class: center, middle, inverse, title-slide .title[ # Lesson 10: Understand, Solve, and Explain ] .author[ ### Fei Ye ] .date[ ### May, 2024 ] --- class: center middle <div> <style type="text/css">.xaringan-extra-logo { width: 110px; height: 25px; z-index: 0; background-image: url(assets/nc_sa.png); background-size: contain; background-repeat: no-repeat; position: absolute; bottom:12px;left:1%; } </style> <script>(function () { let tries = 0 function addLogo () { if (typeof slideshow === 'undefined') { tries += 1 if (tries < 10) { setTimeout(addLogo, 100) } } else { document.querySelectorAll('.remark-slide-content:not(.title-slide):not(.hide_logo)') .forEach(function (slide) { const logo = document.createElement('div') logo.classList = 'xaringan-extra-logo' logo.href = null slide.appendChild(logo) }) } } document.addEventListener('DOMContentLoaded', addLogo) })()</script> </div> <div style="display:none"> `$$\require{cancel}$$` </div> ## Unit 2A： Understand, Solve, and Explain *Primary Source:* PPT for the book "Using & Understanding Mathematics". --- ## Three-Step Problem Solving: Understand-Solve-Explain (U-S-E) (1 of 3) **Step 1:** UNDERSTAND the problem. - Think about what the problem asks you to do. - Draw a picture or diagram to help make sense of the problem. - Ask yourself what the solution should look like. - Try to map a path (either mentally or in writing) that will lead you from your understanding of the problem to its solution. - Continually revisit your understanding of the problem. --- ## Three-Step Problem Solving: Understand-Solve-Explain (U-S-E) (2 of 3) **Step 2:** SOLVE the problem. - Obtain any needed information or data. For multi-step problems, be sure to keep an organized, neatly written record of your work. - Double-check each step as you work to avoid carrying errors through to the end of your solution. - Constantly reevaluate your plan as you work. --- ## Three-Step Problem Solving: Understand-Solve-Explain (U-S-E) (3 of 3) **Step 3:** EXPLAIN your result. - Be sure that your result makes sense. - Recheck your calculations once more or, even better, find an independent way to check your result. - Identify and understand any potential sources of uncertainty in your result. If you made assumptions, were they reasonable? - Write your solution clearly and concisely, using complete sentences to make sure the context and meaning are clear. --- ## Units The **units** of a quantity describe what that quantity measures or counts. **Unit analysis** is the process of working with units to help solve problems. --- ## Example: Cost of Farm Land You are buying 30 acres of farm land at $12,000 per acre. What is the total cost? -- **Solution:** 1. __Understand.__ The question asks about total cost, and one of the given units is dollars, so we expect an answer in dollars. 2. **Solve.** We carry out the calculation; note that the price is given in dollars per acre, so we write the division by acres in fraction form. That allows "acres" to cancel, leaving the final answer in dollars: $$ 30\, \cancel{\text{ac}}\times \dfrac{\$12000}{1\, \cancel{\text{ac}}}=\$360000. $$ 3. **Explain.** We have found that purchasing 30 acres of farmland at a price of $12,000 per acre will cost a total of $360,000. --- ## Example: Running Distance Show operations and units clearly to answer the question: What is the total distance traveled when you run 7 laps around a 400-meter track? -- **Solution:** We could express the same idea as "7 laps of a 400-meter track." Therefore, this problem requires multiplying 7 laps by the 400 meters you run per lap: $$ 7\, \cancel{\text{laps}}\times \dfrac{400\, \text{m}}{1\, \cancel{\text{lap}}}=2800\, \text{m}. $$ --- ## Key Words and Operations with Units | Key word or symbol | Operation |Example| |---|---|---| | per | Division | Read miles `\(\div\)` hours as "miles per hour."| |of or hyphen | Multiplication| Read kilowatts `\(\times\)` hours as "kilowatt-hours."