Algebra 6.1 - Absolute Value

Summary

In this educational video, Matt Disorbo explores the concept of absolute value in the context of the Challenger space shuttle disaster. He explains how the cold temperature on launch day compromised the O-rings, leading to the tragic explosion. The video uses a simple mathematical model to illustrate the relationship between temperature extremes and potential damage, emphasizing the importance of considering both hot and cold conditions in safety assessments. It also critiques NASA's data visualization, suggesting simpler charts could have prevented the tragedy.

Takeaways

• 🚀 The script discusses the Challenger space flight disaster, which occurred on January 28, 1986, killing all seven crew members on board.
• 🌡 The launch took place in unusually cold weather for Florida, at 28 degrees Fahrenheit, which affected the performance of the O-rings.
• 🔍 O-rings are critical rubber seals in the shuttle that prevent fuel leaks; their failure due to the cold led to the explosion.
• 💡 The script introduces a model using absolute value to predict damage to the O-rings based on temperature extremes, both hot and cold.
• 📉 NASA had a damage index for O-rings, with 27 or higher indicating dangerous conditions; the model helps predict when this threshold might be exceeded.
• 📈 The model is represented graphically, showing a 'V' shape where damage increases as temperatures move away from the ideal 70 degrees Fahrenheit.
• 🔢 The script provides a method to solve for temperatures that would result in a damage index of 27, finding critical temperatures of 43 and 97 degrees Fahrenheit.
• ❗ The Challenger disaster was partly due to a focus on high temperatures instead of recognizing the risk of extreme temperatures, including the cold.
• 🛑 The aftermath led to the grounding of the space fleet for nearly three years and highlighted the importance of safety and risk management.
• 📊 The script critiques NASA's data visualization and presents an alternative by Edward Tufte, which simplifies the presentation for clearer understanding.
• 🤔 The video ends with a call to reflect on the presented model and data, questioning its perfection and usefulness in decision-making processes.

Q & A

What was the main topic of the video script?

The main topic of the video script was the discussion of absolute value functions in the context of the Challenger space flight disaster.

What was the date of the Challenger space flight disaster?

The Challenger space flight disaster occurred on January 28, 1986.

How many crew members were on board the Challenger during the disaster?

There were seven crew members on board the Challenger during the disaster, including a teacher.

What was the cost of building the Challenger space shuttle?

The Challenger space shuttle cost 3.2 billion dollars to build.

What was the role of the O-rings in the Challenger disaster?

The O-rings, which are rubber rings that seal the joints of the shuttle together, failed due to the unusually cold temperature on the day of the launch, leading to a fuel leak and the subsequent explosion.

What is an absolute value function and how does it relate to the Challenger disaster?

An absolute value function measures the magnitude of a number without considering its sign. In the context of the Challenger disaster, it was used to model the potential damage to the O-rings from both extremely high and low temperatures.

What was the room temperature at which the O-ring parts were built and assumed to be strong?

The O-ring parts were built at room temperature, which was assumed to be 70 degrees Fahrenheit.

What was the temperature on the day of the Challenger launch and why was it significant?

The temperature on the day of the Challenger launch was 28 degrees Fahrenheit, which was unusually cold. This cold temperature caused the O-rings to become brittle and fail, leading to the disaster.

What is the damage index and how was it used in the video script?

The damage index is a point system used by NASA to track the damage to the O-rings during launches. In the video script, it was used to model the potential damage from both hot and cold temperatures, with an index of 27 or higher considered dangerous.

What was the criticism by Edward Tufte regarding NASA's data visualization of the O-ring damage and temperature?

Edward Tufte criticized NASA's data visualization as difficult to read and understand due to its complexity. He suggested a simpler chart that plotted temperature on the x-axis and O-ring damage on the y-axis for better clarity.

What is the significance of the Challenger disaster in current safety and risk management discussions?

The Challenger disaster is frequently used as a case study in safety and risk management to illustrate the importance of considering all potential risks and the consequences of overlooking them.

