At a glance. One lesson of about 60 minutes. It assumes the sums of products and squares from Lesson 3. Students tilt a line about the mean point to see the minimum of the total of squared gaps, compute b and a by formula once, predict, and meet the GDC command.
Calculator: GDC optional. The GDC finds the regression line from two lists and gives r as well. It is offered after the by-hand pass so that the output is something students could have computed, and so that they can check it passes through the mean point.
What students will learn
Know
- The mean point of a data set.
- y = a + bx and how to substitute into it.
- The sums of products and squares from Lesson 3.
Understand
- The regression line minimizes the total of the squared vertical gaps.
- Its gradient is the sum of products over the sum of dx², and it passes through the mean point.
- It predicts y from x, and it exists for any dots, whether or not they are close to a line.
Do
- Find b and a from the sums.
- Predict from a regression equation.
- Enter lists in a GDC and read the line.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: the formulas for b and a, and predicting (questions 1 to 8).
- Criterion B. Investigating patterns: the figure. Ask how the total changes on each side of the lowest point.
- Criterion C. Communicating: explaining why not every line through the mean point is the regression line (the exit ticket).
- Criterion D. Applying mathematics in real-life contexts: plant growth, test scores and predictions from data.
The one thing to do with the figure
Make students place the minimum before they see it. Ask each pair to write a gradient between 1 and 4 for the smallest total, then run the slider from 1 to 4 slowly and watch the marker on the small graph.
Ask why the total rises on both sides, and what the lowest point looks like. Then give the formula and check it against the lowest point: 27 ÷ 10 = 2.7.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 7 min | Hook: write down a gradient for the smallest total. Collect on the board. |
| 10 min | The figure and key idea: tilt, find the lowest total, then the formula. |
| 10 min | Warning card, worked examples and the faded example in pairs; GDC section for those with one. |
| 5 min | The check. If many choose A, ask what a gap of −4 becomes when it is squared. |
| 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 | 23; 1.5; 12. |
|---|---|
| Hook | The total is smallest at gradient 2.7 (5.1), for the line y = 2.7x + 10.9; the formula gives 27 ÷ 10 = 2.7. |
| Worked examples | y = 1.3x + 3.5; and y = 90 − 2x. |
| Faded example | y = 8.2 + 1.2x. |
| 1. The total of the squared gaps | C. A adds signed gaps; B adds sizes; D squares the total. |
| 2. Gradient from the two sums | 2.5. 90 ÷ 36. |
| 3. Intercept from the mean point | 35. 47 − 1.5 × 8. |
| 4. Intercept from the raw data | 20. b = 600 ÷ 1000 = 0.6; a = 38 − 18. |
| 5. Predicting from a GDC line | 28. 1.6 × 25 − 12. |
| 6. Line from r and the spreads | 61. 55 + 1.2 × (25 − 20). |
| 7. Why the gaps are squared | D. A thinks squaring shrinks gaps; B reverses cause and effect; C mixes squaring with direction. |
| 8. Choosing the least-squares line | A. B adds signed gaps; C counts dots on the line; D adds sizes without squaring. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A, B or D in question 1 | Adds the signed gaps, adds the sizes, or squares the total. | Write the five squares in a row, then add. |
| 0.4 or 54 in question 2 | Inverts the quotient or subtracts. | Write b = (sum of products) ÷ (sum of dx²) before substituting. |
| 59 or 12 in question 3 | Adds the gradient term or stops at it. | Substitute (8, 47) into y = a + 1.5x and solve. |
| 38, 56 or 0.6 in question 4 | Gives the mean of y, adds the gradient term, or gives the gradient. | Write the two steps: b first, then a. |
| 52 or 40 in question 5 | Adds the 12, or forgets the intercept. | Write 1.6 × 25 first, then the 12. |
| 56.5, 30 or 58 in question 6 | Inverts the spreads, forgets the mean point, or uses r as the gradient. | Put spread of y on top and move from (20, 55). |
| A, B or C in question 7 | Thinks squaring shrinks gaps, reverses cause and effect, or mixes squaring with direction. | Ask: what do the signed gaps add to for every line through the mean point? |
| B, C or D in question 8 | Added signed gaps, counted dots on the line, or added sizes without squaring. | Square each of P's and Q's gaps and add. |
Questions worth asking
- Why does the total of squared gaps rise on both sides of the best gradient? Tilting either way moves the line away from the dots on one side more than it moves it toward the others.
