Common Core Standards: Math
5. Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. For example, compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed.
When your students have a chance, tell them to take a look at the speedometer in the family car. They'll see the normal range of speeds in miles per hour or mph. Speed is always calculated in terms of the distance you could cover (miles) at that rate if you drove for a certain amount of time (one hour). So if you drove at a speed of 55 mph for one hour, you would cover 55 miles.
If they take a closer look, they'll notice that most speedometers will also have another scale: km/h or kilometers per hour. A speed of 55 mph is approximately equal to a speed of 88.5 km/h. (Since a kilometer is a bit shorter than a mile, you can cover more of kilometers than miles in an hour of driving the same speed.)
Draw your students' attention to this fact: the same speed can, and often is, represented in a number of different ways. Sometimes it's a speedometer and sometimes it's a graph. Sometimes it's an equation and sometimes it's a speeding ticket. Oops.
Students should realize that ultimately, slope is a quotient: the change in something (miles or vertical units) divided by the change in something else (time or horizontal units).
When we graph lines on the x-y plane, we say that the slope is "rise over run," or the number of squares we move up ("rise") for every square we move to the right ("run"). If we let the y-axis represent miles, and the x-axis represent hours, suddenly a slope of 55 miles per hour becomes a slope of .
Students should understand both the concept of slope and how to find the slope when given a verbal description or the graph of a line. You may also want to teach them how to find the slope given two points, as well as slowly introduce them to the concept of a linear equation. Not necessary quite yet, but it'll prove useful soon.
- Understanding Graphs of Proportional Relationships: Representation
- Understanding Graphs of Proportional Relationships: Identification
- Direct and Inverse Variation
- Unit Rate as Slope
- Comparing Properties of Functions
- Find Linear Function Given Intercepts