Information Processing — Class 5 Mathematics (Samacheer Kalvi)
TN State Board (Samacheer Kalvi) Class 5 Mathematics, Unit 6. Systematic listing, graphical representation of data, pictographs, tally marks and bar graphs. Taught in September.
1. What this unit is for
The aim is to be able to count, compare and make sense of information — how many guests are coming to a birthday party, how the class library should be sorted, how much food a district produces, how many students travel by bus.
Raw information is confusing. Organised information answers questions.
2. Systematic listing
In how many ways can the numbers 1, 2 and 3 be arranged?
Guessing will miss some. Instead, fix the first number and list every way of finishing:
| First number | Arrangements |
|---|---|
| 1 | 123, 132 |
| 2 | 213, 231 |
| 3 | 312, 321 |
Six arrangements in all. And the count could have been predicted: 3 choices for the first place, 2 left for the second, 1 for the last, so 3 × 2 × 1 = 6.
3. Sudoku
A 3 × 3 sudoku is filled with the numbers 1 to 3 so that each number appears exactly once in every row and every column.
| 1 | 2 | 3 |
|---|---|---|
| 2 | 3 | 1 |
| 3 | 1 | 2 |
Check any row and any column and you will find 1, 2 and 3 exactly once.
A 4 × 4 sudoku uses 1, 2, 3 and 4 under the same rule. The method is to look for the row or column that is nearly complete and fill in the number that is missing, then work outwards from there.
4. Magic squares
A 3 × 3 magic square uses the numbers 1 to 9 so that every row, every column and both diagonals add to the same total of 15.
| 2 | 9 | 4 |
|---|---|---|
| 7 | 5 | 3 |
| 6 | 1 | 8 |
Rows: 2 + 9 + 4 = 15, 7 + 5 + 3 = 15, 6 + 1 + 8 = 15. Columns: 2 + 7 + 6 = 15, 9 + 5 + 1 = 15, 4 + 3 + 8 = 15. Diagonals: 2 + 5 + 8 = 15 and 4 + 5 + 6 = 15.
The number 5 always sits in the centre.
5. Collecting and tabulating data
Any collection of information in the form of numerical figures is called data.
Data gathering is very old. Long before writing, people counted their livestock using stones, one stone for one animal. That was the first data-gathering method.
Dinu asked his classmates which sports article they liked:
| Sports article | Number of students |
|---|---|
| Cricket bat | 7 |
| Football | 10 |
| Carrom board | 8 |
| Hockey stick | 10 |
From this single table you can read off which is most liked (football and hockey stick, tied at 10), which is least liked (cricket bat, 7), and the total number of students (35).
6. Tally marks
Counting a long list by writing numbers is slow and error-prone. Tally marks solve it.
Draw one stroke for each item. On the fifth item, draw the stroke across the previous four, making a bundle of five. Counting bundles of five is much faster than counting single strokes.
Balu asked 20 classmates their favourite snack:
| Snack | Tally | Number |
|---|---|---|
| Chocolate | Five and one | 6 |
| Cake | Four | 4 |
| Biscuit | Four | 4 |
| Apple | Three | 3 |
| Banana | Three | 3 |
The numbers add to 20, which is a useful check: the totals must match the number of people asked.
7. Pictographs
A pictograph represents data using pictures.
It is the simplest way of showing information, because you can compare rows at a glance without reading any numbers.
Every pictograph needs a key, telling you what one picture stands for. If one symbol stands for 10 kg of paddy, then 4 symbols mean 40 kg, not 4. Read the key first, always. Half a symbol means half of whatever the key says.
8. Bar graphs
A bar graph shows data as bars whose heights stand for the numbers.
It needs a scale up one side, a label under each bar, equal bar widths, and a title.
Suppose a bar chart shows how students travel to school, with bars for cycle, walking, car and bus against a scale running 0 to 50. You can read off which mode is most used (the tallest bar), which is least used (the shortest), and how many more use one than another by reading both and subtracting.
