I love maths that challenge us to think, but are deceptively easy at first glance so that they don't put children off. This week, I gave Tiger one of these simple and elegant maths riddles that require a good understanding of:
mental calculation
addition
place value
numbers
The
idea is to use the ten digit cards, as set out above, to lay out an
addition equation visually in the vertical format. A few rules to bear in mind:
The digit zero (0) cannot be used on its own.
The digit zero (0) cannot be placed in front of any number, i.e. it can't be used as "081" and such like.
The addition sign (+) is assumed.
The final number on the last line is the sum.
No carry-overs are to be shown.
Easy,
no? For example, 76 + 80 = 156. Each digit in the equation is
selected from the set above. The solution would be shown as follows on
the table:
76 80 156
Or, if the equation is 25 + 66 = 91, the solution would be presented as:
25 66 91
All clear? Ready to go?
The idea is to start with any combination using the set of ten cards as shown in the photo above, then slowly build up to using all ten cards.
Tiger quickly found numerous solutions using three to nine cards, but he struggles to use all ten cards in a single solution. The game is too much fun to just sit and watch so I used a second set to play alongside Tiger, at which point it turned into a competition between us to see who could find the solution to the ten-cards problem first. I am pleased to report that I have uncovered one solution to the ten-cards problem, although it did take me quite a few attempts. I'm sure there is more than one answer to this problem. Tiger hasn't found his solution yet. His challenge is to find the solution by Christmas eve, while mine is to find at least one more solution by then.
Join in the fun if you feel inclined! I'll post our solution(s) on Christmas eve.
I am going against the grain this year: I am NOT planning any specific activities this December. It has been liberating and eye-opening for me to see how and how much unscheduled learning happens when we just 'go with the flow'.
2. Mothers Who Embody Courage and Determination
Tiger has been attending a weekly hands-on science class since September. The class is held in a small church hall so the mums always sit at the back of the class quietly until the lesson is over, unless they are helping their children with a particular tricky part of the experiments. As such, there is very little chance each week for the mums to get to know one another beyond recognising one another's faces.
The sessions for this year have ended so one of the mums has kindly invited everyone to her house for a social gathering. It was at this gathering that I felt I have gotten to know these ladies much better, and I am very glad for that.
For example, I had no idea that one mother has managed to guide her son from being diagnosed as fully autistic (i.e. no speech, no eye contact, mentally delayed) at age two to being mildly socially awkward at age 14. This boy is currently preparing for GCSEs in biology, physics and chemistry. After listening to her amazing experience, I felt a profound sense of respect for this woman. Behind her meek, quiet appearance is a spirit of steel and courage.
It was only in the relaxed environment of the gathering that I got to know that the majority of the mothers here are single parents bringing up children on their own with little financial help from their ex-partners. Most of them are also having to cope with children who have special needs, e.g. diabetic, Aspergers, dyslexic. I don't know the ins and outs of how these mothers find themselves in their situation of becoming a lone parent, but I listened with respect and admiration to every story of the battles they have had to fight to get the support their children need -- financially, educationally, and medically.
Since Tiger has never been to school, I thought he would not share this childhood experience with the 95% of children his age who go to school. Although it is not a big deal to us whether he takes part in a Nativity play, I sometimes wonder whether he might have liked, when he becomes an adult, to be able to say something like, "Yes, I have fond memories of being the donkey lying in the manger in my Year One Nativity play." Well, you never know, do you?
Thanks to a new set of homeschooling friends that we've made this year, we were invited to take part in a Nativity play -- well, it's Tiger who has the part of a shepherd/narrator. My job is to take him to the premise, be part of a supportive audience, and socialise with everyone.
We had a group of 15 children, aged two to 12. The children memorised their lines beautifully from a script written by one of the very talented mums. The play was interspersed with Christmas carols played by a few children in their various instruments (trumpet, flute, recorder, oboe).
Waiting for their turns during the full-dress rehearsal.
Despite only meeting a few hours before the actual performance, the children were ready for the actual performance after three rehersals. The children's level of concentration and dedication to this project, their ability to work together, and the older children's ability and willingness to look out for the younger ones during the 25-minute production really shone through. What a fantastic group of children! I feel very privileged to have witnessed all the goodness in these children with my own eyes.
4. A Victorian Christmas
We are about half way through reading and listening to A Christmas Carol.
There are usually many Victorian Christmas activities available in various parts of the country around this time, and since the era ties in with the book that we are reading, we went to one to experience the jovial atmosphere of a Victorian Christmas.
There were diplays of Victorian toys, Victorian games, street musicians, street performers, and Victorian crafts. We found the hand-turned busker organ to be absolutely fascinating:
The event being of a Victorian theme in December, there was also a two-person street performance of A Christmas Carol:
As it was a two-person production team, they had to adapt much of the story and summarise it into a 20-minute play. Tiger enjoyed watching the performance in spite of its obvious adaptations. It's interesting for me to see him respond differently to adaptions made at a live performance than those made for films. I'd be curious to see his response to a film version of the story, when we watch it at some point.
