Ratios and Proportional Relationships 6.RP.A.2
2. Understand the concept of a unit rate a/b associated with a ratio a:b with b ≠ 0, and use rate language in the context of a ratio relationship. For example, "This recipe has a ratio of 3 cups of flour to 4 cups of sugar, so there is 3/4 cup of flour for each cup of sugar." "We paid $75 for 15 hamburgers, which is a rate of $5 per hamburger."
Ever play the game Gas Station Hunt? It's the one where we'll sputter on fumes past the station advertising $3.63 per gallon because $3.55 per gallon is just a little further down the road. It might seem like we're wasting lots of time and gas for a measly $0.08, but there's a method to the madness. And that method is unit rates.
Start students off by identifying rates that compare quantities of different unit measures. They should be familiar with common rates like miles per gallon, price per pound, and feet per second. The word "per" is unique to rates, and we can just straight-up replace it with the words "for every." In other words, the word "per" means "divide."
When we're dealing with cost per gallon, for instance, have students take the cost and divide that value by the number of gallons. That'll give them the unit rate of cost per 1 gallon, which leads them right into the main point: unit rates always have a 1 on the bottom.
The concept is clearest when we compare prices (unit cost). After all, who doesn't want more for less? This is why the gas station game is so much fun; $3.55 per gallon is the better buy by $0.08 per gallon.
How much moolah do we save when we fill the tank with 13 gallons of gas? If we fill up twice a week, what are the weekly savings? Monthly savings? Yearly savings? This is how we apply unit cost to show why tiny savings are really just big savings in disguise.
But hey, it's not just about numbers. Students should also spend just as much time understanding the context of the problem they're solving. For example, when looking for fuel efficiency, we'd choose the vehicle with the higher mpg. But when we're looking for the better buy, we want the lower cost per unit. Students also need to recognize that they can only compare unit rates when the units are the same (i.e., we can't compare 40 mpg to 33 km/L). Apples and oranges, and all that.
Ready for the big secret? There are always two unit rates hiding inside each rate. If Tom runs 20 miles in 4 hours, that means Tom can run 5 miles in an hour—but we can also see that it takes Tom ⅕ of an hour to run 1 mile. That could definitely come in handy if Tom is trying to improve his time, impress his friends, or reinforce concepts of fractions and work with time (hint, hint).
Bust out some picture diagrams, double number lines, and tape diagrams for this standard. The double number line and tape diagram make things a whole lot easier to compare visually. Tape diagrams are great for identifying fractional relationships (like the ⅕ of an hour it took Tom to run).
This standard slides right into 6.RP.3 like a batter stealing third, where students work with conversions, percents, and graphing linear relationships.
- ACT Math 4.4 Pre-Algebra
- CAHSEE Math 4.4 Algebra and Functions
- CAHSEE Math 4.4 Mathematical Reasoning
- CAHSEE Math 4.4 Measurement and Geometry
- CAHSEE Math 4.4 Number Sense
- CAHSEE Math 6.5 Mathematical Reasoning
- Solving Proportions Using Cross Products
- Unit Rate