**Learning Objective: **

- Students will be able to explain information presented to them in a graphical format.
- Students will be able to convert information into a mathematical graph.

**Background:**

Qualitative graphs allow us to mathematically interpret and draw conclusions from real life phenomena. Graphs can be used to represent almost anything which is measurable. They are an essential tool in understanding and conveying mathematical information.

Graphs often consist of an independent axis (x-axis) and a dependent axis (y-axis). It is useful to think of the “dependent axis” as depending on the “independent axis”.

The graphs in these activities are all line graphs.

**Activity:**

There are two parts to these activities. The first will ask you to indicate which graph corresponds to a given situation. The second part will ask you create your own graph for a given situation.

**A)** Indicate which graph matches the given situation.

**B)** For the following examples, draw a graph with the given y-axis and x-axis to represent the given information.

**B1.** A plane takes off from New York City and lands in Los Angeles 7 hours later.

x-axis: Time elapsed

y-axis: Altitude

**B2.** A man rides his bike from his house to a store at a constant speed down the street, stops, then heads back to his house at a slower constant speed.

**a)**

x-axis: Time elapsed

y-axis: Distance from home

**b) **

x-axis: Time elapsed

y-axis: Speed

**B3.** A snowboarder goes up a ski-lift then rides down a mountain trail accelerating. She stops herself at the end of the slope.

x-axis: Time elapsed

y-axis: Speed

**B4.**

A faucet is turned on at a constant rate and the vase, pictured above, is placed below it.

x-axis: Time elapsed

y-axis: Height of water

**B5.** In a national park there is a population of foxes and a population of rabbits. Rabbits are prey to the foxes.

The population of foxes is initially very low, it rises over 10 years to a high number, stays constant for 1 year, then in 10 years shrinks again to its original small size. This whole process repeats three times.

Think about how the population of foxes and rabbits are related. Use two different colors to graph two lines. One representing the population of foxes, and the representing what you believe the population of rabbits at that time should be.

x-axis: Elapsed Time

y-axis: Population

**Suggested Solutions/Discussion: **

A detailed solution to problems in part A are provided at: http://www.teachertube.com/video/qualitative-graphs-korncast-47936

A1) b

A2) b

A3) a

A4) a

A5) c

B1)

B2a)

Note how the first sloped line is steeper than the second.

B2b)

B3)

B4)

Notice how the height of the water increases faster when the vase is thinner.

B5)

“Volterra lotka dynamics”. Licensed under CC BY-SA 3.0 via Commons – https://commons.wikimedia.org/wiki/File:Volterra_lotka_dynamics.PNG#/media/File:Volterra_lotka_dynamics.PNG

The solution shown is not the exact solution to the problem. Rather, it is the solution to a model actually representing predators and prey. The key thing to note is that when the prey population is high, after a slight time delay, the predator population grows. After the predator population is high, again after a time delay, the prey population falls.

**Download Activity:**

Powerpoint: Qualitative Graphs

Word Document: Qualitative Graphs

**Sources/Further Reading: **

Information on reading qualitative graphs: (Part A was drawn from this resource)

http://www.teachertube.com/video/qualitative-graphs-korncast-47936

Reading line graphs:

More information on predator-prey models (more advanced):

https://en.wikipedia.org/wiki/Lotka%E2%80%93Volterra_equations