# Reflections from Teaching: Fraction Talks

“We need to increase the explicit instruction on unit fractions and we need to explore fractions continually rather than as a single unit of study.” -Cathy Bruce

In this post, I want to describe my experience with the Fraction Talks Activity I originally borrowed from Nat Banting’s site. He has since become a curator to www.fractiontalks.com, a website devoted to various templates teachers may use – if you haven’t seen it yet, it is very worthwhile to check out!

After pondering about the original blog post for a few minutes, I was struck with just how versatile these templates are at connecting certain mathematical aspects together, and providing a visual representation of certain operations/ideas. The fraction talk template that I used when I worked with two classes of Grade7/8 was the following:

Concept of Fraction

Perhaps one of the most basic uses of the template is to connect it to the general concept of a fraction: a shaded portion of a whole, in which the whole has been divided into equal parts. To do this, we can shade in a certain area of the overall square (taking the big square to be the whole) and ask what fraction of the whole we have. For example, we may shade in the rectangle below and ask students to determine what fraction of the whole is shaded in.

In this case, you can reinforce that the regions under consideration must be of equal size. Be on the lookout for answers of 1/16 since this means the student has partial understanding of what a fraction means in terms of shaded regions.

Extension questions (try on your own):

*“What fraction of the whole is not shaded in?”*

*“Shade in a region of the whole representing 1/4.”
Shade in multiple different-sized regions and ask “What fraction of the whole is shaded in?”*

Of importance is which region we use as our whole. For example, let’s use the same shaded region, but restrict our whole to a smaller square.

Here, you can check to see if students recognize that the shaded region depends on what region we agree to be the whole unit.

Extension questions (try on your own):

*“Shade in a region representing 1 and 1/2.”*

Fraction Multiplication

Once students are comfortable with the big ideas up top, you can start to play a little bit! When I used this activity for the Grade 7/8 classes, I had two end-goals in mind: I wanted them to utilize multiplication and addition of fractions to help them determine certain shaded regions. Let’s first look at how we can connect multiplication into this.

First, identify a region you want to work with. Using the big square as the whole, I have shaded in the purple region below as the area of interest to start:

Next, we want to identify a sub-region with our original 1/4:

Finally, we can view this sub-region in terms of fraction multiplication. The sub-region is 1/2 of the original 1/4. But we already know that this region is 1/8 of the whole. Hmmm…

From here, once students can see the visual argument, you can shade in various smaller regions and have them write a multiplication statement in order to determine the fraction of the whole.

Extension question (try on your own):

*“Shade in a region representing 3/4 of 1/4.”*

Fraction Addition

Fraction addition can be visualized in a similar way, and may potentially be easier than multiplication. Here we identify two disjoint regions we want to work with. Using the big square as the whole, I have shaded in the purple region below as the area of interest to start:

Most students were happy dividing the overall shape into small squares to get a fraction of 3/16.

Great! However, my end-goal was connection to fraction addition, so I went back to the shaded region and had the students think about what the fractions were as separate entities. We agreed:

I asked the students to add the two fractions, and then two observations were noted. The first was that 1/8 + 1/16 = 2/16 + 1/16 = 3/16, which is what they had noted before. The second was that when the grid of red lines was added originally, it shows the connection to the equivalent fractions 1/8 = 2/16 (can you see it?). The teachers helping me facilitate noticed that the grid the students formed gave a visual of the LCM of 8 and 16, as well.

Extension questions (try on your own):

*“Shade in a different region that also represents 1/8 + 1/16.”
“Shade in a region representing 1/32 + 3/64. What fraction of the whole is this?”
Shade in multiple different-sized regions and ask “What fraction of the whole is shaded in? Use an addition statement to help you determine your answer.”*

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*Conclusions

I see many other applications of the fraction talks templates, such as perimeter or area, and I am sure they will come as you explore them a bit more. You will notice that the template I chose was slightly challenging. I actually gauged my audience with a simpler template (to decrease extrinsic cognitive load since the extra lines would add to the overall cognitive load) and moved on to the more challenging one since they were comfortable talking about the simpler template. The students I was working with had already studied fractions and fraction operations quite extensively, so this was essentially an extension activity for them to help solidify some of the bigger ideas they had been working with. This said, I still think some of the templates are very accessible and could be used as an introductory activity or as an intermediate activity to gather information on how your students are perceiving fractions. The versatility of the activity is certainly one of the major selling features for me. In closing, check out www.fractiontalks.com and try it our yourself – it makes for a pretty interesting and informative activity.

Here is the BLM from my fraction talk: Fraction Talks BLM

Feel free to use it for your own class.

As a person who teaches students entering High School, I have started to doubt if this is the best way to teach fractions to kids in Grades 7 to 8.

The kids arrive in my class, Grade 9, able do this sort of exercise, and they quite like it (because they are familiar with the context).

But they understand the use of fractions outside the “division of a whole” barely at all. They react to improper fractions with disgust, precisely because they aren’t divisions of a whole — and insist on changing them to mixed numbers no matter how many times I ask them not to. This is because, I suggest, we over-teach in the context of division of a whole, so 5/4 is not a real fraction and must be turned into 1 1/4 so that it is a “real’ fraction.

When you suggest to them that a fraction is an uncompleted division they look at you stunned. They have not learned that 3/5 = 3 ÷ 5 and no more. So when you get a solution to an algebra problem, say 3x + 4 = 8 that is a fraction, but done as a division, they struggle to effectively come to terms with the fact that the final step of division means x = 4/3 and they can stop. They don’t relate fractions to division, and they feel compelled to divide and get 1.3 (the poor rounding is typical, incidentally).

So can I make a plea to move away from this sort of thing please. I know it feels like it is helping, but I’m convinced that it isn’t (at least at those Grades, obviously it is an introduction for younger students).

We need to move towards things that let them deal with fractions of larger groupings, with fractions as the solution to divisions, with fractions more than one, and with fractions like 8/17 that we can’t really sort out with pretty squares, Otherwise they are trapped in a template that prevents them actually using fractions in any but the cutest divisions by powers of 2.

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