Social media had a thread about “frontloading” and “backfilling” lessons, and I was trying to wrap my brain around it… and got provided a great example of it.
Students are supposed to figure out the area of the shaded part of the circle.
Student comes in: coudln’t figure it out last night. “I know I need the common denominator.”
Here’s the frontloading that was missing: I wrote a similar circle with a quarter of it shaded and marked the rest of the circle “3/4.” No, the student didn’t reall grasp quarters of circles. That’s another “some of them don’t bring that to the task” thing. They also … had no idea how to figure that out.
I dind’t have manipulatives or the time to build “discovery” of that… so… “4/4 is 1 whole” (showed on a number line, too… and “four quarters… are a whole dollar… fourths are quarters…”)
Made another circle w/ 4/5 not shaded… a little thought and .. “what would be the whole?” …. oh, 5/5… oh, 1/5 is shaded… anotehr example… then
“what if it’s 3/5… what would the shaded area be?” Hmmm… Hmm…. 2/5, hooray!
Then… to the problem in question: “what do we need to know to figure out the shaded area?”
“The common denominator!”
Well, yes… but let’s think about understanding. Why do you need that? No idea. Some looking and thinking and talking and that oh, we need to know what the unshaded are is, and it’s split into two parts… so to find the whole unshaded area we need to add fractions and that’s why we need that common denominator…
And then I turned it over to student who got those next 3 problems independently.
Now, without the “frontloading,” they might get it. Or, they might memorize the procedure and hope they didn’t drop a step when it came to the test. They’d “get it,” but … not really. The “what is 1?” idea would not be reinforced. Oh, and I talked about “yes, 5/28 is a reasonable answer because 14/28 wouldbe half and this is a lot smaller than half.”
Resource Room Dot Net Blog 
Opus the Poet
October 15, 2019
I had to solve this one, and I have to say that the shaded area in your example is much too large. After solving for the common denominator and converting and adding I came up with the shaded area value of 1/24.
And I had to solve this one because you triggered my OCD.
xiousgeonz
October 19, 2019
I had to turn mine off (for starters, it’s not even there — I have to work *hard* to activate anything like “it has to look right” ;))….
I got to do it again with another kiddo who said “what do I do?? Isn’t there something I have to subtract?” — totally already k nowing from common denominators. So when I asked him the 3/4 question he knew right off… as iwth the other ones… took a lot less time but still needed the bridge. (for him it was more connecting things and less building….)
and now I’m resisting fixing that thing — but what I think I shall do is actually *make* a “thing” walking through it with good drawings (hey, when I took Java we learned how to make slices of circles) and share it w/ the math teacher in charge of the course. The previous teacher-in-charge liked my “things.”