Here is some shaded images. Notice I have some of the parts are shaded and some are not. What I want you to do is count the number of parts that are shaded and count the number of parts in total in the whole. You can enter your answers in these boxes. So for example, for the first one I know I have 1, 2, 3 parts that are shaded out of whole.
Okay. For this one, you had five parts shaded, one, two, three, four, five, out of 12 parts total. If you count up all the triangles, you'll get 12. Here, I have three boxes that are shaded out of ten boxes in the whole temple. And in this cross, I have seven triangles that are shaded out of 20 triangles total. I got 20 really quickly, because I noticed that there are four triangles in one box and I had five different boxes, so I just did
These numbers are actually really important. They can represent a fraction, they can represent parts of a whole. So for example for the circle I have trhee parts out of eight or 3/8. The three represents my numerator, it's how many parts I have. The eight represents my denominator, it's how pieces or parts are in the whole figure. Fractions don't always have to refer to shapes, they can refer to other things, and we'll look at this later in the course, but for now we're just going to focus on these images and shapes. So, 3/8 is a fraction for this circle, I want you to find the fraction for these others. You can enter your answer in these boxes. 3/8 can also be written as 3/8. It's another way of writing a fraction. That's how you should type it in when you type it into these boxes.
Okay, the first one should have been 5/12, or 5 / 12. The next one, 3/10, and the last one, 7/20.
Let's take a look at some other images. Here, I have some circles. I've already written the fractions below each circle. I want you to check the images that have the same shaded area as the one over here. Check the circles that had the same area as 1/2.
I know 2/4 is the same thing as 1/2 because I can see visually that their areas are the same. 2/3 is not equal to here that's already shaded, so this extra little bit would be more than 1/2. 3/4 is also not equal to 1/2. I know 1/2 is this much and here I already have 1/2 and a little bit more. 3/6 is equal to 1/2. I can think about moving this one piece here, over to the empty space here. Here's what it would look like. There, now I have equal to 1/2. I know I'm missing one extra piece to get to my 1/2, so that won't work.
Okay, so based on these answers, I know that 1/2 is equal to 2/4. I also know that do that? How did I change this into this? Well, if I look at this shape and this one, I notice that I just doubled the number of pieces that are in this shape and I doubled the number of pieces that I have shaded. So I doubled both the numerator and the denominator and I got the 3/6. This time, I took my shaded pieces and I tripled them, so I multiplied them by three. I also took the total number of pieces, I had two, and now I have a total number of pieces of six. So I also tripled my number of pieces, 13 is three and 23 is 6. But let's think about why we can do this. I know 2/2 is really just one, so, I'm just multiplying 1/2 by a form of one. I'm not changing the value of 1/2, I'm just changing what the fraction looks like. The same is true down here, I'm taking 1/2 and multiplying it by a different form of on And a'll get 3/6. So we know 1/2 is equal to 2/4, which is equal to 3/6. These are all equivalent fractions. Okay. But let's look a little bit more closer at this 2/2 and 3/3.
Here in this first picture, I have two out of two parts. Notice that it's really one whole circle. I have 1/2 and 2/2. Here I have 1/3 of the circle, 2/3 of the circle, and altogether would be 3/3 of the circle. Here I have 1/4. 2/4, 3/4, and 4/4. The same is true with the fifths and sixths. You can just count up the pieces. So, the numerator counts how many pieces we have, the denominator counts the types of pieces we have. So for example with six, I have whole. I know all these circles really represent one. They're all equal, they represent one whole circle.
So, let's put all this together with a little quiz. What would the numerator would have been for 1/2 to be in tenths? And also, I want you to write in your own equivalent fraction for 1/2. You can write any fraction in this box. You want to put in an answer like 3/4, or 3/4, 3/4. You can put your answer in as a number, a slash, and then another number. Just make sure that it's equal to 1/2.
If you put 5 here, great work. You should have multiplied this number by 5/5. I know that 5/5 is still 1, 25=10, 15=5. For the equivalent fraction over here, you could have put a bunch of different things. Maybe you put 50/100, or 30/60, or multiply 1/2 by the same number. Any of these would have worked.
Alright, let's make this a little bit more challenging. I'm going to show you some shaded images. And then, I want you to write the missing parts of the fraction. So, for example, I know this image has 3/4 or three out of four pieces shaded. This other image has 1, 2, 3, 4, 5, 6 out of 8 pieces shaded. I know that these two fractions are actually equal. I want to think about moving this piece here over into this empty region here. And then, my figures would match. And really, what I did is I doubled the number of shaded pieces I have. I split up each piece into two parts. Also, I doubled every piece I have because in total, I have 8 new pieces here. So, I multiplied the total number of pieces, 4, by 2, to get 8.
Here's some images that have been shaded. What I want you to do is figure out, what parts of the fractions are missing? What numbers should go in the numerators and denominators? You can put your answers in these boxes. Go ahead.
For this first one, you have four pieces that are shaded out of 16 total. 4/16 is the same as 1/4. I can think about moving these three pieces over into this region. If I move this one here, this one here, and this one here, I'll get one triangle, just like I have over here. This is what it would look like. For the circles, I have two out of three circles shaded, so four, five, six pieces shaded out of three, six, nine. So, I know 2/3 is the same as 6/9. I could also think about moving this piece over here, and moving this piece over here, and I'd have two full circles shaded, and one circle left empty. It would look like this. Okay, on this one I know I have two pieces shaded out of six. Here, I have four pieces shaded out of 12. I know 2/6 is the same as 4/12. I can think about shading this piece and moving it here, this one here, and this one here. That way I'd have 2/6 and 4/12. What's also interesting is that both 2/6 and 4/12 is really one out of three or 1/3. These are all equivalent fractions. For the last one, I have three out of seven pieces, and in this last one, I have 6 out of 14 pieces. So, 3/7 is the same as 6/14. Nice work on that quiz.
Okay, for this quiz what I want you to do is simplify these fractions. You're going to simplify these fractions and then pair it with the image that it matches. For example, when I reduce 6/8, I got 3/4. I would look at my images and then I see, oh, f. And I have 1, 2, 3, out of 4. So f would be the answer for 6/8. You can put your answer in these boxes.
Let's finish this last section with a quiz. What do you think 2+5/4+10 is? Is it
I know 2 + 5 makes 7 and 4 + 10 makes 14, so I really have 7/14. We need to remember that when working with fractions we can only simplify factors. We can only take out numbers that appear both in the numerator and in the denominator. So I can take out a 7. I know 7 is the same as 7 the 7 is reduced to give me 1. And I'm left with 1/2. If you chose 1/2 nice work. We want to be careful in carrying out this math, there are some other options that you might have thought about. You might have thought about dividing 2 and 4 by 2 and 5 and 10 by 5. If you would've cancelled out the terms, you would've gotten 1 + 1 / 2 + 2. This simplifies to thinking isn't correct, we know that we can only simplify factors, that is numbers that appear in the numerator and the denominator as multiplication. We can't simplify terms, we did get the correct answer using this root, but it's not the correct mode of thinking when we simplify fractions. Some students work from here and then split up this fraction into 1/2 + because we have division bar, we have to keep our numerator together and our denominator together. Besides we were cancelling terms and not factors.