Chapter 1.1

Objectives

Review the lesson material
Defeat the area challenge
Complete the lesson reflection

Key Vocabulary

Units

Measurement terms for quantifying variables (e.g., meters, seconds).
Dimensional Analysis

Method to convert between units using multiplication by conversion factors.
Conversion Factors

Ratios that represent the relationship between different units for unit conversion.

Lesson Outcomes & Standards

Students will solve multi-step problems with at least 90% accuracy by using consistent units.
Students will correctly apply dimensional analysis to convert between units in at least 2 complex problems.
Students will analyze the impact of unit conversion errors in real-world scenarios.

Common Core:

N.Q.1 Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.

The story continues...

Welcome to Central Park, the vibrant heart of the city. Here, lush greenery meets bustling pathways, creating a haven where everyone can gather and find respite from the urban rush. Towering trees line the walkways, and colorful flowers bloom in every corner, making it a place of peace and harmony. All roads in the city lead back to this central hub, symbolizing unity and connection.

What is typically the beautiful center of our town has been destroyed by the Fraction Faction. A group of Fraction Faction members came through here yesterday and threw the whole place out of balance.

Our town flag used to fly in the center of the park, it was the cornerstone of the park. But now it is missing. I’m worried the Fraction Faction has stolen it and I need your help to get it back.

The Power of Precision

**Figure 1.** *Before you get precision driving, you need precision calculations.*

In the world of racing, every millisecond counts. Imagine you're an engineer working for a Formula 1 team. Your job is to ensure that the car's performance is optimized down to the last detail. The speed, fuel efficiency, and even the tire wear depend on precise calculations. Using units correctly and consistently in formulas allows you to predict how the car will behave under different conditions. One tiny mistake in these calculations could mean the difference between winning the race and crashing into the wall. Understanding and interpreting units isn't just about math; it's about having the power to make critical decisions with confidence.

Dimensional Analysis

Back in elementary school, you learned that any number divided by itself gave you 1. When we talk about units, the same thing applies. This is why you can "cross off" units when you have one in the numerator and one on the denominator . Take this example from your physics class:

If you are traveling at 50 miles per hour for 2 hours, how far will you travel?

$LaTeX: d=50\,\frac{miles}{hour}\times2\,hour$

Notice how we have hours in the numerator and the denominator? Since $LaTeX: \frac{hour}{hour}=1$ , we can cross them out of the equation.

$LaTeX: d = 50 \frac{\text{miles}}{\cancel{\text{hour}}} \times 2 \cancel{\text{hours}} = 100 \text{miles}$

$LaTeX: d=50\:miles\times 2=100 miles$

This type of work is called dimensional analysis. In other words, we arrange the numbers so their dimensions can be removed, leaving us with what we are looking to find.

Sometimes the conversion isn't so straightforward. Say we wanted to find out how many kilometers we traveled in two hours. Since we only have our information in miles, we would need to apply a conversion factor. A conversion factor is a ratio of a numerator and denominator of the same amount but in different units.

For example, 1 mile is the same as 1.6 kilometers. So we can write that as $LaTeX: 1\; mile= 1.6 \;Km$

To use that as a conversion factor, we need to rearrange the equation. We can do that in two ways:

$LaTeX: \frac{1\;mile}{1.6\;Km} =1$

$LaTeX: \frac{1.6\;Km}{1\;mile} =1$

So, to convert 100 miles to kilometers, we can perform dimensional analysis. Since we want to end up with kilometers as our final unit, let's choose the version that has that unit in the numerator.

$LaTeX: 100\,\cancel{\text{miles}} \times \frac{1.6\,\text{Km}}{1\,\cancel{\text{mile}}}$

Before solving, carefully analyze your units to ensure that the conversion factor you chose allows the units to cancel out correctly. We planned ahead and made sure to set up our conversion factor so our desired unit would be in the numerator. Then we could cancel out the miles and solve.

