Sunday, April 19, 2009

Singapore Math: Speed without Models

Between now and October (if we do go ahead with the PSLE route), I'll need to figure out how to help Brian present his math working for Speed questions better. Trouble is I'm pretty bad at Speed questions myself.

The working he's done here is entirely self-concocted, he hasn't learnt this in school, I've not taught him any particular method cos I don't really know how to present answers for Speed questions either and obviously, he is not following the model method taught in Singapore assessment books.

The thing is he gets the answers right, it's just the working that needs refining, and some discipline on his part not to skip steps. Anyone else finds his working confusing? I've included his verbal explanation in Italic, but look at his working without reading his explanation and let me know how many marks he'd lose even if he gets the answer right, assuming each question is 5 marks. Suggestions for improvement, with method mostly intact, would be much appreciated.

1. Fandi took 3h to travel from town A to town B. On the way, he passed a motorcyclist who was travelling in the opposite direction at a uniform speed of 48km/h. 1 and 3/4 h later, Fandi reached town B while the motorcyclist was still 6km away from town A. Find the distance between the two towns.

Distance between meeting point and Town A
= (48 x 1 3/4) + 6
= 84 + 6
= 90

Fandi's speed
= 90km/ 5/4h
= 72km/h

Distance between both towns
= 72km/h x 3h
= 216km

Brian: The motorcyclist took 1 and 3/4 of an hour to travel from their meeting point to a point 6km away from Town A. So the distance between their meeting point and Town A is the motorcyclist's speed times 1 3/4h plus 6 which equals 90. 90km is also the distance that Fandi travelled from Town A to the meeting point. He took 1 and 1/4 of an hour to travel to the meeting point, so his speed is 90km per 5/4 of an hour, which equals 72km/h. He took 3 hours to travel from Town A to Town B, so the distance between the two towns is 72 x 3 = 216.

2. Richard left town A for town B at 10am. 3/4h later, Linus left town A for town B and overtook Richard at 1230pm. Linus arrived at town B at 105pm. At what time did Richard reach town B?

Let Richard's speed and Linus's speed be R and L respectively.

2 1/2 R = 1 3/4 L
5/2 R = 7/4 L
20 R = 14 L
L = 10/7 R
7/3 L = 10/3 R

Richard took 10/3 hours to reach Town B.
Richard reached Town B at 120pm.

Brian: If you put it algebraically, you're only given one piece of information which is 2 1/2 R = 1 3/4 L (2 1/2 and 1 3/4 are the number of hours R & L travelled respectively when they met) so all you can do is compare the two variables. Since Linus took 2 and 1/3 hours (or 7/3 h) to reach town B, you have to find out how long Richard would take to get there.

3. Two towns, X and Y, were 156 km apart. Tom started to travel from town X to town Y at an average speed of 54km/h. At the same time, Peter also started to travel from town Y to town X. After travelling for 1 and 1/3h, they passed each other on the way. Find the average speed of Peter.

Tom's speed
= 54km/h
= 72km/ 1 1/3h

Peter's speed
= (156 - 72)km/ 1 1/3h
= 84km/ 1 1/3h
= 63km/h

Brian: I'm trying to find out the distance Tom has travelled when Tom and Peter met. Then subtract that distance from 156. That would be the distance that Peter had travelled by the time they met, and from this you can find Peter's speed.


Lesley-Anne said...

Hi Aunty Lilian, Mummy ask moi to reply directly. I think that his working for no. 1 and 2 is ok. Just that the final statement has to be short and has to just give the answer. I don't think they would accept the long explanation.

As for no.3, I don't know if it would be accepted as we do not do it this way. We draw what looks like a timeline. if you like, I can draw it for you after school tomorrow and my mummy can email you.

monlim said...

Oops! I just realised the long statement is Brian's verbal explanation. L-A asked me if he'd actually write that, I said yes. Didn't read properly lah. Sorry!

Lilian said...

Hi Lesley-Anne,

I'll let Brian see what you wrote. And yes, please draw No3 for us.

As your mummy noted belatedly :), that was Brian's verbal explanation to me cos I didn't understand his working. Is a final statement required or can he just end with the answer? So you think No1 and No2 can get full marks? Looks rather iffy to me.

Thanks for your help dear!


Anonymous said...

Your son's ability to visualise everything in his mind is amazing. His brain seems to be a screen for him to get his answers, this is probably the difference between genius and layman. And he is very concise in his answers.

I have worked out the layman visual view of the solution here for all 3 questions, just to understand how his mind works:

I learned something interesting from him in question 3. When he put Tom's speed=54=72/(1 1/3), that is an interesting way of presenting the answer by working backwards to get the distance.

No wonder geniuses always leave people speechless! :D


Lilian said...

Hey thanks QX, your answers look really good.

Yup, I find his answers very concise, which makes me worried that marks might be deducted if the marker deems that steps have been skipped. If the final answer is correct, the marker might just give him benefit of the doubt, but if wrong how?

It's difficult though for him to see that steps are missing cos to him the flow is entirely logical.

Anonymous said...

I believe he will be fine. The marker can probably tell he/she has hit a genius when they mark his paper.

Since he thinks fast, he probably has alot of time after the paper. I am sure he can finish the paper in half the time. Perhaps you can encourage him to write more statements and present the solution as if he is teaching someone how to do it as most people cannot visualise things as well as he does. Pai seh, even an adult like me also cannot... LOL..


Lilian said...

He's not at all a genius, but anyway, it's good to know that even someone as math-mad as you are finds it hard to visualise, I don't feel so alone! Will convey to him your tip on writing more statements if he has extra time.

Anonymous said...

Hi Lilian, I've been a silent lurker until now. :)

My take for Q3:

Distance covered by Tom
54 x 4/3hr = 72

Distance covered by Peter
156 - 72 = 84

Therefore, his speed is
84 / 4/3hr
84 x 3/4 = 63km/h

Essentially what I did is the same as what Brian did, but I just did it with more explaining statements. Will definitely be easier for the marker :)

Lilian said...

Cool Silent Lurker. Yes definitely easier for the marker, and for me too. You wanna do the same for Q1 and Q2 LOL! Thanks btw :)

Anonymous said...

Hello Lilian :)
I came across your blog while searching for tips about some math concepts. I'm lousy at maths myself so no contribution to solutions lah. But your amazing son reminds me of an article I read, about showing math workings. Here's the link:

Looking forward to reading your blog :)

Mummy CH

Lilian said...

Hi CH, welcome and thanks for the link. Showing steps is definitely a skill that some kids find hard to master.

Perhaps even harder is to write in words one's thought process in arriving at a solution. In my son's school, the math curriculum isn't rigorous but they do emphasise Math language, ie communicating your Mathematical knowledge in words. Can be frustrating to a child BUT nevertheless a skill worth gaining.

He actually has to think harder when writing his reasoning down (in a way that can be understood by a child one grade lower, that's the test) than solving the question!

Anonymous said...

Hi again... my method for Q1 and Q2is exactly the same as Brian's. :) Looks like he is ready for the Speed questions!

Lilian said...

Thanks anon! Gotta say your No3 method was really good, it was streamlined in a way that even I can understand easily :)

Anonymous said...

did you check the Road to Psle link which you have on the sidebar? There are a few speed problems with solutions using models, timelines, etc. Have a look.

Lilian said...

Oh thanks for reminding me! I've gotta take a closer look at that again, haven't gone to that site in months. It's a fantastic site.