Scaled graphs & frequency
Interpret & construct scaled bar charts, pictograms, time graphs, line graphs, histograms; read timetables; absolute/relative frequency.
Once you have counted data, the next step is to draw it so a small chart can hold big numbers and show how things change — that means choosing a scale, and knowing when a shape means “count”, “how much”, or “over time”.
What it means
You already know that a bar chart turns each count into a column: the taller the bar, the bigger the count. A scaled chart adds one idea — each step up the side is worth more than one. If the scale climbs 0, 5, 10, 15, then a bar reaching the third mark means 15, not 3. The scale is the ruler up the side (the axis); reading a value means lining your eye across to it. Scales let a short chart show large numbers: one square can stand for 2, 10 or 100.
The same “one symbol = many” trick drives a pictogram, where a key tells you what each picture is worth (one apple symbol = 4 apples), so half a symbol means 2.
Different data wants different pictures:
- A bar chart compares separate categories (apples vs pears). Bars sit apart.
- A line graph (or time graph) tracks one thing as it changes over time — temperature through a day, plants growing over weeks. You plot a dot for each time and join them; the line’s slope shows rising or falling.
- A histogram looks like a bar chart but the bottom axis is a number line cut into ranges (heights 100–110 cm, 110–120 cm…). Its bars touch, because the ranges join up.
A timetable is a table of times — buses, trains, lessons — where you cross a row (which bus) with a column (which stop) to read the time in that cell.
Finally, two ways to say “how often”: absolute frequency is the plain count (7 children like blue). Relative frequency is that count as a share of the whole — 7 out of 20, or 7/20, which is 35%. Relative frequency lets you compare groups of different sizes fairly.
Worked examples
1 — Reading a scale. A bar chart of pets goes up in 2s: 0, 2, 4, 6, 8. The “dogs” bar stops at the fourth mark. Value = 4 × 2 = 8 dogs.
2 — Pictogram key. Key: one book = 10 books. A shelf shows 3½ symbols → 3 × 10 + (½ × 10) = 30 + 5 = 35 books. Choosing an even key value keeps half a symbol a whole count.
3 — Timetable. Read the bus that leaves at 9:15:
| Bus stop | Bus A | Bus B |
|---|---|---|
| Market | 9:00 | 9:15 |
| School | 9:12 | 9:27 |
Bus B leaves Market at 9:15 and reaches School at 9:27 — a 12-minute ride.
4 — Relative frequency. In a class of 25, 10 walk to school. Absolute = 10. Relative = 10/25 = 40%.
The generative-art connection
A scaled chart is generative art with one adjustable rule: height = count × scale. In mathartcademy a child sets the scale and watches every bar restretch — turn the dial from “1 per square” to “5 per square” and the whole picture shrinks while the data stays the same, making the axis itself visible.
A line graph is a moving version: feed in a value each tick and the app extends a growing curve — the slope you see is the change you measured. Our dot-multiplier tool makes the pictogram key tangible: one dot bursts into many, so “one symbol stands for N” becomes something you count on the screen. Toy Theater’s graphing tools let a learner drag real objects into columns and watch the scaled bars assemble.
Common misconceptions
- Counting squares, not the scale. On a scale of 5s, a bar at the second mark is 10, not 2. Always read the numbers on the axis.
- Line graph vs bar chart. Only join points with a line when the between-times are real (temperature exists at 10:30). Separate categories like fruit must not be joined — there is no “between apple and pear”.
- Confusing absolute and relative. “More children” (a bigger count) is not the same as “a bigger share”. A group of 8 out of 40 is a smaller share (20%) than 6 out of 20 (30%), even though 8 is more than 6.
What a learner should be able to solve
The benchmark for mastering this concept — the problems a child at this stage is expected to be able to work through.
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A bar chart of favourite fruits climbs the side in 4s: `0, 4, 8, 12, 16, 20`. The "banana" bar reaches exactly the fourth mark. How many children chose bananas? How do you know?
Answer
The fourth mark is `4 × 4 = 16`, so **16 children**. Read the number the bar lines up with on the scaled axis, not how many marks it passes.
Art hook Draw 5 category bars on a Canvas. A slider sets the scale (1, 2, 4, or 10 per square); every bar's pixel-height recomputes as `count × scale × squareSize` live, so the same data restretches and the axis labels renumber as you drag.
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A pictogram shows how many books each class read. The key says one book symbol = `8` books. Class Red shows `3` symbols and half a symbol. How many books did Class Red read?
Answer
`3 × 8 = 24`, and half a symbol is `½ × 8 = 4`, so `24 + 4 =` **28 books**. An even key value keeps half a symbol a whole count.
Art hook A pictogram builder where each symbol is a cluster of N dots (the key). Clicking adds a symbol; a half-click draws a half-cluster. Total dots animate in so 'one symbol = many' is countable on screen, dot-multiplier style.
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A line graph shows the temperature in a garden through one day. At 9 a.m. it reads `12 °C`; at noon `18 °C`; at 3 p.m. `15 °C`. Between which two times did it rise the most, and by how much? Should the points be joined with a line? Why?
Answer
It rose from 9 a.m. to noon by `18 − 12 = 6 °C`, more than any other gap (noon→3 p.m. it *fell* by 3). **Yes, join the points** — temperature exists at every moment between the readings, so the line is meaningful.
Art hook A live line graph: each tick pushes a new temperature value and the curve extends, colouring each segment by slope (warm hue rising, cool hue falling) so the change you measured is the colour you see.
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In Class A, `9` of `30` children walk to school. In Class B, `8` of `20` children walk. Which class has the bigger *share* of walkers? Give each as a relative frequency out of 100.