| |square|Raising to second power|Read ft `\(\times\)` ft, or `\(\text{ft}^2\)`, as "square feet" or "feet squared."| |cube or cubic|Raising to third power|Read ft `\(\times\)` ft `\(\times\)` ft, or `\(\text{ft}^3\)`, as "cubic feet" or "feet cubed."| --- ## Conversion Factors A **conversion factor** is a statement of equality that is used to convert between units. Some conversion factors: $$ 1\, \text{foot}=12\, \text{inches}\quad\text{or}\quad \dfrac{12\, \text{inches}}{1\, \text{foot}}=1\quad\text{or}\quad \dfrac{1\, \text{foot}}{12\, \text{inches}}=1 $$ $$ 1\, \text{day}=24\, \text{hours}\quad\text{or}\quad \dfrac{24\, \text{hours}}{1\, \text{day}}=1\quad\text{or}\quad \dfrac{1\, \text{day}}{24\, \text{hours}}=1 $$ --- ## Example: Unit Conversions Convert a distance of 9 feet into inches. **Solution:** $$ 9\, \cancel{\text{ft}}=9\, \cancel{\text{ft}}\times \dfrac{12\, \text{in}}{1\, \cancel{\text{ft}}}=108\, \text{in}. $$ --- ## Example: Seconds in Days  **Solution:** $$ 94\, \text{d}=94\,\cancel{\text{d}}\times \dfrac{24\, \cancel{\text{h}}}{1\,\cancel{\text{d}}}\times \dfrac{60\, \cancel{\text{min}}}{1\,\cancel{\text{h}}}\times \dfrac{60\, {\text{s}}}{1\,\cancel{\text{min}}}=8121600\, \text{s}. $$ --- ## Conversions with units raised to powers .pull-left[  ] .pull-right[ $$ `\begin{aligned} 1\, \text{yd} =& 3\,\text{ft}\\ 1\, \text{yd}^2 =& 1\, \text{yd}\times 1\, \text{yd}\\ =&3\, \text{ft} \times 3\, \text{ft}\\ =& 9\, \text{ft}^2 \end{aligned}` $$ ] --- ## Example: Cubic yards of soil How many cubic yards of soil are needed to fill a planter that is 20 feet long by 3 feet wide by 4 feet tall? **Solution:** The volume is `\(20\, \text{ft}\times 3\, \text{ft}\times 4\, \text{ft} = 240\, \text{ft}^3\)` `\(1\, \text{yd} = 3\, \text{ft}\)`, so `\(1\, \text{yd}^3 = (3\, \text{ft})^3 = 27\, \text{ft}^3\)` Therefore, $$ 240\, \text{ft}^3=240\, \cancel{\text{ft}^3} \times \dfrac{1\, \text{yd}^3}{27\,\cancel{\text{ft}^3}}\approx 8.9\, \text{yd}^3. $$ --- ## Example: Furlong vs mile The length of the Kentucky Derby horse race is 10 furlongs. How long is the race in miles? **Solution:** The conversion factor between furlong and mile is $$ 1\,\text{furlong} = \dfrac{1}{8}\,\text{mi}=0.125\, \text{mi} $$ Using the quotient form of the conversion factor, 10 furlongs is $$ 10\, \text{fur}=10\, \cancel{\text{fur}}\times\dfrac{0.125\,\text{mi}}{1\,\cancel{\text{fur}}}=1.25\,\text{mi}. $$ --- ## Metric System The international metric system was invented in France late in the 18th century for two primary reasons: (1) to replace many customary units with just a few basic units and (2) to simplify conversions through use of a decimal (base 10) system. The basic units of length, mass, time, and volume in the metric system are - the meter for length, abbreviated m - the kilogram for mass, abbreviated kg - the second for time, abbreviated s - the liter for volume, abbreviated L --- ## Common Metric Prefixes  --- ## Example: Centimeters to Meters Convert 2759 centimeters to meters. -- **Solution:** The conversion factor is `\(1\,\text{m}= 100\, \text{cm}\)`.Therefore, 2759 centimeters is the same as $$ 2759\, \text{cm}=2759\, \cancel{\text{cm}}\times\dfrac{1\,\text{m}}{100\,\cancel{\text{cm}}}=27.59\,\text{m}. $$ --- ## Example: Miles to Kilometers The marathon running race is about 26.2 miles. About how far is it in kilometers? **Solution:** The conversion factor between miles and kilometer is `$$1\, \text{mi} = 1.6093\, \text{km}.