Make Math Moments Academy › Forums › MiniCourse Reflections › The Concept Holding Your Students Back › Lesson 61: Two Types of Relative Thinking – Discussion

Lesson 61: Two Types of Relative Thinking – Discussion
Posted by Kyle Pearce on December 6, 2019 at 6:27 amBased on your current understanding, what is the difference between thinking of a ratio as a multiplicative comparison and a composed unit?
Share your reflection below along with any wonders you still have.
Luke Albrecht replied 1 year ago 20 Members · 31 Replies 
31 Replies

I have never thought of a multiplicative comparison as a ratio before.
At this point, I am thinking that a multiplicative comparison is about “how many” whereas a ratio that compares the size of the trucks, for instance, is about “how big or small”.

OK. I just watched part of the next lesson. Apparently my definitions in my last comment are wrong!! Obviously, I did not know what multiplicative comparison and a composed unit are!

That is AOK! The whole purpose of these reflections is to flesh out some of the muddyness that is proportional reasoning and in particular ratios and rates… so much more awesome learning to come! Get ready for it!


I’m not sure yet which way it would go…but I’m seeing one as being more like cases. If there are this many in case 1, how many would be in case 10? Case 100?

Interesting! That certainly could be the “case” if it is a proportional relationship where that multiplicative comparison is present.


A composed unit allows for scaling to solve problems; i.e. if 3 ice cream cones cost $4.50, how much would 30 ice cream cones cost?
A multiplicative comparison is focused more on looking at the relationship between two or more quantities; i.e. Sally is twice as old as Amy.

Thinking of a ratio as a multiplicative comparison involves asking, ‘How many times greater is one thing than another?’ or ‘What
part or fraction is one thing of another?'”Thinking of a ratio as a composed unit involves grouping two quantities to form one new unit. For example if apples are priced at 6 for
$1.20, then 3 for 60c, or 12 for $2.40 would also be true. 
I thought of one set of ratios as having to do with scaling up or down, and the other set as scenarios that could be described with rates because they each involve two different units (i.e. wheels and trikes).

Interestingly enough, we can use scaling or ratio reasoning as well as rate reasoning for both situations!
Some more fun is to be had in this area moving forward 🙂


Multiplicative comparison as I now understand, is comparing two like units such as the height of a building in relation to the height of a flag pole. Whereas a composed unit comparison is a unit represented by two different things like number of ice cream cones to cost: if two ice cream cones cost 4.50 what is the cost of 10 ice cream cones and then the rate would be the cost of one.

Multiplicative comparison looks at the relationship between a shared attribute (length) that two similar items have (truck) and then compares them.. A ratio as a composed unit, the price of an ice cream cone, is iterated and increases as you buy more ice cream cones.
Something like this.

I felt one compared two of the same units for the same object and one compared two different units of different objects.

While multiplicative comparisons typically involve comparing objects with the same unit, we can still contrive a rate if we wish.
The idea of a rate is when we reveal it through partitive division – typically more naturally through thinking of a ratio as a composed unit.


Not sure how I would respond to this yet!
In the sorting activity, I put the marbles with the tricycles and the giraffe with the trucks. (All the others I sorted the same as Kyle.)

It definitely takes time to construct this thinking. Wondering if you have any new thoughts now?


Units
One half of the screen shows quantities being related to one another and the units of both quantities are the same.
The other half of the screen shows multiplicative or fractional relationships between quantities with different units.

Agreed! Nice job!
Something else one might notice is that while there is a multiplicative comparison in both situations / categories, the different units lend themselves to revealing a rate through partitive division. In the category to the left, this is less natural (but could still be contrived).


I got the first sort wrong, but then I started to see a pattern. I can’t articulate it, but am looking forward to the next module to help me do just that. As of this moment, I sorted them based on actually being given the relationships of both and comparison within one another, but that seems like it maybe could work for each. So…not sure yet, I will get back to you!

Loving it! The suspense is too hard to handle! 😉


I definitely need to know more about composed units before answering this one. Not sure.

There is nothing wrong with that at all! Glad you’re looking to dive deeper here.


It seems to me that in the composed unit you may need to know or figure out a unit rate to see the relationship between the two units of measure, so you can apply it to another quantity. For example, if 3 ice cream cones are $4.50 then one cone is $1.50. You divide both by the same unit3. now I can multiply both parts of that answer to find any other equivalent relationship. In a multiplicative relationship you are only looking at the relative relationship of one thing to another without requiring any other changes.

Hmmm I’m going to need to think more on this. I don’t know if you must find a rate in the composed unit case… however it lends itself more to finding a rate than a multiplicative comparison.


I’ve always thought of it as comparing two related quantities (length to length, height to height) or comparing two distinct quantities (wheels to bikes, popsicles to towers, etc…).

So I feel like this question is using math language to say what is the difference in the truck scenario and the tricycle scenario. That being said I saw one as a growth opportunity and one as a stagnate one place one time comparison.

As it seems to me right now, a multiplicative comparison is something that can expand at a predetermined relationship…as one grows the other one does so at the same proportion as the defined relationship. The word that keeps wanting to slide out is “rate”, but I’m not sure about that! I’m not exactly sure what a composed unit might be, but I wonder if it might not be more of a static multiplicative ratio. Not sure about terms and it still seems fuzzy.

Yes this is definitely foggy when you just start exploring / thinking about this.
I think of a multiplicative comparison as scenarios where finding a scale factor makes sense… for example, two different length pieces of wood. I can find a multiplicative comparison between them, however scaling them doesn’t seem as natural / necessary.
A composed unit, on the other hand, has two or more quantities that feel natural to scale in tandem. For example, scoops of hot chocolate mix and glasses of milk. You could find a multiplicative comparison, however it wouldn’t be a scale factor, that would actually be a rate!
Whew! Crazy stuff, right?


I am looking forward to the next lesson to learn more about this but right now it feels like a multiplicative comparison is taking 2 objects and evaluating one object in terms of the other. You are looking at one attribute.
With a composed unit you are looking at 2 attributes that have a relationship. I sorted these problems by thinking about which problems had information that could be put into a table: number of trikes and number of wheels, number of the ice cream cones and price….

I’m feeling like the two groups are separated by what is being compared. One group is talking about how many times bigger or smaller and the other group is comparing with different units.

Love this! Yes! How we are working with the quantities can help us to identify the type of comparison!


I am thinking about the difference between finding a constant of proportionality and using a unit rate. Sort of like asking what’s the unit rate, okay, now that you know the unit rate, tell me how you can use that information to find any relative amount.