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Contents

- 1 Chocolate Math 2
- 2 Mixed Numbers and Improper Fractions
- 3 Octagons
- 4 Octagons
- 5 An Easier Problem
- 6 Dividing Fractions
- 7 Dividing Fractions
- 8 Predicting Batches
- 9 Predicting Batches
- 10 Sugar as Two Fractions
- 11 Chocolate Math 2 Again
- 12 Another Prediction
- 13 Another Prediction
- 14 Extra Sugar
- 15 Extra Sugar
- 16 Exact Batches
- 17 Exact Batches
- 18 chocolate math 2 solved
- 19 question 3
- 20 question 3
- 21 question 4
- 22 question 4

Now we're going to get into a little bit more fractions. We're going to tackle another chocolate problem. This is going to be chocolate math 2. So here's our question, how many batches of brownies can you bake if a recipe requires 4/5 cup of sugar and you have 3 1/2 cups of sugar at home? Write your answer as a mixed number and I'll provide the boxes. Use any math reason that you know. Remember, the important part is just to try to reason, it doesn't matter if you're right, right now. We'll be coming back to this problem later. Just give your best guess now, and you can always change it and revise it later.

Before we attempt our problem, let's
review mixed numbers and improper
fractions. Mixed numbers are a way to
count objects when we have a couple of
objects and a part of that object. So
here, I have 2 doughnuts and then I have
whole doughnut into 3 equal pieces or
thirds, and I'd have 2/3. So, our entire
mixed number would be 2 and 2/3. Notice we
have a mixture of whole numbers and
fractions. That's why we call these mixed
numbers. If I split all of my doughnuts up
into thirds, I could count the pieces and
I would have 8/3. That would be the
improper fraction. It's improper because
the number in the numerator is higher than
the number in the denominator. So, let's
see how we can go from the mixed number to
the improper fraction. If I take my whole
doughnuts and split them up into thirds, I
would have six of them, 6/3. I can get
that by multiplying 3*2.
I still have 2/3 left over, so I need to*
add this 2/3 to my 6/3. 6+2 makes 8, so I
get 8/3. Let's try going the other way. I
can think about splitting 8/3 into 6/3 and
whole doughnuts. I know 6/3 makes 2. and
then I have my 2/3 left over. So, 2 and
would have a remainder of 2 or 2/3 left
over.

Now, let's look at mixed numbers with octagons. An octagon is just a shape with eight sides. What would be the mixed number and the improper fraction for these octagons? You want to think about splitting these octagons up into equal parts. You will also want to reduce your answer. So, for example, if I had 5 6/12, I'd really want to write that as 5 and I know six goes into 12 two times. So 6/12 reduces to 1/2.

If you got 3 3/4 that's great. I know I have three entire octagons and then I need to figure out this fraction. This is actually going to be 3/4. Let's see why. I could have split my octagons into eighths. So I know I have three whole octagons. And then I have 1/8, 2/8, 3/8, 4/8, 5/8, 6/8 of a hexagon. Remember, I know I can reduce 6/8. I know 2 goes into 6 and 8. So I can simplify it. And then I have 3/4. To find an improper fraction, we just need to count up all the 8ths. If I counted up all the 8ths, I would find out that I would have 30 over 8, or 30/8. Remember, I can reduce this fraction. So I get fifteen over four, or 15/4. But this doesn't quite make sense. My picture has eighths, but my answers have fourths. Well, that means I could have actually done this a little differently. I could have split up my octagons into fourths. This would be 1/4 of the octagon. Notice, when I have fourths, I can easily see that I have 3 have 3/4. 1/4, 2/4, 3/4 of this last hexagon, so 3 3/4. Counting the improper fractions is pretty easy, too. I can actually just use my mixed number. Because this mixed number is in fourths, I can figure out the improper fraction in fourths. I could multiply 4 times 3, which is 12. That makes sense because I have three octagons, and each of them has four pieces. Then, I have to add 3 more pieces, so, I add 12 + 3 and I get 15 over 4,

Let's try answering an easier form of the
problem. This might help us think about
what we should do. How many batches of
brownies could you bake if a recipe
requires 1/2 of a cup of sugar and you
have 1 cup of sugar at home? This seems a
little bit easier. This square is going to
represent my 1 cup of sugar. I know 1/2 of
a cup of sugar will make 1 batch of
brownies, so I'm going to fill up half of
my cup and make 1 batch of brownies. So,
here's a 1/2 cup of sugar and I know I've
made 1 batch of brownies. If I use another
batch of brownies. Now, I've used my 1
whole cup of sugar and I don't have any
sugar left at home. Well, I made 2 batches
of brownies. Let's pair this with some
math. I had 1 cup of sugar. And I
wondered, how many times did 1/2 go into
it? I knew I was able to make 2 batches of
brownies, so 1 divided by a 1/2 must be 2.
Remember, when we divide by a number, we
can actually multiply by the reciprocal,
we learned that from before. So, I could
change this to look like 1*2/1.*
Well, remember, I could just put any
number over 1. And then, multiply
fractions like before. 1*2 is 2, and 1 1
is 1. 2/1 is just 2. Let's change this*
problem a little bit, and see what we can
find out. What if, instead, a recipe
called for 1/3 cup of sugar? I know if I
use 1/3 cup of sugar, I'll make 1 batch of
brownies. If I use another third of cup of
sugar, I'll make another batch of
brownies. And if I use my last third of
cup of sugar, I'll make one more batch of
brownies. So, let's put this with some
math. I was wondering, how many times does
I know that for every third cup of sugar,
I get 1 batch of brownies, so I should get
number, it's the same thing as multiplying
by the reciprocal. I could change this to
multiplication, 1

So here's your quiz. How many batches of brownies can you bake if a recipe requires one-fourth of a cup of sugar and you have one cup of sugar at home? You can put your answer in this box here.