Answer
Class A: `9/30 = 30%`. Class B: `8/20 = 40%`. **Class B** has the bigger share, even though Class A has more walkers (9 vs 8). Relative frequency compares shares fairly across different-sized groups.
Art hook Two rings of dots (30 and 20). Fill the walker-dots in one colour; below each ring a bar grows to its percentage on a shared 0-100 axis, so a bigger count can visibly make a smaller share.
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A bus timetable has stops down the side and buses across the top. Bus 2 leaves Harbour at `10:40` and reaches the Library at `11:05`. Bus 3 leaves Harbour at `10:55` and reaches the Library at `11:15`. Which bus makes the quicker journey, and by how many minutes?
Answer
Bus 2 takes `11:05 − 10:40 = 25` min. Bus 3 takes `11:15 − 10:55 = 20` min. **Bus 3 is quicker, by 5 minutes.** Read across a row and down a column to find each time, then subtract.
Art hook A grid where rows are stops and columns are buses; clicking a cell draws that leg on a number-line clock as an arc, and arc length shows the journey duration so quicker journeys are literally shorter arcs.
Training exercises
Practice problems, easier first, that build toward the bar above. Each doubles as a seed for an interactive artwork.
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A bar chart of pets goes up in 1s: `0, 1, 2, 3, 4, 5`. The "cats" bar reaches the third mark. How many cats?
Answer
**3 cats.** With a scale of 1s, the mark number is the count.
Art hook A single bar on a grid you grow by clicking squares; the count label updates as `squares × 1`, the gentlest first version of height = count × scale.
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A bar chart climbs in 5s: `0, 5, 10, 15, 20`. The "stickers" bar stops at the third mark. How many stickers?
Answer
`3 × 5 =` **15 stickers.** Multiply the mark position by the scale step of 5.
Art hook Same bar tool as before but a dial switches the step to 5; watch one short bar now mean 15 as every axis label multiplies by 5.
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A pictogram uses this key: one star symbol = `2` points. A player has `4` full star symbols. How many points?
Answer
`4 × 2 =` **8 points.**
Art hook Each star bursts into 2 dots when tapped; the running dot-total is the score, making 'one symbol = 2' visible.
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A pictogram key says one leaf symbol = `4` leaves. A branch row shows `2½` symbols. How many leaves?
Answer
`2 × 4 = 8`, plus `½ × 4 = 2`, so **10 leaves.**
Art hook A half-symbol renders as a half-cluster of dots; the total counts up so learners see why an even key value makes the half a whole count.
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A line graph shows a plant's height each week: week 1 = `2 cm`, week 2 = `5 cm`, week 3 = `9 cm`. How much did it grow from week 2 to week 3?
Answer
`9 − 5 =` **4 cm.** Read the two heights off the up-axis and subtract.
Art hook A growing sprout drawn on Canvas: each week a new dot at (week, height) joins the line and the stem redraws to that height, so the graph and the plant grow together.
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Spot the mistake. A friend draws a bar chart of favourite fruits (apple, pear, plum) and joins the tops of the bars with a line to show a 'trend'. Why is joining these bars wrong?
Answer
There is **no 'between' apple and pear** — they are separate categories, not moments in time. Only join points on a line graph when the in-between values are real (like temperature between two clock times).
Art hook Toggle button: 'categories' keeps bars apart with no line; 'over time' snaps the same tops onto a joined line. Seeing it flip teaches when a line is allowed.
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In a class of `20`, `5` children chose pizza as their favourite lunch. Write this as a relative frequency and as a fraction of 100 (a percentage).
Answer
`5/20`, which is `25/100` = **25%.** Relative frequency is the count as a share of the whole.
Art hook A 20-dot grid; the 5 pizza-dots light up and a bar beside it fills to 25% on a 0-100 axis, linking the fraction to the share.
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A histogram shows children's heights in ranges: `100–110 cm`, `110–120 cm`, `120–130 cm`. Why do the bars in a histogram touch, while bars in a fruit chart sit apart?
Answer
The bottom of a histogram is a **number line cut into ranges that join up** (110 ends where the next begins), so the bars touch. Fruit categories are separate with gaps between them.
Art hook A number line along the x-axis with draggable range dividers; bars snap edge-to-edge to fill each range, showing why a histogram has no gaps.
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A pictogram of books read uses key: one symbol = `6` books. Class Blue read `30` books. How many full symbols should you draw?
Answer
`30 ÷ 6 =` **5 symbols.** Going backwards from the total: divide the count by the key value.
Art hook Type a total and the app auto-draws the right number of 6-dot symbols, dividing before your eyes so the key works in reverse.
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A train timetable: Train 1 leaves Central at `2:10` and reaches Park at `2:35`. Train 2 leaves Central at `2:20` and reaches Park at `2:56`. Which train's journey is longer, and by how many minutes?
Answer
Train 1: `2:35 − 2:10 = 25` min. Train 2: `2:56 − 2:20 = 36` min. **Train 2 is longer, by 11 minutes.**
Art hook Two arcs on a clock-face number line, one per train; longer duration = longer arc, so the slower journey is visibly the bigger sweep.
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Open-ended. You surveyed `20` classmates on their favourite season. Choose a scale for a bar chart if the biggest count is `12`, and explain your choice. Then say which season data would suit a *line graph* instead — is 'favourite season' one of them?
Answer
A scale of `2s` (0,2,4,…,12) fits `12` neatly in few marks; `1s` also works but is taller. **Favourite season is NOT line-graph data** — the four seasons are separate categories, not a value changing over time. A line graph suits things like daily temperature across the seasons.
Art hook A survey sandbox: enter counts, pick a scale from a menu, and the bar chart rebuilds; a locked 'line graph' mode greys out for category data and unlocks only for time data.