Just SCRATCHing the Surface of Fractions

Fractions are always fun, right?  I can hear the sighs and cringes even as I mention the word fractions.  For some reason, even children who enjoy math are not fond of this concept.  I wanted to share one of our adventures into the mysterious realm of fractions.

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The Problem:

Ben had 50 more coins than Joan.  After Joan had given 29 of her coins to Ben, she had 1/3 as many coins as Ben.  How many coins did Ben have at first?

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This problem came from the Singapore Math Intensive Level 5A and was included after studying operations with fractions.  In typical fashion, our daughter completely ignores the accompanying diagram and sets out to solve this problem armed with her new math skills.  She quickly found herself at an impasse and extremely frustrated.

I approached her and tried to see into her thinking strategies.  Then, we realized the diagram! 

 

She knew Ben's amount was equal to Joan's amount plus fifty.  Also, that after Joan gave Ben 29 coins, her amount was equal to 1/3 of Ben's.  Good start!  Then, the trouble began.  She kept adding 29 to 50 to get 79 and tried to divide this value by 3.  Really close, but not quite!  She kept running in to the problem that 79 is not divisible by 3.  I asked her what she was trying to do with the amount of  "79."  She thought if she could divide this by three it would represent the amount in each rectangle for Ben.  Still, close but not exactly correct.

After struggling with this for a while, she decided to put it away and move on to other problems.  As a high school math teacher, I kept wanting to set up two equations with two unknowns and just solve it for her.  I knew, though, she needed to keep struggling with it and work it out.  So, she went to her dad and they wrote a Scratch program to solve it.  Not exactly, the approach I would have taken, but it provided a different type of learning experience. 

For those of you who are unfamiliar with Scratch, it is a computer programming language developed by the Life-Long Kindergarten group out of MIT University.  Our daughter has been using this for a couple of years to animating her drawings.  She has been wanting to write a program that served a different purpose.  So, she and her dad decided to tackle the dreaded "fractions problem".  To quote my husband directly, he refers to their process as a "brute force" method of solving.  They created a screencast to explain what they did and posted it to Vimeo.  I also need to add that Abby was unimpressed with the names and decided to change them to something more creative like Percy and Ashley Kit.  Check it out!

Brute Force Solving in Scratch (screen cast) from Rhett Allain on Vimeo.

Actually, they did a slightly different problem in that they are trying to determine how many coins Joan has.  Nevertheless, this was and is a really cool learning experience for our daughter. 

Back to the problem...

From the program you see that Percy must have had 133 coins in the beginning and Joan would have had 83. Now, I still thought it was extremely important for her to know how to solve this problem without technology.  So, after some time, she and I revisited the problem.  This time, we decided to pay attention to the given diagram.

After a break, she realized her mistake.  She had forgotten to add the 29 to Ben's total.  So, really, we know that 2/3 of Ben's total could be found by adding 29+50+29.  This equals 108.  If you divide 108 by 2, it becomes apparent that each rectangle represents 54 coins.  So, after receiving the 29 coins from Joan, Ben had (54*3) or 162 coins.  To find the amount he had in the beginning, you simply subtract 29 from this total.  So, 162-29 = 133.  Therefore, Ben had 133 coins in the beginning.  How many did Joan have?  Well, you would subtract 50 from Ben's total.  So, 133-50 = 83.  This matches what Abby and her dad found using Scratch. 

I have to say I had to fight the urge to just do things for her.  That would be easier!  I feel it is so important to allow students the opportunity to wrestle with concepts regardless of the discipline.  Ultimately, if you do this, the student takes ownership of the problem and gets on the track towards developing genuine understanding.

All in all, this was a valuable learning experience for everyone.  Here's what we discovered:

• Technology is fabulous and really cool!
• In order to use technology effectively, you still need a good foundation of the concept being studied.
• Never ignore the diagrams!
• Sometimes it helps to get a fresh start.  Put it away for a while and revisit the problem later.

Our approach to education is multi-dimensional: exploratory, creative, hands-on, and integrated with technology. The ultimate goal is to create an environment that fosters inquiry, creativity and a genuine love of learning.  I feel this experience has provided our daughter the opportunity to "think outside of the box" while having a meaningful learning experience.  I think this is a problem she will not soon forget.

P.S.  It's okay to have a "P.S." on a blog post, right?  I really am excited that our children have such a wonderful dad who is extremely involved in their education.  Not many kids have a dad who is a physics professor!  It has really opened many doors to them and given them many wonderful opportunities.  I can't wait to share what we have been doing with circuits!  Anyway, please check out his blog if you are interested in physics and how things work in general.  Lots of cool stuff at Dot Physics!

Comments

[David Muir]

I really liked this. Thanks for sharing - especially the video that shows you building the solution in Scratch. I've used a similar approach but with a spreadsheet. With a spreadsheet you can end up with all the numbers laid out in a table which can be a bit daunting, however, it does mean that sometimes you can spot a pattern. Occasionally, we will write a formula and then just plug in numbers, using heuristics to guide which numbers we try next.

The nice thing about the Scratch solution is its visual nature and that you can adjust values as it runs to see what happens. To that end, I would have gone with your daughter's original idea and used Percy and Ashley Kit in the if condition and had separate variables to report the answer. That way you could have watched the amounts of money change and looked to see if you were getting closer to or further from an answer... or if there didn't seem to be a pattern at all.

I agree with your desire to help her to a solution that doesn't involve technology. The technology solution, however, can help there too. The calculation in the if statement condition is a great starting point for producing an algebraic solution and you can use it to work backwards to a more general approach to solving this type of problem.

Sorry for the long comment but my mind is buzzing with possibilities. I'll say it again - this is brilliant. Thank you for sharing.