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The key is in the phrase “of what you have left”. The one-third is being taken from a different whole (3/4 of the original whole)–but the amount remaining is defined by the original whole. In a subtraction problem, the size of the 1/3 would be in relation to the whole which is what this student did.

In my opinion the best way to model this is using bar models – I have an example but I’m not sure how to post it here. I’ll try to describe what this looks like: First draw a bar partitioned into 4 equal-size parts. Second, one part is given to the first friend. What’s left is a bar with three equal sized pieces. Third, friend two takes one of these three parts and you are left with two equal size parts. In order to describe the quantity left, this remaining part is placed on top of the original bar and it’s obvious that it equals one-half of the original bar.

Bar models (also known as Singapore Bar Models) really help kids see what is going on in these types of problems. The next step is to make a similar model that shows what the student did in his/her solution and compare the solutions. It’s not a guarantee that the student will connect this model to the symbolic process right away but at least students can conceptually visualize and think through such problems.

If this is not clear and you would like to see an example, please e-mail me.

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