Outlines

00:00

🚀 Introduction to the Challenger Space Flight Tragedy

This paragraph introduces the topic of the video script, which is the Challenger space flight tragedy. It sets the stage by mentioning the launch date, January 28, 1986, and the presence of a teacher among the seven crew members. The paragraph describes the catastrophic event where the Challenger broke apart 73 seconds into the flight, resulting in the loss of the entire crew. It also touches on the emotional impact of the tragedy, as many children were watching the launch live due to the teacher's involvement. The cost of the Challenger and its successful flights are mentioned, leading into the main question of how the tragedy occurred, and inviting viewers to follow along with the lesson.

05:01

🔍 Investigating the Cause of the Challenger Disaster

This paragraph delves into the technical aspects of the Challenger disaster, focusing on the role of the O-rings, which are critical components in sealing the joints of the shuttle. The script explains that the unusually cold weather on the day of the launch affected the O-rings' performance, leading to a fuel leak and subsequent explosion. The paragraph also discusses the testing of the O-rings and the concept of 'one directional thinking' that may have contributed to NASA's decision to proceed with the launch despite the risks. It introduces a graph that correlates outside temperature with the O-ring damage index, which is used to analyze the potential for damage during launches at different temperatures.

10:03

📊 Modeling Temperature Impact on O-ring Damage

The script introduces a model to predict O-ring damage based on temperature, using a simple formula that calculates damage as the difference from a 'normal' temperature of 70 degrees Fahrenheit. The paragraph explains how the model works for both hot and cold temperatures, using absolute value to represent the magnitude of temperature differences from the norm. It provides examples of how the model predicts damage at various temperatures, including the critical temperature of 28 degrees Fahrenheit on the day of the launch, which resulted in a high damage index. The paragraph also discusses solving absolute value equations to determine the safe temperature range for launches and the implications of the model for the Challenger disaster.

15:05

📈 Absolute Value Functions and Their Role in Risk Management

This paragraph continues the discussion on absolute value functions, emphasizing their importance in modeling the potential damage to O-rings from extreme temperatures, both hot and cold. The script presents a graphical representation of the model, showing a 'V' shaped graph that illustrates how damage increases with temperature extremes. It also discusses the identification of safe and dangerous temperature ranges for shuttle launches based on the model. The paragraph concludes with a reflection on the Challenger disaster, highlighting the consequences of not considering both high and low temperatures as risks and the impact of the tragedy on NASA's operations and safety protocols.

📚 Data Visualization and Lessons Learned from the Challenger

The final paragraph of the script discusses the importance of data visualization in understanding and communicating risks effectively. It contrasts NASA's original charts, which were criticized for being cluttered and difficult to read, with a simpler alternative presented by Edward Tufte. The paragraph invites viewers to consider whether simplicity in data visualization is indeed better and to reflect on the adequacy and usefulness of the theoretical model presented alongside the actual data. It concludes with a prompt for viewers to consider their own decision-making based on the data and to think critically about the lessons learned from the Challenger disaster.

Keywords

💡Absolute Value

Absolute value refers to the non-negative value of a number, regardless of its sign. It is used to measure the magnitude of a difference without considering direction. In the video, absolute value is crucial in understanding the impact of temperature extremes on the O-rings' performance. The script uses absolute value to model the damage index caused by temperatures deviating from the normal 70 degrees Fahrenheit, both above and below.

💡Challenger Space Flight

The Challenger Space Flight refers to the space mission of NASA's Space Shuttle Challenger, which tragically exploded shortly after liftoff on January 28, 1986. The video uses this historical event to discuss the concept of absolute value in the context of the disaster, highlighting how the cold temperature on the day of the launch affected the O-rings, leading to the explosion.

💡O-rings

O-rings are rubber seals used to prevent leakage in mechanical assemblies. In the context of the Challenger disaster, they were critical components that failed due to the cold weather, causing a fuel leak and subsequent explosion. The video explains the role of O-rings and how temperature affected their integrity, leading to the tragedy.

💡Temperature Extremes

Temperature extremes refer to conditions where the temperature is significantly higher or lower than normal. The video emphasizes the importance of considering both hot and cold extremes when assessing the risk of damage to the O-rings. The script illustrates this with a model that calculates potential damage based on the temperature deviation from 70 degrees Fahrenheit.

