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Related Rates Stations Activity

Station 1 - Blowing Up a Balloon. Blow 5 big breaths into a balloon. After each breath, measure the circumference of the balloon and calculate radius. What is the volume of the balloon? How are radius and volume related?

Station 2 - The Sliding Ladder. A metre stick sliding down the wall. The bottom of the metre stick is moving away from the base of the wall at a constant rate. How fast is the top of the stick sliding down the wall? How are they related?

Station 3 - Building Fences. Build several “fences” (rectangles made with multi-link cubes) such that the length is twice the width. Put them in order of size. Measure each rectangle’s length and width. What is the rate of change of the area? How are perimeter and area related?

Station 4 - Driving Cars. Two toy cars are traveling at different rates in perpendicular directions. How fast are each of the cars travelling? How fast is their distance apart changing? How are they related?

“The problem of finding a rate you cannot measure easily from some other rates that you can is called a related rates problem.” - Thomas Calculus

A nice discussion about the power of observing related rates. https://thalestriangles.blogspot.com/2016/04/

The Lengthening Shadow: The Story of Related Rates https://www.jstor.org/stable/2691482

“In the 1941 edition, Granville laid out a method for solving related rates problems, but these problems had now become an end in themselves rather than an exciting and pedagogically important method by which to introduce calculus.”

“An informal survey of ours suggests that for many teachers these problems have lost their significance. One often hears teachers say that related rates problems are contrived and too difficult for contemporary students. It is thus ironic that such problems entered calculus through reformers who believed, much as modern reformers do, that in order for calculus to be accessible, concrete, apt illustrations of the derivative are necessary”

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Related Rates Observations

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Demonstration Station - Shadows

An object is moving away from a wall and towards a light source at a constant rate. How quickly is the object’s shadow growing?

Place a measuring tape on the floor perpendicular to a wall.
Starting from the wall, move the object 5 cm towards the light each unit of time. Then measure the height the shadow makes on the wall.

Time	Distance from wall	Height of shadow	Δdistance	Δheight
0	0 cm	cm	0	0
1
2
3
4

50 cm

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Time 0

0 cm from wall

50 cm

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Time 1

5 cm from wall

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Time 2

10 cm from wall

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Time 3

15 cm from wall

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Time 4

20 cm from wall

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Time 5

25 cm from wall

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Related Relationships

50 cm

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Station 1 - Blowing Up!

Take a new balloon and fill it with one breath of air (a comfortable lung full). Measure its circumference and calculate the radius and volume of the balloon. Record these values.
Continue this process for several more breaths. Try to make each breath of air the same while blowing up the balloon.

Time	Circumference	Radius	Volume	Δradius	Δvolume
0	0 cm	0 cm	0 cm³	0	0
1
2
3
4
5

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Station 2 - Slip Sliding Away

If the base of a ladder is pulled away from a wall at steady rate, will the top of the ladder slide down the wall at the same rate?

Use a metre stick as your “ladder.” Place it flat against a wall.
Each unit of time, move the bottom 10 cm away base of the wall. Then measure the height the top of the meter stick is on the wall.

Time	distance of base from the bottom of the wall	Height the ladder reaches on the wall	Δbase	Δheight
0	0 cm	100 cm	0	0
1
2
3
…
9
10	100 cm	0 cm

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Station 3 - Don’t Fence Me In

A rectangular fence made of cube-a-links is expanding so that its length is always twice its width.
Build 5 different cube-a-link fences with this ratio and measure the length and width (in cm) of each. Then calculate the area in cm².

Rectangle	Length	width	area	Δlength	Δwidth	Δarea
1 (4x2)	cm	cm	cm²	0	0	0
2 (6x3)
3 (8x4)
4 (10x5)
5 (12x6)

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Station 4 - You Got a Fast Car

Set up up two metre sticks with ends touching at right angles. Place the two matchbox cars outside the metre sticks at their meeting point so the cars’ distance from each other is zero.
For each unit of time, car A will move 8 cm and car B will move 12 cm. Measure the actual distance between the two cars (from the back of the cars) and record all three lengths. Continue this process for at least 5 units of time.

Time	A distance from start	B distance from start	distance between cars	ΔA	ΔB	Δdistance
0	0 cm	0 cm	0 cm	0	0	0
1
2
3
4
5