At a glance. One lesson of about 60 minutes. It assumes means and scatter graphs. Students find deviations from the mean point, multiply them, average the products, read the sign by quadrant, and see that the size depends on units, which prepares the correlation coefficient.
Calculator: no calculator. No calculator: five small dots, whole-number deviations and one division. The structure of the products is the point, and technology would hide it.
What students will learn
Know
- The mean of x and the mean of y of a data set.
- Multiplying negative numbers.
- What a scatter graph shows.
Understand
- A dot's product dx × dy is positive in two quadrants and negative in the other two.
- Covariance is the mean of those products: positive, negative or near zero.
- Its size depends on the units, so it cannot say how strong a link is.
Do
- Find deviations and products for each dot.
- Find a covariance from a small data set.
- Say what the sign shows and why the size cannot be compared.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: deviations, products and the covariance of a small set (questions 1 to 7).
- Criterion B. Investigating patterns: the first figure. Ask which pattern makes the positive products win, and what the zero case looks like.
- Criterion C. Communicating: explaining in words why the sign shows direction but the size does not show strength (the exit ticket).
- Criterion D. Applying mathematics in real-life contexts: market stalls, class measurements and savings.
The one thing to do with the figure
Make students sort the quadrants before they look. In the first figure, ask each pair to predict the sign of the product for a dot in each quadrant, then step through the five dots and check.
In the second figure, ask for the new covariance before pressing the button. Then ask what has not changed (the picture, the pattern) and what that means for reading the number. To compare strength, the units have to come out of the number; Lesson 3 does that by dividing by the two spreads.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 7 min | Hook: signs of products by quadrant. Collect predictions on the board. |
| 10 min | First figure and key idea: step through the dots and switch patterns. |
| 10 min | Second figure, the warning card and the worked examples; pairs finish the faded example. |
| 5 min | The check. If many choose A, ask whether covariance counts dots or adds products. |
| 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 7. |
The answers
| Warm-up | 6; positive and strong; more values are averaged and extremes are diluted. |
|---|---|
| Hook | Top right and bottom left give positive products; top left and bottom right give negative. For the five stalls the products are 16, 2, 0, 2 and 20, so the covariance is 8. |
| Worked examples | 5.5 and −20 (negative because y falls as x rises). |
| Faded example | 3.5. |
| 1. Adding products, not counting dots | C. A counts dots; B expects any mix to cancel; D thinks the coordinates are still needed. |
| 2. Product of deviations, same side | 18. (15 − 12) × (46 − 40). |
| 3. Product of deviations, opposite sides | −36. (9 − 12) × (52 − 40). |
| 4. Covariance from four dots | 5. Products 9, 0, −1, 12; total 20; 20 ÷ 4. |
| 5. Changing the unit | 18. 6 × 3. |
| 6. A missing value from the covariance | 6. k − 34 = 4 × (−7). |
| 7. Comparing covariances | B. A reads size as strength; C reverses it; D uses the sign only. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A, B or D in question 1 | Counts dots instead of adding products, expects any mix to cancel, or thinks the coordinates are needed. | Add the five products on the board. |
| 9 or 690 in question 2 | Adds the deviations, or multiplies the coordinates. | Mark the mean point on the graph and measure from it. |
| 36, 9 or 468 in question 3 | Drops the sign, adds the deviations, or multiplies the coordinates. | Ask: is the dot left or right of the mean x, above or below the mean y? |
| 20 or 5.5 in question 4 | Forgets to divide by 4, or gets the sign of one product wrong. | Make a table with columns dx, dy and dx × dy. |
| 9, 2 or 54 in question 5 | Adds the factor, divides by it, or applies it twice. | Write one product before and after the change. |
| 27 or 62 in question 6 | Uses the covariance as the total, or loses its sign. | Write: total of products = 4 × (−7). |
| A, C or D in question 7 | Reads size as strength, reverses it, or uses the sign only. | Ask: if height were in metres, what would happen to the number? |
Questions worth asking
- Why is a dot on the dashed vertical line worth nothing to the covariance? Its dx is 0, so its product is 0.
- If every dot is in the top-right or bottom-left quadrant, what can you say about the covariance? It is positive, and the dots lie on a rising trend.
- What does a covariance of zero mean? No straight-line trend: the positive and negative products cancel. The dots might still follow a curve.
- What would happen to the covariance if x were measured in a different unit? It would be multiplied by the same factor; the picture would be unchanged.
Differentiation
A way in for anyone stuck
- Give a table with columns x, y, dx, dy and dx × dy already ruled.
- Print the first figure's axes with the mean point and quadrants shaded, so students sort dots by quadrant first.
- Use a sentence frame: the dot is ___ the mean x and ___ the mean y, so the product is ___.
Stretch
- Show that if every x is increased by 10 and every y is left alone, the covariance does not change.
- Find two different data sets of four dots with covariance 0 where the dots are not all in a line.
Students learning in English as an additional language
“Deviation” means how far a value is from the mean, with a sign. “Product” means the result of multiplying. “Covariance” combines “co” (together) and “variance” (how much values vary): it says how two variables vary together. “Quadrant” means one of the four parts the two dashed lines make.
| Word | Plain meaning | Example |
|---|---|---|
| mean point | the point whose coordinates are the mean of x and the mean of y | the point where the two dashed lines cross |
| deviation | how far a value is from its mean, with a sign | a stall that is open 2 hours less than the mean has dx = −2 |
| product | the result of multiplying two numbers | dx multiplied by dy for one dot |
| covariance | the mean of the products dx × dy over all the dots | the number that is positive when the dots rise |
| quadrant | one of four regions made by the lines through the mean point | the top-left region |
Sentence starters for explaining aloud or in writing:
- This dot is ___ the mean x and ___ the mean y, so its product is ___.
- The covariance is positive because the products in the ___ and ___ quadrants outweigh the others.
- The size of the covariance depends on ___, so I cannot ___.
Two things not to say
“The bigger the covariance, the stronger the relationship.” Size depends on units. Strength needs the units removed.
“A covariance of zero means the variables are unrelated.” It means no straight-line trend; a curve can have covariance zero.
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.
- ProductThe mean point is (8, 20). A dot is at (11, 17). Find dx × dy.
3 × (−3) = −9. - SignA dot is below the mean y and to the right of the mean x. Is its product positive or negative?
Negative.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextDots (1, 9), (2, 7), (4, 5) and (5, 3). Find the covariance and say what it shows.
Mean point (3, 6); products −6, −1, −1, −6; total −14; covariance −3.5. Negative: y falls as x rises. - Earlier work (Lesson 1)A population is 2, 6 and 10. What is the highest mean of a sample of 2?
8.
3Challenge
Reasoning, generalizing and working backwards.
- UnitsA covariance is 15 with x in minutes. What is it with x in seconds?
900. - ReasoningExplain why adding 100 to every y value does not change the covariance.
Every dy is unchanged because the mean of y also rises by 100.
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
Both figures work with the keyboard (tab to a slider or a button, then the arrow keys or the space bar) and by touch. Under each one, “Show me the steps” gives the same states in words.
No device? Give each pair a printed grid with the mean point and dashed lines drawn, and five dots to sort by quadrant.