Note: As usual, the comments section contains insightful contributions from Denise, mathercize et al. I expounded briefly on a ratio approach and provided a link to some other bar diagram solutions. By the way, I've been using the misnomer, 'fraction bars' instead of the correct phrase bar diagram. I have a lot to learn here...
Lots of traffic from people viewing this placement test. The following is Question #11 from this test and I think it can lead to fruitful discussion of methods and strategies for middle school and secondary students. Older students will usually use an algebraic approach, prealgebra students might use 'Guess-Test-Revise' with a calculator. However, Singapore students apparently use a 'fraction bar' model for many of these, which I see as a ratio approach. How would you solve it? How would you present this to your students? What methods would you expect your students to use? What percent of your students would feel they don't even know where to begin and give up quickly? Would group work help here?
My first inclination was to set up an equation in one variable, however, because it was on the Singapore test, I tried a ratio approach which enabled me to solve it mentally. Have fun! By the way, to personalize it for your students, you may want to change the names to two students in your class and place it in the context of spending money at the mall!
Also, consider that if one were to use 'Guess-Test', it would make sense to start with a number divisible by $48 (since $240 is a multiple of $48, an intuitive student with strong quantitative skills might easily guess the answer!).
The Question:
Peter and Paul each had an equal amount of money. Each day Peter spent $36 and Paul spent $48. When Paul used up all his money, Peter still had $240 left. How much money did each of them have at first?
Monday, September 10, 2007
Another Problem from Singapore Grade 6B Placement Test
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7 comments:
I've only been introduced to Singapore Math & 'fraction bars' recently, but I tried to ignore algebra and set this up using them:
__________________________
| 36 | 36 | 36 |... |240 |
--------------------------------
__________________________
| 48 | 48 | 48 | ... | 48 |
--------------------------------
Now, my thinking was that since we have the same number of 36s as 48s, all the extra 12s must equal 240. 240/12 is 20, so, they shopped 20 times. I checked my work, 36*20+240=48*20=960. So, they started with 960.
It worked, but it doesn't seem trustworthy as a method (too much like magic). I realize its the same as solving 36x+240=48x, which I am happy to accept.
I wonder if I set the bars up correctly and what am I missing to trust the bar method.
When I read the problem, I too went for the algebraic approach. I guess that is because it is what I'm most comfortable with.
I have to learn more about this bar method. It might help my pre-algebra students. I just don't feel fluent enough to introduce their use.
I'm assuming these students would tackle the problem with a guess & check method. As for how many would give up due to not knowing where to begin...I would hope that group or partner work would help with that.
Having been trained by several years of teaching Singapore math to my kids, I almost never jump to algebra as a first resort. Used to do that, but now I look at relationships between numbers. I guess that might be like the ratio approach Dave mentions. Many times, a bar diagram makes the math much easier.
For this problem, I didn't actually draw any bars, but I had a mental image like this:
[- - - -48 - - - - -]
[- - 36 - -][- 12 -]
(Bars may not line up perfectly, depending on how the font settings work.)
I realized that Peter spending $36 is the same as if he had spent $48 (like Paul) but also saved a new $12 each day. In other words, Peter has a net gain of $12 per day, compared to Paul. In the end, Peter is ahead by $240, so there were 20 days in all. (And, as the others have said, this means they started with 20 x $48 = $960.)
Mathercize, you might have felt more comfortable about your diagram if you had included the intermediate step of breaking each of those |48| chunks into |36|12|. Then you could collect the 12's and show that they have to add up to the |240|. But to do an actual drawing with these numbers is a nuisance.
denise--
thanks for coming to the rescue! I was hoping someone who knows far more about bar diagrams than I do would rescue me!
mathercize--
I liked your approach and you're definitely ahead of me in using this model, but it's still a learning curve for all of us.
A few thoughts...
(1) I'm using the incorrect terminology, 'fraction bars', instead of the correct phrase 'bar diagrams'. Just shows you how much I know about this! I'm simply posing these questions to open up a dialog about different approaches to solving these excellent problems, the bar diagrams being one of these.
(2) From what I've seen in studying a few solutions, the central notion in the Singapore approach is to train children to choose a UNIT in each problem and represent the different quantities in terms of that UNIT: |---|. The following is how I solved the problem (most likely NOT what a Singapore student would have used:
Peter: |---|---|---|
Paul: |---|---|---|---|
In my twisted way of thinking, Peter spent THREE-FOURTHS of what Paul spent each day. After, Paul expended all his money, Peter would have spent only three-fourths of his money, which means he would have ONE-FOURTH of his money left. If one-fourth of the total equals $240, then the total would be $960. This is what I meant by using RATIOS to solve it mentally. I was only able to do this when I made the paradigm shift away from an algebraic approach.
(3) Look at http://singmath.com/sol/s.htm
for additional solutions using bar diagram. This link was provided by Jenny who works for the US distributor of Singapore math materials. She provided excellent insights for me in my previous Singapore posts and I am indebted to her for sharing this.
Yeeks. I would have problems teaching this, I think, since my own habits are strange. I don't have a standard algorithm for ratio problems. I guess a few years in upper elementary or middle school would cure me.
The gap is $12/day, so we know we have 20 days to get 240 (division, not algebra).
Actually, I think in Singapore they call these diagrams "models." The term "bar diagrams" evolved on a homeschooling forum, and it has stuck with me.
I like your ratio method! It's always cool to see a different way to solve a problem, and that solution is very efficient. Peter spends 3/4 as much as Paul and has $240 left:
Peter:
[------][------][------][-240-]
Paul:
[------][------][------][------]
So one unit is $240, and you're home free.
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