At a glance. One lesson of about 60 minutes. Reading r, a rough guide to strength, what r cannot see, units, and how an outlier changes r and the line.
Calculator: no calculator. r is read from a spreadsheet or a statistics mode, so no hand calculation is expected.
What students will learn
Know
- r is between −1 and 1.
- The sign gives the direction and the size gives the strength.
- r measures straight-line fit only.
- An outlier can change r and the line a lot.
Understand
- A graph and r should be used together.
- r = 0 means no linear relationship, not no relationship.
Do
- Read and interpret r.
- Compare strengths by size.
- Describe the effect of an outlier.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: reading r and its sign (questions 1 to 4).
- Criterion B. Investigating patterns: students vary the ninth dot and generalize its effect on r and the line.
- Criterion C. Communicating: r is interpreted in words with direction and strength.
- Criterion D. Applying mathematics in real-life contexts: height and arm span, screen time and sleep.
The one thing to do with the figure
Ask for r before moving the ninth dot to 10. Most students guess a small change; r falls from 0.97 to 0.40.
Then move it to 60 and ask why r falls again. Finish with Hide the ninth dot: this is the shape of the criterion B investigation in the unit.
A possible order
| 5 min | Warm-up, then answers. |
|---|---|
| 8 min | Hook: guess the change in r. |
| 12 min | The figure; predict r at each height. |
| 12 min | Worked examples; compare strengths in pairs. |
| 5 min | Question 1, the check. |
| 13 min | Your turn, Levels 1 to 3. |
| 5 min | Exit ticket, question 9. |
The answers
| Warm-up | Positive and strong; 28; 0.85. |
|---|---|
| Worked examples | Strong negative linear; graph B is stronger. |
| Faded example | Almost no linear relationship. |
| 1. Reading r | B. A reads the sign wrongly; C reads the size wrongly. |
| 2. Size of r | 0.74. |
| 3. Sign of r | 3. |
| 4. Strong correlations | 3. |
| 5. Effect of an outlier | 0.39. |
| 6. What r cannot see | B. A and C assume r = 0 means random. |
| 7. Units and r | 0.6. |
| 8. Predicting an extra dot | 28.5. |
| 9. Interpreting a change in r | A. B says r cannot change; C confuses removing an outlier with cheating. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A or C in question 1 | Misreads the sign or the size. | Ask: what does the minus sign tell you, and what does 0.9 tell you? |
| 0.68 or 1.42 in question 2 | Picks the weaker graph or adds the sizes. | Cover the signs and compare. |
| 2 or 5 in question 3 | Counts negatives or all values. | Circle values without a minus sign. |
| 2 or 4 in question 4 | Ignores negative r or rounds up 0.79. | Ask: is the size, not the sign, at least 0.8? |
| 1.49 or 0.55 in question 5 | Adds, or reports the new value. | Ask: old minus new. |
| 6 or 0.06 in question 7 | Applies the unit change to r. | Ask: does the shape of the cloud change? |
| 18.7 or 11.8 in question 8 | Leaves out the constant or does not multiply. | Write 2.08 × 9 first. |
Questions worth asking
- What does the sign of r tell you, and what does the size tell you? Listen for “direction and strength”.
- Why does a dot at height 10 hurt r more than one at height 30? Listen for “it is far from the line the other dots make”.
- Should an outlier always be removed? Listen for “not without a reason, such as a recording error, and it should be reported”.
Differentiation
A way in for anyone stuck
- Give a number line from −1 to 1 with the words strong, moderate and weak marked.
- Provide r already read from the spreadsheet and ask only for interpretation.
- Use a sentence frame: r = …, so the correlation is … and …
Stretch
- Sketch eight dots and a ninth that would give r close to 0.
- Why can r never be greater than 1 or less than −1? Think about what the closest possible fit to a line looks like.
Students learning in English as an additional language
Pair “coefficient” with “a number that measures something” and show a number line from −1 to 1.
| Word | Plain meaning | Example |
|---|---|---|
| correlation coefficient | a number r between −1 and 1 for how closely dots follow a straight line | r = 0.97 |
| strong | r close to 1 or −1 (0.8 or more in size) | r = −0.9 |
| weak | r close to 0 (below 0.5 in size) | r = 0.2 |
| outlier | a dot far from the pattern that can change r and the line | the ninth dot at height 10 |
| linear | following a straight line | dots in a band |
Sentence starters for explaining aloud or in writing:
- r = …, so the correlation is … and …
- The outlier changes r from … to …, which means …
- r does not tell us …
Two things not to say
“r = 0.5 is half as strong as r = 1.” r is not a percentage of strength.
“Delete any dot that spoils r.” Removing data needs a reason.
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.
- Readr = −0.55. Describe the correlation.
A moderate negative linear correlation. - CompareWhich is stronger, r = 0.7 or r = −0.75?
−0.75.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- Countr values 0.9, −0.2, 0.82, −0.95. How many are strong?
3. - Fallr goes from 0.88 to 0.52 after one dot is added. By how much did it fall?
0.36.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningWhy can two graphs with the same r look different?
r measures only straight-line fit; one may have an outlier or a curve. - From Lesson 4A line is y = 3x + 8 for data with x from 2 to 10. Predict y at x = 40 and comment.
128; it is far outside the data.
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
The slider and the button work with the keyboard and by touch; “Show me the steps” describes four states in words.
On paper: the same nine pairs in a table for use with a spreadsheet.