If they want to show their data through a pictograph, where they also use one symbol for each student, as Lakhanpal did, what are the challenges they might face?
Answer
Lakhanpal's data was small — only 3, 5, 4, 2, 0, 1, 5, 7 students, that is 27 symbols altogether.
Lakhanpal's pictograph: students absent in each class, one symbol for one student. Class V has no symbol at all — nobody was absent.
So with one symbol per student they would face these challenges:
Far too many symbols — 215 of them have to be drawn, one by one.
It takes very long, and the hand gets tired; the last symbols start looking different from the first ones.
It will not fit — a row of 35 symbols runs off the edge of the notebook page.
Counting becomes hard — to read “35” the reader must count 35 tiny pictures, and may easily miscount. The picture then answers no faster than a plain table.
The cure is a bigger scale. Jarina used one symbol for 5 students, so only 43 symbols were needed:
Jarina's pictograph with the key “1 symbol = 5 students”. Class II, with 35 students, needs 7 symbols.
Sangita went further and used one symbol for 10 students, with a half symbol for 5:
Sangita's pictograph with the key “1 symbol = 10 students”. 25 students are shown as 2 symbols and a half.
The lesson: the scale must suit the size of the data. Small numbers → 1 symbol = 1 unit. Large numbers → 1 symbol = 5, 10, 100 … units. But the scale must also divide the data neatly, otherwise the next problem appears — see the next question.
Q2.
What could be the problems faced in preparing such a pictograph, if the total number of students present in a class is 33 or 27?
Answer
Sangita's key is “one symbol = 10 students”, and a half symbol = 5 students. Now try 33 and 27:
33 = 3 × 10 + 3 → 3 symbols and 3 more students
27 = 2 × 10 + 7 → 2 symbols and 7 more students
To show the extra 3 students we would need three-tenths of a symbol, and for 7 students seven-tenths of a symbol. Such tiny fractions of a picture cannot be drawn — and certainly cannot be read back correctly.
A half symbol is no help: half of 10 is 5, which is neither 3 nor 7.
Two different readers would guess two different numbers from the same part-symbol. The pictograph stops being accurate.
What to do instead:
Choose a scale that divides all the values — with 33 and 27 present, 1 symbol = 1 student is exact (but long); if all the numbers were multiples of 3, a scale of 3 would work.
Or give up the pictograph and draw a bar graph, where a bar can be drawn to any height, so 33 and 27 are shown exactly.
Math Talk: Which scale would you pick for 12, 18, 24 and 30? (A scale of 6 works perfectly — 2, 3, 4 and 5 symbols.) Now try it for 12, 19, 24 and 30. Discuss what goes wrong.