A bar graph and a pictograph show the same data. The bar graph is better when the numbers are large or awkward, since you would otherwise be drawing hundreds of little pictures.
9. Modelling: breaking a task into steps
Modelling means taking a job that is too big to start on and splitting it into smaller tasks that can be done one after another.
Poovizhi is asked to arrange her brother's birthday party and does not know where to begin. Her father's advice is to break the event up: fix the date and time, make the guest list, write the invitations, plan the food, buy the decorations, and set out the chairs. Not one of those is difficult on its own.
The same habit works in arithmetic. Multiplying 243 by 132 is one large task, so split it into three small ones and add the results.
| Step | Working | Result |
|---|---|---|
| Multiply by the 2 ones | 243 × 2 | 486 |
| Multiply by the 3 tens | 243 × 30 | 7,290 |
| Multiply by the 1 hundred | 243 × 100 | 24,300 |
| Add the three parts | 486 + 7,290 + 24,300 | 32,076 |
Two rules make a model work. The steps must come in a sensible order, since you cannot write the invitations before you have the guest list. And every step must be small enough to finish without having to split it again.
10. Worked examples
Example 1. In how many ways can 1, 2 and 3 be arranged? List them.
Solution: 3 × 2 × 1 = 6 ways: 123, 132, 213, 231, 312, 321.
Example 2. What must every row and column of a 3 × 3 sudoku contain?
Solution: Each of the numbers 1, 2 and 3 exactly once.
Example 3. What is the magic total of a 3 × 3 magic square built from 1 to 9?
Solution: 15, along every row, column and diagonal.
Example 4. In Dinu's table, how many more students like football than the cricket bat?
Solution: 10 − 7 = 3 students.
Example 5. How do you record the fifth item in tally marks?
Solution: Draw the fifth stroke across the previous four, making a bundle of five.
Example 6. A pictograph key says one symbol is 10 kg of paddy. What do three and a half symbols mean?
Solution: 3 × 10 = 30, plus half of 10 which is 5, giving 35 kg.
Example 7. In Balu's snack survey the counts are 6, 4, 4, 3 and 3. Check the total.
Solution: 6 + 4 + 4 + 3 + 3 = 20, which matches the 20 students asked, so the tally is complete.
11. Practice
- What is data?
- How did primitive people first record how many animals they had?
- In how many ways can the numbers 1, 2 and 3 be arranged?
- Complete the rule for a 3 × 3 sudoku.
- What is the magic total of a 3 × 3 magic square using 1 to 9, and which number sits in the centre?
- Why is the fifth tally mark drawn across the other four?
- What must every pictograph have besides the pictures?
- In Dinu's table, which sports articles were equally liked?
- Name three things every bar graph needs.
- Split 156 × 23 into steps and find the product.
12. Answers
- Any collection of information in the form of numerical figures.
- By counting with stones, one stone for each animal.
- Six ways.
- Each of 1, 2 and 3 must appear exactly once in every row and every column.
- The total is 15, and 5 sits in the centre.
- Because bundles of five are much quicker and safer to count than single strokes.
- A key, saying what one picture stands for.
- Football and hockey stick, both at 10.
- A scale, a label under each bar, and equal bar widths (also a title).
- 156 × 3 = 468 and 156 × 20 = 3,120. Adding, 468 + 3,120 = 3,588.
13. Summary
- Organised information answers questions that raw information cannot.
- List possibilities systematically; 3 different things can be arranged in 3 × 2 × 1 = 6 ways.
- In a sudoku each number appears once in every row and column.
- A 3 × 3 magic square from 1 to 9 totals 15 on every line, with 5 in the centre.
- Data is information in numerical form; counting with stones was the first method.
- Tally marks bundle in fives, with the fifth stroke drawn across the other four.
- A pictograph shows data as pictures and must carry a key.
- A bar graph needs a scale, labels, equal widths and a title, and suits larger numbers.