💡Damage Index

The damage index is a hypothetical point system used in the video to quantify the damage to the O-rings during launches. An index of 27 or higher is considered dangerous. The video uses this index to demonstrate how different temperatures can affect the O-rings' condition and the likelihood of a catastrophic event.

💡Risk Management

Risk management is the process of identifying, assessing, and controlling risks to minimize harm and loss. The video discusses how NASA's decision to launch despite the cold weather was a failure in risk management, leading to the Challenger disaster. It serves as a case study for the importance of considering all potential risks.

💡Data Visualization

Data visualization involves presenting data in a graphical format to make it easier to understand and analyze. The video critiques NASA's original data visualization for the Challenger disaster as difficult to read and presents an alternative by Edward Tufte, which is simpler and more effective in conveying the relationship between temperature and O-ring damage.

💡Modeling

Modeling in this context refers to the creation of a mathematical representation to predict outcomes based on certain variables. The video uses a simple model to predict the damage to O-rings based on temperature. It demonstrates how a model can help understand the relationship between temperature extremes and the potential for disaster.

💡Directional Thinking

Directional thinking is a cognitive bias where one direction of an effect is considered more likely or important than another, leading to oversight. The video mentions this concept as one of the reasons NASA failed to prevent the Challenger disaster, as they were overly concerned with high temperatures and did not adequately consider the risks of cold temperatures.

💡Memorial

A memorial is a tribute or monument to honor those who have died. The video mentions the Challenger crew memorial in Arlington, which serves as a reminder of the tragedy and the importance of safety and risk management in space exploration.

Highlights

Introduction to the concept of absolute value in the context of the Challenger space flight disaster.

The Challenger shuttle broke apart 73 seconds into the flight, resulting in the tragic loss of all seven crew members.

The cost of building the Challenger was $3.2 billion, highlighting the financial impact of the disaster.

Explanation of the role of O-rings in sealing the joints of the shuttle and their failure due to cold temperatures.

The use of a damage index to quantify the damage to O-rings during launches, with a value of 27 or higher considered dangerous.

The establishment of a simple model to predict O-ring damage based on temperature, with zero damage at 70 degrees Fahrenheit.

The model's application to predict damage on hot days, showing an increase in damage as temperature rises.

Introduction of absolute value to model damage from both high and low temperatures, emphasizing the magnitude of temperature differences.

Demonstration of solving absolute value equations to find temperatures that would result in a dangerous damage index.

Graphical representation of the absolute value function to model potential damage on both hot and cold days.

Identification of the safe temperature range for shuttle launches based on the damage index model.

Analysis of the launch day temperature of 28 degrees Fahrenheit and its catastrophic impact on the O-rings.

Discussion on the aftermath of the Challenger disaster, including the grounding of the space fleet and the establishment of a presidential commission.

Critique of NASA's data visualization by Edward Tufte, advocating for simplicity and clarity in presenting information.

Comparison of NASA's original chart with Tufte's simplified version, emphasizing the importance of effective data presentation.

Reflection on the question of whether the launch should have been stopped based on the presented data and models.

Final thoughts on the usefulness and limitations of the theoretical model in predicting O-ring damage.

Transcripts

hello mathematicians my name is matt

disorbo covering the allisborough

lessons for

skew the script today we will be

discussing absolute value in the context

of the ill-fated

challenger space flight without further

ado

let's skew it

[Music]

