Topic A: Equivalent fractions
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So I've got a whole pizza here, and let's say that I were to cut it into two equal pieces. Let me cut it right over here into 2 equal pieces. And let's say that I ate one of those 2 equal pieces. So let's say I ate all of this right over here. What fraction of the pizza have I eaten? Well, I took the whole and I divide it into two equal pieces, and then I ate one of those pieces. So I ate 1/2 of the pizza. Now, let's imagine that instead of cutting that pizza into only 2 equal pieces, let's imagine instead that decide to cut it into 4 equal pieces. So let's draw that. So 4 equal pieces. So I could cut once this way and then I could cut it once this way. And so here I have 4 equal pieces. But let's say that I want to eat the same amount of pizza. How many of these 4 equal pieces would I have to eat. I encourage you to pause the video and think about that. Well, I would eat this piece. You could imagine me eating this piece and this piece right over here. I've eaten the same amount of the pizza. Each of these pieces you could imagine got cut into 2 pieces when I cut the whole pizza this way. And so now I have to eat 2 slices of the 4, as opposed to 1 slice of the 2. So I just ate 2 out of the 4 slices. I'm using different numbers here. Here I'm using a 1 in the numerator and 2 in the denominator. Here, I'm using a 2 in the numerator and a 4 in the denominator. These two fractions represent the same quantity. I ate the same amount of pizza. If I eat 2/4 of a pizza, if I eat 2 out of 4 equal pieces, that's the same fraction of the pizza as if I eat 1 out of 2 equal pieces. So we would say that these two things are equivalent fractions. Now let's do another one like this. Instead of just dividing it into 4 equal pieces, let's divide it into 8 equal pieces. So now we could cut once like this. So now we have 2 equal pieces. Cut once like this. Now we have 4 equal pieces. And then divide each of those 4 pieces into 2 pieces. So I'll cut those in-- So let's see. I want to make them equal pieces. Those don't look as equal as I would like. So that looks more equal, and that looks reasonably equal. So now how many equal pieces do I have? I have 8 equal pieces. But let's say I wanted to eat the same fraction of the pizza. So I could eat all of these pieces right over here. Well, how many of those 8 equal pieces have I eaten? Well, I've eaten 1, 2, 3, 4 of those 8 equal pieces. And so once again, this fraction, 4 of 8, or 4/8, is equivalent to 2/4, which is equivalent to 1/2. And you might see a little bit of a pattern here. Going from this scenario to this scenario, I got twice as many equal slices. Because I had twice as many equal slices, I needed to eat two times the number of slices. So I multiply the denominator by 2, and I multiply the numerator by 2. If I multiply the numerator and the denominator by the same number, then I'm not changing what that fraction represents. And you see that over here. Going from 4 slices to 8 slices, I cut every slice, I turned every slice into 2 more slices, so I had twice as many slices. And then if I want to eat the same amount, I have to eat twice as many pieces. So all of these, 1/2, 2/4, four 4/8, and I could keep going. I could do 8/16. I could do 16/32. All of these would be equivalent fractions.