Consider subtracting 5 7/11 – 6 1/2 (five and 7/11 minus 6 and 1/2). Try working through the situation with paper and pencil and then think about how you might represent this problem conceptually.

In a recent exchange with some middle school teachers, this issue arose. How do you represent this situation conceptually? If students understand the meaning of subtraction and the meaning of fraction (mixed number), could they work to develop an algorithmic strategy to subtract mixed numbers? I think so! This video might show how…

Subtracting Mixed Numbers

Scott

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This is terribly confusing. Negative circles? Gah. How about a number line instead. I think that is how the common core suggests teaching all of this stuff. It eliminates the junk that was introduced with the red circle, which is quite muddled. You may announce to students that it makes sense, but does it really help with understanding?

Make a number line. Divide units into 22nd’s. Done.

http://math.berkeley.edu/~wu/CCSS-Fractions.pdf

Andrew,

While I agree with Wu and the CCSS that number lines are powerful ways to make sense of mathematics, I must respond to your comments about the video being “…terribly confusing.”

1. This was made for a particular group of teachers who had been using red and yellow chips to help their students to make sense of integer operations. Also, they had been using fraction models (circles cut up into smaller pieces) to help to make sense of fractions. We wanted to find a way to make sense of this particular computation using the models that had previously been used.

2. Making a number line, divide units into 22nds, done???? We most certainly would not be done after your seemingly simple two steps. Let’s look into how the computation would play out using the number line and also see why the number line model also has flaws and can become “muddled.”

In this case, we could start at 5 7/11 (aka 5 14/22 aka 124/22) and step to the left (where each step is equivalent to 1/22) 6 1/2 (aka 6 11/22 aka 143/22) and this would take us right past zero to -19/22.

The number line model gets “muddled” (in my opinion) when you have something like:

5 – (-3)

So, start at 5…subtract by moving to the left…but then go backwards since the 3 is negative…thus moving to the right…

This is where I like the idea of the “zero pair” and why we implemented that strategy in the video.

3. I agree that telling students that something makes sense does not make “it” make sense! I propose that teachers create tasks that allow students to try to make sense of fractions/integers using multiple models. Some may latch onto the number line…some may latch onto the chip model.

I have had great success with the chip model with students. I have found that the number line model (move left, move right, turn around, go backwards) has more rules than the rules that I want the students to discover!

All I can tell you is that I understand the concepts and the mechanics and I was thoroughly confused, although I’m glad it works for your students.

One of the key benefits of a number line over pies is you don’t need to “make up” some concept like “negative pies.” If you’re trying to be concrete by using something like pies, introducing negative pies just throws that out the window, doesn’t it?

Wu would get you out of the subtraction of negative problem by referring to the definition of subtraction, I think: A – B = C. C is then the number that, when added to B, give A:

A – B = C -> A = C + B -> 5 = C + -3

This is easily accomplished on a number line.

Personally, by the time you are subtracting negatives, I would hope that students would understand inverses and inverse operations. A – (-B) = A + (-(-B)) = A + B. One can apply these concepts (which are true by definition) without resorting to number lines or pies.