$LaTeX: 100\times 1.6\,Km=160\;Km$

Master Essential Skills

The following questions will test your prior knowledge to see if you have the skills required to finish today's challenge.

Question 1: You find an ancient treasure map. The map says to travel 200 cubits directly north. How many meters is that?

Write out the formula to solve this equation. (Bonus! Solve it if you can!)

Conversion Factors
From	To
1 Cubit	18 inches
1 inch	2.54 cm
1 meter	100 cm

Question 2: Solve for x: $LaTeX: 15=\frac{3}{2}\left(x+6\right)$

Hint: Remember your order of operations!

As always, don't check your work until you have attempted the question.

Check Your Work

Question 1:

We know we want meters on the top, since that is our desired final unit. So start with that conversion, then choose the next conversion so the centimeters cancel, and so forth.

$LaTeX: \frac{1meter}{100cm}\times\frac{2.54cm}{1\:in}\times\frac{18in}{1\:cubit}\times200cubits$

That's it! Check to make sure you have each conversion factor.

Bonus: Cancel out the units and solve

$LaTeX: \frac{1\,\text{meter}}{100\,\cancel{\color{red}{\text{cm}}}} \times \frac{2.54\,\cancel{\color{red}{\text{cm}}}}{1\,\cancel{\color{blue}{\text{in}}}} \times \frac{18\,\cancel{\color{blue}{\text{in}}}}{1\,\cancel{\color{green}{\text{cubit}}}} \times 200\,\cancel{\color{green}{\text{cubits}}} = \text{result in meters}$

Now simplify

$LaTeX: \frac{1m\times 2.54\times18 }{100} =91.44\;meters$

Question 2: $LaTeX: 15=\frac{3}{2}\left(x+6\right)$

Step 1: Multiple into the parenthesis. $LaTeX: 15=\frac{3}{2}x+\frac{18}{2}$

Simplify to get $LaTeX: 15=\frac{3}{2}x+9$

Step 2: Isolate the terms. We want the x to be alone (and positive), so we will do the opposite of adding 9 to both sides. By subtracting 9 from both sides, we keep the equation equal while making it simpler.

$LaTeX: 15-9=\frac{3}{2}x+9-9$

And simplify to get: $LaTeX: 6=\frac{3}{2}x$

Step 3: Isolate the x. Since we are multiplying by 3 and dividing by 2, we will do the opposite on both sides.

$LaTeX: 6\times \frac{2}{3} =\frac{3}{2} x\times \frac{2}{3}$

And simplify:

$LaTeX: \frac{12}{3} =\frac{\cancel{3}}{\cancel{2}} x\times \frac{\cancel{2}}{\cancel{3}}$

LaTeX: 4 =x

Therefore, x=4.

Note: Some conversions (such as temperature) don't have simple conversion factors. If you struggled with this example, you should practice some linear algebra until it becomes second nature.

Common Conversions

There are some common conversions that are worth memorizing, as you'll use them a lot in this course. These are rounded for easier memorization.

Click on each tab to view the conversions.

Length Conversions

Length
Conversion	Formula
Inches to Centimeters	1 inch = 2.54 cm
Feet to Meters	1 foot = 0.3 meters
Miles to Kilometers	1 mile = 1.6 km

Mass/Weight Conversions

Mass/Weight
Conversion	Formula
Pounds to Kilograms	1 pound = 0.45 kg
Ounces to grams	1 ounce = 28.3 grams

Volume Conversions

Volume
Conversion	Formula
Gallons to Liters	1 gallon = 3.79 liters
Cubic Inches to cubic centimeters	1 cubic inch = 16.4 cm³

Temperature Conversions

Temperature
From	Formula
Fahrenheit to Celsius	C = (F - 32) × 5/9

Time Conversions

Time
From	Formula
Hours to minutes	1 hour = 60 minutes
Minutes to seconds	1 minute = 60 seconds

Practice

Example 1 – Moving Dirt

A construction worker needs to move 240 ft³ of dirt by using a wheelbarrow. One wheelbarrow load holds 8 ft³ of dirt and each load takes her 10 minutes to complete. How many hours would she need to complete this job?

