Graphing stories
A graphing story is a classroom routine that can be used to develop secondary school students' knowledge of graphs, relationships and mathematical models. Students see short films of everyday events, and then use graphs to describe them. In the ensuing discussion, there is a natural need to introduce mathematical concepts such as linear relationship, proportionality, exponential function and derivatives.
The following graphing stories are organized in two categories: linear- and non-linear relationships.
Linear relationships
Glass
In this graphing story, students sketch how the water level in a glass increases with time, when the glass is filled. The graph is a straight line through the origin.
Glass vs vase
Students compare the graph that shows how the water level in a glass changes over time, with the corresponding graph when a wider vase is filled with water. It provides an opportunity to compare the slope of the graphs.
Glass with a pause
Just like in the graphing story Glass, students draw a graph that shows how the water level in a glass changes over time, but this time the water flow pauses for a few seconds (at a height of 5 cm). How does that affect the graph?
Snickers on a scale
In this graphing story, students plot how the total weight changes when Snickers pieces are placed on a scale. Each Snickers weighs 50 g. The plotted points lie on a straight line through the origin.
Snickers removed
In this graphing story, students plot how the total weight changes when Snickers pieces are removed from a scale. Each Snickers weighs 50 g. The points lie on a straight line with a negative slope.
Snickers in a bowl
In this graphing story, students plot how the total weight changes when Snickers pieces are placed in a bowl. Each Snickers weighs 50 g and the bowl weighs 400 g. This means that the points lie along a straight line with a positive slope, which does not go through the origin.
Petrol
In this graphing story, students see how the total cost changes when you fill up a tank with petrol. It is one of the few graphing stories, where the independent variable is not time. The graph is a straight line through the origin.
Cake candle
Students sketch a graph that describes how the length of a cake candle decreases with time when it burns. The graph is a straight line with a negative slope.
Walk, then run
In these graphing stories, students see a person walking a few meters and then starting to run. Their task is to draw the graph that describes the distance traveled as a function of time and the person’s speed as a function of time respectively. By contrasting these graphing stories, students can make connections between the graph describing distance and the graph describing speed. This in turn can lead to students making connections between the graph of a function and the graph of its derivative.
Walk, turn, run
In these videos, students see a person who walks up to a cone, stops for a few seconds and then runs back. The students are to draw a graph of how the person's distance, speed and distance to the starting point, changes with time. By working with all three videos, students can make important connections between the different graphs. Follow up with the graphing stories Walk then run.
Postage table
In this graphing story, students are prompted to draw the graph that describes how the number of stamps depends on the weight of the letter. The graph is a so-called step function.
Non-linear relationships
Irregular vase
In this graphing story, students draw a graph that shows how the water level in a vase changes over time, when water is poured in at a steady speed. Because the vase has an irregular shape, the graph is not a straight line.
Feather
Students are prompted to sketch a graph that shows how the height of a feather changes over time as it sails towards the ground.
Tea cup
In this graphing story, students sketch the graph that describes how the temperature of a cup of tea decreases over time. The graph is a curve that can be described with a function of the form f(x) = Ca^x + D. This graphing story can be used to introduce the concept of mathematical model.
Boiling water
In this graphing story, students sketch a graph that describes how the temperature of water changes when you heat it to boiling. The resulting graph is an S-curve.
Salmon in the oven
Students are prompted to describe how the temperature of a piece of salmon increases with time when it is placed in the oven. The resulting graph is an S-curve. This graphing story can be used to introduce the concept of mathematical model.
Ice cubes in a glass
In this graphing story, students graph how the temperature of a glass of water changes with time after some ice cubes are added to the glass. It can be used to introduce the concept of a minimum.
Toy car
Students get to see several different experiments where a toy car is pulled back and then goes forward. The students' task is to sketch how they think the distance the car travels depends on how far back you pulled the car. It turns out that the distance can be modeled with a quadratic function, but also a linear function can be adapted to the measured values.
Pulse up a hill
Students get to see a person run up a steep hill and are prompted to sketch the graph that describes the person's heart rate. The resulting graph has a maximum point. At the end of the graphing story, the difference between the measured values and the mathematical model is shown.
Bike speed and distance
In these graphing stories, students sketch the graph that describes the speed or distance of a bicycle as it goes down a hill and continues up another hill until it stops. The graph of the speed is a quadratic curve with a maximum point. The graph that describes the distance has the shape of an S-curve and is describes by a third degree polynomial. Contrasting the two graphing stories provides an opportunity to discuss concepts such as derivative and primitive function.
Swing’s height and speed
In this graphing story, students get to sketch how a swing's height changes over time when it is in motion. The resulting graph is periodic and can be used to introduce trigonometric functions.
Swing speed
In this graphing story, students get to sketch how the swing's speed changes over time when it is in motion. The graph is periodic.
Wheel of fortune
Students get to see a wheel of fortune that spins and are prompted to sketch the graph that describes how the height of the star changes over time. The graph is periodic and corresponds to a sine function.
Shadow on a wall
Students describe how the height of a shadow, cast by a toy figurine, depends on the figurine’s distance to the wall. The graphing story can be used to introduce concepts such as rational function and vertical asymptote.
Emptying a bottle
Students describe how the height of water decreases with time, as the water is let out of a bottle. The resulting graph can be used to discuss slope and derivative.