Visualizing equivalent fractions

let's think about what fraction of this grid is actually shaded in pink so the first thing we want to think about is how many equal sections do we have here well this is a 1 2 3 4 5 by 1 2 3 grid so there's 15 sections here you could also count it 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 so there are 15 equal sections here there are 15 equal sections here and how many of those equal sections are actually shaded in this kind of pinkish color well we have 1 2 3 4 5 6 so 6 15 is shaded in but I want to simplify this more I have a feeling that there's some equivalent fractions that represent the exact same thing as 6 15 and to get a sense of that let me redraw this a little bit where I still shade in 6 of these rectangles but I'll shade them in a little bit in one chunk so let me throw in another grid right over here and let me attempt to shade in let me attempt to shade in the rectangles as fast as possible so that is one one rectangle now I'll even make my I can make my thing even bigger all right one rectangle two rectangles two rectangles three rectangles three rectangles halfway there four rectangles four rectangles five rectangles shade it in and now six rectangles six rectangles shaded in so this this right over here what I just did this is still six rectangles of 15 of the 15 rectangles are shaded in so this is still 6 15 s these are representing the same thing but how can I simplify this even more well when you look at it numerically you see that both 6 and 15 are divisible by 3 in fact the greatest common factor is 3 so what happens if we divide the numerator and the denominator by 3 if we do the same thing to the numerator and the denominator we're not going to be changing the value of the fraction so let's divide the numerator by 3 and divide the denominator by three and what do we get we get to two over five now how does this make sense how does this make sense in the context of this diagram right here well here we start off with six shaded in you divide by three if two shaded in so what you're essentially saying hey let's group these into sections of three so let's say that this is this right over here is one section of three this is one section of three right over here so that's one section of three and then this is another section of three right over here and so you have two you have two sections of three and actually let me color it in a little bit better so you have two sections of three and if you were to combine them it looks just like this it looked just like this notice this is covering the exact same area as your six smaller ones did and then how many equal sections of this size do you have on this entire thing well you have five equal sections so let me go because this is one section of three all right over here this is another section of three and then this is another section this is another section of three so notice you're covering the exact same area of the original thing you're carrying your covering two out of the five equal sections so two-fifths and 615 are equivalent fraction so if you want to say what fraction of this is covered in the simplest form you would say two-fifths