welcome in to lesson 6.1 of our algebra

skew the script series today discussing

absolute value functions specifically

we'll be talking about the challenger

on january 28 1986 the challenger took

off from the coast of florida with a

crew of seven aboard including a teacher

uh here is a photograph of the

challenger's final crew

very unfortunately 73 seconds into the

flight the shuttle broke

apart and sadly none of the crew

survived

in addition many children were watching

live because of the teacher on board

making this an extremely traumatic

experience for america

challenger was very costly to build it

cost 3.2

billion dollars and successfully logged

nine flights and completed 62 days in

space

so the question remains and that's

today's key analysis how did the

challenger tragedy

happen if you'd like to follow along

with our lesson check out the link below

feel free to print or download the

guided notes and work through them as we

talk through the video

to begin our discussion we'll be talking

about absolute value

so on the day of the launch in florida

generally these shuttles are launched

from florida because it's a warm climate

state

but on this specific day the morning was

actually unusually cold 28 degrees

fahrenheit

there uh it was ice on the launch pad

which you can see in the picture in the

left

nasa did decide to proceed anyways

um so to understand more about the crash

we have to understand

o-rings which are rubber rings that seal

the joints of the shuttle together you

can see

um them in the diagram on the right

here's some common examples of o-rings

this is something that you'll see in a

sink or a faucet

obviously for the rocket they are much

much larger

um on the morning of the launch again it

was very cold and the cold temperature

actually broke the seal of an

o-ring which in turn leaked fuel and the

leaked fuel actually caused the

explosion

these two photographs uh here in the

photograph on the left you can see the

red circle

black smoke at the start of the launch

from the o-ring that failed and on the

right

obviously the catastrophic explosion

so like everything else in the shuttle

the o-rings were meticulously tested

so how could nasa make this mistake

um there were many reasons among uh

many many reasons that this happened but

one among several was

one directional thinking um so let's

visually explore the problem of the

o-ring and the challenger with the scrap

here

graph here on the x-axis is outside

temperature and the uh

y-axis is damage index so the o-ring

damage index

it's a point system used by nasa tracks

the damage to the o-ring during launches

a larger index means uh more damage and

we slightly modified this for the lesson

you can see the

original index scale and the link on the

right side

so let's say damage of index of 27 or

higher is dangerous that's our kind of

danger threshold

most of these engineer o-ring parts are

built at room temperature about 70

degrees fahrenheit

so we can assume that they are strong at

this temp so we'll assume zero damage at

the launch with an outside temperature

of 70 degrees fahrenheit

rocket fuel of course is hot so nasa

worries about heat

um so for example when things are hot

o-rings may expand and not seal properly

on a launch pad during a hot day

so combined with the heat created by a

launch hot days could be extremely

dangerous

uh we can actually build a model for the

potential damage on hot days

uh in this case we'll say predicted

damage equals temperature

minus 70. so we can put in our table of

temperature and damage

70 for our temperature um

and if we plug this into our very simple

model we get predicted damage 70 minus

70

or zero so on a date with 70 degrees

fahrenheit

um the damage is expected to be zero

which makes sense because uh

the the this is the temperature of the

rings were built at and you can see

we've we've dotted that out here

what about if we have uh 80 degrees an

80 degree temperature if we plug that

into our simple model

and again we can just use x for

our outside temperature to be more

general um we plug in 80 for x we get 80

minus 70

so predicted damage of 10 and we can

plot that on our chart as you can see

here

um similarly we continue with 90 degrees

our damage becomes 20

plot it in our chart 100 degrees becomes

a damage of 30

and we can draw our nice line connecting

these dots here

um as you would expect as the heat gets

more extreme the potential for damage

rises

so nasa may not launch on extremely hot

days because they'll be

in excess of our our threshold our

damage index

however again on the launch day the uh

opposite actually happened it was

unusually cold 28 degrees fahrenheit

so we if we return to our model what

happens at extremely cold temperatures

um this is where absolute value can be

useful because it turns numbers into

