Monday, January 5, 2009

Maths models revisited

After over a month of holiday blogging, it's back to normal transmission. Come on! Out of your stupor and back to work! (You just experienced being my kids for 3 seconds).

In the past few weeks, I've been trying to familiarise Lesley-Anne with some of the new p6 maths topics so that she'll find it easier to keep up in class. I'm again reminded of how blessed the gifted mind truly is. When I introduced a new topic like circles or speed, I only had to spend about 10mins explaining the basic concepts and she very quickly grasped them and was able to tackle the even less than straightforward sums. And this is taking into account the fact that she is hardly the most gifted child in maths.

In contrast, I often have to explain everything in slo-mo to Andre and even then, it takes multiple repetitions before he eventually sees the light. So a gentle reminder to all parents that it's really different strokes for different folks - please adjust your methods and your expectations according to the ability of your child.

So I discovered that algebra is in the p6 syllabus afterall - sorry if I've misled any parent in the past! But I still love me models (if you've missed my early posts on maths models, do check them out, especially if you're unsure about the rules to drawing models - Parts 1,2 and 3 or click on the "mathematics" label).

I came across a couple of problem sums that stumped both me and Lesley-Anne. In the end, she solved them using algebra but the calculation was so complex that with her penchant for careless mistakes, it's not really a good strategy. I only went "ohhh, I see..." after I saw the solutions, which used models. Really rusty lah, after a month of holidaying.

Anyway, here are the sums and the solutions using the model method. I suggest you try them first without looking at the answers.

Question 1:

Joanne and Bernard had a certain number of stamps each. If Joanne gave 90 stamps to Bernard, they would have the same number of stamps. If Bernard gave 10 stamps to Joanne, she would have 5 times as many stamps as Bernard. How many stamps did Joanne have at first?

The key to getting this right is being able to draw both sets of transferrence onto one model, which was what gave us the mental block. The first part is easy - first, draw the model for what happens if Joanne gives Bernard 90 stamps.

Then using the same model, draw what happens when Bernard gives Joanne 10 stamps. Basically, Bernard will be left with the shaded part and Joanne's shaded part represents 1 unit, the rest of her stamps represent 4 units (since she would have 5 times as many stamps as Bernard).

We then need to work out how many stamps are represented by Joanne's unshaded part, which is easy since it's the same number as the corresponding parts in Bernard's below, ie Joanne's unshaded part is 100 stamps (10+90 in Bernard's portion).

So 4 units = 100 + 90 + 10 = 200
1 unit = 200 ÷ 4 = 50
At first, Joanne didn't have the extra 10 stamps given to her by Bernard, so the number of stamps Joanne had at first is 5 units - 10 stamps.
5 units = 50x5 = 250
250-10 = 240

Answer: Joanne had 240 stamps at first.

Question 2:
Jeffrey spent $187 on a pair of shoes. He spent 1/5 of his remaining money on a pen. He still had 1/4 of his money left.

a) How much money did he spend on the pen?
b) How much money had he at first?

First, draw the model for how what Jeffrey spent - $187 and the amount for a pen which represents 1/5 of his remaining money.

Next we know that the remaining 4/5 of his money is the same as 1/4 of his original amount of money. Since 1/4 is 4 units, 3/4 = 4x3 = 12 units.

The pen is 1 unit, so $187 = 12 units - 1 unit = 11 units.
1 unit = $187 ÷ 11 = $17

Answer for a): Jeffrey spent $17 on a pen.

Jeffrey had 16 units of money at first - 3/4 (12 units) + 1/4 (4 units)
16 x $17 (1 unit) = $272

Answer for b): He had $272 at first.


elan said...

Dear Monica,
Thanks so much for your Math posts! Models have been the bane of my older son's existence (as well as my husband's and mine. We used to literally spend hours trying to do his Math homework. Chris would try on his own for more than 1 hour, then I would try another hour then we would wait for Daddy to come home and scratch his head. Finally Dan (the younger one) would come and peer between our shoulders, and say (without need for reams of paper for working)" easy-peasey lemon squeezy, the answer is 128" and lo and behold, if you work backwards with his answer it would be correct but when asked how he got it he would give a very brief and impatient explanation and run off to play with his Lego....that's how we knew he would also get into GEP the next year with no problem!
Yes every child is different.
Your posts have also been making me feel guilty, we "wasted" the entire school holiday, not touching a single school book. I let them play with all their toys that they had not touched for months and infested all the public libraries near our house for 6 whole weeks when I should have been preparing Chris for the PSLE!!!! YIKES!!! We've got some catching up to do I think.

monlim said...

Elan: Don't worry about it, actually I think the kids deserve a good holiday before the PSLE (anyway, you went library right? Reading is counted!!) Now, the grind has started for the year and you'll be so wishing for the holidays real soon (I know I will be!)

Good luck with all the preparations!

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