Writing digits
Lee was writing all the counting numbers from 1 to 20. She stopped for a rest after writing seventeen digits. What was the last number she wrote?
Age
5 to 7
Challenge level
Problem
Writing Digits printable sheet
Lee was recording some numbers:
Image
What do you think she wrote next?
Why do you think that?
Click on the 'Show' button to find out.
Image
She continued writing the counting numbers in order but she stopped for a rest after writing seventeen digits.
What was the last number she wrote?
She carried on until she got all the way up to the number 20.
How many digits has she written altogether now?
Can you find a way to work this out without writing the counting numbers yourself?
Getting Started
You could use a number line to help.
Student Solutions
Junsu from Ms. Brown's class at The Bangkok Patana School in Thailand sent in the following:
The numbers under 10 have a single digit, so when you write it down it counts as writing one digit.
Numbers starting from 10 have two digits, so it counts as writing two digits.
From 1-9 it is 1 x 9 = 9 digits
She wrote a total of 17 digits so 17 - 9 = 8. She has eight digits left, and from 10-20 each number counts as writing two digits 8/2=4, so she wrote four two-digit numbers which is 13
First answer 13.
If she writes the numbers up till 20, 1 to 9 = 9 digits, 10 to 20 = 2 x 11 = 22 digits
22+9=31, total 31 digits Answer: 31
Hakuya from The Maadi British International School in Egypt sent in his thoughts:
I started by writing the numbers and then counting the digits. I noticed that the 17th digit meant that Lee finished at the number 13. I then carried on writing to 20 and counted the digits again, there were 31 digits.
The problem asked me to think of a way of working it out without writing the numbers. I decided that you should count in 1's to 9 and then count in 2's. So number 10 would have 11 digits.
Majid from Essex Primary School wrote this very thorough account:
Well, in order to figure this out, you need to know your numbers. The first two questions are easy to figure out. The next number after 5 is 6 as the sequence's rule is +1.
Writing seventeen digits doesn't necessarily mean the number she stopped at is 17. The question asks how many digits. In that case, 1-9 is 9 digits as they don't have more than one digit in their number. So, 17 take away 9 is 8. If numbers up to 99 have two digits, the number will not go that high as we only have 8 more digits. 8 divided by 2 is 4. So four more numbers. 10, 11, 12,13. The number she stopped at is 13.
The next question, how many digits up until 20? Well, we know seventeen digits are up until 13. So 13-20 is seven numbers. We know the numbers are two-digit numbers, 7 x 2 is 14 so fourteen digits from 13-20 (not including 13). 14 + 17 is 31. So thirty-one digits up until 20.
You don't need to count the digits one by one to work it out. But of course, if you find it easier to do so, it's all right. Another way to work it out is knowing how many one-digit numbers are included, how many two-digit numbers are included and so on. Knowing that you have to multiply the number of numbers by how many digits it has.
An example:
1-200.
1-9 is nine digits. Nine numbers times 1= 9
10-99 is 90 numbers (including 10 and 99). So 90x2 is 180. So 180 digits.
100-200 is 101 numbers (Including 100 and 200). So 101x3 is 303 digits.
9+180+303=492. Between 1 and 200 (including 1 and 200), there are 492 digits.
I found a file, an answer from the internet to prove my answer.
Shree from The International School Frankfurt, Germany sent in a picture of her work:
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Thanks for these really good examples of how it may be worked out.
Teachers' Resources
Writing Digits
Writing Digits printable sheet
Lee was recording some numbers:
Image
What do you think she wrote next?
Why do you think that?
Click on the 'Show' button to find out.
Image
She continued writing the counting numbers in order but she stopped for a rest after writing seventeen digits.
What was the last number she wrote?
She carried on until she got all the way up to the number 20.
How many digits has she written altogether now?
Why do this problem?
This problem provides meaningful practice of writing and ordering the numbers from $1$ to $20$. Learners might be curious to find out about the number of digits used in the counting sequence, which provides a different focus within a familiar context.
Possible approach
Show the class the first picture and ask them to say what they see, perhaps to a partner first and then take some ideas to share with the whole group. Try not to dismiss any comments that do not seem relevant - children will need to get all of their observations 'off their chests' before they can focus on a more directed task. Ask them what Lee might write next and take some
ideas. They may well make all sorts of suggestions - again try to welcome everything, encouraging learners to explain why they think that.
Reveal the second picture and discuss what Lee is doing, drawing out the fact that she is writing the counting numbers in order. You could invite a learner to the board to write up to ten and then pause to ask how many digits have been written. This is a good opportunity to clarify the meaning of 'digit', as opposed to 'number'.
Reveal the second picture and discuss what Lee is doing, drawing out the fact that she is writing the counting numbers in order. You could invite a learner to the board to write up to ten and then pause to ask how many digits have been written. This is a good opportunity to clarify the meaning of 'digit', as opposed to 'number'.
You can then lead into the suggested questions, giving children time to work in pairs on the task. The last question probes a little further by challenging learners to explain how they could work out the number of digits written without doing any writing themselves. Listen out for those who have a good understanding of the structure of our number system in terms of the first nine
numbers being one-digit numbers and then the numbers all the way up to and including 99 being two-digit numbers.
Key questions
Tell me about what you're doing.
What do we mean by 'digit'?
What do we mean by 'digit'?
How might you do that without writing all the numbers?
What do you know about the number of digits in the counting numbers?
What do you know about the number of digits in the counting numbers?
Possible extension
You could ask similar questions with learners counting in twos, for example.
Possible support
Children could be encouraged to use a number line.