5. A Bedtime Lecture
Tiger
is a very animated child and his bedtime routine has always taken
forever because he always has a lot to say when he is in bed. After
I've read him stories and kissed him goodnight, he always has something
he's excited about to tell me. It may well be his delaying tactic, but I
feel obliged to stay for an extra 15 minutes because he does often have
genuinely interesting things to tell me just before bed.
This week's bedtime topic of interest is computer networks.
Mum: (kissed him on the forehead and about to switch the lights off) Goodnight, darling. Tiger:Mum, do you want to know about computer networks? Mum: (hesitatingly) Umm... I don't know. Do I? Tiger: Yeah, they're very interesting. I'll show you a few diagrammes quickly and you'll understand them in no time. Mum: Well, ok.
That was his cue to sit up in bed in his dressing gown and start explaining to me how different networks work.
In
the 30-minute session, Tiger educated me about the different switches,
routers, how LANs (local area networks) and WANs (wide area
networks) can be designed and how they work, as well as how they can be combined in various ways to make
efficient cross-border networks. The mini lecture was delivered with
numerous hand drawn diagrammes to illustrate his examples:
I
vaguely remember studying two IT-related modules at university, so
Tiger's diagrammes and explanations made some sense to me. What I
enjoyed most in the session was his enthusiasm to share what he has
learnt so far.
Much to my own surprise, despite my non-plan it has been a very busy week filled with rich experiences to nourish our hearts and minds. I wonder what is in store for us next week.
I feel as though I've been running since September, and I feel I need to slow down and take a breather. Traditionally I would plan lots of activities for the month of December to take us nicely to Christmas. This year, I don't feel that I have the energy to repeat the advent activities that we did last year, so I am going to have a slightly more relaxed December this year by not taking on ambitious new projects this time of the year.
Having said this, we aren't going to sit around doing nothing -- although that will probably be nice for a change -- we're just going to not try to do 24 different activities in December. Plans are not concrete yet, but here is what I am thinking of at the moment, in addition to continuing with a few ongoing projects:
1. Christmas-related literature
Reading a Christmas-related book and enjoying some activities related to it has become one of our family's traditions in December. One year we looked at The Legend of the Poinsettia. For two years running we looked at The Nutcracker. This year's book selection is A Christmas Carol.
2. Food, Crafts, and Gifts
These three items come together for us in December. I always encourage Tiger to make his own Christmas cards and bake his own foodie gifts for people. At a time of rampant consumerism, I'd like Tiger to understand that the true value of a gift has more to do with the sincere efforts of the giver/maker rather than with the monetary value attached to commerical products.
Besides making things for others, we will also spend some time making simple crafts to keep ourselves amused. Nothing too fancy or complicated. I am still getting ideas from last year's craft activities, such as the Christmas lanterns, the paper chains, and maybe sewing a few simple tree ornaments.
3. Friends and Family
We have an exceedingly busy social calendar this December, thanks to a number of new homeschooling friends we've made, as well as plans to see some old friends as well. Tiger and I are out and about so often that it's hard to imagine having a more active social life, but it seems that way only because we used to spend December more inwardly focused on our own little family. This year, my mother-in-law is in charge of Christmas celebrations so there will be a big three-day family gathering from Christmas eve to Boxing Day with plenty of time with immediate and extended family. Free from the responsibility of planning and preparing Christmas Lunch, I am able to accept invitations to enjoy the jovial company of our friends, homeschooling and otherwise.
During our visit to the Tower of London, we saw a reenactor dressed up as a medieval priest who
gave us an overview of the history of the Tower as a royal palace, a
prison, and a fortress. A full-sized replica trebuchet was on site to
demonstrate the power of this medieval siedge weapon. It took five adults
to launch the trebuchet.
The trebuchet is of special interest because the weapon has been traced back to being first used by Edward I during his siege of Stirling Castle.
While Tiger and I discussed about medieval siege weapons, I was given a quick education by Tiger of the differences between a trebuchet and a mangonel. If your knowledge of medieval weaponry is like mine, i.e. that you think all catapulting machines work in the same way, you will benefit (as I have) from watching the two clips below:
1) Trebuchet
2) Mangonel
We learn much about the trebuchet from the following clip:
Why do we want to know so much about the trebuchet? The main reason is that Tiger is passionate about everything that has to do with warfares and weaponry, historical and otherwise. After hours of listening and watching experts talk about this medieval war machine, we decided it was time to build one ourselves.
It took us a few days to put the trebuchet together...
and when it was fully assembled, all of us thought it looked beautiful!
Naturally, we had to test whether it to see how good it is.
Unfortunately, our first few attempts at launching the trebuchet didn't go too smoothly.
