1. Warm up
Biff the Builder is having trouble with the current house that he is working on. Explain the causes of Biff’s problems.

2. Significant Figures, Uncertainty, Accuracy and Precision
Focus Question: How are significant figures useful?

3. = 5,000,000 has six zeroes. 5 x 106 also has six zeroes.
What is Scientific Notation? For example:
Scientific Notation is a way of compressing large or small numbers so that they are easier to write. This is done by representing zeroes at one end of a number as powers of ten.
5,000,000 has six zeroes. =
5 x 106 also has six zeroes.

4. How do we put numbers in scientific notation?
First we move the decimal behind the front non-zero digit. Ex: for 403,000. we write 4.03 or for we write 5.8 Then we count the number of spaces the decimal moves left or right and drop any zeroes on the end. 403,000→ left →5.8 4 right

5. What is the coefficient?
The coefficient is the left hand side of scientific notation. This represents the numbers that we consider important. 403,000→ →5.8 For very long numbers without any zeroes on the end we often round the number before putting it in the coefficient.

6. What is the exponent?
The exponent is the right hand side of scientific notation and is written as ten to a power determined by the number of places the decimal moved.
403,000→ left 4.03 x 105
→ right 5.8 x 10-4
If the decimal moved left, the exponent is positive. If the decimal moved right the exponent is negative.

7. What is Accuracy?
A measurement’s accuracy is how close it is to a value that is accepted as being true.
Inaccurate measurements make our calculations and thus our predictions unreliable.
We typically refer to inaccuracy of a measurement as the error value of the measurement.

8. Ea = |X – A| How do we calculate error values?
Absolute Error is the difference between a measurement we make and the accepted value.
Ea = |X – A|
X -the measured value
A -the accepted value
Ea -the absolute error.

9. Er = (Ea/A) x 100 What is Relative Error?
The relative error expresses the absolute error as a percentage of the accepted value.
Er = (Ea/A) x 100
A -the accepted value
Ea -the absolute error.
Er -the relative error.

10. What is Precision?
The precision of a measurement reflects how close it is to the average of our measurements.
Imprecision in our measurements indicates difficulty in reproducing the outcome of an experiment.
We refer to how imprecise a measurement is as the measurement’s deviation from the set of data.

11. Da = |X – M| How do we calculate deviation?
The absolute deviation is the difference between the measurement and the average of all the measurements.
Da = |X – M|
X -the measured value
M -the mean value
Da -the absolute deviation.

12. Dr = (Da/M) x 100 What is Relative Deviation?
The relative deviation expresses absolute deviation as a percentage of the mean value.
Dr = (Da/M) x 100
Da- Absolute deviation
M- Mean measurement
Dr- Relative deviation

13. What can be done about inaccurate or imprecise data?
Results that are far from the accepted value lack accuracy.
Problems with accuracy are usually addressed by eliminating or identifying sources of error.
Results that vary widely from other measurements lack precision.
Problems with precision are usually addressed by standardizing the experimental procedures.

14. Accuracy and Precision

15. What are significant figures and uncertainty?
Significant figures (or significant digits) represent how small of a measurement we are able to make before it loses accuracy. This is because our instruments usually have a minimum useful measurement they can make.
We can't tell the exact amount of water so we round to 50 ml and have 1 significant figure.

16. How do we determine significant figures?
Because numbers are rounded to end with zeroes as placeholders, non- zeroes are always treated as significant while zeroes often aren’t.
Non-zero digits each count as one significant figure. so
594 has 3 significant figures: one for each digit in the number.

17. Why are some zeroes not considered significant?
Zeroes that result from rounding are not significant. If we round this number to the tens we have: 594→590 sig figs= 2 Rounding to the hundreds we get: 594→600 sig figs= 1

18. The Atlantic- Pacific Rule

19. When else do zeroes count?
When they are in the coefficient of scientific notation (before the “x”)
5.30 x 106

20. Adding and subtracting significant figures
1.Line up the decimal points of both numbers. 2. Calculate and round to the common number.
412.8
=449.06
=449.1
26.1
=17.607
=17.6

21. The answer will never have more significant figures than the starting numbers.
The 27 only has two sig figs so our answer is rounded to 2.
480. has 3 sig figs but only has 2, so we have 2.
Multiplying and dividing significant figures.
243.5
x 27
=6574.5
=6600
480.
¸ 13
= 36.9
= 37 21

22. Don’t be fooled by sig figs!
Sometimes this can lead to some funny looking answers.

23. What is uncertainty?
Uncertainty represents the limit to how well we can estimate a number. We assign uncertainty based on the smallest unit we can accurately measure with our equipment. Here we have a cylinder that measures in 10’s of milliLiters, but we can estimate the volume to individual milliliters.

24. How do we include uncertainty values into a measurement?
We know the water in the cylinder has a volume between fifty and fifty-five mL and estimate it to be 53 mL but we might be off by 2 mL more or less which we call the uncertainty, ∆x. 53 ±2 mL We can also represent this uncertainty as a percentage of the value. 2mL / 53mL = 3.8% 53 mL ±4%

25. How do we determine the uncertainty of measurements in the first place?
Uncertainty is determined by making multiple measurements and averaging the maximum and minimum values. ∆𝑥 = 𝑥 𝑚𝑎𝑥 − 𝑥 𝑚𝑖𝑛 2

26. Whiteboard Session
A student takes a series of measurements of the speed of a ball as it falls from a shelf. If the student gets values of 4.65 m/s, 4.61 m/s, m/s and 4.63 m/s and 4.62 m/s. Express the mean value with the appropriate uncertainty, ∆x. Then express the same value but with the uncertainty expressed as a percentage.

27. Warm Up
Biff the Builder is having trouble with the current house that he is working on. Using the vocabulary terms covered in class, explain what is the source of Biff’s problem?