Tally marks

Statistics and data · level 2, lesson 1 of 6

A question to start

You sit by the window for ten minutes and count the vehicles going past: cars, trucks and buses. A car. Another car. A bus! Three cars in a row. A truck. They come fast, sometimes two at once.

How can you keep count of three different things at the same time, without losing your place? You cannot write "17" and then rub it out and write "18" every time a car goes by. You would miss the next one.

People solved this problem thousands of years ago, long before anyone wrote numbers the way we do. The answer is a mark for each thing: a tally mark. It is so simple that it is still used today, by scorekeepers, scientists counting birds, and anyone running a survey.

The idea

One mark for each thing

A tally mark is a short line. You draw one line each time you see the thing you are counting. You never rub anything out, and you never have to remember a number. You just add a line.

Car: |. Another car: ||. Another: |||. The marks are the count.

The word "tally" comes from an old word for a stick with notches cut in it. Bones and sticks with rows of cut notches have been found that are tens of thousands of years old, and many people think they were used for keeping count. Tally marks may be one of the oldest kinds of math there is.

Bundles of five

Here is the problem with plain lines. What is this?

||||||||||||||||

You have to count every line, one by one, and it is easy to lose your place. So we bundle the lines into fives. Draw four lines, and then draw the fifth line across them. That makes a little gate, and the gate means 5.

That is one bundle of five and two more lines: 7.

Why five? Look at your hand. Four fingers and a thumb across them: five. Five is small enough to see at a glance, without counting. You do not count the lines in a gate; you just see "a gate, that's 5".

Counting a tally fast

To read a long tally, count the gates by fives, and then count the single lines by ones.

Gates: 5, 10, 15, 20. Then single lines: 21, 22, 23. The tally says 23.

Counting by fives is much quicker than counting 23 single lines, and much harder to get wrong.

Making a tally chart

A tally chart is a table with one row for each choice and a space for the tally marks. You use it while the answers are coming in.

Here is how a class of 26 children answered the question "Which season do you like best: winter, spring, summer or fall?" The teacher read the answers out one at a time, and a child made one mark in the right row for each answer.

When all the answers are in, you count each row:

• Winter: one gate and one line: 5, 6. 6.
• Spring: one gate: 5.
• Summer: two gates and one line: 5, 10, 11. 11.
• Fall: four lines: 4.

And check the total: 6 and 5 make 11, and 11 make 22, and 4 make 26. There are 26 children in the class, so every answer was marked once.

Working through a list

Sometimes the answers are already written down, in a long jumbled list. To make a tally chart from a list:

1. Take the first answer. Make a mark in its row. Cross it out on the list.
2. Take the next answer. Make a mark. Cross it out.
3. Keep going until every answer is crossed out.

Crossing out is the secret. You always know which answers you have done, so you never count one twice and you never skip one.

Other ways to tally

The four-lines-and-a-gate tally is common in Canada, the UK and the US, but it is not the only way.

• In some places, especially in South America and parts of Europe, people draw a square one side at a time (four marks) and then a line from corner to corner for the fifth.
• In China, Japan and Korea, people often write the character 正, which has exactly five strokes. Each stroke is one mark, and a finished character is 5.

All three ideas are the same: one mark for each thing, and a finished picture for every five.

Worked examples

Example 1. Reading tallies. A tally chart shows: red, three gates and two lines; blue, one gate and four lines; green, two gates. How many of each?

1. Red: gates 5, 10, 15, then lines 16, 17. 17.
2. Blue: gate 5, then lines 6, 7, 8, 9. 9.
3. Green: gates 5, 10. 10.
4. Total: 17 and 9 make 26, and 10 more make 36.

Example 2. From a list to a tally chart. Here are the snacks 15 children chose: apple, crackers, apple, yogurt, crackers, apple, apple, yogurt, crackers, apple, cheese, apple, crackers, yogurt, apple.

1. Make a chart with four rows: apple, crackers, yogurt, cheese.
2. Go through the list in order, one mark each, crossing out as you go.

Snack	Tally	Count
apple	one gate and two lines	7
crackers	four lines	4
yogurt	three lines	3
cheese	one line	1

Check: 7 and 4 make 11, and 3 make 14, and 1 makes 15. Fifteen children. Good.

Example 3. The vehicle count. In ten minutes at the window, the tallies were: cars, four gates and three lines; trucks, one gate; buses, two lines.

1. Cars: 5, 10, 15, 20, then 21, 22, 23. 23 cars.
2. Trucks: 5. Buses: 2.
3. Altogether: 23 and 5 make 28, and 2 make 30 vehicles.
4. How many more cars than trucks? From 5 up to 23 is 18 more.

Notice how the tally let you keep three counts going at once, as fast as the traffic came.

Try it

1. Draw the tally for 12.
2. What number is shown by four gates and one line?
3. How many gates and how many single lines do you need for 19?
4. A tally chart for favourite fruit shows: apple, two gates and three lines; banana, one gate and one line; grapes, three gates. Find each count and the total.
5. Make a tally chart for these answers about pets: dog, cat, dog, fish, dog, cat, dog, dog, fish, cat, dog, cat.
6. Nadia wanted to draw 5. She drew five lines standing up and then a line across them. What went wrong? What number did she really draw?
7. Using the square way of tallying, how many full squares (with their corner-to-corner lines) do you need for 20? For 22?
8. Why is a tally easier than writing numbers when you count cars going past?

Answers

1. Two gates and two lines: 5, 10, 11, 12.
2. 5, 10, 15, 20, then 21. 21.
3. 19 is 5, 10, 15 and then 16, 17, 18, 19. So 3 gates and 4 single lines.
4. Apple: 5, 10, then 11, 12, 13: 13. Banana: 5, then 6: 6. Grapes: 5, 10, 15: 15. Total: 13 and 6 make 19, and 15 make 34.
5. Dog: one gate and one line, 6. Cat: four lines, 4. Fish: two lines, 2. Check: 6 and 4 and 2 make 12, and there are 12 answers in the list.
6. The fifth line should go across the first four, not stand up next to them. Five lines standing and one across is a gate of 6 lines. It looks like a 5 but it is really 6. To show 5 she needs four standing lines and the fifth across them.
7. 20 is four fives, so 4 squares. 22 is 4 squares and then 2 more marks (two sides of a new square).
8. With a tally you only add one line each time, quickly, without stopping to think of the next number or rub anything out. And bundles of five make it easy to count up at the end.

Common mistake

Drawing five standing lines before the gate. The gate line is the fifth mark. Four standing lines, then the fifth goes across. A gate drawn over five standing lines holds 6, not 5.

Counting every line one by one at the end. That is slow and easy to get wrong. Count the gates by fives first, then the single lines by ones.

Where it leads

• Next lesson: Tables, where the tallies become numbers with a total.
• Back to level 1: Asking a question. This level: Level 2: Tallies and bar graphs. The whole topic: Statistics.
• Tally marks were the first way of writing numbers; see them beside Egyptian, Roman and Maya numerals in Old ways to write numbers.
• Counting by fives: Multiplying as equal groups.