positives we use it when we care about

the magnitude

of a difference so in this case what's

the difference between 86 degrees

fahrenheit and 70 degrees fahrenheit

it's 16 degrees fahrenheit um 86 degrees

is 86 degrees fahrenheit is 16 degrees

higher than

normal or we define normal as 70 degrees

fahrenheit or the room temperature

essentially

how about the difference between 54

degrees fahrenheit and 70 degrees

fahrenheit

well in this case you have a difference

of negative 16 degrees

fahrenheit so 54 degrees is 16 degrees

lower than normal um so

an example if we can turn to this

natural example how different

are 86 degrees fahrenheit and 70 degrees

fahrenheit this is a question about

distance not direction we don't care

about if 86 is higher or lower

we only care about the size of the

difference essentially what we're saying

is what is the magnitude or size of the

difference

in this case it's 16 degrees fahrenheit

86

is 16 degrees away from normal we don't

care about the direction

extending the example how different are

54 and 70 degrees

we see that 54 minus 70 is negative 16

degrees fahrenheit

which is also 16 degrees away from

normal

even though it's lower than 70 whereas

86 is greater so

absolute value measures the magnitudes

of differences

you can see these little vertical lines

we put here these are absolute value

marks

for our notation we get the absolute

value of 16 on the left

absolute value of negative 16 on the

right and what absolute value

essentially does

is it makes it positive so 16 is already

positive it just

stays 16 negative 16 we make it positive

and it turns into

16. so negative 16 turns into 16. um

this shows us that the sizes or the

magnitudes of the differences

are the same so 54 and 86 are both

equally far

from 70. now let's turn to actually

solving for absolute values

again we'll return to our simple model

where predicted damage is x minus 70

x being the outside temperature um again

we saw that if temperature

temperatures are much higher than 70

degrees they're predicted to cause

large damage so nasa will not launch on

extremely hot days

however temperatures that are much lower

than 70 degrees could also be dangerous

o-rings could become brittle in the cold

and break

so how can we model uh predicted damage

from both extremely high

and extremely high extremely low

temperatures both extremely high and

extremely low um in this case we can get

the magnitude

which we can use the absolute value for

of the difference from a normal

70 degree fahrenheit day so our

difference of x the outside temp

from a normal 70 degree day um so here

we can

use our model and actually test it we

have a very hot day of 110 degrees

fahrenheit

we plug it into our model 110 minus 70

we get the absolute value of 40.

40 is already positive so we get a

predicted damage of 40 very high

predicted damage so our model works for

a very hot day

how about a freezing cold day of 30

degrees fahrenheit

we plug in 30 minus 70 the absolute

value of negative 40. remember absolute

value makes things positive we get a

predicted damage of 40.

also very high predicted damage

um we can discuss more specifically

solving inside the actual absolute value

marks so remember that an o-ring damage

index value of 27 is considered

dangerous

so what temperature would create a

predict predicted damage index

of 27. again a predicted damage is what

we're modeling for

in our model so we can set these two

sides equal 27 equals the absolute value

of x minus 70.

um how do we actually solve for that

temperature x because we have these kind

of pesky absolute value markers on the

right side of our equation

remember that the size or magnitude of

the difference between x and 70

has to be equal to 27. so

basically we're saying what numbers have

the same magnitude

as 27 that's going to be positive 27

or negative 27 if we get these two

values

we absolute value them we're going to

get 27.

so we're basically saying that x minus

70 must equal

either 27 or negative 27. so we can have

a positive and negative version of the

side without the absolute value

and that actually allows us to uh write

out the equation

27 equals x minus 70 and negative 27

equals x minus 70. and again we just see

where x minus 70 equals these two values

so simply we can just solve both

equations

um we'll look at both of these equations

in tandem or

separately we'll start with the equation

on the left add 70 to both sides

we get the right side cancels out the

left side becomes 97

equals x um the second equation we add

70 to both sides

right side cancels out left side becomes

43 equals x

and this essentially says we are we

successfully solved our absolute value

equation

this says that temperatures of 97

degrees or or

43 degrees would lead to a prediction

predicted dangerous damage index of 27.

