Math Antics

Ratios and Rates, 2014

https://youtu.be/RQ2nYUBVvqI

Hi! Welcome to Math Antics.

In this lesson, we’re gonna learn about ratios. Well…what in the world is a ratio?

Well, let’s look it up in a math book to find out.

It says here that a ratio is a “comparison of two numbers by division”.

Well that’s true, but it’s also a little confusing.

It’s confusing because most of us think of comparing numbers as

trying to decide if a number is greater that, less than, or equal to another number.

But with ratios, we’re not trying to compare numbers like that.

Instead, we’re really trying to see how two numbers relate to each other,

and so at Math Antics, we like to think of ratios as a “relationship between two numbers by division”.

Okay, but how do you compare (or show how two numbers are related) by division?

Well, To see what the “by division” part really means, let’s look at an example of a ratio.

Uh, excuse me… That’s not a ratio. That’s a fraction.

Oh, it’s a ratio alright. Mathematically, ratios and fractions are basically the same thing.

It’s just that when we use a fraction in a particular way, we call it a ratio.

Well sure, everybody knows that!

Well like I was saying… Ratios are basically just like fractions.

The difference is how we use them to describe things in the real world.

To see what I mean, let’s look at examples of how we could use the fraction 1 over 2 and the ratio 1 over 2.

Mathematically, these are both the same thing.

They’re just the division problem: 1 divided by 2.

But in the case of the fraction, we usually treat it as if it’s just a single number.

For example, at lunch time, you might eat 1 sandwich.

Or if you’re really hungry, you might eat 2 sandwiches.

But, if your not very hungry, you might just have 1/2 sandwich.

We can use the fraction 1/2 just like we use 1 or 2 to show how many sandwiches you eat.

It’s just that in the case of 1/2 we know that it’s only part of a sandwich; just a fraction of one.

Now let’s see how we can use the ratio 1 over 2.

With a ratio, we don’t treat it as if it’s just a single number.

Instead, we pay close attention to the top and bottom numbers because we use them to refer to different things.

For example, let’s say we’re planning to go on a picnic.

And for every two people that are going on the picnic, we’re only bringing 1 sandwich.

In that case, we'd say that the ratio of sandwiches to people is 1 to 2, or 1 sandwich per 2 people.

Do you see the difference between our fraction and our ratio?

The math part of each of them is the same.

But, with the fraction, both the top and bottom numbers are referring to the same thing; the sandwich.

However, with the ratio, the top and bottom numbers are referring to different things: sandwiches and people.

The fraction shows a part of something, but the ratio shows a relationship (or a comparison) between two different things.

And you can see that they’re the same mathematically because

if you did have the ratio of 1 sandwich per every 2 people on a picnic, guess how much of a sandwich each person would get?

Yep… half a sandwich.

Alright, so now you know that fractions and ratios are basically the same thing.

But, since they’re used differently in math, sometimes they're also shown differently.

Once in a while, instead of seeing the standard division form, a ratio might be represented with this symbol. ( : )

When you see a ratio written this way, it just means “1 to 2” or “1 per 2”.

For example, in this picture, you could say the ratio of dogs to cats is 3 to 2, (3 dogs to 2 cats).

And you could also write it in the standard division form: 3 dogs over 2 cats.

They’re just different ways to write the same ratio.

Ratios are used all the time to represent all sorts of things in real-world situations,

so let’s see a few more examples to help you really understand what ratios are.

Have you ever wanted to compare apples to oranges, but someone told you you couldn’t?

Well you can with a ratio! Let’s say a fruit stand sells 5 apples for ever 3 oranges they sell.

The ratio of apples to oranges would be 5 to 3.

Or, have you ever helped someone bake cookies?

The recipe might tell you that for every 2 cups of flour, you need 1 cup of sugar.

That means that the ratio of flour to sugar is 2 to 1.

Or, what about your TV screen or your computer monitor.

Have you ever hear someone say that the size (or aspect ratio) is 16 to 9?

16 to 9 is the ratio of the screen’s width to its height.

So if the screen is 16 inches wide, then its height would be 9 inches tall.

Ah, here’s another good ratio that you might use in your car: 40 miles per hour.

Ah ha! Didn’t you said that a ratio was a relationship between TWO numbers?

But 40 miles per hour is just one number!

…looks like someone’s got some explainin’ to do.

Actually, there ARE two numbers.

Do you remember how any number can be written like a fraction just by writing ‘1’ as the bottom number?

Well, 40 miles per hour is the ratio, 40 miles per ONE hour.

Well I guess you have an answer for everything, don’t you?

40 miles per 1 hour is a type of ratio that we call a rate.

A rate is just a ratio that usually involves a period of time.

Here are some common examples of rates:

10 meters per second

$12 per hour

3 meals per day

50 games per year

Notice that the bottom numbers in each of these ratios relate to a period of time.

…seconds, hours, days, years. And that’s why we call them a rate.

Alright, so that’s simple enough.

But you might be wondering, why are the bottom numbers of all these rates 1?

Couldn’t you have a rate like, 90 meters per 9 seconds, or $60 per 5 hours?