Solution:

Step 1

Write down all the conversions available:

1 wheelbarrow load = 8 ft³

10 minutes = 1 wheelbarrow load

1 hr = 60 minutes

Step 2

Find the number of loads needed to move 240ft³ of dirt:

$LaTeX: =\frac{240ft^3}{1}\times\frac{1\:load}{8ft^3}$

Save simplifying towards the end to make the process easier.

Step 3

Find the amount of time taken (in minutes) to complete the loads:

$LaTeX: =\frac{240ft^3}{1}\times\frac{1\:load}{8ft^3}\times\frac{10\min}{1\:load}$

Step 4

Now find the time taken in hours to complete the loads

$LaTeX: =\frac{240ft^3}{1}\times\frac{1\:load}{8ft^3}\times\frac{10\min}{1\:load}\times\frac{1hr}{60\min}$

Step 5

Cancel out same units in numerator and denominator

$LaTeX: =\frac{240}{1}\times\frac{1\:}{8}\times\frac{10}{1\:}\times\frac{1hr}{60}$

Step 6

Simplify:

LaTeX: =5hr

Conclusion

It takes LaTeX: 5hr for the construction worker to move 240 ft³of dirt.

Example 2 – Convert to ºC

Jess moved from Canada to the USA and wants to know the temperature in degrees Celsius (ºC). Her friend gave her the following formula to convert degrees Fahrenheit (ºF) to degrees Celsius: $LaTeX: C=\frac{5}{9}\left(F-32\right)$

If Jess calculated 68ºF, what would her result be in degrees Celsius?

Solution:

Step 1

Given information is ºF = 68, and we need to find the temperature in ºC.

Step 2

The formula used to convert ºC to ºF is

$LaTeX: C=\frac{5}{9}\left(F-32\right)$

Step 3

Plug in the value of ºF into the equation

$LaTeX: C=\frac{5}{9}\left(68-32\right)$

Step 4

Simplify to find ºC

LaTeX: C=20

Conclusion

Jess converted 68 ºF to 20 ºC.

Example 3 – Gas Money

Two friends plan a road trip covering 300 miles. They need to calculate the amount of fuel their car will use for the trip to budget appropriately. They have a spending limit of $40 for fuel. Their car's fuel efficiency is given as 25 miles per gallon. If the price of gasoline is $3.50 per gallon, will they have enough money to complete the trip? Justify your reasoning.

Solution:

Step 1

Given information is:

Total distance = 300 miles

Fuel efficiency = 25 miles/gallon

Price per gallon = $3.50/gallon

Spending limit = $40

Step 2

Find the gallons of fuel needed for the trip:

$LaTeX: Fuel\:needed\:in\:gallons=\:\frac{total\:distance\:in\:miles}{fuel\:efficiency\:in\:\frac{miles}{gallon}\:}$

Step 3

Plug in the values on the right side of the equation and simplify

$LaTeX: Fuel\:needed\:in\:gallons=\:\frac{300\:miles}{25\:\frac{miles}{gallon}\:}=12\:gallons$

Step 4

Find the total cost of fuel:

$LaTeX: Cost\:of\:fuel=fuel\:needed\:\:in\:gallons\times price\:of\:fuel\:per\:gallon$

Cost of fuel = 12 gallons x $3.50/gallon

Cost of fuel = $42

Conclusion

The two friends are $2 short of completing their trip. They have $40 but they need $42. Hence they cannot complete their trip on this budget.

Example 4 Stock Price changes

Consider the graph below that shows the stock price of a company over a week. The y-axis starts at $52 and ends at $56, and does not show the origin (0, 0).

Stock price vs days. Cart starting at $52 and ending at $56

Choose the best option from below and provide an explanation for your choice.

