
The activities we did at home were supplemented with a hands-on workshop at a water treatment centre where the children did additional experiments related to water and recycling.

A simple experiment is to drop a few drops of water onto different types of materials to find out which material is the most absorbent.

The next experiment investigate three factors at once:
  1. which type of earthy material (gac, sand, or gravel) is the most effective for filtration;
  2. which type of earthy material (gac, sand, or gravel) is the most abosrbent.
  3. which type of earthy material (gac, sand, or gravel) is the most permeable.

6) Recycling
As the facility we went to is a part of a water treatment centre, recycling naturally forms part of the education programme that was on offer.

One of the experiments the children did was to find out why they should only put toilet paper (rather than other paper materials) down the toilet.
  1. To do this, three pots with plain water were prepared.  A piece of different material (toilet paper, tissue paper, wet wipe) was put into each pot.
  2. The pot was then sealed and the children shook each one vigourously for 5 minutes.
  3. The children then examined the contents of each pot to see the different degrees of disintegration that has occurred.  We found that the toilet paper broken down the best, followed by tissue paper.  The piece of wet wipe did not break down at all.

The children also had a go at paper making using torn up pieces of waste paper.

This post is linked up to:
  1. Entertaining and Educational -1700's Wig
  2. Collage Friday - Infinite Value and Learning in Each Moment
  3. Weekly Wrap-up: The One Where We Got Back to School
  4. Homeschool Mother's Journal {January 18, 2014}
  5. Science Sunday: What Do Your Students Needs to Know Before High School Chemistry
  6. Hip Homeschool Hop - 1/21/14

Saturday, 11 January 2014

Rain, Rain, Go Away

Has anyone else in the UK noticed that there's been a lot of rain around us lately?  In my typical fly-off-the-seat-of-my-pants approach to learning what catches our interest at any given moment, I thought it would be most relevant to learn more about rain.

For example, why does it rain?

We also learned that rainfall can be measured using the high-tech way via radar, or the low-tech way of a rain gauge:

Since the low-tech rain gauge looked simple enough to do, we decided to make one ourselves following the instructions here:

This is our version, which according to the explanation here, may not be yield the most accurate measurement since the top and bottom widths of our rain gauge are not exactly the same.  Nonetheless, it serves as a rough and ready version to give us a sense of how much rainfall there has been.

Since it has been so wet, we thought it would be interesting to learn how to measure the humidity in the air at any given moment:

Compared to the natural, pinecone hygrometer we made a few years ago, the bottle hygrometer we've made this time offers us a more accurate reading of the relative humidity in the air.

By putting the readings that we got off of the wet bulb (3 degrees Celcius) and dry bulb (11 degree Celcius) into the computation website found here, we can see immediately that the relative humidity of that day was 17%.  We think that is pretty cool.

This post is linked up to:
  1. Nature Study Monday: January 2014 NSM! Link Up
  2. Geography and History linkie #1
  3. Entertaining and Educational - Atoms
  4. Collage Friday - No Spend Month, Marriage, Menus, & Miscellany
  5. Weekly Wrap Up: The One Where Dad Was Home
  6. The Homeschool Mother's Journal {January 11, 2014}
  7. Science Sunday: The Difference Between a Science Demonstration and a Science Experiment
  8. Hip Homeschool Hop - 1/14/14

Thursday, 9 January 2014

When Everything Adds Up

Now that we are very familiar with straightforward addition, let's try our hand at finding the sum of all the numbers in a range, say from 26 to 765, without using a spreadsheet or calculator?

In order to learn the technique to work out sums of the above nature, let's start with the basics.

Q1: Find the sum of all whole numbers from 1 to 10.
The answer (55) can be easily worked out mentally or by writing the sums out on a piece of paper, but we are trying to learn a logic that will enable us to work out sums of any range.

To do this, I used a piece of cotton string and attached each individual number card 1 to 10 with paper clips.

I then asked Tiger what he would do, besides using mental calculation, to find the sum of all the numbers from 1 to 10.  He removed all the numbers from the number line, so that all ten numbers were accounted for, and paired them up into equations with the same sum total of 11 each:

            1 + 10 = 11
          2 +  9  = 11
          3 +  8  = 11
          4 +  7  = 11
          5 + 6   = 11

Once Tiger set the five equations out on the floor, it was obvious that the sum total would be 5 x 11 = 55, which tallied with his mental calculation.

