Answers, and why this lesson is worth teaching before the Higher Level expectation algebra rather than after it.
The two sliders do something the rest of this sub-topic does not: they set up Var(aX + b) = a²Var(X) a few lessons early, without any algebra.
Students meeting that result cold are surprised the b disappears, and spend the HL lesson arguing with it. Students who have pushed the top slider and watched the standard deviation refuse to move already know it is true, and only need the notation. If you teach one of these four lessons before starting HL content, make it this one.
Move the add slider and ask what happened to the gaps between the points. Nothing. Then move the multiply slider and ask again. Everything stretched. Adding moves, multiplying stretches, and spread only notices the one that stretches.
| Question | Answer |
|---|---|
| 1. Add 8 to every mark | 12. The standard deviation is unchanged. |
| 2. Multiply by 1.5 | 81. The mean scales: 54 × 1.5. |
| 3. Skewed data | B, the median, because extreme values do not move it. |
Worked example on the student page: the set 4, 6, 8, 10, 12 has mean 8, variance 8 and standard deviation about 2.83.
| They enter | What it means |
|---|---|
| 20 (Q1) | The one that matters. They added 8 to the standard deviation, which means they are treating spread like position. The page sends them back to the slider rather than restating the rule, because watching it is more convincing than being told. |
| 62 (Q1) | They gave the new mean. Method is sound, they answered a different question. |
| 55.5 (Q2) | Added 1.5 instead of multiplying. Usually a student working too fast rather than one who does not know. |
| 18 (Q2) | They gave the new standard deviation. Again the right work, the wrong question asked. |
| "The mean" (Q3) | They justified it with "it uses every value", which is true and is exactly the problem. Worth saying out loud: the property that makes the mean good normally is the property that makes it bad here. |
Notice how many of these are reading errors rather than misconceptions. Worth telling the class: on transformation questions, write down what is being asked for before you calculate anything.
Transformation questions are cheap marks, lost constantly. One line stating which transformation is happening prevents nearly all of it.
Standard deviation is never negative. If a student produces one, the useful response is not to correct the arithmetic but to ask what quantity they rooted.
Justifying a choice of average needs a feature of the data. "The median, because the data is skewed" earns the mark. "The median, because it is more accurate" does not, and is what most of them will write first.
| What is happening | |
|---|---|
| 1 | Both sliders, and the question about the gaps. Do not state the rule before they have seen it. |
| 2 | Which average and when. Quick, because they have met it before; this is revision. |
| 3 | The standard deviation worked example by hand. Worth doing once even though the calculator does it, because it explains the squared units. |
| 4 | Questions 1 and 2. Expect the reading errors above and name them as reading errors. |
| 5 | Question 3, then forward-link to Var(aX + b) explicitly. Tell them they have already proved it. |
Do not introduce Var(aX + b) = a²Var(X) as a formula to memorise in this lesson. Let the slider be the whole argument. The notation can wait, and it lands far better when they already believe the result.
Do not say "the mean is affected by outliers" as the only reason to prefer a median. It is true and it is thin. Push for which direction and by roughly how much, which is what an examiner is actually after.
Three tiers, ramping the way practice should: the method on its own, then the method inside something real, then a challenge. Set the tier the class in front of you needs rather than one undifferentiated sheet. Answers are given so these can go straight onto a board.
The method on its own, with friendly numbers. Set these first and move on quickly once they are secure.
The same skill inside a real situation, where the first job is working out what is being asked.
Reasoning, working backwards, or spotting an error. These are where the top grades are decided.
Works on a phone. Nothing a student types is saved or sent anywhere. The standard deviation band drawn on the number line is one standard deviation each side of the mean, so it is a useful thing to point at when the formal definition arrives.