Common Core State Standards support video: this is Grade 6. The math standard is 6.RP.2. This standard reads: understand the concept of a unit rate a over b associated with the ratio a to b, with b not equal to zero, and use rate language in the context of a ratio relationship.
A unit rate involves a ratio where the denominator is 1. Now, the foundation for a unit rate is that a numerical comparison to 1 is the easiest comparison that there is to make. If you went to hamburger place, the posted prices wouldn’t have something like 11 dollars and 25 cents for three deluxe burgers. Instead, it would read: deluxe burger, 3 dollars and 75 cents, with the understanding that it’s $3.75 for one deluxe burger.
The problem is that the idea of comparing to 1 is lost in most contexts because of how the ratio’s expressed. We typically say 20 dollars an hour or 4 dollars a gallon rather than 20 dollars for every 1 hour or 4 dollars for 1 gallon. Students probably have a lot more experience with unit rate than we give them credit for because of how unit rates are expressed verbally and in writing.
If a over b is a unit rate, then, if you stop and think about it, it makes sense that the b has to be a 1. So, any task to find a unit rate can be set up as a proportion with the known rate on one side and the unknown unit rate on the other. So, for example, let’s say we know some ratio c over d and we want to know what the unit rate is, so then, we would set up. Again, it has to be over 1. So, this is what we’re looking for. We’re looking for the a.
Let’s take a unit rate context. Let’s say a farmer used 340 pounds of seed to plant 40 acres. And let’s say, in the future, he wants to plant that same crop again, but he’s not going to plant 40 acres the next time. So, he wants to know how much seed, how many pounds does he use in 1 acre? So, the setup here would be that we would want to know how many pounds of seed there would be for 1 acre. So, in this context, it was 340 pounds that he used for 40 acres. Now it’s just a matter of doing the computation, and we end up with the solution that we would use 8 1/2 pounds of seed for every 1 acre. So, now, the next time that the farmer’s going to plant this crop, he knows that, a nice easy ratio, that it’s a unit rate of 8 1/2 pounds of seed for every 1 acre that he wants to plant.
In this context, we have two different cans of some type of item, and we want to figure out which is the better deal. This is a situation when a unit cost would be advantageous because that enables you to compare apples to apples. It’s hard to tell here, just by looking, if brand w at 12 ounces for 84 cents would be a better deal than 20 ounces for $1.90. So, again, let’s figure out what the unit rate is for each of the two different brands. In this case, you always want to have your students write out what it is that they’re comparing to what. In this case, we want a unit rate of cost for every ounce. So, again, that’s the first, that is the important thing to set up right off the bat.
For the first situation, brand w, we have a total cost of 84 cents in that scenario, for 12 ounces. If we do the computation, crunch the numbers, and so forth, we get that this simplifies to the unit rate of 7 cents per ounce. Then, in the case of brand z, again, we want to figure out the cost for every 1 ounce. In this case, the cost is $1.90 for 20 ounces. We take $1.90 divided by 20 ounces, we will get 9 1/2 ounces, I mean, sorry, 9 1/2 cents per ounce. So, now we can make a valid comparison that brand w is the better deal because that one is 7 cents an ounce, and brand z is 9 1/2 cents an ounce.
There are a lot of other contexts where a unit rate would be most advantageous. Again, in most situations, it would be a context where you want to be able to make a side-by-side comparison, and the unit rate enables you to do that.