THE PROBLEM
When mom was 40, her son was twice as old as her daughter. Now her daughter is 28 and mom is twice as old as her son. How old is mom now?
Answer: 64
REFLECTIONS
Not exactly an authentic real world assessment but there's probably a reason why these are "Problems for the Ages"!! I don't believe harm is done by having students develop these kinds of relationships...
Of course for years I used to teach this using a chart and setting it up algebraically.
For younger students I would encourage a Guess-Test-Revise approach. Say, girls was 3 so boy was 6. Now it's 28-3 = 25 years later, so boy would be 6+25=31 and mom would be 40+25=65 which is a little more than twice 31. Revise to girl was 4, son was 8. 24 years later, girl would be 28, boy would be 32 and mom would be 40+24=64, Bingo!
Ah but children in Singapore perhaps as early as Grade 4 are representing these relationships using unit lengths and bar models. I simply don't have enough experience doing this so those out there with more knowledge please improve upon this so I can learn!
Then
Mom |-----40------|
Son |----||----|
Daughter |----| (some unit length)
Note: [//////] below represents the additional years from Then to Now
Now
Daughter |----| [//////] = [--28--]
Son |----||----| [//////]
Mom |----40---| [//////] = |----||----| [//////] |----||----| [//////] = |----||----||----| [//////] [--28--]
So 40 = 3u + 28 or u = 4 yrs, etc.
I'm limited by my graphics but I believe there are different, better and more efficient models used in Singapore Math. If you're thinking I retrofitted an algebraic solution to make it look like a bar model you're probably right! I need wiser and more experienced 'bar modelers' to help me!
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3 comments:
Did you base this on an actual Singapore Math problem? This is more convoluted than most bar model problems I've seen. It's probably easier to work it out with regular algebra, although the models do give a visual representation of what the variables are doing. But the models are only meant as a stepping-stone to regular algebra, and when problems start to get too complex, it's time to pull out the real thing.
As you say in the last paragraph, what you have drawn isn't really a bar model because you haven't used the size and alignment of the blocks to visually demonstrate the number relationships. So you're just doing regular algebra with some rather awkward symbols.
One rule for drawing bar models is to keep each bar all on one line. The fact that your brackets-and-dashes bar jumps to a new line makes your model look wrong at first glance and very confusing even when I see what you mean. This is easier done with a graphic:
Bar models for "When Mom was 40"
Thanks Denise! I made that crude attempt to bring the'experts' to the fore!
I'm anxious to see your model for this problem.
And yes I found this problem among comments from an old post of mine. I believe it was posted by someone from Singapore who was demonstrating harder problems tackled by Grade 6B students there. I'll have to verify the details but I didn't construct this problem.
My instincts are to use algebra of course with one or more variables because that's how I was taught. But I'm always open to learning new things. It's just heard to teach an old dog like me new tricks!
Wow! I just saw your models for this on your website and I commented there. My attempt was pathetic but I am limited these days by posting via email on my Nexus 7. No pretty graphics! But then I figured someone with more talent will do it for me!
Interestingly though the essential ideas are fairly close.
Thanks for sharing...
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