so

uh uh damage index dangerous enough to

not

launch therefore our temperature should

be between

these two temperatures for a safe launch

um turning to an absolute value function

um

again continuing our theme of modeling

potential damage on hot days

here we have predicted damage equals x

minus 70. we have

our chart from earlier um and we're

going to just replace uh

y uh with our predicted damage with y

and

in this case we want to model extreme

days not necessarily just hot days but

hot

or cold days so we have these absolute

value bars around x

minus 70. we can have our handy dandy

table

and plugging in uh 70 for x again we get

y equals absolute value of 70 minus 70

y equals the absolute value of zero

which is zero so our damage is still

zero

which agrees with our point from earlier

if we plug in 80 minus 70

that is the absolute value of 10 which

comes out to 10

damage of 10 agrees with our point from

earlier so so far so good

um we're going to get the same results

we plug in 90. absolute value of 20 is

20

same point from earlier and the same

result if we plug in 100

absolute value of 100 minus 70 is the

absolute value of 30

which is just 30 and we get our same

result familiar so for the hot days

our model works just as well but what

about for cold days

so let's plug in 70 to our value or to

our model again we get absolute value 70

minus 70 that's zero

zero damage on 70 degree days what about

60 days

if we plug in 60 minus 70 we get the

absolute value of negative 10

which comes out to 10. remember absolute

value makes negative numbers

positive so damage of 10. if we plug in

50 we get absolute value of 50 minus 70

absolute value of negative 20 that comes

out to 20

so damage of 20 and finally if you plug

in 40 we get 40 minus 70

absolute value of negative 30 comes out

to 30.

so now we have our full model for y

equals the absolute value of x

minus 70. and you can kind of see the

shape it

takes basically a v shape um

and this happens because the absolute

value absolute value keeps things

positive it keeps the y

axis positive y is positive as x

decreases it's also positive as

x increases and it moves um

higher and higher as x either decreases

or it's actually there

as x increases more um and again we see

that more predicted damage

uh more damage is predicted at extreme

temperatures in either direction it

doesn't matter if it's hot

or cold the more extreme we get the

higher damage is predicted

um now we can actually turn to finding

the temperature boundaries graphically

so again 27 is our dangerous damage

index if we extend that

line we can look at where our graph

crosses we draw our cold point and our

hot point there

extend those down to the x axis and we

see 43

and 97 are where it crosses the the axis

which actually agrees with our model

from earlier

and in between there we have our safe

temperature zone where the damage index

is below

27. however remember that on the day of

the launch

it was 28 degrees fahrenheit

if we put that on our graph on launch

day you can see the 28 over there

we can draw a vertical line extend our

damage function and we see that we get a

pretty high

damage when we have 28 degrees

fahrenheit we can actually solve

for what the damage would be we use our

model y equals absolute value of x minus

70.

plug in 28 for x 28 minus 70. get the

absolute value of negative 42

which of course just becomes 42. so a

damage of 42 when it's 28 degrees

outside that's far far above the danger

well in excess of the danger boundary of

27

um for our damage index so the result

again

uh very unfortunately was tragedy uh

partly because engineers are worried

about high temperatures instead of just

extreme temperatures hot or cold

the launch proceeded the o-rings became

so brittle uh from the cold they failed

to seal

and the result was uh catastrophic

uh the aftermath um was definitely very

challenging the space place the space

fleet was grounded for

nearly three years it was a presidential

commission established by president

reagan to investigate nasa

and this uh challenger is now frequently

used as a case study

for safety and risk management and you

can see the memorial for the challenger

crew

in arlington on the right um finally

we're going to turn to data

visualization and we'll actually

uh dovetail this into our discussion so

during testing and prior launches nasa

actually did chart the o-ring damage in

the outside temperature

at launch you can see it here on the

left however uh data visualization

expert edward tufte

uh very famously criticized these charts

as difficult to read and understand

and i will let you decide for yourself

if that chart on the left

is very easy to comprehend or not um

tufti uh gave a uh alternative to this

chart

which just plots the temperature on the

x-axis first the o-ring damage on the

y-axis

definitely much simpler and looks like

sort of the chart we were working on

um we've created here with the same

chart and data but with using the damage

scale that we actually

created in this lesson again much easier

to to visualize

so um if you had this data would you

have stopped the launch

and you want to justify your answer

using the data that you've seen here

um tufty was famous for arguing that

these data visuals should be simple

and avoid drawings figures and other

chart junk you can see how tough to easy

example

was much simpler than nasa's which had

sort of a lot of

diagrams and numbers all over the place

and it's more difficult to read

and our second discussion question is do

you agree with tuffy that simpler is

better

and explain your reasoning and address

these two charts tupt's chart

versus nasa's charter finally you can

look at our theoretical model alongside

the data

and ask yourself is our model perfect is

it useful

explain why or why not that's all for

today on skew the script thanks for

coming and we'll see you next time

[Music]