One-to-One and Many-to-One Correspondence in Graphs
Warm-up
Show two bar graphs of the same data with different scales. Graph A: scale 1=1, bars reach 20+ squares. Graph B: scale 1=5, bars reach 4 squares. Both correct. Which is easier to read? Why? When would you choose a larger scale? (When data values are large.)
Explore
Scale selection and graph creation: each group receives a data set with values between 20 and 80. They must (1) choose an appropriate scale, (2) build the graph, (3) verify the scale works for all values. Compare: did different groups choose different scales? Are both valid?
Formalize
Critical reading exercise: show a misleading bar graph (y-axis starting at 50, making a small difference look huge). Ask: does this graph show a big difference? Look at the numbers. Is it actually big? What would the graph look like if the axis started at 0? Redraw it.
One-to-One and Many-to-One Correspondence in Graphs
Three-level questioning with the class graph at scale 1=5: Level 1: how many students chose hockey? (Read: 4 squares x 5 = 20.) Level 2: how many more chose hockey than swimming? (20-5=15.) Level 3: what does this data tell us about recreational preferences in our class?
Practice
Students create one bar graph and one pictograph for the same data set using chosen scales, then critically evaluate 2 provided graphs for potential misrepresentation. Exit ticket: a bar graph has scale 1=10. A bar is 7 units tall. What value does it represent?
Exit ticket
Students create one bar graph and one pictograph for the same data set using chosen scales, then critically evaluate 2 provided graphs for potential misrepresentation. Exit ticket: a bar graph has scale 1=10. A bar is 7 units tall. What value does it represent?
The graph: cars in the parking lot by colour; KEY: each ⬤ = 5 cars. Rows show: white ⬤⬤⬤, black ⬤⬤, red ⬤ and a HALF symbol.
Step 1: Read with the key, never by symbol-count: white = 3 × 5 = 15 cars. Black = 10. Red = 1½ symbols → 5 + half of 5 = 7½?? — pause. Can half a car exist? The half-symbol here means about 2 or 3… no: by convention exactly HALF the key value. Key = 5 → half-symbol ≈ 2.5, which is impossible for cars — so a half-symbol with an odd key was a DESIGN ERROR. Real graphs choose keys that divide their data (key of 2 or 4).
Step 2: Fix the graph: with key ⬤ = 2, red's 7 cars = 3½ symbols ✓ (half of 2 is a whole car).
Step 3: The two-question drill for any pictograph: What's the key? Do fractional symbols make sense with it?
The deep point: scaling compresses big data into small pictures, and the KEY is the contract. Break the contract (or forget to read it) and the picture lies.
The data: bottles collected — Mon 24, Tue 36, Wed 18, Thu 42, Fri 30.
Step 1: Try a one-to-one graph on grid paper first — 42 squares tall for Thursday. It won't fit the page. THIS is why scale exists: necessity, not decree.
Step 2: Choose a scale that fits AND divides the data kindly: try 1 square = 6 bottles: Mon 4 squares, Tue 6, Wed 3, Thu 7, Fri 5. Everything lands on whole squares ✓.
Step 3: Draw with the non-negotiables: axis labelled "number of bottles," scale marked (0, 6, 12, 18…), title, even bar widths.
Step 4: Read it back through the scale: "Thursday's bar is 7 squares → 42 ✓." Then interrogate: how many more Tuesday than Wednesday? (3 squares × 6 = 18 — read the DIFFERENCE in squares, then scale it.)
Step 5: Compare a partner's graph who chose 1 square = 4: taller bars (Thu = 10½ squares — half-squares appear!). Same truth, different grain. Scale choice trades compactness for precision — a real design decision, made by the grapher.
The plan: survey the whole school (about 400 students) on lunch preference, then graph it. You expect categories of roughly 60–150 votes.
Step 1: The planning question that reverses the usual order: what scale will the graph need? With counts near 150 and a page 25 squares tall, 1 square = 1 won't fit; 1 square = 10 puts the biggest bar at 15 squares — comfortable. Decide: key = 10.
Step 2: Anticipate the rounding cost: a category with 87 votes will draw as 8.7 squares — awkward. Options: round bars to the nearest square (and SAY the graph shows approximate values), or pick key = 5 and accept taller bars. Every choice is a trade; the grapher must know which they made.
Step 3: Collect (tally), then graph with the pre-chosen scale.
Step 4: Debrief the inversion: amateurs graph whatever they collected and fight the page; planners size the container first. Estimating the data's RANGE before collecting it is a genuine statistical habit — Grade 4 edition.
Exit: you expect data from 0 to 80 on a 20-square grid. Defend your scale choice in one sentence.