I then reset the number line with numbers 1 to 10 again, and asked Tiger whether he could solve the problem in a slightly different way, i.e. using a total that is not 11.  When Tiger couldn't think of any other way after a while, I suggested changing the total to 10.  This way, the logic behind the solution is the same as before, except that there are a few extra steps:

When we set the total to be ten, the number 10 is removed first from the number line.  That is followed by the pairing up of numbers that make up ten.  When the pairing is completed, you will find that the number 5 is left unpaired.

The pattern will look like this:
          1 + 9 = 10
          2 + 8  = 10
          3 + 7  = 10
          4 + 6  = 10

Therefore, the final summation to find the answer is:
           (5 x 10) + 5 = 55

Between the two methods (sum to 11 versus sum to 10), I personally lean towards the one with sum to 10, but Tiger prefers what he considers to be the more straightforward method of summation to 11.  As you've seen from the above, both are equally valid so the choice depends on what makes the most sense to the child.

Q2: Find the sum of all whole numbers from 12 to 30.
We tried another exercise to check whether Tiger has understood the logic/technique to apply it to another question.

Tiger had a choice of using whatever method that worked best for him.  He chose to do the first method, i.e. the one where the first number (12) was added to the last number (30), and so on.

By doing so, he found that he had the 'middle number' (21) left unpaired, so he put it at the top of his layout so to ensure that he did not forget to add it to the final answer later on.

The pattern that he found was this:
          12 + 30 = 42
          13 + 29 = 42
          14 + 28 = 42
          15 + 27 = 42
          16 + 26 = 42
          17 + 25 = 42
          18 + 24 = 42
          19 + 23 = 42
          20 + 22 = 42

Therefore, the final summation to find the answer is:
           (9 x 42) + 21 = 399

At this point, I introduced another concept to Tiger: how to find the number of items (in this case, the number of pairing) from one number to another.

By counting or just looking quickly at the layout above, we determined that there were nine sets of pairings that summed up to 42.  Counting is fine until the range gets too large.  Therefore, I asked Tiger how he would determine the number of pairings without counting.  When he started to experience some difficulty, I asked him to think about how he would determine the number of pages he has read in a book, say from page 1 to 10.  After first he said the asnwer would be nine pages since 10 - 1 = 9, but when he quickly realised that the correct number was ten.  That was a bit puzzling.  I then asked him how many pages he would have read had he read from pages 2 to 11.  Again, answer is ten, but if he had applied the simple subtraction method of 11 - 2, he would get only nine.

By now, Tiger saw the logic behind this: that the number of items in a range is the simple subtraction of the last and first numbers followed by adding one more to it.  I helped him with the next step by putting his explanation into a simple formula:
        the number of items in a range = last number - first number + 1

We then verified our formula against the two examples above by first applying the numbers to the formula followed by verifying our answers with visual counting.

Q3: Find the sum of all whole numbers from 27 to 68.
Now it's time to see whether Tiger really knows what to do, so I set him the following exercise and left him to solve it himself:

Tiger proceeded with it methodically:
First, he laid out all the numbers in pairs so that every number on the number line was accounted for.  The sum of each equation was 95.  The pattern that he made was:
           27 + 68 = 95
          28 + 67 = 95
          29 + 66 = 95
          30 + 65 = 95
          31 + 64 = 95
          45 + 50 = 95
          46 + 49 = 95
          47 + 48 = 95

Using the formula he learned above, Tiger found out that there were 21 sets (48 - 68 + 1 = 21) of equations that each added to the sum of 95.  Therefore, the final answer was
      21 x 95 = 1995.

Once Tiger has got the hang of it, he is now able to find the sum of all whole numbers from different ranges.  I gave him a few extra questions to work on:

Q4: Find the sum of all whole numbers from 5 to 100.

Q5: Find the sum of all whole numbers from 36 to 205.

Q6: Find the sum of all whole numbers from 12 to 156.

Q7: Find the sum of all whole numbers from 4 to 237.

Having said all of the above, it makes sense to acknowledge that we are dealing with elementary maths, which is essentially concrete, so perhaps not...

Watching the clip above led us to ponder more with about the following:

If one gets into higher, more abstract maths (as the following videos show), then the concrete number sense that we have just spent so much time learning are nullified, similar to how conditions of quantum physics defy many common laws of physics.

But I'm not worried about that just yet.  I'm happy that Tiger is enjoying the concrete aspects of maths at the moment.

This post is linked up to:
  1. Entertaining and Educational - Atoms
  2. Collage Friday - No Spend Month, Marriage, Menus, & Miscellany
  3. Weekly Wrap Up: The One Where Dad Was Home
  4. The Homeschool Mother's Journal {January 11, 2014}
  5. Hip Homeschool Hop - 1/14/14
  6. Math Teachers at Play #70
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