Physical Modeling and Measurement
  Necessary / Sufficient
 A condition can be necessary for your assumption to be true, but it may not be sufficient to prove that it is in fact true.
 Multiplication: For a number to be divisible by 4, it is necessary for that number to be even, but evenness is not sufficient to prove that the number is divisible by 4.
 Addition: When you add two positive integers greater than zero, it is necessary that the resulting number is greater than either original numbers. However, just because the sum is greater is not sufficient to say you have the right answer. However, if the sum is smaller, it is sufficient to say you have the wrong answer.
 Multiplying Fractions: When you multiply two proper fractions together, it is necessary that the resulting number is smaller than either of the original fractions. In fact, the resulting number is only "some of" the smaller fraction, so it must be smaller than the smaller fraction.
 Activity: Water Jar Activity
 Consistent / Conclusive
 A condition can be consistent with the observed situation, but it may not be conclusive evidence that it is the cause of the observation.
 Life Example: Your little brother running around may be consistent with the broken lamp, but it may not be conclusive evidence that he was the only possible cause for the breakage.
 Example: There is evidence that people who drink more coffee get less sleep. Coffee drinking is thus consistent with lack of sleep. However, people could be drinking more coffee because they need to stay up anyway, so the evidence is not conclusive to prove that coffee drinking keeps people awake.
 In the News: A scientific finding is not "proof" of something, but evidence that was found to be consistent with the observation.
 Uncertainty / Certainty
 Observations always come with a certain level of uncertainty. When analyzing your evidence, you must take this uncertainty into account. When you give evidence of something, you know it to a reasonable level of scientific certainty.
 Ambiguity: Uncertainty can arise from information that is not completely clear.
 3 + 2 * 6 ... answer could be 15 or 30 depending on the order in which you do the calculations. Parentheses take away ambiguity.
 "Count from ten to one backwards." ... Ambiguous statement, could mean to count "1, 2, 3, 4..." or to count "net, enin, thgie, neves..." or to turn around and count "10, 9, 8, 7...".
 Randomness: Uncertainty can arise from factors that affect your results and cause random variations.
 Spin a coin on a desk, try to get the coin to hit an object at the end of the desk. Use the same set of steps to spin the coin, and observe the direction the coin goes. It won't always spin the same way due to variations you can't measure.
 Systematic Error: Uncertainty can arise from the architecture you are using for your calculations. Even if your algorithm is correct, you can end up with round off error.
 Definition
 Sometimes you know something is true because it is defined to be so.
 Geometry:
 How many angles does a triangle have? 3. How do you know? That's the definition of a triangle.
 How many degrees are there in a circle? 360. How do you know? That's the definition of a degree and a circle.
 Convention
 Sometimes you know something is true because people have agreed to do things a certain way.
 Convention can reduce uncertainty by ensuring a consistent way of doing things.
 Order of Operations: We know that 3 + 2 * 6 should be 15 because we have defined that multiplication is done before addition.
 Algorithm: When a particular set of steps is followed every time in the absence of randomness, you will have the same answer.
 Deterministic Card Tricks
 Construction
 Sometimes you know something is true because you can actually build or "construct" something that proves it's true.
 Geometry: How do you know that all of the angles of a triangle add up to make a straight line (180 degrees)? You can cut up the triangle and put the angles next to each other. Or, you can use the Triangle Sum of Angles Activity.
 Measurement
 Sometimes you know something is true because you can measure the results.
 Geometry: How do you know that all of the angles of a triangle add up to make a straight line (180 degrees)? You can measure the angles and add them together.
 Evidence
 When you find something that is consistent with your observations, it can be called evidence.
 This can be related back to the consistent / conclusive argument.
 Proof
 When you have conclusive evidence, it can be considered proof that your hypothesis is true.
 The word proof is often used incorrectly when people think it is synonymous with evidence.
 It is easier to give a proof for something in mathematics, through measurement, construction, and convention. In science there is the challenge of uncertainty.