We sure could, but most of the time when we have rates like that,

we want to convert them into an equivalent rate that has 1 as the bottom number.

That’s because whenever the bottom number represents only one unit of time (like one hour or one day)

it makes comparing different rates much easier.

For example, imagine two cars driving at two different rates.

The first car’s rate is 120 miles per 3 hours

and the second car’s rate is 150 miles per 5 hours.

Which car is going faster?

Well, it’s not all that easy to tell when the rates have different bottom numbers.

Fortunately, it’s really easy to change a rate so that it has 1 as the bottom number.

All you have to do is divide the top number by the bottom number.

The answer you get is the top number of the new (equivalent) rate, and the bottom number is just 1.

Rates like this are called “unit rate” because “unit” means “one”.

Alright, let’s convert the rates of speed for our two cars into unit rates so that we can compare them easily.

The first car’s rate was 120 miles per 3 hours.

So if we take 120 and divide it by 3, we get 40.

That means that the unit rate for the first car is 40 miles per hour.

The second car’s rate was 150 miles per 5 hours.

So if we divide 150 by 5, we get 30.

So the unit rate for the second car is 30 miles per hour.

And now, you can easily tell that the first car is going faster.

And you can tell why unit rates are so helpful.

Okay, so that’s it for this lesson.

We’ve learned that a ratio is basically just like a fraction.

But instead of showing what part of something you have, it shows the relationship between two different things.

We also learned that when one of those two things is time, we call the ratio a rate.

And last of all, we learned how to convert a rate into a unit rate for easy comparison.

As always, thanks for watching Math Antics and I’ll see ya next time.

Learn more at www.mathantics.com

The Best Hands-on Fractions Activity Ever.

Mashup Math

https://www.youtube.com/watch?v=PmWnMtLBJZM

Hey guys Anthony here from mashup math. If you're new welcome and if you've been with us for a while, thanks again for stopping by. This is something different - something we haven't done before but something I've wanted to do since we first launched at the beginning of this school year and that is to do a weekly vlog. So normally we do animated math video lessons that teach a particular topic or focus on a particular question but in this case, I'm going to cover a hands-on and exploratory math activity. These are great activities for you to do at home with your kids, in your classroom with your students or if you are a student and you have permission and use the materials for this activity then you should go for it. This is a great way to build a strong foundation of math understanding. Now in this episode I'm going to show you how to create a fraction kit. And the reason why I chose this activity for the first episode it's pretty simple but it's very powerful. I used to do this with my middle school students and I wish I would see more teachers doing this. But so maybe there’s something you want to add to your repertoire of teaching activities, especially if you work with elementary or middle school students. So, we want to get a good strong foundation of fractions as soon as possible and activities like this helped us to achieve that goal. I just want to quickly remind you before we start that there is a worksheet that accompanies this lesson. You can download it for free, it's on mashupmath.com, the link is in the description for this video. You don't need the worksheet, but if you want something to follow along as you go through the activity, then you can definitely do that. I've also written a blog post which is also on mashupmath.com, where you can get some more insight into this activity and the research behind why activities like this help students gain a conceptual understanding of a math topic and why we want to avoid things like just simply memorizing definitions and doing a lot of robotic repetition and practice. It's about understanding and activities like this help us to do that.

Okay so that's the activity. It's very simple. One thing I would recommend for a teacher or a parent doing this, where the student is, to hold back as much as you can. This is an exploratory activity and what I mean by this is that if the student is struggling or making mistakes, allow them to make those mistakes. It's okay. This is an exploratory activity. Let them explore. Let them mess up, think whatever they're doing wrong. Let it happen, okay? Don't rush in to correct them because they might correct themselves and if they don't, you can wait til later on to have a discussion or to explore why what they were doing was incorrect and then let them explore more in the right directions. And again, remember, there is a worksheet that goes along with this lesson. You can be really creative with this - you can add new questions, new fractions, throw some curveballs in there. Customize it any way you want! Take it and run with it. And if you have some cool ideas, I would love to hear them in the comment section below this video. Finally, I want to share with everyone that mashup math is committed to sharing free and accessible math resources because we want to reach as many students, schools, teachers, and parents as possible. So with that in mind, if you want to support us, if you find what we're doing to be helpful, you can do a few things that take very little effort and would really help us out. One of those things is to LIKE this video. A second thing is to leave a comment in the comments below if you like the video, let us know why, or if you have questions, comments, concerns. The third thing you can do is subscribe to our YouTube channel, if you're not already. And the fourth thing you can do is share this video on whatever social media outlet that you prefer to use. So, Facebook, Pinterest, Instagram, Twitter, or wherever! Sharing it helps to boost our exposure and makes it easier for us to attain our goal of reaching as many students as possible. So, one of those four things, all those four things, or any combination would mean a lot to helping us out. So, on behalf of the whole team here at mashup math, thank you all for checking us out and come back! Next Thursday will be episode number 2 which is going to focus on area models for analyzing probability outcomes, another pretty cool activity, so you won't want to miss that! So thanks a lot and we'll see you next time!

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