### Thinking and Re-thinking about Fractions

Last month, I was passing through the hallway one of our high schools, when I overheard a teacher delivering a lesson. He was discussing equations of some sort, where a fraction was multiplied by a variable.

In this particular case, he was asking about 2/3 x 6. He checked the class for understanding: "Do we all know how to multiply a fraction by a whole number?"

Think about that for a second. What is your go-to method for solving 2/3 x 6?

Prior to this year, if someone asked how to multiply a fraction by a whole number, I would have given them the “algorithmic“ method of doing so: create a fraction out of the whole number by placing a numerator of six over a denominator of one.

Then multiply my 6/1 fraction by 2/3: multiplying the numerators together and then multiplying the denominators together. This would give me 12/3, which could then be reduced to 4.

For the first time as a math teacher myself, as I walked by that classroom, I thought about the multiplication of a fraction and a whole number in a completely different way: I visualized

In my mind I saw the whole number of six broken down into 6 equal parts.

I then thought about breaking those six equal parts into three equal groups.

I then imagined two of those groups (2/3), giving you total of four pieces.

This may not seem like a big deal, but this represented a big shift for me. Growing up, math was always taught to be procedural. Multiplying numerators and denominators feels like math to me: I can see the calculations on the page in front of me.

But even as I did the calculations, I would never have visualized what the answer actually meant. I knew the final answer had to be less than the original six, but other than that, I wouldn’t have proven my answer.

Coming at the problem from a visual perspective allows me to not only arrive at the answer, but also see how the answer makes sense. This is not to say that we should all abandon the algorithm. 2/3 of 6 is very easy fraction-multiplication to visualize. 2/3 of 1397, however, is not as easy. Knowing the algorithm definitely makes the latter question more manageable.

### Linking it to our students

This revelation made me think a lot about the intentionality of choosing numbers when creating problems for students. Do we choose numbers so that they can visualize what they’re doing? Or do we choose numbers that nudge them towards a more general procedure?

As a high school teacher, this way of thinking - this “new math“ thinking - is becoming ever-increasingly important for me to wrap my head around. The students now arriving in grade 9 have been introduced to more authentic problem-solving, mental math strategies, and visualization than the previous generation of students.

And it's not just for fractions - no longer is the standard algorithm for multiplication students' go-to method when multiplying 2-digit numbers. And if my students are thinking in new ways, and I want to help them as best I can, I have to start thinking in new ways as well.

1. Love this! Visualization of the math is new for many, I think. Do secondary teachers have manipulatives available for students to use?

1. Hi Lisa! Secondary teachers do often have access to manipulatives, but it's often not our first instinct to pull them out. The tide is turning though as we get more comfortable with some of these concrete ways of conceptualizing!

2. Interesting! I’m thinking a lot about them. I find my students (Primary) still use them as prompted in some way, but don’t usually ask for them. They wait for one brave soul to ask, and work without them otherwise...even if they are out in plain sight!

3. Same in Intermediate/Senior... students won't select them on their own. But if we place the bin directly in front of them, and initially instruct EVERYONE to use them, maybe we can help reduce the stigma? Working on it!