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8.F Bike Race

### Alignments to Content Standards

Domain
F: Functions
Cluster
Use functions to model relationships between quantities.
Standard
Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally.

### Tags

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Antonio and Juan are in a 4-mile bike race. The graph below shows the distance of each racer (in miles) as a function of time (in minutes).

1. Who wins the race? How do you know?
2. Imagine you were watching the race and had to announce it over the radio, write a little story describing the race.

### Commentary

The purpose of this task is for students to interpret two distance-time graphs in terms of the context of a bicycle race. There are two major mathematical aspects to this: interpreting what a particular point on the graph means in terms of the context, and understanding that the "steepness" of the graph tells us something about how fast the bicyclists are moving. Technically, students do not study average rate of change until high school, but they can understand that if a bicycle has gone 1/2 mile in one minute it is moving about 30 miles per hour (or if its distance is unchanged for a minute, it is standing still). This task adds to the complexity by having two graphs, so the relative positions and speeds of the bicycles can also be compared.

This task represents the culmination of work that students do in both the 8.F and 8.EE domains, and represents the kind of work that transitions them to high school algebra and functions. As such, it would work well at the end of 8th grade or beginning of a high school algebra or modeling class. Students should have opportunities to do similar kinds of tasks before this, starting with linear and then piece-wise linear graphs.

An excellent variation of this task is to give the graph without describing the context or labeling anything but the axes. The teacher could ask what kinds of objects might be described by these graphs? By noting that in the last minute of the green graph one of the objects is moving at its fastest at about 30 mph, we can rule out hikers or turtles, but also rockets or meteorites.

This task can be used to generate a classroom discussion, small group discussion or as a formative assessment item. If used in this last way, teachers should be on the lookout for students who interpret the graphs as representing the paths of two objects through two-dimensional space as opposed to distance-time graphs.

#### Solutions

Solution: Interpreting graphs
1. Juan wins the race. He will have finished the race in 13 minutes while Antonio takes 15 minutes to ride the same distance.
2. Juan and Antonio are off for a 4-mile bike race. Antonio has the early lead. He is picking up speed, pulling away from Juan who seems to have some trouble finding his stride.

Antonio looks like the clear favorite 4 minutes into the race. But wait, he is in trouble now – oh no, he ran off the road and fell off his bike. He seems a bit dazed sitting at the side of the road. Okay, he is getting up and checking that all limbs are still working. He lost two minutes getting up and dragging his bike out of the ditch, but now he is back on the bike.

Juan had a slow start but he was picking up speed and now is virtually flying by Antonio, just as he is getting back on this bike. Can Juan keep up his newfound speed or will Antonio catch up again? Juan passed Antonio 7 minutes into the race and with 2 miles to go. He is pulling ahead now of Antonio who looks like he is hurting a little bit after his fall and has trouble finding his old speed.

Oh no, Juan is loosing steam and slowing down. Is his advantage big enough to get him over the finish line ahead of Antonio? Yes! He has won the race in 13 minutes and is collapsing into the grass trying to catch his breath after his epic win. Where is Antonio? Ah, here he is coming now, he is speeding up, going really fast as he crosses the finish line at the 15-minute mark, but too late to make a difference in the race.