At a glance. One lesson of about 60 minutes. It assumes covariance from Lesson 2 and square roots. Students build the three sums from deviations, divide, read the sign and size, and see that r is unchanged by units and shifts but is blind to curves.
Calculator: scientific calculator. A scientific calculator is used for the square root in questions where it is not exact. All the main data are chosen so that the product under the root is a perfect square, so the arithmetic is by hand and the structure of the formula stays visible.
What students will learn
Know
- Covariance as the mean of the products dx × dy.
- How to square a deviation and add squares.
- How to take a square root.
Understand
- r is the sum of products divided by the square root of the two sums of squares multiplied.
- The sign of r is the sign of the sum of products, and |r| ≤ 1.
- r measures straight-line strength only and is unchanged by changes of unit.
Do
- Build a table of dx, dy, products and squares.
- Find r for a small data set, by hand.
- Judge the direction and strength from the sign and size of r.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: the formula and its sums (questions 1 to 8).
- Criterion B. Investigating patterns: the figure and the arch example. Ask what r would be if the arch were tilted.
- Criterion C. Communicating: explaining what r says and does not say, in a sentence with both direction and strength (question 7 and the what-if).
- Criterion D. Applying mathematics in real-life contexts: market stalls and measured variables.
The one thing to do with the figure
Make students predict r, then build it. Ask each pair to say close to 1, about a half or close to 0 before step 1. Then run the five steps without comment, and press “Flip to a falling pattern” at step 5.
Ask: which totals changed, and which did not? Students should see that the dy² column holds the same numbers in a new order, so both sums of squares are unchanged and only the numerator changes sign.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 7 min | Hook: close to 1, a half or 0? Vote by show of hands. |
| 10 min | The figure and key idea: five columns, then flip. |
| 10 min | Warning card, the arch example and the faded example in pairs. |
| 5 min | The check. If many choose A, ask which of the three sums can change sign. |
| 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 | 9, 0, 9 and 18; 5.5; 12 and 1.5. |
|---|---|
| Hook | r is close to 1: 40 ÷ √1760 = 0.95 for the stalls. |
| Worked examples | 0.8; and 0 for the arch. |
| Faded example | 0.5. |
| 1. What changes the sign of r | B. A looks only at the denominator; C expects cancelling; D confuses the multiplier with r. |
| 2. r from the three sums | 0.75. 36 ÷ √(64 × 36). |
| 3. r from a small data set | 0.9. 18 ÷ √(40 × 10). |
| 4. r from covariance and spreads | 0.6. 12 ÷ (4 × 5). |
| 5. A negative r | −0.9. −27 ÷ √(50 × 18). |
| 6. The sum of products from r | 28. 0.7 × √(25 × 64). |
| 7. What r does not depend on | B. A lets the factor reach r; C adds the shift to r; D thinks unlike units break r. |
| 8. The denominator and strength | C. A reads a bigger sum as stronger; B looks at the numerator only; D thinks r carries units. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A, C or D in question 1 | Looks only at the denominator, expects cancelling, or confuses the multiplier with r. | Write one row of dx, dy and dx × dy before and after. |
| 0.36, 4.5 or 0.5625 in question 2 | Adds the sums of squares, uses one sum only, or divides by the sum of dx² only. | Write the denominator as one root: √(64 × 36). |
| 0.45 or 1.8 in question 3 | Divides by one sum of squares only. | Build the table and count three totals before dividing. |
| 3 or 2.4 in question 4 | Divides by only one standard deviation. | Write r = covariance ÷ (spread of x × spread of y). |
| 0.9, −1.5 or −0.54 in question 5 | Loses the sign, or divides by one sum. | Write the sign first, then the size. |
| 1120, 9.1 or 44.8 in question 6 | Leaves out the root, adds the roots, or uses one sum. | Work out √(25 × 64) = 40 first. |
| A, C or D in question 7 | Lets a factor reach r, adds the shift, or says unlike units break r. | Rewrite r for x in grams and watch the 1000 cancel. |
| A, B or D in question 8 | Read a bigger sum as stronger, used the numerator only, or thought r carries units. | Work out both values of r. |
Questions worth asking
- Why can r never be greater than 1? Expanding Σ(dx ÷ A − dy ÷ B)² ≥ 0 gives 2 − 2r ≥ 0 (the closed note under the key idea).
- Which of the three sums can be negative? Only the sum of products. The squares are never negative.
- What does r = 0 tell you about the arch? Only that there is no straight-line trend; the dots follow a curve exactly.
- What happens to r if every y is doubled? It is unchanged: dy and the sum of dy² scale together and cancel.
Differentiation
A way in for anyone stuck
- Give a table with the five columns already ruled and the first row filled in.
- Choose data in which the product under the root is a perfect square, as in the worked examples.
- Use a sentence frame: the sum of products is ___, the sums of squares are ___ and ___, so r = ___.
Stretch
- Show that if every x is multiplied by 3 the sum of dx² is multiplied by 9, the sum of products by 3, and r is unchanged.
- Find five dots that have r = 1 exactly, and five dots with r = 0 that are not in a line.
Students learning in English as an additional language
“Coefficient” means a number that measures something. “Correlation” means how two things change together. “Deviation” means the distance of a value from its mean. “Square root” means the number that multiplied by itself gives the number under the root.
| Word | Plain meaning | Example |
|---|---|---|
| correlation coefficient | a number between −1 and 1 that measures how closely dots follow a straight line | r = 0.3 for dots in a loose, rising cloud |
| sum of squares | the total of the squares of the deviations | 1 + 0 + 1 + 9 = 11 |
| standard deviation | the typical size of a deviation: the square root of the mean of the squares | the spread of x in hours |
| strength | how closely the dots follow a straight line, the size of r without the sign | 0.8 is stronger than 0.3 |
| direction | whether y tends to rise or fall as x rises, shown by the sign of r | negative: y falls as x rises |
Sentence starters for explaining aloud or in writing:
- The sum of products is ___ and the denominator is ___, so r = ___.
- The sign shows ___ and the size shows ___.
- r is close to 0 but the dots ___, because r only measures ___.
Two things not to say
“r is the percentage of dots on the line.” r is a measure of straight-line closeness, not a count.
“r = 0 means the variables are unrelated.” It means no straight-line trend.
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.
- ComputeThe sums are: products 30, squares of dx 25 and squares of dy 36. Find r.
30 ÷ √(25 × 36) = 30 ÷ 30 = 1: the dots lie exactly on a rising line. - SignThe sum of products is −12 and the squares add to 16 and 9. Find r.
−12 ÷ √(16 × 9) = −12 ÷ 12 = −1.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextDots (1, 1), (2, 3), (3, 4), (4, 5) and (5, 7). Find r to two decimal places.
Means 3 and 4; dy = −3, −1, 0, 1, 3; products 6, 1, 0, 1, 6, total 14; squares 10 and 20; r = 14 ÷ √200 = 0.99. - Earlier work (Lesson 2)Dots (0, 0), (2, 6) and (4, 6). Find the covariance.
Means 2 and 4; products 8, 0 and 4; total 12; covariance 12 ÷ 3 = 4.
3Challenge
Reasoning, generalizing and working backwards.
- Reasoningx is in seconds and y in metres. Explain why r has no units.
The units of the products cancel with the units under the root. - Working backwardsr = −0.6, the sum of dx² is 16 and the sum of dy² is 9. Find the sum of products.
−0.6 × 12 = −7.2.
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 the slider or the button, then the arrow keys or the space bar) and by touch. Under it, “Show me the steps” gives the same states in words.
No device? Give each pair the five-column table with five empty rows and a small data set to complete.