$$` We use the conversion in the form with miles in the denominator to find $$ 26.2\, \text{mi}=26.2\, \cancel{\text{mi}}\times\dfrac{1.6093\, \text{km}}{1\, \cancel{\text{mi}}}\approx 42.2\, \text{km}. $$ --- ## Temperature Units (1 of 2) - The **Fahrenheit** scale, commonly used in the United States, is defined so water freezes at 32 `\(^\circ\)` F and boils at 212 `\(^\circ\)` F. - The rest of the world uses the **Celsius** scale, which places the freezing point of water at 0 `\(^\circ\)` C and the boiling point at 100 `\(^\circ\)` C. - In science, we use the Kelvin scale, which is the same as the Celsius scale except for its zero point, which corresponds to -273.15 `\(^\circ\)` C. A temperature of 0 K is known as absolute zero , because it is the coldest possible temperature. (The degree symbol `\(^\circ\)` is not used on the Kelvin scale.) --- ## Temperature Units (2 of 2)  --- ## Temperature Conversions  --- ## Example: Body Temperature Average human body temperature is 98.6 `\(^\circ\)`F. What is it in Celsius and Kelvin? **Solution:** Convert from Fahrenheit to Celsius by subtracting 32 and then dividing by 1.8: $$ C=\dfrac{F-32}{1.8}=\dfrac{98.6-32}{1.8}=37.0^\circ \text{C}. $$ We find the Kelvin equivalent by adding 273.15 to the Celsius temperature: $$ K = C + 273.15 = 37 + 273.15 = 310.15 K. $$ --- ## Currency Conversions Converting between currencies is a unit conversion problem in which the conversion factors are known as the **exchange rates**. The following tables shows a typical table of currency exchange rates: - Use dollars per foreign when you buy dollars. - Use foreign per dollars when you buy foreign.  --- ## Example: Euros to Dollars At a French department store, the price for a pair of jeans is 45 euros. Suppose the exchange rate is 1 euro= $1.320. What is the price in U.S. dollars? **Solution:** As usual, we can write this conversion factor in two other equivalent forms: $$ \dfrac{1\, \text{euro}}{\$1.320}=1\qquad\text{and}\qquad \dfrac{\$1.320}{1\, \text{euro}}=1. $$ Then $$ 45\, \text{euros}=45\, \cancel{\text{euros}}\times\dfrac{\$1.320}{1\,\cancel{\text{euro}}}\approx \$59.4. $$ --- ## Example: Canadian Dollars to US Dollars A gas station in Canada sells gasoline for CAD 1.34 per liter. (CAD is an abbreviation for Canadian dollars.) What is the price in dollars per gallon? Suppose the currency exchange rate is $1.005 per CAD. **Solution:** We use a chain of conversions to convert from CAD to dollars and then from liters to gallons. Note that the conversion factor between liters and gallon is 3.785 liters per gallon. A gas station in Canada sells gasoline for CAD 1.34 per liter. Then the price in US dollars is $$ \dfrac{1.34\, \cancel{\text{CAD}}}{1\,\cancel{\text{liter}}}\times\dfrac{\$1.005}{1\,\cancel{\text{CAD}}}\times\dfrac{3.785\,\cancel{\text{liters}}}{1\,\text{gallon}}\approx\dfrac{\$5.10}{1\,\text{gallon}}. $$ --- ## Practice: Length Conversion Use the conversion factors `\(1\,\text{ft}=0.3048\, \text{m} = 1/3\,\text{yd} = 12\, \text{in}\)` to answer the following questions 1. Convert 45 feet into inches. 2. Convert 45 feet into yards. 3. Convert 45 feet into meters. --- ## Practice: Area conversion 1. A field is 30 yards long and 40 yards wide. Find its area in square feet. 2. A field is 30 yards long and 40 yards wide. Find its area in square meters. --- ## Practice: Volume conversion Use the conversion factors `\(1\,\text{gal}\approx 0.3.785\, \text{L}\)` and `\(1 L=0.001\,\text{m}^3\)` to answer the following questions. 1. Convert 12 liters into gallons. 2. Convert 12 liters into cubic meters. 3. Convert 12 liters into cubic centimeters. --- ## Practice: Temperature conversion Use the conversion formulas `\(C=\frac{5}{9}(F-32)\)` to answer question. 1. Convert 16 degrees Celsius into degrees Fahrenheit. 2. Convert 16 degree Fahrenheit into degrees Celsius --- ## Practice: Currency exchange Suppose the exchange rate between USD and EUR is 1 USD = 0.85 EUR today. 1. A liter of gas costs 1.23 Euros in Paris, how much would it cost in US dollars? 2. A cup of coffee from a local restaurant in NYC costs $3.56, how much would it cost in Euros. --- ## Practice: Mileage conversion .iframecontainer[ <iframe src="https://www.myopenmath.com/embedq2.php?id=33004&amp;seed=2021&amp;showansafter" width="100%" height="400px" data-external="1"></iframe> ] .footmark[ <a href="https://www.myopenmath.com/embedq2.php?id=33004&seed=2021&showansafter" target="_blank">Click here to open the practice in a new window</a> ]