Lesson 6-4: Types of Ratios – Discussion

[marianne aamodt]

Milk costs $4.50 for 4 L. What is the unit price?

COST | Litres

$4.50 X 1/4 | 4 X 1/4

|

$1.13 | 1

Ratio as a quotient:

$4.50 divided by 4 L of milk

Ratio as a rate:

Milk is $1.13 per litre of milk

I have to get used to thinking of multiplying by a fraction instead of dividing. I’m curious as to why this is easier for kids. Don’t they still have to think about division to “do the math”?

I am interested in moving farther and seeing how to do this algebraically. I’ve always found the input/output “machines” (table of values) that are found in MMS texts starting about gr. 4 (?) to be very confusing – not only for kids! I wonder: how does this turn into graphing linear relations? Looking forward to finding out!

• This reply was modified 2 years, 8 months ago by  marianne aamodt. Reason: formatting of chart did not work when posted

[Jeanette Cox]

Thinking of the piano keyboard the following ratio problem represents a multiplicative relationship of black keys to white keys per octave. Then reframed the ratio to a composed unit by comparing octaves to black keys and then determining the unit rate of black keys to octaves.

• This reply was modified 2 years, 4 months ago by  Jeanette Cox.

[Heidi Peterson]

13 out of 52 cards are hearts. 13/52 of the cards are hearts.

13:39 “13 hearts to 39 non-hearts cards”

[John Sasko]

It costs John $30 to fill his car’s gas tank (12 gallon capacity).

What is John’s cost for gas?

Ratio as Quotient: $30 dollars divided by 12 gallons.

Ratio as Rate: $2.50 per 1 gallon of gas

[Brent Sturtevant]

currently one way I have been looking at with dilation and slope. If the width is 15 and height is 10, then similar one is 30 and 20 respectively. So the ratio of heights is 15 to 30 and the rate(slope or scale factor) is 2:1. Breaking it down in this manner and showing it this manner has helped tremendously.

[Aaron Davis]

Part to Part Ratio

7 Canadian based NHL teams to 24 US based NHL teams. 24:7 or 7:24

Part to Whole Ratio

7 Canadian teams to 31 total NHL teams. 7 thirty oneths of the group are Canadian based teams.

[Nicole Jackson]

5 out of the 12 eggs in the carton were cracked.

Multiplicative Comparison:

5/12

Composed Unit:

number of cracked eggs number of eggs total

5/12 1

5 12

10 24

Rate:

1 cracked egg per 12/5 eggs total

5/12 cracked egg per 1 egg total

Part to Part Ratio:

5:7 or 5/7

Part to Whole Ratio:

5:12 or 5/12

Ratio as Quotient:

5 cracked eggs divided by 12 eggs total

Ratio as Rate:

5/12 cracked eggs per 1 egg

🙈

[Kyle Pearce]

Love the emoji! Ha!

So much to think about, right?

Lots of love put into these examples, which is great!

The beauty is that you can really contrive a lot of these ideas so sometimes one of the ideas might not feel as natural, but you can still force it.

How do you feel overall about your understanding here?

[Nicole Jackson]

Thanks for the feedback.

I am on my way to really understanding each ratio type. And I agree with you, some ratios fall into place quickly and easily while others are a bit more complicated. However, learning to notice and name each ratio type is fascinating.

More than likely, I will watch this video at least once more before moving on.

The introduction of quotient as ratio and rate as ratio is where things became a bit shaky for me.

Overall, I feel as though I am beginning to make sense of the information. I just need to spend a little more time unpacking it.

[Kyle Pearce]

So happy for you!

You are approaching things from a very helpful perspective as it is impossible to truly build that understanding by simply watching the video and moving along. These reflections are going to pay off for you in the long run! Nice job!

[Robin Bergen]

A wildlife conservatory maintains a ratio of 2 squirrels for every 8 birds. How many birds would there be if there were 15 squirrels?

Multiplicative comparison: 2 squirrels: 8 birds

Composed unit: 2:8, 4:16

Rate: 1squirrel per4 birds

[Kyle Pearce]

Nice work here.

A couple thoughts to consider:

While there is a multiplicative comparison for every ratio, the type of ratio “feels” more naturally as a composed unit. Again, there is no hard rule here.

For the rate, remember that is the result of partitive division, so it would be ___ birds per squirrel or ____ squirrels per bird.

Little nuances that will develop as you continue through the course ! 🙂

[Anne Sheeter]

In my example with llamas 75 pounds per one llama is a rate. As a part to part ratio it could be represented as 75:1, 75 to 1 or 75/1. One can scale it by multiplying both parts by 4 to get 300 pounds for 4 llamas making this a composed unit.

