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Math OER
 Error Propagation Playground 

When people tell you something's wrong or doesn't work for them, they are almost always right.

When they tell you exactly what they think is wrong and how to fix it, they are almost always wrong.

- Neil Gaiman

Tacoma Narrows bridge collapse Sometimes a small initial wobble gets really out of control.

Which type of arithmetic is most vulnerable? Let's find out!

We will take pairs of numbers, make them wobble slightly, and see how much error is produced.

Why? We use wobbly numbers all the time in real life!

Maybe you are trying to plan for a car trip, and know the price of gas and lodging will vary from town to town.

Maybe you are baking a cake with a friend's recipe, and know that measuring cups and oven temperatures actually vary quite a bit.

Maybe you are running a day care, and know the numbers of children and the hours they attend vary a lot.

And so on. We often make decisions using numbers that have some wobble. How much can we trust the results? What kinds of wobble are worst?

An Empty Play Structure

This math playground is a sturdy piece of equipment. It is not a slide or swing or bars, but a piece of math equipment. It looks like this:

worksheet part 1

Start by picking two numbers. Put them in the middle, dark gray boxes. Then add some wobble by changing the first number by 5%, and the second number by 10%.

If needed, the Percent Of... part of our online notes can help remind you how to find 5% or 10% of a number.

Now some light gray boxes and equal signs swing into view.

worksheet part 2

Pick one arithmetic operation (+, −, ×, or ÷) and write it in all three smaller, light gray boxes.

Then do the three arithmetic problems. The middle one is the right answer. The top and bottom ones are wobbly and wrong.

The amount they are wrong is slightly interesting. But it is very dependent on context. For example, an answer being wrong by $5 would be amazing accuracy if comparing the bid and actual cost for a large construction project, but disappointing accuracy if predicting the cost of a breakfast pastry.

So more important is their percent change from the right answer, for the top and bottom problems. That better measures how much the wobble affected the results.

If needed, go to the percent change part of our online notes for help.

Finally, some pieces of the equipment flip around. Now the second number is shrunk on top, and grown on bottom.

worksheet part 3

That probably changes how the wobbles affect the results.

Watch Two Kids

Here is an example. As you approach this piece of playground equipment, two kids are just climbing onto it.

They are 12 and 10.

For 12, we do 12 × 0.05 = 0.6, and then on the top 12 + 0.6 = 12.6 and on the bottom 12 − 0.6 = 11.4.

For 10, we do 10 × 0.1 = 1, and then on the top 10 + 1 = 11 and on the bottom 10 − 1 = 9.

worksheet part 1

This time addition is the chosen operation.

worksheet part 2

The top result is incorrect by 23.6 − 22 = 1.6

This is a percentage change of change ÷ original = 1.6 ÷ 22 = 0.07 = 7%.

The bottom result is incorrect by 22 − 20.4 = 1.6, which is again a 7% percentage change.

So that time the wobbles made a 7% error. That seems okay, considering our 5% and 10% wobble.

But wait!

What are they doing?

Those kids decide to switch to subtraction for the the chosen operation. Also, some pieces of the equipment flip around, to swap where the 10 is shrunk or grown.

worksheet part 3

Now the top result is incorrect by 3.6 − 2 = 1.6

This is a percentage change of change ÷ original = 1.6 ÷ 2 = 0.8 = 80%.

Now the bottom result is incorrect by 2 − 0.4 = 1.6, which is again an 80% percentage change.

It would be hard to trust any decision made with 80% error. What happened to make things so much worse?

Your Turn

Your turn to play around with this equipment.

• Pick two numbers.

• Pick one arithmetic operation (+, −, ×, or ÷).

• Pick the normal or flipped version of the equipment.

Then see how badly the right answer gets wrecked by the wobbles!

Sadly, this is not as viscerally satisfying as wrecking a physical stack of blocks or sand castle. But it can still be fun, and oddly enjoyable.

Try that a few times.

Do you discover any patterns?

Does it matter if the two numbers are near each other, or much different?

Are bigger or smaller numbers especially vulnerable or resistant to wobbles?

Are any of the arithmetic operations especially vulnerable or resistant to wobbles?

When is the flipped version of the equipment better or worse than the normal version?

XKCD's Turn

Is this xkcd cartoon correct? How many of its claims can you double-check?