Finding the Balance

Algebra.  Sounds like a really sophisticated maths idea, doesn't it?  The truth is, most children already intuitively know the basics of algebra without calling it that.  The concept of single-variable algebra can be very simply introduced using the following tools:

1. a balancing scale
2. several one-pence coins (or the equivalent of the smallest monetary denomination in any currency)
3. modeling clay of different colours

To prepare the materias, I first made a clay ball with the equivalent weight of a 1p coin.

I often like to get Tiger involved in making his own learning tools, so I got him to help me make the equivalent-weight clay balls.

That is, until it occured to me that I had to make the clay balls myself to make the game work, so I diverted his attention to a few slices of cake while I carried on with making the following clay balls:

• orange clay balls ("O") - each is equivalent to the weight of one 1p coin
• purple clay balls ("PB") - each is equivalent to the weight of two 1p coins
• green clay balls ("GB") - each is equivalent to the weight of three 1p coins

Now we're ready to play.

The first step is to give the child an idea of balance, i.e. the one side should be the same as the other.  Without Tiger seeing, I placed five 1p coins in the right-side tray and covered it with a tissue paper.  Then I asked him to find the equivalent number of coins to balance the scale.  When he found the scale to balance with five 1p coins, I lifted the tissue paper to reveal the answer.

Next, I repeated the step above, but this time I used two orange clay balls and four 1p coins on the covered, right-side tray.  I then asked Tiger to find the number of 1p coins it would take to balance ths scale.  He found that it took seven 1p coins to balance the scale.

Tiger immediately knew then that each orange clay ball represents one 1p coin.  I asked him how he would represent the information on the scale as an equation.  This is his equation:

         7 "1p" = 5 "1p" + 2 "O"
==>    1 "O" = 1p

I did another exercise with him to make sure that he understood the concept.  This time I used three orange clay balls and three 1p coins, asking him to find the equivalent number of 1p coins to balance the scale.

After he has found it to be six 1p coins, I asked him to write the equation down again:

         6 "1p" = 3 "1p" + 3 "O"
==>    1 "O" = 1p

Just as Tiger was starting to think the game was too easy, I changed it slightly.  For the next go I used one purple clay ball and five 1p coins (covered up using tissue paper) in one tray, and asked Tiger to find the number of 1p coins to balance the scale.  He found it took seven 1p coins.

When I removed the tissue paper, Tiger was briefly stumped by the number of items on the right-side tray.  He quickly figured out that the purple ball must weight more than the orange balls that were used earlier.  He then went to to figure out that the purple ball must be equivalent to two 1p coins.  When asked for the equation for this, he said:

              7p = 5p + 1 "PB"
==>   1 "PB" = 2p

More variations on the same theme followed.  Next up, I used two green clay balls and five 1p coins.  It took eleven 1p coins to balance that. Using the same reasoning as the example above, Tiger worked out that each green ball is equivalent to 3p.  Obviously there have been some mental calculations that involved addition, subtraction, and division to arrive at the answer but Tiger doesn't like to write out all the intermediate steps.  Therefore the equation for this is shown as:

             11p = 5p + 2 "GB"
==>   1 "GB" = 3p

For the last exercise, I used two purple clay balls and three 1p coins, which Tiger found to need seven 1p coins to balance.

 

Once again, he worked out that each purple ball is equivalent to two 1p coins, equation as below: 

              7p = 3p + 2 "PB"
==>   1 "PB" = 2p

This time, Tiger wanted to show his 'proof' of how he arrived at his answer, so he demonstrated by removing three 1p coins from both trays, leaving four 1p coins in the left-side tray and the two purple balls on the right-side tray.  He did not write out the equation for this, so every step was done visually and orally.  For those who would like to see his demonstration in mathematical form, this is how it would look:

                  7p = 3p + 2 "PB"
==>     7p - 3p = 3p + 2 "PB" - 3p
==>             4p = 2 "PB"

From here, it is then a matter of applying simple division to find what each purple ball is.

                4p = 2 "PB"
==>     4p / 2 = 2 "PB" / 2
==>      1 "PB" = 2p


This post is linked up to:

1. Hip Homeschool Hop - 12/17/2013
2. Entertaining and Educational - Christmas
3. Collage Friday - Remembering Ecuador
4. Weekly Wrap Up: The One that will be Hard to Top for a While
5. The Homeschool Mother's Journal {December 21, 2013}
6. Math Teachers at Play #70 

Simple Elegance

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:

1. The digit zero (0) cannot be used on its own.
2. The digit zero (0) cannot be placed in front of any number, i.e. it can't be used as "081" and such like.
3. The addition sign (+) is assumed.
4. The final number on the last line is the sum.
5. 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.


This post is linked up to:

1. Hip Homeschool Hop - 12/10/2013
2. Math Activity Thursday
3. Entertaining and Educational: Travel is One of the Best Ways to Learn
4. Collage Friday - A New Camera and Birthday Happenings
5. Weekly Wrap Up: The One Where I'm Still Not an Aunt
6. The Homeschool Mother's Journal {December 14, 2013}

How did he do it?

We resumed lessons this morning with maths.  Usually we do a little bit of mental warm up with me asking Tiger to solve a few maths equations just to start his cogs turning, especially after last week's cold and total absence of any form of formal lessons...

This morning's warm up question was: what is £9.87 x 32?

I asked the question verbally while Tiger wrote it out in his exercise book.  He wrote it out below:

And stared at it for a minute (I don't have the patience to wait for more than a minute for a maths answer) before I suggested to him that maybe it would be easier to write the question out vertically and use the long multiplication method.  He accepted my suggestion and dutifully wrote it out as requested:

As you can see from the above, my suggestion didn't go very far to help Tiger.  After another 30 seconds of staring at it, he gave up on my suggestion and said he wanted to 'think about it'.  He took 30 seconds before telling me the answer: £315.84

I had to double check the answer with a calculator.

When asked how he did it, Tiger couldn't explain his mental process to me, although I could guess it from the way he gave me the answer.  He gave me the answer in two parts.  First, he told me it was '£315 and something', then in the next few seconds, he had worked out the rest was 84p.

Tiger was very pleased that he got the correct answer, but quickly told me that he doesn't want to do anymore today since his brain is 'worn out from all that calculating'.

If he were in school where everyone is expected to show their maths workings on paper in a step by step manner, Tiger's unusual mental process would probably be labeled as some kind of 'learning difficulty' or 'special need', whereas being taught at home means that he is in an environment with a parent who recognises and supports his unique way of learning.

There has been much debate in the UK about how maths is best taught in schools.  There doesn't seem to be any consensus yet (I don't expect there to be), but Tiger has seldom been taught to memorise anything (we tried to do that with grammar following the WTM recommendations and we both found that to be excruciatingly dull and ineffective), not least learning time tables by heart.  I don't believe in rote learning.  We are more inclined towards gaining an understanding and appreciation of how things work, including mathematics.


This post is linked up to:
1) Math Monday Blog Hop #84
2) Hip Homeschool Hop - 1/29/13