Reducing Fractions (8/10 to 4/5) with photographs [3rd grade]

Maybe 8/10 to 4/5 is a bit hard to grasp? I think starting with 1/2 is better. For visual aids, I would print out each of these circles with the correct portions and telling a story something like this:

Jimmy, let's pretend you're having a pizza party with a friend. You have this whole big pizza to share so what do you do? Yes, you cut it in half. This [show pizza card with single dividing line] is a pizza cut in half. See, you and Billy can both have 1/2 of the pizza.

Except, two more of your friends are coming over for the party. You need to make room for them too! So, we have to cut the pizza into fourths now. This [show the two dividing lines card] is a pizza cut into fourths. See? 1, 2, 3, 4. You each will get one or 1/4 of the pizza.

Another call just came in. It turns out two more of your friends are coming over for the pizza party. You will have to share the pizza of course, so we have to cut it again. This [show the card with 3 dividing lines] is a pizza cut into 6ths. Now all of you will get 1/6 of a pizza.

Now, don't be sad Jimmy. It turns out that your friends can't make it. That's OK though because you and Billy can now each get 1/2 of the pizza again.

Wait... the pizza is cut into a bunch of pieces. How can we do this? Well, [show the 1/2 again] This line makes 2 parts of the pizza. We have 1/2 here. So, how many slices will you two each get?

Look, [show the 2/4]. This is ALSO 1/2. If this pizza was cut into fourths, you each would get 2 slices.

Hmm, [show the 3/6]. This is ALSO 1/2! You two will get 3 slices each.

So, 1/2, 2/4, 3/6. Notice the pattern? We simply multiplied 2 by the number of cuts we made because our original cut is 1/2.

When we had two cuts, it was 1/22. When we had three cuts, it was 1/23. This pattern continues... 1/2*4.

Ah, but that just tells you how much to cut. What we want to instead do is be able to change the scale. To do that, we increase the top and the bottom equally. Think of eating the whole pizza though. You have 2/2 and two friends come over. It is now 2+2/2+2 or 4/4. Another two friends come over. It's now 4+2/4+2 or 6/6 for the whole pizza.

Ok, but you can't eat it all. So, you take your part. You want to take half of the pizza, so you divide the numerator by 2 and leave the denominator the same. You get 3/6 slices or 2/4 or 1/2. It's all the same.

Well, sometimes fractions are a bit messy and there's more cuts than people showing up. To make it easier, we can just simplify the fraction (OR make less cuts in the pizza).

So, let's look at 3/6 first. Notice that 3 goes into both the numerator and the denominator. as (3/3)/(6/3). We know that 3/3 is the whole pizza so that part becomes 1. And we know 6/3 = 2. So our fraction is really 1/2 in disguise.

Hmm, what about 2/4? Well, they both are divisible by 2 so, (2/2)/(4/2) which gives us 1 (because 2/2 is 1) over 2 or 1/2 again.

The same thing happens with this problem: 8/10. We can see that both are divisible by 2. So, we have (8/2)/(10/2) or 4/5. This is the same quantity but just using less cuts.

We can even do this with a big number like with money. Let's say you have 25 cents. Well, how much of a dollar is that? A dollar is 100 cents right? So, you have 25/100. But wait! Both numbers are divisible by 25, so you have (25/25)/(100/25) or 1/4. And it does take 4 quarters to equal a dollar.


So, I don't know how helpful that was. I also hope the whole pizza party thing isn't insensitive to his condition. I know that some autism leads to the person being unable to discern story from reality and so it's possible he might think he's going to go on a pizza party with friends. Anyway, I would build up to this idea of 8/10 equaling 4/5 through the interesting properties of 1/2 first.

I might also be wrong about some things. I just made it up as I went along. It was very Socratic.

/r/learnmath Thread