Concrete Graphs Using One-to-One Correspondence
Warm-up
Display a ready-made concrete graph (weather for the week). Ask Level 1: How many sunny days? Level 2: How many more sunny than rainy? Level 3: What does this graph tell us about our week? Run through all three levels quickly to establish the expectation.
Explore
Class survey: what is your favourite season? Each student places one cube in their column. In table groups, students work through a question card set ranging from Level 1 to Level 3. Groups share answers and compare reasoning.
Formalize
Highlight one-to-one correspondence. Each cube represents exactly one person. What would happen if I put 2 cubes for each person? Demonstrate: the graph looks twice as tall but represents the same data. Is that fair? Would a reader get the right idea?
Concrete Graphs Using One-to-One Correspondence
Transfer to a pictorial graph: students draw a seasonal symbol on a sticky note and rebuild the graph on a grid. Is this showing the same data as the cube graph? Yes, because the one-to-one correspondence is preserved. The same data, a different medium.
Practice
Students pose their own survey question, survey 6 classmates, build a concrete graph, and write 2 sentences about what they found. Exit ticket: teacher points to two columns and students write one comparison sentence.
Exit ticket
Students pose their own survey question, survey 6 classmates, build a concrete graph, and write 2 sentences about what they found. Exit ticket: teacher points to two columns and students write one comparison sentence.
The graph: birthday seasons — Summer 8, Fall 3, Winter 5, Spring 6, built from one cube per child.
Level 1 (read the data): "How many birthdays in Winter?" Count that tower: 5. Every student should get these.
Level 2 (read between the data): "How many MORE birthdays in Summer than Fall?" Don't count both towers separately — stand them side by side and count only the part of Summer's tower that sticks up past Fall's: 5. Comparison means looking at the DIFFERENCE, not the two totals.
Level 3 (read beyond the data): "How many children are in our class altogether?" The graph knows: 8 + 3 + 5 + 6 = 22 — every child appears exactly once because each placed exactly one cube.
A graph that's been READ at all three levels has done its job; stopping at level 1 wastes it.
Step 1: Build the class graph live: each child places ONE cube on the row for their favourite fruit. Say the rule as it happens: "one person, one cube."
Step 2: Sabotage it (theatrically): let a puppet vote twice for apples. Ask: "The apple tower says 9 — is that how many children like apples best?" No — 8 children, 9 cubes. The graph now LIES.
Step 3: Fix it and state the principle: a concrete graph only tells the truth if each object stands for exactly one person or thing. That matching — one child ↔ one cube — is called one-to-one correspondence, and it's the entire foundation the graph stands on.
Step 4: Quick check for transfer: "If two towers are the same height, what does that MEAN?" (The same number of children chose each — without counting either tower.)
What you hear: a student announces that Summer WON, or that Summer is BEST, because its tower is tallest.
Step 1: Honour the correct part: the student read the graph right — Summer's tower IS tallest, so MORE CHILDREN in our class have summer birthdays. That's a true graph sentence.
Step 2: Probe the leap: "Does the graph say Summer is better — or just that more of us were born then?" Help them hear the difference between what the data SHOWS and the opinion they added on top.
Step 3: Practice careful graph sentences as a class: "Most children in OUR class…", "Fall has the fewest…", "Nobody's birthday…" — each one checkable against the cubes.
Step 4: Ask the generalization question: "Would the Grade 2 class's graph look the same?" (Maybe not!) Data describes the group that made it — a gentle first meeting with the idea of a sample.