# Time versus Distance

Rate is a very important type of ratio, used in many everyday problems, such as grocery shopping, traveling, medicine--in fact, almost every activity involves some type of rate. Miles per hour or feet per second are both **rates **of speed. Number of heartbeats per minute is called "heart **rate**." If you ask a babysitter, "What is your **rate**?", you are asking how many dollars per hour you will be charged. The word "**per**" is always a clue that you are dealing with a **rate**. Many everyday problems involve rates of speed, using distance and time. We can solve these problems using proportions and cross products. However, it's easier to use a handy formula: rate equals distance divided by time: R = D/T. Actually, this formula comes directly from the proportion calculation -- it's just that one multiplication step has already been done for you, so it's a shortcut to learn the formula and use it. You can write this formula in two other ways, to solve for distance (D = R xT) or time (T = D/R).

In this lesson, the teacher introduces the basic D = R x T equation and asks students to use this one equation to solve for whichever quantity is missing. For example, if we traveled for 3 hours at a speed of 60 mph, we traveled a distance of

3 hr x 60 mi/1 hr = 180 mi.

Ms. Barney then asks her students to identify what is being asked for and then to solve the equation, including the appropriate use of units. Notice that even when being asked for *distance*, some students seem to think that the answer should be given in *time *units, such as minutes. That's one reason she has students write 60 mph as 60 mi /1 hr.

Let's peek in on Ms. Barney as she walks students through this deceptively simple equation.

**Classroom Clip Reflection:**

- How might you introduce the concept of "rate"?
- How would you determine whether all students understand the concept of rate during the lesson?
- How would you explain to students why you multiply Rate x Time to get Distance (versus adding or dividing or subtracting)?