At a glance. One lesson of about 60 minutes. Students find the effect of adding a constant and of multiplying by a constant on each measure, apply it to unit conversions, work backwards, and prove the rule for σ.
Calculator: GDC optional. List arithmetic makes the pattern appear quickly, and the proof is by hand. The optional section shows both lists side by side and warns that a display is evidence, not a proof.
What students will learn
Know
- The mean and median transform with the data: kμ + c.
- The range, IQR and σ are multiplied by |k|.
- Adding a constant leaves every spread unchanged.
Understand
- Centre is a position and spread is a distance.
- The proof: each deviation is multiplied by k, so σ is multiplied by |k|.
- A negative multiplier reflects the data but cannot make a spread negative.
Do
- Predict and test the effect of kx + c.
- Convert a mean and σ to new units.
- Work back from a transformed σ and prove the result.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: applying kx + c to the mean and σ, including a negative k (questions 1 to 7).
- Criterion B. Investigating patterns: the investigation. Students vary k and c in a table, describe the pattern and justify it with the formula.
- Criterion C. Communicating: a results table with a column for each measure and a precise statement of the rule.
- Criterion D. Applying mathematics in real-life contexts: converting between units, such as centimetres to millimetres and Celsius to Fahrenheit.
The one thing to do with the figure
Have students write a prediction table before touching a slider. Rows: add 5, multiply by 2, multiply by −2. Columns: mean, median, range, IQR, σ. Test one row at a time, and have students correct the entries they got wrong.
Then ask what stays the same when k = −1 and c = 0: the spreads. The data are the same shape, reflected.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 5 min | Hook: the quiz marks. Collect predictions in a table. |
| 12 min | The figure with predictions; the key idea; settle the hook. |
| 10 min | Warning card and worked examples (including the proof); pairs finish the faded example. |
| 5 min | Question 1, the check. If many choose C or D, return to which part of kx + c reaches the spread. |
| 18 min | Your turn, Levels 1 to 3, at each student's own pace. |
| 5 min | Something to think about, then the exit ticket, question 8. |
The answers
| Warm-up | 6.25; 8; 3x² − 12x + 12. |
|---|---|
| Hook | Adding: mean and median change, spreads do not. Doubling: mean and median double, and every spread doubles. |
| Worked examples | Mean 14 and σ 7.35; 287.2 K and 3.1 K; σ′ = |k|σ. |
| Faded example | 164 cm and 8 cm. |
| 1. New standard deviation after kx + c | B. A ignores the multiplier; C adds the 3; D uses the 3 as the multiplier. |
| 2. Median after adding a constant | 19. 11 + 8. |
| 3. Standard deviation after scaling | 45. 10 × 4.5. |
| 4. Standard deviation after a unit conversion | 5.4. 1.8 × 3. |
| 5. Mean after a negative multiplier | −6. −2 × 8 + 10. |
| 6. Original standard deviation from the new one | 4. 12 ÷ 3. |
| 7. Standard deviation after a negative multiplier | 6.7. √45. |
| 8. Centre and spread after a negative multiplier | C. A makes a spread negative; B drops the sign of k; D adds the constant to a spread. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A, C or D in question 1 | Ignored the multiplier, added the constant to σ, or used the added number as the multiplier. | Separate the roles: k multiplies, c adds. |
| 88 or 8 in question 2 | Multiplied the median by the shift, or gave the shift. | A shift adds the same to every value, so to the median. |
| 4.5 or 14.5 in question 3 | Thought scaling leaves the spread, or added the multiplier. | Draw two dots 1 apart; after ×10 they are 10 apart. |
| 37.4, 3 or 71.6 in question 4 | Added the 32, ignored the multiplier, or gave the mean. | Which part of kx + c reaches the spread? |
| 26 or −16 in question 5 | Dropped the sign of k, or forgot c. | Apply the whole rule to the mean in two lines. |
| 36 or 8 in question 6 | Multiplied instead of dividing, or undid the shift. | State the forward rule for σ before reversing. |
| 45 or −6.7 in question 7 | Gave the variance, or kept a negative sign. | A square is positive: k² then root gives |k|. |
| A, B or D in question 8 | Made a spread negative, dropped the sign of k, or applied the added number to the IQR. | Two lines: the mean follows the whole rule; a spread uses |k| only. |
Questions worth asking
- Why does the mean change with c but the standard deviation does not? The mean is a position; deviations are differences, and c cancels in a difference.
