Double Bar Graphs and Many-to-One Correspondence
Warm-up
Show a completed double bar graph comparing two schools' favourite sports. Ask all three levels: Level 1: how many students at School A chose hockey? Level 2: how many more students at School B chose soccer than at School A? Level 3: which sport showed the biggest difference between the two schools? What might explain this difference?
Explore
Compare and create: each group receives two related data sets (e.g., monthly temperatures in two BC cities, or book loans by genre in two grade levels). They choose a scale, create a double bar graph, and write 3 comparison sentences (one at each level).
Formalize
Scale selection for double bars: data set A max = 85, data set B max = 110. The graph must accommodate 110. Scale 1=10: max bar = 11 units. Clean. Scale 1=5: max bar = 22 units (workable but crowded). Choose 1=10. Is 85 a problem? 85/10 = 8.5: draw the bar between the 8 and 9 marks, or round to 90.
Double Bar Graphs and Many-to-One Correspondence
Environmental data double bar graph: rainfall (mm) in Vancouver and Victoria for 6 months. Students create the graph, apply the comparison analysis, and write a paragraph about what the data reveals about BC climate differences. This is data-to-text writing using mathematical evidence.
Practice
Students create one double bar graph from given data, choose scale, draw legend, and answer 5 questions at all three levels. Exit ticket: in a double bar graph with scale 1=10, one bar is 7.5 units. What value does it represent?
Exit ticket
Students create one double bar graph from given data, choose scale, draw legend, and answer 5 questions at all three levels. Exit ticket: in a double bar graph with scale 1=10, one bar is 7.5 units. What value does it represent?
The data: favourite sport by group — Soccer: boys 8, girls 10. Basketball: boys 6, girls 5. Swimming: boys 4, girls 7. Track: boys 5, girls 3.
Step 1: Build the double bar graph: each sport gets a PAIR of bars (boys' colour, girls' colour) side by side with a small gap between pairs. Legend mandatory — without it the colours are noise.
Step 2: Scale: counts reach 10 → one square = 1 works on grid paper.
Step 3: Read WITHIN a category: "which sport has the biggest boy–girl gap?" Swimming (4 vs 7 → gap 3) and Soccer (8 vs 10 → gap 2) — Swimming wins the gap contest.
Step 4: Read ACROSS categories: girls' favourite overall? Soccer (10). Boys'? Soccer too (8). Totals by sport (stack the pairs mentally): Soccer 18, Basketball 11, Swimming 11, Track 8.
Step 5: The double graph's whole purpose, stated: ONE graph answers comparison questions that two separate graphs would make you flip between. Paired bars = built-in comparison. If the question is only about totals, a single-bar graph of sums would be honest and simpler — match the display to the question.
The data: class donations — 3A: 150, 4A: 100, 5A: $250.
Step 1: Why many-to-one is forced: one symbol per dollar means 250 symbols. Choose a key. Candidates: 25 (175 = 7 exact symbols ✓✓).
Step 2: Test the key against EVERY value before committing (the professional habit): 175 ÷ 25 = 7 ✓, 150 ÷ 25 = 6 ✓, 225 → 9 ✓, 100 → 4 ✓, 250 → 10 ✓. All whole — key $25 wins.
Step 3: Draw: rows per class, coin symbols, key box stating "⊙ = $25."
Step 4: Read through the key both directions: "4A shows 9 symbols → $225." "Which classes together match 5A's total?" 3B + 4B = 150+100 = 250 ✓ (6 + 4 symbols vs 10 symbols — the KEY makes symbol-arithmetic legal).
Step 5: The half-symbol wrinkle on purpose: a late 160 = 6.4 symbols. Options: round the DISPLAY (6½ symbols ≈ 20 (8 exact). Every key is a contract with trade-offs; renegotiate when data changes.
Setup: two classes graphed identical juice-sale data (Grape 48, Orange 40) — Graph A with key 1 symbol = 4 bottles; Graph B with key 1 symbol = 8.
Step 1: Read both: A shows 12 vs 10 symbols (difference LOOKS like 2 symbols but means 8 bottles); B shows 6 vs 5 (difference looks like 1 symbol — same 8 bottles).
Step 2: The observation that matters: bigger keys COMPRESS differences visually. In B the gap looks trivial; in A, modest; with key = 1 it would tower. None of the graphs lies — but each choreographs a different first impression.
Step 3: Connect to the axis-trickery lesson from bar graphs: scale choices are rhetorical choices. A juice company preferring Grape would publish… which? (A — or even key = 2.)
Step 4: The defence kit, one question long: "what is one symbol worth?" — ask it BEFORE letting a pictograph impress you. Then translate two bars into numbers and re-judge.
Exit: take our class's real data and produce two honest graphs — one that makes the winner look dominant, one that makes the race look close. Explain the choreography you used. (Making bias visible is the best vaccination against it.)