[Kyle Pearce]

Love me some lamas!

One nuance that I try to keep up with is avoiding using ratio language with rates. A rate has a compound unit, so a single quantity therefore I try to avoid using the number 1 and rather say “pounds per lama”

This really drives home the idea that a rate is the result of partitive division and a single quantity.

[Amy Johnson]

I’ll refer back to my homemade shoe deodorizer: 2 oz. rubbing alcohol to 2 oz. apple cider vinegar to 2 drops of tea tree oil

Part to Part

2 oz. rubbing alcohol : 2 oz. apple cider vinegar

Part to Whole

2 oz. rubbing alcohol : 4 oz total liquid

Quotative Division

How many oz of rubbing alcohol in the total liquid or How many ounces total in the rubbing alcohol?

Proportional Relationship

Partitive Division: Ounces of rubbing alcohol per ounces of total liquid which reveals the rate

2 oz. / 4 oz =

Rate: 2/4 rubbing alcohol per total fluids.

[Kristin Snell]

A selling factor for auto makers is mpg (miles per gallon). A new Toyota is hyped to get 45 mpg, which is a rate. As a ratio, it can be represented as 45:1 or 45 miles to 1 gallon. This can be expanded onto a table with 2 columns, one for gallons and one for miles. From here you can ask questions like how many miles can you go with 9 gallons of fuel?

These are so intertwined, it’s easy to see how we get lost. I think I’m getting better at separating the different parts.

[Kyle Pearce]

Great reflection. This clearly articulates the difference between a ratio and a rate. Your example also reminds us that the same composed unit can be represented as a ratio or divided partitively to reveal a rate.

[Lavon Heath]

When cooking my turkey for Thanksgiving I planned on 1 1/2 pounds of turkey per person, which is the rate. The ratio would be 1.5:1 and I would scale that up to buy a 12 pound turkey for the 8 people coming to dinner. I can see how this could easily be framed as a composed unit. The part/part would be 1.5:1, but I am not exactly sure how I would make this into a part/whole comparison….

[Merrillee Reboullet]

I previously mentioned that the rate of carrot consumption for our home is 1/2 bag per week. 1:2, 1/2. For four weeks, it would be two bags or 2:4 I have a little trouble turning the ratio around because in real world situations we don’t really talk like that so it is more abstract. It isn’t very usual to talk about the number of weeks per bags of carrots which I believe would be 2:1 or 4:2…or 52 to 26. But there I just almost created a rate using partitive division as a reflex because all of a sudden I got curious about how many bags I would need for the year. 52/2 (because the relationship is 1:2) to get 26 bags…I think I have way more so I’m probably in trouble.

Anyway, I’m not sure I actually did what I was supposed to in this post. Some of what you were talking about in this video didn’t feel very applicable so I got lost in the terms and the round-about way of expressing things that seemed like they should be more straight-forward than all that. I did appreciate your comment not to expect students to be able to express their understanding to this level or to imagine that we should have them memorize all these terms for different ways of looking at the same thing.

One thing I wonder about is at what level would students begin thinking like this. I shudder to think of introducing a discussion of ratios like this to my Grade 5s. We barely touch on ratios and those are generally part-whole as they relate to fractions…I feel that they wouldn’t be quite ready to think about 9/7 of a person to 1 person who likes the arts!

[Kyle Pearce]

I think your post is actually bang on. The idea is to reflect on the learning and all of your post shows you building flexibility with different ratios and rates. The intent of the video was to get you thinking about building flexibility with ratios and rates … we want to do this with students as well, but as you mentioned, not all at once. Since we are all educators, we can handle the amount of complexity that you are hit with in a single video, however students might not. So use your judgement and you will always come out on top!

[Luke Albrecht]

There are 2 oranges in the fridge and 7 apples on the table. 2 apples to 7 oranges is not a fraction!! 2 oranges to 9 pieces of fruit is a fraction!! Really dig this. I also was thinking 1 to 9/7 is getting a little up in theoretical head space and then to hear you say–this then takes us to percent–wow, just wow. Thanks.

[Kyle Pearce]

💥crazy right?

Remember, we can represent a ratio with a fraction bar, but it is still a ratio. Glad you’re digging the learning!

[Michael Duguay]

My students have solved the Sr-71 problem using ratio tables in the past

The North American X-15’s top speed is about 4,520 mi/h.

How much time would it take for each to travel across the United States, a distance of about 2,800 miles?

Hours 1 1/2 1/4 1/8 ??

Dist 4520 2260 1130 560 2800

So in 1/2 hr (30 min)+ 1/8 hr (7.5 min) 37 min 30 sec the plane travels 2820miles.

• This reply was modified 1 week ago by  Michael Duguay.