1. Introduction to Quantitative Reasoning Unit 0 Applications of Advanced Mathematics

2. Learning Objective: Understanding If-Then Logic  I can break down an argument into a series of conditional statements and identify their hypotheses and conclusions.  I can describe which situations are allowed/not allowed according to a conditional statement and can identify situations that show a conditional is false.  I can identify hidden assumptions in an argument.

3. Learning Objective: Working with Real Numbers and Units  I can round numbers to specified place values.  I can find the smallest and largest possible values of rounded numbers.  I can change a quantity from one unit to another through a series of conversions.  I can explain why using one unit is better than another in certain situations.

4. Learning Objective: Understanding the Role of “Averages”  I can give examples where a single number is used to summarize data.  I can explain why averages can sometimes be misleading.

5. What in the World?!?!

6. Real World Example: Thin Films in Technology

7. Real World Example: Deposition Chamber

8. Learning Objective: Understanding If-Then Logic CONDITIONAL  If the sensor measures 0.15 Å/s and film is deposited for 2000 seconds, then the film will be 300 Å thick. HIDDEN ASSUMPTION  The film on the sensor and the film on the substrate grow at the same rate. How might this statement turn out to be FALSE?

9. Real World Example: Some Authentic Data SAMPLE NOMINAL THICKNESS FINAL THICKNESS MT10 509 Å59 nm MT2 353 Å41 nm MT5 354 Å41 nm MT7 360 Å42 nm MT3 339 Å40. nm AP1 8550 Å998 nm Data courtesy of Antony Jans, Clemens Research Group, Department of Materials Science and Engineering, Stanford University.

10. THINK! Does our “hidden assumption” appear to be true?  The film on the sensor and the film on the substrate grow at the same rate. SAMPLE NOMINAL THICKNESS FINAL THICKNESS MT10509 Å59 nm MT2353 Å41 nm MT5354 Å41 nm MT7360 Å42 nm MT3339 Å40. nm AP18550 Å998 nm

11. Learning Objective: Working with Real Numbers and Units SAMPLE NOMINAL THICKNESS FINAL THICKNESS MT10 509 Å59 nm MT2 353 Å41 nm MT5 354 Å41 nm MT7 360 Å42 nm MT3 339 Å40. nm AP1 8550 Å998 nm

12. THINK! Does our “hidden assumption” appear to be true?  If the film grows at x Angstroms/second on the sensor, then the film also grows at x Angstroms/second on the substrate. Is the sensor data still useful? What is the relationship between nominal and final thickness? Can the nominal thickness be used to predict the final thickness?

13. TRANSFORM… SAMPLE NOMINAL THICKNESS FINAL THICKNESS FINAL − NOMINAL MT10 51 nm59 nm8 nm MT2 35 nm41 nm6 nm MT5 35 nm41 nm6 nm MT7 36 nm42 nm6 nm MT3 34 nm40. nm6 nm AP1 855 nm998 nm143 nm

14. TRANSFORM… SAMPLE NOMINAL THICKNESS FINAL THICKNESS FINAL / NOMINAL MT10 51 nm59 nm1.16 MT2 35 nm41 nm1.17 MT5 35 nm41 nm1.17 MT7 36 nm42 nm1.17 MT3 34 nm40. nm1.18 AP1 855 nm998 nm1.17

15. TRANSFORM…

17. Learning Objective: Understanding the Role of “Averages” In the most general sense, the term “average” can be used to describe any number which summarizes data.  How can we summarize the relationship between nominal and final thickness with a single number?

18. USE FINAL/NOMINAL?  The ratio of final to nominal thickness ranged between 1.16 and 1.18.  The average (mean) for the six samples is (1.16 + 1.17 + 1.17 + 1.17 + 1.18 + 1.17)/6 which is equal to 1.17.  So final thickness --------------------------- = 1.17 nominal thickness

19. USE BEST FIT LINE?

20. BACK INTO CONTEXT!  Either method leads us to the prediction formula final thickness = 1.17 * nominal thickness  The number 1.17 is an average which summarizes the relationship between what was measured by the sensor and the actual thickness of the film.  This number is called the TOOLING FACTOR. The tooling factor for each deposition set-up can be input into the device so that sensor readings provide more accurate thicknesses.

21. Yes, this is an example of a FUDGE FACTOR