Teacher guide — Grade 10, Unit 3: Relationships and the line of best fit¶

Time: 3 lessons + project (about 5 periods). Track: B (code).

#	Lesson	Time
1	Scatter plots and correlation	40–45 min
2	The line of best fit	40–45 min
3	Correlation is not causation	40–45 min
P	Unit project: Two variables, one honest line	2 periods

Before you start¶

Materials: Notebook notebooks/best_fit.ipynb; ruler and string for the by-hand fit.
Read docs/pedagogy/lesson-anatomy.md and docs/pedagogy/ai-use-policy.md.

If you have no computers / no internet¶

Do the line of best fit with a ruler on a printed scatter plot; estimate slope by two points.

Common misconceptions¶

A high correlation proves one causes the other.
A line fits any data; check the scatter.

Differentiation¶

Support: give a partially filled table / sentence starters; pair with a data-steward role.
Stretch: ask "what would change your conclusion?" and "who is missing?"

Answer key (check questions)¶

Lesson 1: Scatter plots and correlation 1. r = −0.9 means…
Answer: A strong negative straight-line relationship. 2. r = 0 means…
Answer: No straight-line relationship (curves possible). 3. Does r show cause?
Answer: No.

Lesson 2: The line of best fit 1. Slope 1.6 for PM10 vs PM2.5 means…
Answer: PM10 is about 1.6 µg/m³ higher for each 1 µg/m³ of PM2.5. 2. What is a residual?
Answer: Observed − predicted. 3. Why squares?
Answer: So positive and negative misses don't cancel.

Lesson 3: Correlation is not causation 1. What is a confounder?
Answer: A hidden factor that affects both variables. 2. Give an example.
Answer: Summer heat drives both ice-cream sales and swimming accidents. 3. What design supports causal claims?
Answer: A randomised experiment.

Assessment¶

Starter quiz each lesson (3 questions, one recalling the previous lesson); unit quiz in quiz.yml; project marked with the rubric in docs/pedagogy/assessment.md; 2-minute oral check per group.

Alignment¶

Optional Computer Science Grade 10: Python programming, databases and AI & contemporary technology; compulsory Maths data handling/statistics.

Answers: practice exercises¶

(a) hours studied (b) altitude (c) rainfall.
ŷ = 25; residual = 22 − 25 = −3.
Any curve such as a U-shape (points at (−2, 4), (−1, 1), (0, 0), (1, 1), (2, 4)) has r = 0.
b = 0.8 × 10 ÷ 4 = 2.0; a = 50 − 2 × 20 = 10 → ŷ = 10 + 2x.
Slope −0.006 °C per metre (−0.6 °C per 100 m). 25 − 0.006 × (1,400 − 70) ≈ 17 °C. Wrong because of local effects: valleys, slope aspect, cloud and rain shadow, latitude; the lapse rate varies by season.
Jan–Jun slope ≈ −17.4 µg/m³ per month (falls into the monsoon); intercept ≈ 119. Over 12 months the pattern goes down then up (U-shape), so a straight line fits badly.
r² ≈ 0.97: the line accounts for about 97% of the day-to-day variation in PM10 in this synthetic data.
Sleep improves concentration (cause); students with fewer worries/better home conditions both sleep and study more (confounding); students who do well feel less stress and sleep better (reverse). An experiment (randomly encourage one group to sleep earlier) or comparing the same students across weeks helps.
Regressing y on x minimises vertical misses; x on y minimises horizontal misses — different lines unless r = ±1.
r = 0.90 (by hand and pandas).

Lesson 1 step 3: one fake point (10, 400) drops r from 0.986 to 0.953. Lesson 3 step 1: all months r = 0.67; monsoon months only 0.33; dry months only 0.02 — the season explains most of the overall correlation. Lesson 2 predictions: PM2.5 40 → PM10 ≈ 66; 100 → ≈ 165; typical residual (SD) ≈ 9 µg/m³.

Real-data exercises: notes¶

R1 Roughly +5 to +8 years of life expectancy for each tenfold income increase in recent data; Nepal typically lives longer than its income predicts (positive residual) — a nice discussion of public health, remittances and other factors.
R2 Expect around −0.5 to −0.65 °C per 100 m; Jomsom (rain-shadow, windy valley) and Kathmandu (valley basin) may have notable residuals. Open-Meteo's grid smooths mountains, so high towns may look warmer than station data.
R3 Strong positive relationship. Plausible: lower child mortality → parents choose fewer children; also education, income and contraception access drive both (confounding).