Resources for Unit Rates

Question

I am currently mentoring my little brother in mathematics. There is an issue with the pedagogy of unit rates. For example when given the following concept " 11.00 U.S. Dollars to 20 Planet X Dollars Find the Unit Exchange Rate." I am aware that does eventually leads to unit rates which is the name of the lesson the math teacher is on. Which the individual has also explained. The issue is are there any resources on how to understand the unit rate, cost, or etc. that is being presented in this lesson outline? My attempts to help him with problem has been to grasp that the problem ultimately has to do with something like this $\frac{\mathrm{rate}}{1}=\frac{11.00}{20}$. Or are there any examples, or methods that can help drive the concept home?

Edit

This is arbitrary " 11.00 U.S. Dollars to 20 Planet X Dollars Find the Unit Exchange Rate." Any rate either from U.S dollars to Planet X dollars or reverse is fine. Here is the specific one "11.00 U.S Dollars to 20 Planet X Dollars Find the Unit Rate Exchange for U.S Dollars to Planet X Dollars." Again none of these specific problems are actually meaningful if the general concept is what is the underlying issue. Firstly, the definition of ratio, proportion, fraction, and decimal are all given before hand if the assessment of these problems are being presented. This is a stack exchange about Math Educators so math educators in the domain of middle/elementary school would ideally know the issue. The Unit Rate might not make a lot of sense it college educator level, but it does to me a student who passed Calculus 1, 2, 3, and Differential Equations and is aiming to major in Mathematics, and finds uses to its applications.

Answer 1

Firstly, the definition of ratio, proportion, fraction, and decimal are all given before hand if the assessment of these problems are being presented.

If they have been given before, it would be nice to know those specific definitions.

This is a stack exchange about Math Educators so math educators in the domain of middle/elementary school would ideally know the issue.

One would hope so.

The Unit Rate might not make a lot of sense it college educator level, but it does to me a student who passed Calculus 1, 2, 3, and Differential Equations and is aiming to major in Mathematics, and finds uses to its applications.

The word "rate" is such an overloaded word and everyone bends it differently. To me there is a ratio, which can be written using a colon or division sign or as a fraction. Equality of two ratios is a proportion.

Ratio can be between same-dimension values, it this case the resulting ratio is dimensionless, like widescreen TV is 16 cm by 9 cm or 16 inches by 9 inches or just 16:9 or 1.78. Side note: every time I see how compression ratio of an internal combustion engine is expressed in American magazines, I smirk: it is something like 9.5:1 or 12:1, but never just 9.5 or 12.

When ratio links two values having different dimensions, then the word rate comes up, and we can say "per", like pressure is force per area, although to me it is a bit weird to call it rate. If the unit in denominator is time, then temporal rate comes up and we can have miles per hour. In colloquial speech people remove "temporal" and just say "rate". Speed is rate. I am puzzled why so many people prefer saying "rate" instead of "speed" or "velocity". But very often when people mean speed they say "rate of speed", which is wrong, because rate of speed is acceleration, speed is rate of change of position. In elementary school they do it all the time.

Exchange rate is a special fiscal term. Do you expect middle schoolers to know what exchange rate is? Ok, supposedly they know, they were given a definition. It would be nice to see it, but you don't want to provide it. Now you introduce yet another term, unit rate. Judging by your proportion, it is how much of one thing corresponds to just one item of another thing. Actually, it is how much of another thing corresponds to one item of chosen thing for which you are calculating this "unit rate".

In your proportion $\frac{\mathrm{rate}}{1}=\frac{11.00}{20}$ what you wrote as "rate" is not rate, it is number of USD per 1 PlanetXD. It is money. It has dimension, USD. It is x: $\frac{\mathrm{x\ USD}}{\mathrm{1\ PlanetXD}}=\frac{\mathrm{11\ USD}}{\mathrm{20\ PlanetXD}}$. Rate is $\frac{\mathrm{x\ USD}}{\mathrm{1\ PlanetXD}} = \mathrm{x\ USD/PlanetXD}$. Rate has dimension of USD/PlanetXD. American dollars per one Planet X dollar. Numerically the same, but conceptually different.