The projectile either didn't go in an arc path, or was trapped in the launch net, or was launched backwards (gasp!). It was slightly frustrating to have to work out what went wrong. Not wanting to give up now, after we had spent so much time putting the model together, we pressed on with more tweaking, taking into consideration the following factors that might affect the success of our launch:
tension of string
length of string
weight of projectile
size of projectile
weight of counter-weight load
It's very exciting when it finally worked!
Tiger spent the rest of the day trying to break the record for the longest distance the projectile would go. So far, his record is at 17 feet.
Most times, when we start on a topic of study, we find it very diffcult to draw a line under it and to move on to something else. I mean, when does one decide that he/she has learnt enough to move on to the next topic? I thought this week we would do a little bit of introduction to fractions then move to the next topic, but we ended up spending the entire week on it as we found more things to learn in fractions, e.g. applying the four operations to fractions, and converting between mixed numbers and improper fractions.
How did we end up doing more and more without intending to? Well, it all came about because I thought it would be a good idea to have a little review of the basics:
That was all I wanted to do. However, the last question in the clip above: Estimate 7/8 + 12/13 led Tiger to find out about conversions, which I explained using on a few pieces of paper. Not very sophicated, I know, but the most basic tools are sometimes the most effective.
Once he has understood my explanations, Tiger wanted to try out some activities to test his own knowledge so I let him test himself here, but not before he has looked through the revision section and the activity section.
We first started looking at fractions about two years ago but hadn't formally done much about it until this week.
I like to introduce maths in a visual, concrete way, so I took out two manipulatives that have been tucked away in the cupboards from Tiger's preschool days and let him play around with them for a bit:
The Equivalency Cubes can also be used to teach concepts of of percentages and decimals. The main purpose of them this week, however, is for Tiger to become familiar with the concept of equivalent fractions.
After Tiger has played with the cubes for a while, we started on our fraction activity. I gave them three sheets of papers, each printed with three circles to be divided into smaller fractions. For example, the first sheet consists of three circles that have been divided into halves by a line drawn through the centre of the circle. The idea is to have the first circle remained as halves, and have the student further divide the other two circles into fourths and eighths. The second set of circles are: thirds (already drawn), sixths, and twelves; the third set consists of: fifths (already drawn), tenths, and twentieths.
After the division of the circles into smaller fraction parts is done, Tiger cut them out and glued them onto a piece of construction paper (together with each circle's corresponding labels) to make "The Dangerous Fraction Garden", as Tiger puts it. This step is not strictly necessary to learn the maths concept, but we wanted to have a laugh so we did it.
We then proceeded to understand Fractions of a Set. Using the printed sheets of incremental stacks of 4x5 squares, we first determined the total of each set. Then, we talked about how much one-fourth of each set would be -- this is quite simple, given that the sets are designed in four rows each. Finally, from one-fourth of each set, Tiger could easily determine three-fourths of each set by multiplying by three.
When the colouring is done, we glued both sheets of paper together to have an overall view of all our three-fourths. It provides a clear visual sense of what three-quarters of an incremental total looks like:
While Tiger has labelled under each set its non-simplified three-quarters fraction (15/20, 30/40, 45/60, 60/80, 75/100, 90/120), I showed him how to simplify them by finding the common factor of both numerator and denominator, so that each fraction eventually reduces to 3/4.
Tiger then worked through some of the problems in the Fractions section at Khan Academy. We found the drills to be adequate for practice.
The word problems on fractions present an interesting way to apply what Tiger has learned so far, so that is well worth a try. When he had enough of drills, he played the Fractions and Coins Game.
Finally, we also learned not to get too hung up about getting perfect scores, especially when in France:
I decided that we should get back to the basics of maths -- addition -- with the following simple exercise.
I gave Tiger a copy of a sheet printed with ones, tens, and hundreds boards. The total of which is 999. His task was to split the sum into four addends. Very easy, isn't it? That was what Tiger thought.
Before Tiger began, we had a quick review to make sure he understood the different terms (addends, sum). I also showed him a quick example of how he was to split the sum into four addends (example on the bottom left)
As you can see from the photo above, Tiger split the total of 999 into four addends immediately (the example on the bottom right)
The next task is to cut out the printed sheet and use the arrays to form the addend in each section on the construction paper. This was where Tiger realised that he had to consider the limitation set by the arrays, i.e. 9 ones, 9 tens, 9 hundreds. With this limitation, the initial answer he gave (251, 199, 151, 398) could not work. This made him work a little harder than merely throwing random numbers out to make the sum total.
After some struggle and when Tiger started showing signs of frustration, I suggested that thinking about place values and the limitation set by the arrays and having to split the sum total into four addends at the same time might help. The light came on when I sketched the array on paper (see pencil mark below):
Once he saw the sketches, Tiger understood how to approach the problem. There are a few ways to solve the problem. Below is Tiger's solution:
