Math for APES students

1. Math for APES students Using the SI System, Scientific Notation and the Factor-Label Method of Conversions (AKA Dimensional Analysis)

2. The SI/metric system • The International System of units • Uses prefixes and base units. • The prefixes are here  • And the base units you need to know are: • Meter (length) • Kilogram (mass) • Joule (energy) • Second (time)

3. Converting within the SI system • Use the stairs to convert 1.5 hm to mm. • How about 1.5 cJ to kJ?

4. Using scientific notation • In science, numbers are often VERY LARGE, or very small. To deal with that, you should consider using scientific notation. • Students new to sci notation tend to think it’s difficult, but that’s because they’ve memorized the rules the wrong way, so they only get the right answer half the time. • If you aren’t excellent at scientific notation, take a second and forget what you’ve learned about it. Trust me, my way’s easier than what you learned.

5. Putting numbers into scientific notation • Step 1: find the decimal. If there’s not one, put one at the right-hand end of the number. • Step 2: Move the decimal so that there is only ONE non-zero number to the left of the decimal. • Step 3: Re-write the number with the decimal in the new spot. If there are zeros to the left of your decimal now (leading zeros), leave them off. If the digit farthest to the right is a zero, leave it off, along with any other “trailing zeros” (at the end). • Step 4: Add “x10” to the right of what you’ve written now. • Step5: Look at the original number. Is it greater than 1 (would you be happy if someone gave you that many dollars)? If so, write a + sign after the 10. If not, write a – sign. • Step 6: In step 2, how many places did you move the decimal? Write that number after the + or – sign.

6. But, I thought you said it would be easy…

7. What about the other way?

8. Math with scientific notation • Multiply: • Multiply the big numbers, add the exponents • Divide: • Divide the big numbers, subtract the exponents • Add/subtract: • If exponents match, add/subtract the big numbers, keep the exponents. • If exponents don’t match, change them so they do, or write the numbers out in normal notation

9. Math with scientific notation practice:

10. Let's start with a sample problem:How many centimeters are in 35.2 miles?Given: 1 cm= 0.394 in1ft = 12 in1mi = 5280 ft Ways of converting between units:

11. Ways of converting between units: • The stair step method – Works great for conversions within the SI system, but ONLY if you know how (remember 3 steps between k, M, G, T and m, μ, n, p) and ONLY applies within SI/metric systems. • Guessing and/or logic – Works well if the problem isn’t too complex and you’re good at it. • Factor-label method – Works right every time, if you know the steps (and follow them).

12. The factor-label method • Warning: This starts out kind of confusing. Be patient. • Comments from many past students: • I didn’t get it at first, but now it’s SO easy! • That factor-labeling thing really came in handy! • I wish I had paid attention when you first explained factor-labeling!

13. You MUST learn this method. • Because: • You will use it if you ever take another chemistry, physics or engineering class ever in your life. • It’s useful outside of science too. • It makes you look smart • I said so! 

14. Steps in the factor-labeling method • Start with the number in the problem with its units. Make it into a fraction (put it over 1).

15. Steps in the factor-labeling method • Multiply by a new fraction so that: • The unit that you don’t want (in the first fraction) cancels out AND • The top of the second fraction is equal to the bottom of the second fraction.

16. Steps in the factor-labeling method • Cancel out the units and check: • If you have the units you want, go to step 4. • If you don’t have the units you want, repeat steps 2 and 3.

17. Steps in the factor-labeling method • Solve: • Multiply across the tops of the fractions and write it down. • Multiply across the bottoms of the fractions and write it down. • Divide: top divided by bottom.

18. Steps in the factor-labeling method • Double check: • Check for significant figures (the APES exam is lenient with this, as long as you show your work. Keep three non-zero digits and round the rest.) • Don’t leave the number naked! (Add the correct unit!) • Check to see if the answer MAKES SENSE.

19. Five steps • No really, that’s all!

20. Sample problem • Since easy problems can be solved other ways, let’s try a complex problem. • Please trust that I will give you the occasional problem that is too complex for you to use logic on. • Use Factor-Labeling.

21. Sample problem • How many centimeters are in 156 miles? • Known conversion factors (use these even if you have access to others, for today): • .394 in = 1 cm • 12 in = 1 ft • 1 mi = 5280 ft

22. 1. Start with the number in the problem with its units. Make it into a fraction (put it over 1). • How many centimeters are in 156 miles? 156 mi 1

23. Multiply by a new fraction so that:The unit that you don’t want (in the first fraction) cancels out AND The top of the second fraction is equal to the bottom. 156 mi x 1 5280 ft 1 mi

24. Cancel out the units and check:If you have the units you want, go to step 4.In you don’t have the units you want, repeat steps 2 and 3. 156 mi x 1 5280 ft 1 mi We do not have the units we want (centimeters). Repeat steps 2 and 3.

25. Multiply by a new fraction so that:The unit that you don’t want (in the first fraction) cancels out ANDThe top of the second fraction is equal to the bottom. 156 mi x 1 12 in 1 cm 5280 ft 1 mi 1 ft .394 in This is the unit we want, so we go on to the last step…

26. Step 5 – SOLVE it! 156 1 12 1 cm 5280 1 1 .394 156 x 5280 x 12 = .394 25,086,700.5 cm

27. So how about this?How many XYZs are in 35.2 LMNs?Given: 18.2 ABC= 1RST1RST = 0.332 DEF128.65 LMN = 1 ABC1DEF = 0.997 XYZ