At a glance. One lesson of about 60 minutes. The mean point, a line through it by eye, equal totals of gaps above and below, and the idea that a good slope makes squared gaps small.
Calculator: no calculator. No calculator is needed: the means are of small whole numbers.
What students will learn
Know
- The mean point is the mean of the x values and the mean of the y values.
- A line of best fit passes through the mean point and follows the trend.
- For such a line the gaps above and below balance.
Understand
- The line need not pass through any dot.
- Balance does not choose the slope; following the trend does.
Do
- Find the mean point.
- Draw a line of best fit through it by eye.
- Use a line through the mean point to find y.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: mean point and using a line (questions 2 to 5).
- Criterion B. Investigating patterns: students turn the line and find that the totals above and below stay equal.
- Criterion C. Communicating: a line is drawn through a ringed mean point and labelled.
- Criterion D. Applying mathematics in real-life contexts: revision hours and scores.
The one thing to do with the figure
Ask for the totals above and below at three slopes before they look. They stay equal; that surprises most students.
Then ask which slope is best and why balance did not help. The squared gaps total is smallest near slope 5: that is what “follows the trend” means.
A possible order
| 5 min | Warm-up, then answers. |
|---|---|
| 8 min | Hook: two lines, one point. |
| 12 min | The figure; predict the totals. |
| 12 min | Worked examples; mean point 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 | 6; 13; positive and strong. |
|---|---|
| Worked examples | Mean point (5, 10). |
| Faded example | (3.5, 7.5). |
| 1. What a line of best fit does | B. A uses two dots only; C chases dots rather than the trend. |
| 2. Mean of x | 6. |
| 3. Mean of y | 21, so the mean point is (6, 21). |
| 4. Using the line | 25. |
| 5. Going left of the mean point | 40. |
| 6. Balancing gaps | 14. |
| 7. Missing gap | 2. |
| 8. Choosing the slope | 2. |
| 9. What must be true | B. A counts dots instead of gaps; C confuses the line with the dots. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A or C in question 1 | Thinks the line must touch dots. | Ask: if the dots are scattered, can any line touch them all? |
| 5, 30 or 10 in question 2 | Reports the count, the sum or the largest value. | Write sum, then divide by the count. |
| 105, 19 or 29 in question 3 | Reports the sum, the median or the largest. | Write the formula first. |
| 16, 23 or 37 in question 4 | Uses the wrong step from the mean point. | Ask how far x = 8 is from x = 6. |
| 50 or 70 in question 5 | Moves one unit, or goes the wrong way. | Ask: is x = 1 to the left or the right of 4? |
| 0, 8 or 28 in question 6 | Sums signed gaps, counts dots, or adds both sides. | Ask what balance means. |
| 16 in question 7 | Adds all gap sizes. | Add above and below separately. |
Questions worth asking
- Why must the line go through the mean point? Listen for “it is the centre of the data, so gaps balance there”.
- Why can the totals be equal even when the slope is wrong? Listen for “balance is true for every line through the mean point”.
- What makes slope 5 better than slope 8? Listen for “its dots are closer to the line overall”.
Differentiation
A way in for anyone stuck
- Provide a table with columns for x, y and a row for the totals.
- Give the mean point already ringed on a printed graph.
- Let students use a transparent ruler or a taut piece of string.
Stretch
- Explain why the signed gaps always add to zero for a line through the mean point.
- Find the line of best fit for the data (1, 2), (2, 4), (3, 3), (4, 7) by eye and test it with the gap totals.
Students learning in English as an additional language
Pair “mean point” with a sketch of a ringed dot at the centre of a cloud.
| Word | Plain meaning | Example |
|---|---|---|
| mean point | the point whose coordinates are the mean x and the mean y | (4, 55) |
| line of best fit | a line through the mean point that follows the trend of the dots | slope 5 through (4, 55) |
| trend | the general direction in which the dots go | rising |
| gap | the vertical distance from a dot to the line | 3 above the line |
| squared gap | the gap multiplied by itself, used to judge how well a line fits | 3 × 3 = 9 |
Sentence starters for explaining aloud or in writing:
- The mean of the x values is … and the mean of the y values is …
- The line passes through the mean point and rises/falls because …
- The gaps above add to … and the gaps below add to …
Two things not to say
“Just join the first and last dots.” Those two dots may be unusual.
“The line must have the same number of dots on each side.” Totals balance, not counts.
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.
- Mean pointFind the mean point of (1, 4), (3, 8), (5, 6).
(3, 6). - UsingA line through (4, 20) has gradient 3. What is y at x = 6?
26.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- UsingA line through (10, 50) has gradient 2.5. What is y at x = 6?
40. - BalanceGaps above add to 12 and gaps below to 9, apart from one dot. How far below is it?
3 below.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningCan the line of best fit pass through no dot at all? Give a reason.
Yes: it follows the trend and may pass between the dots. - From Lesson 2Dots within 4 units of a line and dots within 12 of the same line: how many times as wide is the second band?
3.
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
The slider works with the keyboard and by touch; “Show me the steps” describes four states in words.
On paper: a printed graph and a ruler, with the mean point already marked.