Showing the origin is necessary to understand the absolute value of the stock.
The graph should always show the origin to maintain consistency.
The origin is omitted to confuse the viewer.
Omitting the origin allows for a clearer view of the fluctuations in stock price.

Solution:

Step 1

Analyze the given options one by one.

Step 2

Option A:

The origin would compress the fluctuation visually on the graph, making it harder to observe the changes clearly. Stock traders and analysts are more interested in the relative changes and patterns over days rather than the absolute price from $0.

Step 3

Option B:

In financial graphs, the key is to show meaningful data that aids decision-making. The origin is not necessary if it reduces the graph's effectiveness in showing important trends.

Step 4

Option C:

This is not a valid reason for omitting the origin in professional and academic settings. Graphs are designed to enhance understanding, not to confuse.

Step 5

Option D:

This option is the best for this scenario because it enhances the visibility of fluctuations within a specific range. By focusing on the range from $52 to $56, the graph emphasizes the week's price movements without unnecessary distractions.

Conclusion

Omitting the origin allows for a clearer view of the fluctuations in stock price.

This approach keeps the graph simple and focused on the price range of interest, making it easier to interpret the important changes in stock prices over the week.

Cancel Crew

Read the Lyrics

Verse 1:
Start with the number and unit in hand,
Ask yourself, what’s the final plan?
Find the conversion that you need to use,
Write it down, don’t get confused!

Pre-Chorus:
First, write what you know at the top,
Then find the unit you want and don’t stop,
Flip the conversion if it’s upside-down,
Now you’re ready, let’s break it down!

Chorus:
Flip it, flip it, cancel, cancel,
Multiply to find the answer,
What’s left is what you need,
Unit analysis, it’s guaranteed!

Verse 2:
Need to change inches to feet, here’s the trick,
Write down 12 inches = 1 foot, real quick,
Flip the fraction so feet’s on top,
Cancel inches, and you’re ready to hop!

Pre-Chorus:
First, write what you know at the top,
Then find the unit you want and don’t stop,
Flip the conversion if it’s upside-down,
Now you’re ready, let’s break it down!

Chorus:
Flip it, flip it, cancel, cancel,
Multiply to find the answer,
What’s left is what you need,
Unit analysis, it’s guaranteed!

Bridge:
R-U-M, Remember the plan:
R for "Right unit on top,"
U for "Use what you got,"
M for "Multiply, then stop!"
R-U-M, that’s the jam!

Chorus:
Flip it, flip it, cancel, cancel,
Multiply to find the answer,
What’s left is what you need,
Unit analysis, it’s guaranteed!

Outro:
Now you’ve got the rhythm and the rhyme,
Dimensional analysis, every time,
Flip, cancel, and multiply,
You’ve got this down, so give it a try!

The Mars Climate Orbiter Mishap

Mars vehicle crashing into the surface of the planet — **Figure 2.** *The Mars Climate Orbiter failed due to misuse of units. 10 months of work and $125M were lost due to a simple error.*

In 1999, NASA's Mars Climate Orbiter was on a mission to study the Red Planet's weather and climate. However, the mission ended in failure due to a simple yet catastrophic mistake: a mix-up in units. One team used metric units (newtons) while another used imperial units (pounds-force) in the spacecraft's navigation system. This inconsistency caused the orbiter to descend too low into the Martian atmosphere, leading to its destruction. This real-world example highlights the critical importance of using and interpreting units correctly in complex, multi-step problems. A small error in unit conversion can lead to disastrous outcomes, even in the most advanced scientific endeavors.

Reflection Questions

When performing dimensional analysis, why is it important to cancel out the units systematically? How does this process ensure that your final answer is correct?
Think about the Mars Climate Orbiter mishap. Why is it crucial to use consistent units in calculations? What can we learn from this incident about the importance of precision in scientific work?

What's next?

Complete the practice challenge

Continue to the practice section.

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