- A set is multiplied by −1. What changes and what does not? The mean and median change sign, the spreads are unchanged, and the order reverses.
- How would you convert σ from kg to g, and the variance? σ × 1000; the variance × 1000².
- Which measures are affected by k², and which by |k|? The variance by k²; the standard deviation, range and IQR by |k|.
Differentiation
A way in for anyone stuck
- Give a results table with columns k, c, mean, median, range, IQR and σ, and fill the first row together.
- Use a set of three whole-number values, so the arithmetic is quick.
- Colour-code k and c on the board, so students separate which affects which.
Stretch
- Show that the mean of kx + c is kμ + c, starting from Σ(kx + c) ÷ n.
- Find the transformation that turns a set with mean 40 and σ 5 into one with mean 100 and σ 15, and say whether it is unique.
Students learning in English as an additional language
“Shift” means to add the same number to every value, and “scale” means to multiply every value by the same number. “Transform” means to change in a systematic way. Say “k x plus c” for kx + c, and “the size of k” for |k|.
| Word | Plain meaning | Example |
|---|---|---|
| shift | add the same number to every value | adding 4 to every length |
| scale | multiply every value by the same number | converting metres to centimetres |
| transformation | a change applied to every value in the same way | x → 2x − 7 |
| centre | a measure of where the data sit, such as the mean | the median of 3.5 |
| spread | a measure of how scattered the data are | the IQR of 0.8 |
| invariant | unchanged by a transformation | the range under a shift |
Sentence starters for explaining aloud or in writing:
- The mean changes because …
- The standard deviation is multiplied by … because …
- Adding … does not change the spread because …
Two things not to say
“When you add to the data, everything goes up.” The spread does not.
“A negative multiplier gives a negative standard deviation.” A standard deviation is never negative.
Questions to set
Extra practice in three tiers, including one question from earlier work for spacing. Every answer has been checked.
1Fluency
The idea on its own, with friendly numbers. Set these first.
- ApplyMean 30 and σ 4. Every value is increased by 6. Give the new mean and σ.
36 and 4. - ApplyMean 10 and σ 2. Every value is multiplied by 5. Give the new mean and σ.
50 and 10.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextTimes for a race in Lima are in minutes with mean 52 and σ 3. Convert to seconds. Give the mean and σ.
3120 seconds and 180 seconds. - Earlier work (Lesson 4)Find the variance of a table with n = 10, Σfx = 30 and Σfx² = 110.
2. The mean is 3, 110 ÷ 10 = 11, and 11 − 9 = 2.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningMarks have mean 55 and σ 10. The teacher scales them with y = kx + c to give mean 60 and σ 15. Find k and c.
k = 1.5 and c = −22.5: σ′ = 1.5 × 10 = 15 and 1.5 × 55 − 22.5 = 60. (k = −1.5 and c = 142.5 also work, but reverse the order of the marks, so no teacher would use it.) - ProofShow that the IQR of kx + c is |k| times the IQR of x, for k negative.
A negative k reverses the order, so the new Q₁ is kQ₃ + c and the new Q₃ is kQ₁ + c. Their difference is k(Q₁ − Q₃) = |k| × IQR.
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
The figure works with the keyboard (tab to a slider, then the arrow keys) and by touch. “Show me the steps” gives the same states in words.
The With a GDC section is closed by default and optional. Students without a GDC lose nothing; every answer is reached by hand.