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Lesson 63: Digging Into Thinking Of Ratios As A Multiplicative Comparison and Composed Units – Discussion
Posted by Kyle Pearce on December 6, 2019 at 6:32 amBased on your current understanding, what is a rate and how can we reveal a rate from a ratio relationship?
Share your reflection below along with any wonders you still have.
Luke Albrecht replied 8 months ago 16 Members · 27 Replies 
27 Replies

I hadn’t thought of the rate as partitive and scaling as more quotative. I like the idea because it lends itself to the conversation that there is a difference.
 This reply was modified 2 years, 4 months ago by Ryan Foley.

Love the reflection! I couldn’t agree more… some might ask “why does it matter?” and my thinking is that by being able to “sort” ratio reasoning apart from rate reasoning via the type of division is a huge help when trying to work my way through a proportional situation.

A ratio compares two amounts of the same kind of a thing (eg. people: girls to boys).
A rate is a special kind of ratio that compares different kinds of measures such as dollars per kilogram

Love what Ryan and Valerie stated. Now I’ll try my best to synthesize the two together. Rate is comparing two of the same units through partitive division which is dividing a number into measured groups of the same quantity while quotative–is finding how many of one unit is in another. (yikes! this is mindboggling) not sure I’ve got this right.
 This reply was modified 1 year, 10 months ago by Jeanette Cox.

No, I think you’re on the right track!
It is definitely mind boggling, though!
A rate is revealed through partitively dividing two quantities from a ratio.
While revealing a rate feels more natural when you can look at a ratio as a composed unit, you could technically contrive a rate from a multiplicative comparison as well.


I would say what others have stated that a Ratio is comparing two amounts of the same thing and rate is comparing two different types of measurements.
One thing that I need learn and made sense was the comparison of the two when needed to determine the proportional aspect of a situation. This is clearly seen in my current class as we are getting into dilations and slopes.

Just keep in mind that the rate is revealed through partitive division – it isn’t as simple as comparing two quantities with different units.


A rate is revealed by carrying out partitive division.
A rate is a single quantity that describes the relationship between two quantities, e.g., cost per mile; cost per unit; length per (1) unit.

Connecting this idea of Rate with Partitive Division really helps me understand this concept. Using the language of “How many units of one group are in another?” makes sense to me now, especially using the wand idea. Dividing 8cm Wand B into 10 cm of Wand A clearly shows that there is not enough of Wand A to fit into Wand B (how many 10cm units can fit into 8cm units) so the ratio is less than one, or 8/10 cm of Wand B per cm of wand A.
Language again, is key, and so is the use of Partitive Division.

Love it. Also, as in your wand example, we can contrive a rate even when comparing two quantities of the same unit (ie: wand A to wand B) even if it feels less natural than comparing quantities with different units (miles per hour).


@Kyle Pearce during the video at approximately 21:41 you made a statement that you was dividing by the rate…
I watched this video several times and each time I expected you to say you were dividing by the scale factor. Can you help me better understand the difference.

Love this question! Because we were using quotative division, we are dividing by the rate whereas partitive division, you are dividing by the quota.
Keep in mind, we were contriving a composed unit with the two trucks scenario so it definitely doesn’t “feel” right to be scaling in tandem… so the rate is pretty contrived here, but still technically a quantity per group situation.
Does that help? Feel free to dig deeper on this one…


I see a rate as a ratio (of quantities with the same or different units) simplified by way of partitive division to reveal either of the quantities to a value of 1.

I agree with this. The rate “is” the quantity per group.


The rate is how much per one. You would use partitive division to find the rate. The ratio is any comparison of two units.

I think a rate is what I’ve always thought of as a unit rate namely with a denominator of 1. I think I’m more confused though about partitive division versus quotative. In partitive are you breaking the whole into groups of whatever you are dividing by and quotative you are breaking the whole into the number of groups you divide by?

You’re not alone on that one!
First off – in our eyes (and van de Walls’s) a rate is a single quantity (doesn’t need a denominator of 1…) with a compound unit (pounds per inch, km per hour, etc)
A rate could be:
100 km per hour
But also
98/3 km per hourBoth are rates: a single quantity with a compound unit. One is not a unit rate and the other is not… they are both just rates.
As for partitive and quotative:
Partitive is dividing by number of parts to find how many per part (to find the rate)
“12 cookies divided by 4 kids”Quotative is dividing by how many per part (the rate) to determine how many parts/groups
“12 cookies divided by 3 cookies per kid”Crazy stuff – I know! I got a math degree without ever knowing there were two types of division. Crazy!


For me, I really appreciate the separation of the ratio (composed unit version) as being the linked growth between two (or more) quantities (often distinct or different attributes), while the RATE is the amount of one of these quantities per another. And I particularly appreciate the idea that the RATE doesn’t happen until after the partitive division has be “completed.” That otherwise, it’s not a rate – it just a method to calculate the rate — which can be written as a fraction.
I’m wondering how this idea of multiplicative comparison and composed units can fit into the idea of percents/percent increase and decrease. I am feeling that the increase and decrease problems will lend themselves more towards the multiplicative comparison kind of relationship, but I also have had a lot of success as thinking of percents on a double number line, which would be more connected to composed units (where, for example, $ and % are the two units).

Glad that the learning is resonating!
A percent is really a ratio that you can easily scale in tandem. 20% means 20 per 100 or 20 to 100 which has a rate of 0.2 of one unit per another. The tricky part that is complex is that it also has the same units of measure which makes it more of a multiplicative comparison, yet scaling in tandem still feels helpful/useful. Complex stuff!


A rate is the answer to a division problem with units attached. For example: If I travel 120 miles in 2 hours at what rate did I travel. So how many hours in 120 miles or 120 / 2 which is 60 miles per hour. The rate is the partitive division result not the simplified equivalent ratio of 120/2 =60/1.
So I wonder one of the definitions I have been given, I think by my textbook, is a ratio is a comparison by division. This is the inverse of multiplicative thinking. So should I not use that definition?

Currently, my understanding of rate is ‘how much for one’. For example, if 12 eggs are $3.67, how much is 1 egg? We can reveal a rate from a ratio relationship by dividing one quantity by the other quantity.

A rate is derived from taking 2 different quantities and through partitive division evaluating them to find a single unit measure such as price per one ice cream cone or distance per mile.
Also, I am glad I am not the only math major who made it through college without knowing there were 2 kinds of division. I will look at your resources to understand them better, which will then help my understanding of rates.
Also, I love the language of “scaling in tandem” rather than “x varying per y.” Such a good, visual description!

A rate is the amount of something in relation to another factor or element; a certain amount per another. For example typing speed (numbers of words per minute) or number of bags of carrots consumed per week. If I understand correctly, I can use partitive division to calculate the rate from the ration. So if I eat 1 bag every two weeks, the rate is 1/2 bag per week, but the ratio is 1 bag for every two weeks, 2 bags for 4 weeks, etc. (or 1:2, 2:4 etc) Now I believe this is a composed unit example. Does a rate necessarily come from a multiplicative comparison or is it okay if it comes from a composed unit…that part isn’t really clear to me yet.

Great ideas here which show you’ve clearly differentiated a ratio and a rate.
Rates are more commonly calculated with composed units since it involves quantities with different units. However, a multiplicative comparison can reveal a rate, but it is more contrived or less natural. Ie: a scale on a map of 500:1has a rate of 500 cm per cm on the map. Works, but feels less useful / natural.


I think I like the rate as a saying how much of one goes into another. The wands be 8/10 or 10/8 based on which direction you scale and then thinking of that as the rate to the compare any “small” wand to and “large” wand.