- Why are the gaps squared? So positive and negative gaps cannot cancel, and big gaps count much more than small ones.
- How do you know a GDC answer is plausible? The line passes through the mean point and the gradient has the sign of r.
- Does a regression line exist for scattered dots? Yes: it is always defined. Whether it is a good model is a separate question.
Differentiation
A way in for anyone stuck
- Give a table with columns x, y, dx, dy, dx × dy and dx² with the first row filled in.
- Give the formulas on a card: b = sum of products ÷ sum of dx²; a = mean of y − b × mean of x.
- Use a sentence frame: the gradient is ___ and the line passes through ___, so the intercept is ___.
Stretch
- Show that for the lesson's five dots, the total of the squared gaps at gradient m through the mean point is 78 − 54m + 10m², and find its minimum by completing the square.
- Explain why the regression line of x on y has gradient (sum of products) ÷ (sum of dy²) when drawn with x as the output, and why the two lines are the same only when r = 1 or −1.
Students learning in English as an additional language
“Regression” here means finding a line that describes how one variable depends on another. “Gap” means the distance between a dot and the line. “Minimize” means make as small as possible. “Squared” means multiplied by itself. “Intercept” is where the line crosses the y-axis.
| Word | Plain meaning | Example |
|---|---|---|
| regression line | the line of y on x that makes the total of the squared vertical gaps smallest | the line used to predict a plant's height from its hours of light |
| gradient | how much y changes for a rise of 1 in x | a gradient of 1.5 means y rises by 1.5 for each extra 1 in x |
| intercept | the value of y where the line crosses the y-axis | the height at 0 hours of light |
| residual | the gap from a dot to the line: the dot's y minus the line's y | a plant 2 cm below the line |
| least squares | the method that chooses the line with the smallest total of squared gaps | the rule that picks the regression line |
Sentence starters for explaining aloud or in writing:
- The gradient is the sum of ___ divided by the sum of ___.
- The line passes through the mean point, so the intercept is ___.
- I know the GDC answer is sensible because ___.
Two things not to say
“The regression line goes through the middle of the dots, so any line through the middle works.” Only one gradient minimizes the squares.
“A regression line proves that x affects y.” It describes the pattern in the data; it does not give a cause.
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.
- GradientThe sum of products is 42 and the sum of dx² is 12. Find b.
3.5. - InterceptThe mean point is (5, 30) and b = 4. Find a.
30 − 20 = 10.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextDots (1, 3), (2, 5), (3, 6) and (4, 10). Find the regression line.
Means 2.5 and 6; dx −1.5, −0.5, 0.5, 1.5; dy −3, −1, 0, 4; products 4.5, 0.5, 0, 6 = 11; squares 5; b = 2.2, a = 0.5. - Earlier work (Lesson 3)The sum of products is 24, the squares of dx add to 16 and the squares of dy to 36. Find r.
24 ÷ √576 = 24 ÷ 24 = 1.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningTwo data sets have the same r. Must they have the same regression line? Explain.
No: the gradient also depends on the spreads of x and y, and the line on the means. - Working backwardsA regression line has b = 0.5 and passes through (40, 30). At what x is y = 0?
a = 30 − 20 = 10, so y = 10 + 0.5x is 0 at x = −20, far outside any sensible data.
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, then the arrow keys) and by touch. Under it, “Show me the steps” gives the same states in words.
No device? Give each pair the five dots on a printed grid and a ruler: draw lines through the mean point with different gradients and compute the squares of the gaps.