Make Math Moments Academy › Forums › MiniCourse Reflections › The Concept Holding Your Students Back › Lesson 64: Types of Ratios – Discussion

Lesson 64: Types of Ratios – Discussion
Posted by Jon on December 6, 2019 at 6:34 amCreate or reference a previously shared scenario involving a ratio relationship and attempt to represent it using two (2) or more ratio constructs.
Share your reflection below along with any wonders you still have.
Kyle Pearce replied 10 months ago 16 Members · 22 Replies 
22 Replies

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, 4 months ago by marianne aamodt. Reason: formatting of chart did not work when posted

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 ago by Jeanette Cox.

13 out of 52 cards are hearts. 13/52 of the cards are hearts.
13:39 “13 hearts to 39 nonhearts cards”

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

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.

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.

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
🙈

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?

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.

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!



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

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 ! 🙂


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.

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.


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.

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.

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.


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….

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 roundabout way of expressing things that seemed like they should be more straightforward 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 partwhole 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!

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!


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.

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