If you said 4 batches, way to go. I know for the first 1/4 cup of sugar, I'll make of sugar, I'll make another batch of brownies. Another 1/4 cup of sugar will give me another batch of brownies. And my last 1/4 cup of sugar will give me another batch of brownies. I started with 1 cup of sugar, and I wanted to see how many times did 1/4 divide into 1 cup. I can turn division into multiplication by multiplying by the reciprocal, so 1 times times 1, which is 1. 4 / 1 makes 4. The key idea in all these problems is that dividing by a fraction is the same as multiplying by the reciprocal of the fraction. We're going to use this idea in the next problem.

Let's make this problem a little more challenging. What if the brownie recipe called for 2/3 of a cup of sugar, and you had 1 cup of sugar at home? Before we get into the problem, let's make a prediction. Could we make more than one batch, exactly one batch, or less than one batch?

I had more sugar at home than my recipe calls for. My recipe only calls for 2/3 of a cup of sugar. If I had one cup of sugar at home, I could use 2/3 to make one full batch of brownies. And then, I'd have 1/3 of a cup left over so I know I can make more than one batch of brownies. If I only had 2/3 of a cup of sugar I could make exactly one batch of brownies. But I'll have extra leftover, I'll have a 1/3 of a cup of sugar that I can still use. If you said more than one batch of brownies, nice work.

Remember, I still have 1/3 of a cup of sugar that I can use at home to make brownies. I've drawn it over here. This fraction amounts. I can see that this 1/3 cup of sugar actually represents 1/2 of a batch of brownies. So I made one batch of brownies from 2/3 cup of sugar and I made sugar. And all together I used 1/3 and 2/3 which is 1 cup. I can see all together I made 1 1/2 batches of brownies. I used 2/3 cup of sugar to make one batch and then with the extra 1/3 cup of sugar I made 1/2 of a batch. I saw that, that piece was 1/2 of a batch of brownies. So 1 1/2 all.

Now, that we've covered more information about mixed numbers and improper fractions, let's revise our guess on our second chocolate math problem. How many batches of brownies can you bake if a recipe calls for 4/5 of a cup of sugar and you have 3 and 1/2 cups of sugar at home? You can put your answer here as a mixed number. Put your whole number here and the fraction part here. Remember, use any of your math reasoning. It's okay, you might not have all the tools to solve this problem yet. But just give your best guess and use any reason that you've learned so far.

Let's make another prediction. This time, how many batches of brownies can you bake if a recipe requires 3/4 of a cup of sugar and you have 2 cups of sugar at home. Again, before we solve this problem, let's guess about how many batches we'll make. Will we make less than one batch of brownies, exactly one batch, more than one batch but less than 2, exactly 2 batches or more than 2 batches? Go ahead, choose your answer.

Here's one cup of sugar, and I know 3/4 cup makes one batch of brownies. Here's another cup of sugar and I know this 3/4 cup of sugar makes another batch of brownies. So I know I can at least make 2 batches of brownies. I haven't used two full cups yet. I actually have some sugar that I can still use. So, I know I could've made more than two batches of brownies. If you chose more than two more batches, great job.

But what about the sugar I have left over? I've only used 3/4 of a cup and 3/4 of a cup. I still have 1/4 and 1/4 that I can still use to make more brownies. 1/4 and a cup of sugar that I can use to make brownies. This is the 2/4, so this 2/4 cup of sugar represents what fraction of the batch of brownies? You can enter your answer here.

This extra 2/4 cup of sugar actually represents 2/3 of a batch of brownies. I know one whole batch requires 3/4, so this

Okay. Let's wrap up and figure out exactly how many batches of brownies we can make if our recipe calls for 3/4 a cup of sugar and we have 2 cups of sugar at home. Go ahead and solve it.

If you counted the batches from the last problem we know we have 2 and 2/3 of a batch. To do that math, I would have done many times 3/4 goes into 2. Instead of dividing by a fraction, I'm going to multiply by the reciprocal. So I have 2 x now I can multiply 2 x 4 = 8. 1 3 is 3. So I have 8/3. I'm going to write this as a mixed number. I know 3 goes into 8 evenly 2 times, because 2 3 is 6. So I have 3 going into 8, 2 times. And then I'm left with two extra pieces. So, 2 and 2/3.

So let's go back and answer our Chocolate Math 2 question. How many batches of brownies can you bake if a recipe requires cups of sugar at home? You can put your answer here as a mixed number. Remember, you'll want to use what we've learned about mixed numbers and dividing fractions.

Nice work on practicing your multiplication of fractions. Now, let's try a couple of division problems. The first problem that we want to do is similar to the multiplication problem but we're going to divide. 3/4 / 5/7. You can put your answer here.

*If you got 21/20, great job. Let's see how
we got that. Well, we know that dividing
fractions is the same as multiplying by
the reciprocal. So, 3/4/5/7 is actually
And we know when we multiply fractions*
together, we multiply their numerators
together. So, 3*7=21.*
And we multiply the denominators together,
=20.

Let's try a slight more difficult division problem, let's try 24/35 divided by 40/21. You can put your answer here

If you got 9/25, you got it right. We know
to divide fractions, we want to multiply
by the reciprocal. So, I can rewrite this
problem as a multiplication problem as
fraction, which is 21/40. So, instead of
multiplying 24*21 and trying to reduce
that, I can factor before I multiply. So,*
written as 7*5, 21 can be written as 7 3,*
and 40 can be written as 4